Effects of study duration, frequency of observation, and sample size
on power in studies of group differences in polynomial change, Psychol
Methods 6(4): 387–401.

[Rijcken et al.]Rijcken, Schouten, Weiss, Speizer   van der Lende1987Rijcken:1987
Rijcken, B., Schouten, J. P., Weiss, S. T., Speizer, F. E.  van der
Lende, R.  1987.
The relationship of nonspecific bronchial responsiveness to
respiratory symptoms in a random population sample, Am Rev Respir Dis
136(1): 62–8.

Rochon, J.  1998.
Application of gee procedures for sample size calculations in
repeated measures experiments, Stat Med 17(14): 1643–58.

Schlesselman, J. J.  1973.
Planning a longitudinal study. ii. frequency of measurement and study
duration, J Chronic Dis 26(9): 561–70.

Schouten, H. J.  1999.
Planning group sizes in clinical trials with a continuous outcome and
repeated measures, Stat Med 18(3): 255–64.

Searle, S. R.  1971.
Linear models, Wiley, New York,.

Senn, S.  2002.
Cross-over trials in clinical research, 2nd edn, J. Wiley,
Chichester, Eng.; New York.

Snijders, T.  20033.
Pint, http://stat.gamma.rug.nl/snijders .

Snijders  Bosker1993Snijders:1993
Snijders, T.  Bosker, R.  1993.
Standard errors and sample sizes for two-level research, Journal
of Educational Statistics 18(3): 237–259.

Snijders  Bosker1994Snijders:1994
Snijders, T.  Bosker, R.  1994.
Modeled variance in two-level models, Sociological methods and
research 22(3): 342–63.

[Takkouche et al.]Takkouche, Cadarso-Suarez   Spiegelman1999Takkouche:1999
Takkouche, B., Cadarso-Suarez, C.  Spiegelman, D.  1999.
Evaluation of old and new tests of heterogeneity in epidemiologic
meta-analysis, Am J Epidemiol 150(2): 206–15.

Timm, N. H.  2002.
Applied multivariate analysis, Springer texts in statistics,
Springer, New York.

[Tu et al.]Tu, Kowalski, Zhang, Lynch   Crits-Christoph2004Tu:2004
Tu, X. M., Kowalski, J., Zhang, J., Lynch, K. G.  Crits-Christoph,
P.  2004.
Power analyses for longitudinal trials and other clustered designs,
Stat Med 23(18): 2799–815.

[Van der Lende et al.]Van der Lende, Kok, Peset, Quanjer, Schouten
Van der Lende, R., Kok, T., Peset, R., Quanjer, P. H., Schouten, J. P.
Orie, N.  1981.
Decreases in vc and fev1 with time: Indicators for effects of smoking
and air pollution, Bulletin of European Physiopathology and Respiration
17: 775–92.

[Villamor et al.]Villamor, Msamanga, Spiegelman, Antelman,
Peterson, Hunter  Fawzi2002Villamor:2002
Villamor, E., Msamanga, G., Spiegelman, D., Antelman, G., Peterson, K. E.,
Hunter, D. J.  Fawzi, W. W.  2002.
Effect of multivitamin and vitamin a supplements on weight gain
during pregnancy among hiv-1-infected women, Am J Clin Nutr 76: 1082–90.

[Ware et al.]Ware, Dockery, Louis, Xu, Ferris   Speizer1990Ware:1990
Ware, J. H., Dockery, D. W., Louis, T. A., Xu, X. P., Ferris, B. G., J.
Speizer, F. E.  1990.
Longitudinal and cross-sectional estimates of pulmonary function
decline in never-smoking adults, Am J Epidemiol 132(4): 685–700.

Wolfram Research Inc.2005Wolfram:2005
Wolfram Research Inc.  2005.
Mathematica, version 5.2, Champaign, Illinois.

Yi  Panzarella2002Yi:2002
Yi, Q.  Panzarella, T.  2002.
Estimating sample size for tests on trends across repeated
measurements with missing data based on the interaction term in a mixed
model, Control Clin Trials 23(5): 481–96.

Ziegler, A.  2004.
Geesize, http://www.imbs.uni-luebeck.de/pub/Geesize .

§ APPENDIX

§ VARIANCE FORMULAS

§.§ Proof of formulas (<ref>) and (<ref>)

From equation (<ref>) we have ${\mathbf{\Sigma }}_{\rm B}  = \mathbb{E}^{ - 1} \left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } \right)
$.  Now,

\begin{multline*}
{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i  =\\
\left( {\begin{array}{*{20}c}
1 &  \cdots  & 1 &  \cdots  & 1  \\
{t_{i0} } &  \cdots  & {t_{i0}  + sj} &  \cdots  & {t_{i0}  + sr}  \\
{k_i } &  \cdots  & {k_i } &  \cdots  & {k_i }  \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
{\nu _{00} } &  \cdots  & {\nu _{0r} }  \\
\vdots  &  \ddots  &  \vdots   \\
{\nu _{r0} } &  \cdots  & {v_{rr} }  \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
1 & {t_{i0} } & {k_i }  \\
\vdots  &  \vdots  &  \vdots   \\
1 & {t_{i0}  + sj} & {k_i }  \\
\vdots  &  \vdots  &  \vdots   \\
1 & {t_{i0}  + sr} & {k_i }  \\
\end{array} } \right) = \\
% here
= \left( {\begin{array}{*{20}c}
{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } & {} & {}  \\
{ \begin{gathered} \biggl\{
t_{i0} \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } }  +  \hfill \\
\hfill s\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } }  \biggr\}\\
\end{gathered}  } & { \begin{gathered} \biggl\{
t_{i0}^2 \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } }  + st_{i0} \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {(j + j')v_{jj'} } }  \hfill \\
\hfill + s^2 \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } } \biggr\}  \\
\end{gathered}} & {}  \\
{k_i \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } & {k_i \left( {t_{i0} \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } }  + s\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } } \right)} & {k_i ^2 \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }  \\
\end{array} } \right)
\end{multline*}

Using $\bar t_0  = \mathbb{E}\left( {t_0 } \right)$, $p_e  = \mathbb{E}\left( k \right) = \mathbb{E}\left( {k^2 } \right)$ and

\rho _{\operatorname{e} ,t_0 }  = \frac{{\mathbb{E}\left( {kt_0 } \right) - p_e \bar t_0 }}
{{\sqrt {p_e (1 - p_e )} \sqrt {V\left( {t_0 } \right)} }},

and assuming without loss of generality that $\bar t_0  = 0$ (this can be achieved by centering the initial time), which implies $\mathbb{E}\left( {t_0^2 } \right) = V(t_0 )$, we have that

\begin{multline*}
\mathbb{E}\left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } \right) =\\
{\small \left( {\begin{array}{*{20}c}
{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } & {} & {}  \\
{s\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } } & {V\left( {t_0 } \right)\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } }  + s^2 \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } } } & {}  \\
{p_e \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } & {\left( {\rho _{\operatorname{e} ,t_0 } \sqrt {p_e (1 - p_e )} \sqrt {V\left( {t_0 } \right)} } \right)\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } }  + sp_e \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } } & {p_e \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }  \\
\end{array} } \right)}
\end{multline*}

We are interested in the [3,3] component of the inverse of this matrix, which is

\begin{multline*}
{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = {\mathbf{c'}}\left( {\mathbb{E}\left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i {\mathbf{X}}_i } \right)} \right)^{ - 1} {\mathbf{c}} = \\
\frac{{s^2 \det ({\mathbf{A}}) + \left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)^2 V(t_0 )}}
{{p_e (1 - p_e )\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)\left[ {s^2 \det ({\mathbf{A}}) + \left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)^2 \left( {1 - \rho _{e,t_0 }^2 } \right)V(t_0 )} \right]}}
\end{multline*}

If either $V\left( {t_0 } \right)$ or $\rho _{\operatorname{e} ,t_0 } $ are zero then
{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{1}
{{p_e (1 - p_e )\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)}}.

If we follow Lachin's approach <cit.>, instead of using the asymptotic variance use the variance of ${\mathbf{\hat B}}$ conditional on the covariates, which is

\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1},

and redefine ${\mathbf{\Sigma }}_{\rm B} $ as

\left( {\frac{1}
{N}\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1}

so that the test statistic is still

T = \frac{{\sqrt {N\,} {\mathbf{c'\hat {\rm B}}}}}
{{\sqrt {{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}} }}.

Then, we would take the expected value of the non-centrality parameter under the alternative hypothesis over the distribution of ${\mathbf{X}}_i $, i.e. we would compute $\mathbb{E}\left[ {T^2 |H_1 } \right]$. If we assume that everyone is observed at the same set of time points, then the only random covariate is exposure.

\begin{multline*}
\frac{1}
{N}\sum\limits_i {{\mathbf{X}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X'}}_i }  = \\
\left( {\begin{array}{*{20}c}
{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } & {} & {}  \\
{s\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } } & {s^2 \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } } } & {}  \\
{\left( {\frac{{\sum\limits_i {k_i } }}
{N}} \right)\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } & {s\left( {\frac{{\sum\limits_i {k_i } }}
{N}} \right)\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } } \right)} & {\left( {\frac{{\sum\limits_i {k_i ^2 } }}
{N}} \right)\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }  \\
\end{array} } \right),
\end{multline*}

and the [3,3] component of the inverse is

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \left[ {\left( {\frac{{\sum\limits_i {k_i } }}
{N}} \right)\left( {1 - \frac{{\sum\limits_i {k_i } }}
{N}} \right)\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)} \right]^{ - 1}.

T^2  = \frac{{\hat \beta _2^2 }}
{{Var(\hat \beta _2 )}} = N\hat \beta _2^2 \left( {\frac{{\sum\limits_i {k_i } }}
{N}} \right)\left( {1 - \frac{{\sum\limits_i {k_i } }}
{N}} \right)\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)

\mathbb{E}\left[ {T^2 |H_1 } \right] = \mathbb{E}\left[ {N\beta _2^2 \left( {\frac{{\sum\limits_i {k_i } }}
{N}} \right)\left( {1 - \frac{{\sum\limits_i {k_i } }}
{N}} \right)\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)} \right],

where the expected value is taken over the distribution of $k_i $, so

\mathbb{E}\left[ {T^2 |H_1 } \right] = N\beta _2^2 \left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)\mathbb{E}\left[ {\left( {\frac{{\sum\limits_i {k_i } }}
{N}} \right)\left( {1 - \frac{{\sum\limits_i {k_i } }}
{N}} \right)} \right].

Noticing that $Z = \sum\limits_i {k_i } $ is a Binomial variable we can work out the expected value,

\begin{multline*}
\mathbb{E}\left[ {T^2 |H_1 } \right] = N\beta _2^2 \left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)\mathbb{E}\left[ {\left( {\frac{Z}
{N}} \right)\left( {1 - \frac{Z}
{N}} \right)} \right] \\
= N\beta _2^2 \left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)\left( {\frac{{\mathbb{E}\left( Z \right)}}
{N} - \frac{{\mathbb{E}\left( {Z^2 } \right)}}
{{N^2 }}} \right)\\
= N\beta _2^2 \left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)\left( {p_e  - \frac{{Np_e (1 - p_e ) + N^2 p_e^2 }}
{{N^2 }}} \right)  \\
= N\beta _2^2 \left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)\left( {\frac{{Np_e  - p_e  + p_e ^2  - Np_e ^2 }}
{N}} \right) \\
= \left( {N - 1} \right)\beta _2^2 \left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)p_e \left( {1 - p_e } \right).
\end{multline*}

The non-centrality parameter with the approach we followed in the paper is

N\beta _2^2 \left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)p_e \left( {1 - p_e } \right),

so there is only a $\left( {1 - \frac{1} {N}} \right)$ correction compared with the one obtained with Lachin's method.

§.§ Proof of formula (<ref>)

Following model (<ref>), and our derivations on Appendix <ref>, we now have that

\begin{multline*}
{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i  = \\
\left( {\begin{array}{*{20}c}
1 &  \cdots  & 1 &  \cdots  & 1  \\
{t_{i0} } &  \cdots  & {t_{i0}  + sj} &  \cdots  & {t_{i0}  + sr}  \\
{k_i } &  \cdots  & {k_i } &  \cdots  & {k_i }  \\
{k_i t_{i0} } &  \cdots  & {k_i t_{i0}  + k_i sj} &  \cdots  & {k_i t_{i0}  + k_i sr}  \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
{v_{00} } &  \cdots  & {v_{0r} }  \\
\vdots  &  \ddots  &  \vdots   \\
{v_{r0} } &  \cdots  & {v_{rr} }  \\
\end{array} } \right) \\
\left( {\begin{array}{*{20}c}
1 & {t_{i0} } & {k_i } & {k_i t_{i0} }  \\
\vdots  &  \vdots  &  \vdots  &  \vdots   \\
1 & {t_{i0}  + sj} & {k_i } & {k_i t_{i0}  + k_i sj}  \\
\vdots  &  \vdots  &  \vdots  &  \vdots   \\
1 & {t_{i0}  + sr} & {k_i } & {k_i t_{i0}  + k_i sr}  \\
\end{array} } \right)
\end{multline*}

and using the results in Appendix <ref> we only need to derive the components in the last row. We can derive $\mathbb{E}\left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } \right)$, in which the [4,1] component is equivalent to the [3,2] and therefore it takes the value

\left( {\rho _{\operatorname{e} ,t_0 } \sqrt {p_e (1 - p_e )} \sqrt {V\left( {t_0 } \right)} } \right)\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } }  + sp_e \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } }.

The [4,2] component is

\mathbb{E}\left( {kt_0^2 } \right)\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } }  + 2s\left( {\rho _{\operatorname{e} ,t_0 } \sqrt {p_e (1 - p_e )} \sqrt {V\left( {t_0 } \right)} } \right)\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } }  + s^2 p_e \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } },

the [4,3] component is

\left( {\rho _{\operatorname{e} ,t_0 } \sqrt {p_e (1 - p_e )} \sqrt {V\left( {t_0 } \right)} } \right)\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } }  + sp_e \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } },

and the [4,4] component is the same as the [4,2] component.
An expression for $\mathbb{E}\left( {kt_0^2 } \right) = p_e \mathbb{E}\left( {t_{0,k = 1}^2 } \right)$ in terms of the known parameters is needed. Since we assumed that $\bar t_0  = 0$, then $V\left( {t_0 } \right) = \mathbb{E}\left( {t_0^2 } \right) = (1 - p_e )\mathbb{E}\left( {t_{0,k = 0}^2 } \right) + p_e \mathbb{E}\left( {t_{0,k = 1}^2 } \right)$, which implies

\begin{equation}
\label{eqA121}
\mathbb{E}\left( {t_{0,k = 0}^2 } \right) = \frac{{V\left( {t_0 } \right) - p_e \mathbb{E}\left( {t_{0,k = 1}^2 } \right)}}
{{1 - p_e }}.
\end{equation}

We have from Appendix <ref> that

\mathbb{E}\left( {kt_0 } \right) = p_e \bar t_{0,k = 1}  = \rho _{\operatorname{e} ,t_0 } \sqrt {p_e (1 - p_e )} \sqrt {V\left( {t_0 } \right)},

\bar t_{0,k = 1}  = \rho _{\operatorname{e} ,t_0 } \sqrt {\frac{{(1 - p_e )}}
{{p_e }}} \sqrt {V\left( {t_0 } \right)}

and it can be deduced that

\bar t_{0,k = 0}  =  - \rho _{\operatorname{e} ,t_0 } \sqrt {\frac{{p_e }}
{{(1 - p_e )}}} \sqrt {V\left( {t_0 } \right)}.

\left( {\bar t_{0,k = 1} } \right)^2  = \rho _{\operatorname{e} ,t_0 }^2 \frac{{(1 - p_e )}}
{{p_e }}V\left( {t_0 } \right)

\left( {\bar t_{0,k = 0} } \right)^2  = \rho _{\operatorname{e} ,t_0 }^2 \frac{{p_e }}
{{(1 - p_e )}}V\left( {t_0 } \right).

We assume that the variance of $t_0 $ is the same in exposed and unexposed, i.e. $V\left( {t_{0,k = 0} } \right) = V\left( {t_{0,k = 1} } \right)$. It follows that

V\left( {t_{0,k = 0} } \right) = V\left( {t_{0,k = 1} } \right) \Leftrightarrow \mathbb{E}\left( {t_{0,k = 0}^2 } \right) - \left( {\bar t_{0,k = 0} } \right)^2  = \mathbb{E}\left( {t_{0,k = 1}^2 } \right) - \left( {\bar t_{0,k = 1} } \right)^2.

Plugging in expression (<ref>) we obtain that

\mathbb{E}\left( {t_{0,k = 1}^2 } \right) = V\left( {t_0 } \right) + \rho _{\operatorname{e} ,t_0 }^2 \left( {\frac{{1 - 2p_e }}
{{p_e }}} \right)V\left( {t_0 } \right).

\mathbb{E}\left( {kt_0^2 } \right) = p_e \mathbb{E}\left( {t_{0,k = 1}^2 } \right) = V\left( {t_0 } \right)\left[ {p_e  + \rho _{\operatorname{e} ,t_0 }^2 (1 - 2p_e )} \right].

Now, plugging in this last expression in the formula for $\mathbb{E}\left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } \right)$, and inverting the matrix, it can be derived that its [4,4] component is

\begin{multline*}
{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = {\mathbf{c'}}\left( {\mathbb{E}\left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } \right)} \right)^{ - 1} {\mathbf{c}} = \\
\frac{{\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)}}
{{p_e (1 - p_e )\left[ {s^2 \det ({\mathbf{A}}) + \left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)V(t_0 )\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)^2 } \right]}}.
\end{multline*}

If $V(t_0 ) = 0$, then

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)}}
{{p_e (1 - p_e )s^2 \det ({\mathbf{A}})}},

and if $\rho _{\operatorname{e} ,t_0 }  = 0$ then

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)}}
{{p_e (1 - p_e )\left[ {s^2 \det ({\mathbf{A}}) + V(t_0 )\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)^2 } \right]}}.

If we can assume that $t_0 $ and exposure are independent, then the formula we derived for the case $\rho _{\operatorname{e} ,t_0 }  = 0$ also applies to model (<ref>), which assumes a general form for the relationship between response and time in the unexposed but requires that a main effect of time is in the model, we can rewrite the model as

\mathbb{E}\left( {Y_{ij} |X_{ij} } \right) = \gamma _0  + \gamma _1 t_{ij}  + \alpha _1 f_1 \left( {t_{ij} } \right) +  \cdots  + \alpha _q f_q \left( {t_{ij} } \right) + \gamma _2 k_i  + \gamma _3 \left( {t_{ij}  \times k_i } \right),

where $f_u \left( {t_{ij} } \right),\;u = 1, \ldots ,U$ are arbitrary functions of time. Since the $[m,q]$ term of the matrix $\mathbb{E}\left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } \right)$ can be written as $\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {x_{ijm} x_{ij'q} } \right)} $, where $x_{kijm} $ is the value of the mth covariate for subject i from group k at time $t_j $, and exposure and time are independent, which implies

\mathbb{E}\left( {k_i f_u \left( {t_{ij'} } \right)} \right) = \mathbb{E}\left( {k_i } \right)\mathbb{E}\left( {f_u \left( {t_{ij'} } \right)} \right) = p_e \mathbb{E}\left( {f_u \left( {t_{ij'} } \right)} \right)\;\;\forall u,

we have

\begin{multline*}
\mathbb{E}\left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } \right) = \\
\left( {\begin{array}{*{20}c}
{\sum\limits_{j,j'} {v_{jj'} } } & {\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij'} } \right)} } & {\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {f_1 \left( {t_{ij'} } \right)} \right)} } &  \cdots   \\
{\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} } \right)} } & {\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} t_{ij'} } \right)} } & {\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} f_1 \left( {t_{ij'} } \right)} \right)} } &  \cdots   \\
{\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {f_1 \left( {t_{ij} } \right)} \right)} } & {\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} f_1 \left( {t_{ij} } \right)} \right)} } & {\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {f_1 \left( {t_{ij} } \right)f_1 \left( {t_{ij'} } \right)} \right)} } &  \cdots   \\
\vdots  &  \vdots  &  \vdots  & {}  \\
{\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {f_V \left( {t_{ij'} } \right)} \right)} } & {\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} f_V \left( {t_{ij'} } \right)} \right)} } & {\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {f_1 \left( {t_{ij} } \right)f_V \left( {t_{ij'} } \right)} \right)} } &  \cdots   \\
{p_e \sum\limits_{j,j'} {v_{jj'} } } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij'} } \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {f_1 \left( {t_{ij} } \right)} \right)} } &  \cdots   \\
{p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} } \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} t_{ij'} } \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} f_1 \left( {t_{ij} } \right)} \right)} } &  \cdots   \\
\end{array} } \right.  \\ %end line
\left. {\begin{array}{*{20}c}
\cdots  & {\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {f_V \left( {t_{ij'} } \right)} \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} } } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij'} } \right)} }  \\
\cdots  & {\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} f_V \left( {t_{ij'} } \right)} \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij'} } \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} t_{ij'} } \right)} }  \\
\cdots  & {\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {f_1 \left( {t_{ij} } \right)f_V \left( {t_{ij'} } \right)} \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {f_1 \left( {t_{ij'} } \right)} \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} f_1 \left( {t_{ij'} } \right)} \right)} }  \\
{} &  \vdots  &  \vdots  &  \vdots   \\
\cdots  & {\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {f_V \left( {t_{ij} } \right)f_V \left( {t_{ij'} } \right)} \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {f_V \left( {t_{ij'} } \right)} \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} f_V \left( {t_{ij'} } \right)} \right)} }  \\
\cdots  & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {f_V \left( {t_{ij} } \right)} \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} } } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij'} } \right)} }  \\
\cdots  & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} f_V \left( {t_{ij'} } \right)} \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} } \right)} } & {p_e \sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} t_{ij'} } \right)} }  \\
\end{array} } \right) \\% end line
= {\mathbf{M}} = \left( {\begin{array}{*{20}c}
{\mathop {{\mathbf{M}}_1 }\limits_{(2 \times (V + 2))} }  \\
{\mathop {{\mathbf{M}}_2 }\limits_{(V \times (V + 2))} }  \\
\end{array} } & {\mathop {p_e {\mathbf{M'}}_1 }\limits_{((V + 2) \times 2)} }  \\
{\mathop {p_e {\mathbf{M}}_1 }\limits_{(2 \times (V + 2))} } & {\mathop {p_e {\mathbf{M}}_4 }\limits_{(2 \times 2)} }  \\
\end{array} } \right)
\end{multline*}

Define the matrix

{\mathbf{Q}} = \left( {\begin{array}{*{20}c}
1 & 0 &  \cdots  &  \cdots  &  \cdots  & 0  \\
0 & 1 &  \ddots  & {} & {} &  \vdots   \\
\vdots  &  \ddots  &  \ddots  &  \ddots  & {} &  \vdots   \\
0 &  \cdots  & 0 & 1 &  \ddots  &  \vdots   \\
{ - p_e } & 0 &  \cdots  & 0 & 1 & 0  \\
0 & { - p_e } &  \cdots  & 0 & 0 & 1  \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
{\mathop {\mathbf{I}}\limits_{((V + 2) \times (V + 2))} } & {\mathop {\mathbf{0}}\limits_{((V + 2) \times 2)} }  \\
{\mathop {{\mathbf{Q}}_1 }\limits_{(2 \times (V + 2))} } & {\mathop {\mathbf{I}}\limits_{(2 \times 2)} }  \\
\end{array} } \right),

such that

\begin{multline*}
{\mathbf{QMQ'}} = {\mathbf{B}} = \\
\left( {\begin{array}{*{20}c}
{{\mathbf{B}}_1 } & {\begin{array}{*{20}c}
{0\quad \quad } & 0  \\
{ \vdots \quad \quad } &  \vdots   \\
{\quad \quad \quad \quad \quad 0\quad \quad \quad \quad \quad \quad \quad } & {\quad \quad \quad 0\quad \quad \quad }  \\
\end{array} }  \\
0 &  \cdots  & 0  \\
0 &  \cdots  & 0  \\
\end{array} } & {\begin{array}{*{20}c}
{p_e (1 - p_e )\sum\limits_{jj'} {v_{jj'} } } & {p_e (1 - p_e )\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij'} } \right)} }  \\
{p_e (1 - p_e )\sum\limits_{jj'} {v_{jj'} \mathbb{E}\left( {t_{ij} } \right)} } & {p_e (1 - p_e )\sum\limits_{j,j'} {v_{jj'} \mathbb{E}\left( {t_{ij} t_{ij'} } \right)} }  \\
\end{array} }  \\
\end{array} } \right)  \\ % end line
= \left( {\begin{array}{*{20}c}
{{\mathbf{B}}_1 } & {\mathbf{0}}  \\
{\mathbf{0}} & {{\mathbf{B}}_2 }  \\
\end{array} } \right)
\end{multline*}

Since $\mathbb{E}\left( {t_{ij'} } \right) = \mathbb{E}\left( {t_0 } \right) + sj'$ and $\mathbb{E}\left( {t_{ij} t_{ij'} } \right) = \mathbb{E}\left( {t_0^2 } \right) + s(j + j')\mathbb{E}\left( {t_0 } \right) + s^2 jj'$, and assuming without loss of generality that $\mathbb{E}\left( {t_0 } \right) = 0$ and therefore $\mathbb{E}\left( {t_0^2 } \right) = V(t_0 )$, we have that

{\mathbf{B}}_2  = p_e (1 - p_e )\left( {\begin{array}{*{20}c}
{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } & {s\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } }  \\
{s\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } } & {V\left( {t_0 } \right)\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } }  + s^2 \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } } }  \\
\end{array} } \right).

We are interested in the [V+4,V+4] component of ${\mathbf{M}}^{ - 1} $, which corresponds to $NVar\left( {\hat \gamma _3 } \right)$. Now,

{\mathbf{M}}^{ - 1}  = {\mathbf{Q'{\rm B}}}^{ - 1} {\mathbf{Q}}  = \left( {\begin{array}{*{20}c}
{{\mathbf{B}}_1^{ - 1}  + {\mathbf{Q}}_1 '{\mathbf{B}}_2^{ - 1} {\mathbf{Q}}_1 } & {{\mathbf{Q}}_1 '{\mathbf{B}}_2^{ - 1} }  \\
{{\mathbf{B}}_2^{ - 1} {\mathbf{Q}}_1 } & {{\mathbf{B}}_2^{ - 1} }  \\
\end{array} } \right),

\begin{multline*}
{\mathbf{B}}_2^{ - 1}  = \frac{1}
{{p_e (1 - p_e )\left[ {V(t_0 )\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)^2  + s^2 \det ({\mathbf{A}})} \right]}} \\
\left( {\begin{array}{*{20}c}
{V\left( {t_0 } \right)\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } }  + s^2 \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } } } & { - s\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } }  \\
{ - s\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } } & {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }  \\
\end{array} } \right).
\end{multline*}

Thus, the [V+4,V+4] component of ${\mathbf{M}}^{ - 1} $ is

\frac{{\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)}}
{{p_e (1 - p_e )\left[ {V(t_0 )\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)^2  + s^2 \det ({\mathbf{A}})} \right]}}.

If we follow Lachin's approach <cit.>, instead of using the asymptotic variance use the variance of ${\mathbf{\hat B}}$ conditional on the covariates, which is

\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1},

and redefine ${\mathbf{\Sigma }}_{\rm B} $ as

\left( {\frac{1}
{N}\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1}

so that the test statistic is still

T = \frac{{\sqrt {N\,} {\mathbf{c'\hat {\rm B}}}}}
{{\sqrt {{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}} }}.

Then, we would take the expected value of the non-centrality parameter under the alternative hypothesis over the distribution of ${\mathbf{X}}_i $, i.e. we would compute $\mathbb{E}\left[ {T^2 |H_1 } \right]$. If we assume that everyone is observed at the same set of time points ($V(t_0 ) = 0$), then the only random covariate is exposure and we have

\begin{multline*}
\frac{1}
{N}\sum\limits_i {{\mathbf{X}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X'}}_i }  = \\
\left( {\begin{array}{*{20}c}
{\sum\limits_{j,j'}^{} {v_{jj'} } } & {} & {} & {}  \\
{s\sum\limits_{j,j'}^{} {jv_{jj'} } } & {s^2 \sum\limits_{j,j'}^{} {jj'v_{jj'} } } & {} & {}  \\
{\left( {\frac{{\sum\limits_i {k_i } }}
{N}} \right)\sum\limits_{j,j'}^{} {v_{jj'} } } & {s\left( {\frac{{\sum\limits_i {k_i } }}
{N}} \right)\sum\limits_{j,j'}^{} {jv_{jj'} } } & {\left( {\frac{{\sum\limits_i {k_i ^2 } }}
{N}} \right)\sum\limits_{j,j'}^{} {v_{jj'} } } & {}  \\
{s\left( {\frac{{\sum\limits_i {k_i } }}
{N}} \right)\sum\limits_{j,j'}^{} {jv_{jj'} } } & {\left( {\frac{{s^2 \sum\limits_i {k_i } }}
{N}} \right)\sum\limits_{j,j'}^{} {jj'v_{jj'} } } & {\left( {\frac{{s\sum\limits_i {k_i } }}
{N}} \right)\sum\limits_{j,j'}^{} {jv_{jj'} } } & {\left( {\frac{{s^2 \sum\limits_i {k_i } }}
{N}} \right)\sum\limits_{j,j'}^{} {jj'v_{jj'} } }  \\
\end{array} } \right)
\end{multline*}

and the [4,4] component of the inverse is

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)}}
{{s^2 \det ({\mathbf{A}})\left( {\frac{{\sum\limits_i {k_i } }}
{N}} \right)\left( {1 - \frac{{\sum\limits_i {k_i } }}
{N}} \right)}}.

Following the same steps as in Appendix <ref> we can derive that

\mathbb{E}\left[ {T^2 |H_1 } \right] = \frac{{\left( {N - 1} \right)\gamma _3^2 s^2 \det ({\mathbf{A}})p_e \left( {1 - p_e } \right)}}
{{\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)}}.

The non-centrality parameter with the approach we followed in the paper is

\frac{{N\gamma _3^2 s^2 \det ({\mathbf{A}})p_e \left( {1 - p_e } \right)}}
{{\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)}},

so there is only a $\left( {1 - \frac{1}{N}} \right)$ correction compared with the one obtained with Lachin's method.

§.§ Proof that $N\,Var\left( {\hat \eta _5 } \right) =
{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)}}
{{p_e (1 - p_e )s^2 \det ({\mathbf{A}})}}$ under model (<ref>).

From model (<ref>), we have

\begin{multline*}
{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i  = \left( {\begin{array}{*{20}c}
1 &  \cdots  & 1 &  \cdots  & 1  \\
{t_{i0} } &  \cdots  & {t_{i0} } &  \cdots  & {t_{i0} }  \\
0 &  \cdots  & {sj} &  \cdots  & {sr}  \\
{k_i } &  \cdots  & {k_i } &  \cdots  & {k_i }  \\
{k_i t_{i0} } &  \cdots  & {k_i t_{i0} } &  \cdots  & {k_i t_{i0} }  \\
0 &  \cdots  & {k_i sj} &  \cdots  & {k_i sr}  \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
{v_{00} } &  \cdots  & {v_{0r} }  \\
\vdots  &  \ddots  &  \vdots   \\
{v_{r0} } &  \cdots  & {v_{rr} }  \\
\end{array} } \right) \\ % new line
\left( {\begin{array}{*{20}c}
1 & {t_{i0} } & 0 & {k_i } & {k_i t_{i0} } & 0  \\
\vdots  &  \vdots  &  \vdots  &  \vdots  &  \vdots  &  \vdots   \\
1 & {t_{i0} } & {sj} & {k_i } & {k_i t_{i0} } & {k_i sj}  \\
\vdots  &  \vdots  &  \vdots  &  \vdots  &  \vdots  &  \vdots   \\
1 & {t_{i0} } & {sr} & {k_i } & {k_i t_{i0} } & {k_i sr}  \\
\end{array} } \right)
\end{multline*}

and we can deduce using the following results derived in appendices 1.1 and 1.2, i.e.

\begin{gathered}
\mathbb{E}\left( {t_0 } \right) = \bar t_0  = 0, \mathbb{E}\left( {t_0^2 } \right) = V\left( {t_0 } \right),  \mathbb{E}\left( k \right) = \mathbb{E}\left( {k^2 } \right) = p_e,  \\
\mathbb{E}\left( {kt_0 } \right) = \rho _{\operatorname{e} ,t_0 } \sqrt {p_e (1 - p_e )} \sqrt {V\left( {t_0 } \right)}  = \overline {kt}, \\
\mathbb{E}\left( {kt_0^2 } \right) = V\left( {t_0 } \right)\left[ {p_e  + \rho _{\operatorname{e} ,t_0 }^2 (1 - 2p_e )} \right] = \overline {kt^2 },
\end{gathered}

\begin{multline*}
\mathbb{E}\left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } \right) = \\
\left( {\begin{array}{*{20}c}
{\sum\limits_{j,j'}^{} {v_{jj'} } } & {} & {} & {} & {} & {}  \\
0 & {V\left( {t_0 } \right)\sum\limits_{j,j}^{} {v_{jj'} } } & {} & {} & {} & {}  \\
{s\sum\limits_{j,j'}^{} {jv_{jj'} } } & 0 & {s^2 \sum\limits_{j,j'}^{} {jj'v_{jj'} } } & {} & {} & {}  \\
{p_e \sum\limits_{j,j'}^{} {v_{jj'} } } & {\overline {kt} \sum\limits_{j,j'}^{} {v_{jj'} } } & {sp_e \sum\limits_{j,j'}^{} {jv_{jj'} } } & {p_e \sum\limits_{j,j'}^{} {v_{jj'} } } & {} & {}  \\
{\overline {kt} \sum\limits_{j,j'}^{} {v_{jj'} } } & {\overline {kt^2 } \sum\limits_{j,j'}^{} {v_{jj'} } } & {s\overline {kt} \sum\limits_{j,j'}^{} {jv_{jj'} } } & {\overline {kt} \sum\limits_{j,j'}^{} {v_{jj'} } } & {\overline {kt^2 } \sum\limits_{j,j'}^{} {v_{jj'} } } & {}  \\
{sp_e \sum\limits_{j,j'}^{} {jv_{jj'} } } & {s\overline {kt} \sum\limits_{j,j'}^{} {jv_{jj'} } } & {s^2 p_e \sum\limits_{j,j'}^{} {jj'v_{jj'} } } & {sp_e \sum\limits_{j,j'}^{} {jv_{jj'} } } & {s\overline {kt} \sum\limits_{j,j'}^{} {jv_{jj'} } } & {s^2 p_e \sum\limits_{j,j'}^{} {jj'v_{jj'} } }  \\
\end{array} } \right)
\end{multline*}

The [6,6] component of the inverse of this matrix is

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)}}
{{p_e (1 - p_e )s^2 \det ({\mathbf{A}})}},

as we derived in Appendix <ref> for the LDD case with $V(t_0 ) = 0$.

§.§ Proof that $s\hat \lambda _1  = \hat \eta _5 $ and $s^2 Var\left( {\hat \lambda _1 } \right) = Var\left( {\hat \eta _5 } \right)$ from models (<ref>) and (<ref>)

The GLS estimator has the expression

{\mathbf{\hat B}} = \left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1} \left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{Y}}_i } } \right),

where ${\mathbf{X}}_i $ is the matrix of covariates for participant $i$. To derive $\hat \eta _5 $ from model (<ref>) we only need the sixth row of

\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1},

which we denote

\left[ {\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1} } \right]_{[6]} ,

and then

\hat \eta _5  = \left[ {\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1} } \right]_{[6]} \left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{Y}}_i } } \right),

which we rewrite as

\hat \eta _5  = \left( {\sum\limits_{i = 1}^N {\left[ {\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1} } \right]_{[6]} {\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{Y}}_i } } \right).

Then, by calling

{\mathbf{c}}_\eta   = \left[ {\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1} } \right]_{[6]} {\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1}

we have

\hat \eta _5  = \left( {\sum\limits_{i = 1}^N {{\mathbf{c}}_\eta  {\mathbf{Y}}_i } } \right).

In Appendix <ref> we derived and expression for

\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } } \right)

and from that we can derive

\begin{multline*}
\left[ {\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1} } \right]_{[6]}  =
\frac{1} {{\det ({\mathbf{A}})p_e (1 - p_e )s}} \\
\left( {\begin{array}{*{20}c}
{p_e \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } ,} & {0,} & {\frac{{ - p_e }}
{s}\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } ,} & { - \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } ,} } & {0,} & {\frac{1}
{s}\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }  \\
\end{array} } \right).
\end{multline*}

For convenience, some terms can be rewritten in vector form. We define ${\mathbf{1}}$ as a $(r + 1) \times 1$ vector of ones, and ${\mathbf{t}}$ as a $(r + 1) \times 1$ matrix such that ${\mathbf{t'}} = (0,1,2, \ldots ,r)$, and then

\begin{multline*}
\left[ {\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1} } \right]_{[6]}   = \frac{1}
{{\det ({\mathbf{A}})p_e (1 - p_e )s}} \\
\left( {\begin{array}{*{20}c}
{p_e {\mathbf{t'\Sigma }}^{ - 1} {\mathbf{1}},} & {0,} & {\frac{{ - p_e }}
{s}{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}},} & { - {\mathbf{t'\Sigma }}^{ - 1} {\mathbf{1}},} & {0,} & {\frac{1}
{s}{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}}  \\
\end{array} } \right).
\end{multline*}

We can also derive

{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1}  = \left( {\begin{array}{*{20}c}
1 &  \cdots  & 1 &  \cdots  & 1  \\
{t_{i0} } &  \cdots  & {t_{i0} } &  \cdots  & {t_{i0} }  \\
0 &  \cdots  & {sj} &  \cdots  & {sr}  \\
{k_i } &  \cdots  & {k_i } &  \cdots  & {k_i }  \\
{k_i t_{i0} } &  \cdots  & {k_i t_{i0} } &  \cdots  & {k_i t_{i0} }  \\
0 &  \cdots  & {k_i sj} &  \cdots  & {k_i sr}  \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
{v_{00} } &  \cdots  & {v_{0r} }  \\
\vdots  &  \ddots  &  \vdots   \\
{v_{r0} } &  \cdots  & {v_{rr} }  \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
{{\mathbf{1'\Sigma }}^{ - 1} }  \\
{t_{i0} {\mathbf{1'\Sigma }}^{ - 1} }  \\
{s{\mathbf{t'\Sigma }}^{ - 1} }  \\
{k_i {\mathbf{1'\Sigma }}^{ - 1} }  \\
{k_i t_{i0} {\mathbf{1'\Sigma }}^{ - 1} }  \\
{sk_i {\mathbf{t'\Sigma }}^{ - 1} }  \\
\end{array} } \right).

\begin{multline*}
{\mathbf{c}}_\eta   = \left[ {\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } } \right)^{ - 1} } \right]_{[6]} {\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1}  =
\frac{1} {{\det ({\mathbf{A}})p_e (1 - p_e )s}} \\
\left( {p_e {\mathbf{t'\Sigma }}^{ - 1} {\mathbf{11'\Sigma }}^{ - 1}  - p_e {\mathbf{t'\Sigma }}^{ - 1} {\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}} - k_i {\mathbf{t'\Sigma }}^{ - 1} {\mathbf{11'\Sigma }}^{ - 1}  + k_i {\mathbf{t'\Sigma }}^{ - 1} {\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right) \\
= \frac{{( - p_e  + k_i )\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)}}
{{\det ({\mathbf{A}})p_e (1 - p_e )s}}{\mathbf{t'}}\left( {{\mathbf{\Sigma }}^{ - 1}  - {\mathbf{\Sigma }}^{ - 1} {\mathbf{1}}\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)^{ - 1} {\mathbf{1'\Sigma }}^{ - 1} } \right).
\end{multline*}

Now let us move to model (<ref>). Define the $r \times (r + 1)$ matrix

{\mathbf{\Delta }} = \left( {\begin{array}{*{20}c}
{ - 1} & 1 & 0 &  \cdots  &  \cdots  & 0  \\
0 & { - 1} & 1 & 0 &  \cdots  & 0  \\
0 & 0 &  \ddots  &  \ddots  &  \ddots  &  \vdots   \\
\vdots  & {} &  \ddots  &  \ddots  &  \ddots  & 0  \\
0 &  \cdots  &  \cdots  & 0 & { - 1} & 1  \\

\end{array} } \right).

Note that ${\mathbf{\Delta Y}}_i $ contains the differences of the response from one visit to the next, so ${\mathbf{\Delta Y}}_i $ is the response variable in model (2.10). The covariance matrix of the response for model (<ref>) will then be ${\mathbf{\Delta \Sigma \Delta '}}$. Let us call ${\mathbf{Z}}$ the $r \times 2$ matrix of covariates for model (<ref>),

{\mathbf{Z'}} = \left( {\begin{array}{*{20}c}
1 &  \cdots  & 1  \\
{k_i } &  \cdots  & {k_i }  \\
\end{array} } \right)

and ${\mathbf{X}}_{[3,6]} $ a $(r + 1) \times 2$ matrix containing the third and sixth column of ${\mathbf{X}}_i $ from model (<ref>),

{\mathbf{X'}}_{[3,6]}  = \left( {\begin{array}{*{20}c}
0 & {} & {sj} & {} & {sr}  \\
0 & {} & {k_i sj} & {} & {k_i sr}  \\
\end{array} } \right).

Then, it can be noted that $\frac{1} {s}{\mathbf{\Delta X}}_{[3,6]}  = {\mathbf{Z}}$. Therefore, the GLS estimate of $\lambda _1 $ can be written as

\begin{multline*}
\hat \lambda _1  = \left[ {\left( {\sum\limits_{i = 1}^N {{\mathbf{Z'}}_i \left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{Z}}_i } } \right)^{ - 1} } \right]_{[2]} \left( {\sum\limits_{i = 1}^N {{\mathbf{Z'}}_i \left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta Y}}_i } } \right) =\\
\left( {\frac{1}
{s}\sum\limits_{i = 1}^N {\left[ {\left( {\frac{1}
{{s^2 }}\sum\limits_{i = 1}^N {\left( {{\mathbf{\Delta X}}_{[3,6]} } \right)^\prime  \left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta X}}_{[3,6]} } } \right)^{ - 1} } \right]_{[2]} \left( {{\mathbf{\Delta X}}_{[3,6]} } \right)^\prime  \left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta Y}}_i } } \right) \\
= \left( {\sum\limits_{i = 1}^N {s\left[ {\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_{[3,6]} {\mathbf{\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta X}}_{[3,6]} } } \right)^{ - 1} } \right]_{[2]} {\mathbf{X'}}_{[3,6]} {\mathbf{\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta Y}}_i } } \right) \\
= \sum\limits_{i = 1}^N {{\mathbf{c}}_\lambda  {\mathbf{Y}}_i }.
\end{multline*}

\begin{multline*}
\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_{[3,6]} {\mathbf{\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta X}}_{[3,6]} } } \right) \\
= s^2 \left( {\begin{array}{*{20}c}
1 &  \cdots  & 1  \\
{k_i } &  \cdots  & {k_i }  \\
\end{array} } \right)_{(2 \times r)} \left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} _{(r \times r)} \left( {\begin{array}{*{20}c}
1 & {k_i }  \\
\vdots  &  \vdots   \\
1 & {k_i }  \\
\end{array} } \right)_{(r \times 2)} \\
= s^2 \left( {{\mathbf{t'\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta t}}} \right)\left( {\begin{array}{*{20}c}
1 & {p_e }  \\
{p_e } & {p_e }  \\
\end{array} } \right),
\end{multline*}

\begin{multline*}
\left[ {\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_{[3,6]} {\mathbf{\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta X}}_{[3,6]} } } \right)^{ - 1} } \right]_{[2]}  =\\
\frac{1}
{{p_e (1 - p_e )s^2 \left( {{\mathbf{t'\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta t}}} \right)}}\left( {\begin{array}{*{20}c}
{ - p_e } & 1  \\
\end{array} } \right).
\end{multline*}

Now, by property B.3.5 of Seber (1984, page 536),

{\mathbf{\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta }} = {\mathbf{\Sigma }}^{ - 1}  - {\mathbf{\Sigma }}^{ - 1} {\mathbf{1}}\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)^{ - 1} {\mathbf{1'\Sigma }}^{ - 1}.

\frac{1}
{{\left( {{\mathbf{t'\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta t}}} \right)}} = \frac{1}
{{\left( {{\mathbf{t'}}\left( {{\mathbf{\Sigma }}^{ - 1}  - {\mathbf{\Sigma }}^{ - 1} {\mathbf{1}}\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)^{ - 1} {\mathbf{1'\Sigma }}^{ - 1} } \right){\mathbf{t}}} \right)}}

and with some algebra this expression equals

\frac{{\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)}}
{{\det ({\mathbf{A}})}}.

\left[ {\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_{[3,6]} {\mathbf{\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta X}}_{[3,6]} } } \right)^{ - 1} } \right]_{[2]}  = \frac{{\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)}}
{{p_e (1 - p_e )s^2 \det ({\mathbf{A}})}}\left( {\begin{array}{*{20}c}
{ - p_e } & 1  \\
\end{array} } \right).

Now we need to derive ${\mathbf{X'}}_{[3,6]} {\mathbf{\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right){\mathbf{\Delta }}$, and by using Seber's property again we have

{\mathbf{X'}}_{[3,6]} {\mathbf{\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right){\mathbf{\Delta }} = {\mathbf{X'}}_{[3,6]} \left( {{\mathbf{\Sigma }}^{ - 1}  - {\mathbf{\Sigma }}^{ - 1} {\mathbf{1}}\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)^{ - 1} {\mathbf{1'\Sigma }}^{ - 1} } \right).

\begin{multline*}
{\mathbf{c}}_\lambda   = s\left[ {\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_{[3,6]} {\mathbf{\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right)^{ - 1} {\mathbf{\Delta X}}_{[3,6]} } } \right)^{ - 1} } \right]_{[2]} {\mathbf{X'}}_{[3,6]} {\mathbf{\Delta '}}\left( {{\mathbf{\Delta \Sigma \Delta '}}} \right){\mathbf{\Delta }} = \\
= \frac{{\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)}}
{{\det ({\mathbf{A}})sp_e (1 - p_e )}}\left( {\begin{array}{*{20}c}
{ - p_e } & 1  \\
\end{array} } \right){\mathbf{X'}}_{[3,6]} \left( {{\mathbf{\Sigma }}^{ - 1}  - {\mathbf{\Sigma }}^{ - 1} {\mathbf{1}}\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)^{ - 1} {\mathbf{1'\Sigma }}^{ - 1} } \right) \\
= \frac{{\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)}}
{{\det ({\mathbf{A}})p_e (1 - p_e )}}\left( {\begin{array}{*{20}c}
{ - p_e } & 1  \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
{{\mathbf{t'}}}  \\
{k_i {\mathbf{t'}}}  \\
\end{array} } \right)\left( {{\mathbf{\Sigma }}^{ - 1}  - {\mathbf{\Sigma }}^{ - 1} {\mathbf{1}}\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)^{ - 1} {\mathbf{1'\Sigma }}^{ - 1} } \right) \\
= \frac{{\left( { - p_e  + k_i } \right)\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)}}
{{\det ({\mathbf{A}})p_e (1 - p_e )}}{\mathbf{t'}}\left( {{\mathbf{\Sigma }}^{ - 1}  - {\mathbf{\Sigma }}^{ - 1} {\mathbf{1}}\left( {{\mathbf{1'\Sigma }}^{ - 1} {\mathbf{1}}} \right)^{ - 1} {\mathbf{1'\Sigma }}^{ - 1} } \right)
\end{multline*}

and we can observe that ${\mathbf{c}}_\lambda   = \frac{{{\mathbf{c}}_\eta  }}
{s} $ and therefore $s\hat \lambda _1  = \hat \eta _5 $ and $s^2 Var\left( {\hat \lambda _1 } \right) = Var\left( {\hat \eta _5 } \right)$.

§ BIAS AND/OR INEFFICIENCY OF THE ANCOVA, SLANC, SLAIN TESTS UNDER CS

These tests <cit.> have the form

T = \frac{{N{\kern 1pt} p_e (1 - p_e )\left( {\bar S_1  - \bar S_0 } \right)^2 }}
{{{\mathbf{c'\Sigma c}}}},

where $\bar S_k $ is exposure group $k$'s mean, $k = (0,1)$, of a summary measure, $S_i $, that is a linear combination of the repeated measures of each subject, $S_i  = {\mathbf{c'Y}}_i $. The vector ${\mathbf{c'}}$ defines the summary measures, which could be the within-subject mean of the repeated measures, the within-subject slope, or ANCOVA, SLANC and SLAIN. Let $n_k $ be the number of participants in exposure group $k$. Then,

\bar S_k  = \frac{{\sum\limits_{i = 1}^N {S_i I\left\{ {k_i  = k} \right\}} }}
{{n_k }} = {\mathbf{c'}}\left( {\frac{{\sum\limits_{i = 1}^N {{\mathbf{Y}}_i I\left\{ {k_i  = k} \right\}} }}
{{n_k }}} \right) = {\mathbf{c'\bar Y}}_k,

where $I\left\{ {k_i  = k} \right\}$ is an indicator variable that takes the value one when $k_i  = k$ and zero otherwise, and ${\mathbf{\bar Y}}_k $ is the $(r + 1) \times 1$ vector of sample means for each time in group k. Thus,

T = \frac{{N{\kern 1pt} p_e (1 - p_e )\left( {{\mathbf{c'}}\left( {{\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 } \right)} \right)^2 }}
{{{\mathbf{c'\Sigma c}}}}.

The summary measure approach is appropriate when all subjects in the two exposure groups are observed in the same set of time points, i.e. when $V(t_0 ) = 0$. Otherwise, exposure group and time can be correlated and the summary measure approach would produce biased estimates of the effect. Clearly, $\mathbb{E}\left[ {\bar S_k } \right] = {\mathbf{c'\mu }}_k $, where ${\mathbf{\mu }}_k $ is the vector of true means for each time in group k, and $\mathbb{E}\left[ {\bar S_1  - \bar S_0 } \right] = {\mathbf{c'}}({\mathbf{\mu }}_1  - {\mathbf{\mu }}_0 )$. If

{\mathbf{c'}} = \left( {\frac{1}
{{r + 1}}, \cdots ,\frac{1}
{{r + 1}}} \right),

we are testing the equality of the means of the two groups. The goal is to choose ${\mathbf{c'}}$ so that $\mathbb{E}\left( {T|H_0 } \right) = 0$ (valid) and for which the power of $T$ is at its maximum possible under $H_A $ (efficient). In this paper, interest is in two hypotheses which commonly arise in longitudinal studies, CMD and LDD. The CMD hypothesis specifies that the mean group differences are constant over time, or, equivalently, that the two response profiles are parallel, i.e. $({\mathbf{\mu }}_1  - {\mathbf{\mu }}_0 ) = p_1 \mu _{00} $. The LDD hypothesis specifies that the group mean differences are a linear function of time, or, equivalently, that there is a linear interaction between exposure and time, i.e.

{\mathbf{\mu }}_1  - {\mathbf{\mu }}_0  = (p_1  + \frac{{p_2 p_3 }}
{\tau }t_j )\mu _{00}.

The parameter of interest under CMD is $p_1 $, and one wants to test $H_0 :p_1  = 0
$ vs. $H_A :p_1  \ne 0$. The parameter of interest under LDD is $p_3 $, and one wants to test $H_0 :p_3  = 0$ vs. $H_A :p_3  \ne 0$. To construct the relevant test statistic in each case, we need to identify a vector ${\mathbf{c'}}$ such that $H_0 :{\mathbf{c'}}({\mathbf{\mu }}_1  - {\mathbf{\mu }}_0 ) = 0$ vs. $H_A :{\mathbf{c'}}({\mathbf{\mu }}_1  - {\mathbf{\mu }}_0 ) \ne 0$ provides a test statistic of form $T$ that is the most powerful under the alternative hypothesis.

§.§ Unbiasedness of the ANCOVA, SLAIN and SLANC test statistics under CMD, CS and $V(t_0 ) = 0$

Under CMD, the expected value of the numerator of test statistics of the form discussed above, which include ANCOVA, SLAIN and SLANC, is

\mathbb{E}\left[ {{\mathbf{c'}}({\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 )|\,H_0 } \right] = {\mathbf{c'}}\mathbb{E}\left[ {({\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 )|\,H_0 } \right] = {\mathbf{c'}}\left( {\begin{array}{*{20}c}
{p_1 \mu _{00} ,} &  \cdots  & {,p_1 \mu _{00} }  \\
\end{array} } \right).

Under $H_0 $, $p_1  = 0$ so $\mathbb{E}\left[ {{\mathbf{c'}}({\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 )|\,H_0 } \right]$ is 0 and all vectors ${\mathbf{c'}}
$ produce unbiased estimators, including those given for ANCOVA, SLANC and SLAIN, for which the vectors ${\mathbf{c'}}$ under CS are

\begin{gathered}
{\mathbf{c'}} = \left( { - \rho ,\frac{1}
{r}, \cdots ,\frac{1}
{r}} \right), {\mathbf{c'}} = \frac{6}
{{r(r + 1)(r + 2)}}\left( { - \rho ,\;2 - r, \cdots ,\;2j - r, \cdots ,\;r} \right)
, \\
c_j  = \frac{{12j + 6\rho r(2j - r - 1)}}
{{r(r + 1)\left[ {\rho r(r - 1) + 2(2r + 1)} \right]}},
\end{gathered}

for ANCOVA, SLANC and SLAIN, respectively <cit.>.

§.§ Inefficiency of the ANCOVA, SLAIN and SLANC tests under CMD, CS and $V(t_0 ) = 0$

§.§.§ Inefficiency of the ANCOVA test under CMD and CS and $V(t_0 ) = 0$

Frison:1992 showed that, under CS, ANCOVA is the most powerful test for the CMD alternative hypothesis when the two groups have the same response at baseline, as would be the case in expectation in a randomized clinical trial, and when the post-baseline measures have a constant difference. However, in observational studies, where baseline has no special significance, the difference at baseline is assumed to be the same as the difference at the other time points. We will show here that the ANCOVA test is less powerful than the test we discussed in this manuscript, based on the GLS estimator, which is known to be the best linear unbiased estimator and therefore produces the most powerful test among those unbiased.

To obtain the GLS estimator and its corresponding test statistic, we fit the model $Y_{ij}  = \beta _0  + \beta _1 k_i $, similarly to equation (<ref>) but without including the effect of time. Because $V(t_0 ) = 0$, exposure and time are independent and inclusion or exclusion of the time variable does not affect the estimation of $\beta _1 $, the parameter of interest under CMD. Reparameterizing using $\alpha _0  = \beta _0 $, $\alpha _1  = \beta _0  + \beta _1 $, and $\beta _1  = \alpha _1  - \alpha _0$, the model becomes $Y_{ij}  = \alpha _0 (1 - k_i ) + \alpha _1 k_i $. We derive the ${\mathbf{c}}$ vector for the test based upon the GLS estimator, and show that it is not equal to the ${\mathbf{c}}$ vector for the ANCOVA test. The design matrix for subject $i$ will contain a column of ones and a column of zeros if subject $i$ is unexposed, and a column of zeros and a column of ones if subject $i$ is unexposed. The GLS estimator for $\left( {\hat \alpha _0 ,\hat \alpha _1 } \right)$ is

\left( {\sum\limits_{i = 1}^N {\left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } \right)} } \right)^{ - 1} \left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{Y}}_i } } \right).

\left( {\sum\limits_{i = 1}^N {\left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}^{ - 1} {\mathbf{X}}_i } \right)} } \right)^{ - 1}  = \frac{1}
{{N\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }}\left( {\begin{array}{*{20}c}
{{1 - p_e }}} & 0  \\
0 & {\frac{1}
{{p_e }}}  \\
\end{array} } \right),

where $v_{jj'} $ is the $(j,j'){\text{th}}$ element of ${\mathbf{\Sigma }}^{ - 1}
$, then

\hat \alpha _0  = \frac{1}
{{N(1 - p_e )\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }}\sum\limits_{i = 1}^{N(1 - p_e )} {\left( {1, \ldots ,1} \right){\mathbf{\Sigma }}^{ - 1} {\mathbf{Y}}_i }  = \frac{1}
{{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }}\left( {1, \ldots ,1} \right){\mathbf{\Sigma }}^{ - 1} {\mathbf{\bar Y}}_0

\hat \alpha _1  = \frac{1}
{{Np_e \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }}\sum\limits_{i = N(1 - p_e ) + 1}^N {\left( {1, \ldots ,1} \right){\mathbf{\Sigma }}^{ - 1} {\mathbf{Y}}_i }  = \frac{1}
{{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }}\left( {1, \ldots ,1} \right){\mathbf{\Sigma }}^{ - 1} {\mathbf{\bar Y}}_1,

\beta _1  = \frac{1}
{{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }}\left( {1, \ldots ,1} \right){\mathbf{\Sigma }}^{ - 1} {\mathbf{\bar Y}}_1  - \frac{1}
{{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }}\left( {1, \ldots ,1} \right){\mathbf{\Sigma }}^{ - 1} {\mathbf{\bar Y}}_0.

{\mathbf{c'}}_{GLS}  = \frac{1}
{{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } }}\left( {1, \ldots ,1} \right){\mathbf{\Sigma }}^{ - 1}.

This coincides with the result of Frison:1997 that the optimal ${\mathbf{c'}}$ is proportional $({\mathbf{\mu }}_1  - {\mathbf{\mu }}_0 )'{\mathbf{\Sigma }}^{ - 1} $. Since ANCOVA has a different vector ${\mathbf{c'}}$, ANCOVA could be, at best, as powerful as the GLS under CS.

Under  CMD and with a CS covariance specifically, we show that the test based upon the GLS estimator is explicitly more powerful than the ANCOVA test. The non-centrality parameter (NCP) for the test $T$ is

\lambda  = \frac{{\left[ {{\mathbf{c'}}({\mathbf{\mu }}_1  - {\mathbf{\mu }}_0 )} \right]^2 }}
{{{\mathbf{c'\Sigma c}}}}.

For ANCOVA, the numerator of $\lambda $ is

\left[ {{\mathbf{c'}}({\mathbf{\mu }}_1  - {\mathbf{\mu }}_0 )} \right]^2  = \left[ {\left( { - \rho ,\frac{1}
{r}, \cdots ,\frac{1}
{r}} \right)\left( {\begin{array}{*{20}c}
{p_1 \mu _{00} } &  \cdots  & {p_1 \mu _{00} }  \\
\end{array} } \right)^\prime  } \right]^2  =
\left( {p_1 \mu _{00} } \right)^2 \left( {1 - \rho } \right)^2.

The denominator of $\lambda _{ANCOVA} $ is

\begin{multline*}
{\mathbf{c'\Sigma c}} = \sigma ^2 \left( { - \rho ,\frac{1}
{r}, \cdots ,\frac{1}
{r}} \right)\left( {\begin{array}{*{20}c}
1 & \rho  &  \cdots  & \rho   \\
\rho  & 1 &  \ddots  &  \vdots   \\
\vdots  &  \ddots  &  \ddots  & \rho   \\
\rho  &  \cdots  & \rho  & 1  \\
\end{array} } \right)\left( { - \rho ,\frac{1}
{r}, \cdots ,\frac{1}
{r}} \right)^\prime   \\
= \sigma ^2 \left[ { - \rho ^2  + \frac{1}
{r}\left( {1 + (r - 1)\rho } \right)} \right]
\end{multline*}

The vector ${\mathbf{c'}}$ for the GLS approach is

\begin{multline*}
{\mathbf{c'}} = \frac{{\sigma ^2 (1 + r\rho )}}
{{(r + 1)}}\left( {\frac{1}
{{\sigma ^2 (1 + r\rho )}},\frac{1}
{{\sigma ^2 (1 + r\rho )}}, \cdots ,\frac{1}
{{\sigma ^2 (1 + r\rho )}}} \right)  \\
= \left( {\frac{1}
{{r + 1}},\frac{1}
{{r + 1}},...,\frac{1}
{{r + 1}}} \right).
\end{multline*}

Thus, the numerator of the $\lambda _{GLS} $ is

\left[ {{\mathbf{c'}}({\mathbf{\mu }}_1  - {\mathbf{\mu }}_0 )} \right]^2  = \left[ {\left( {\frac{1}
{{r + 1}},\frac{1}
{{r + 1}}, \cdots ,\frac{1}
{{r + 1}}} \right)\left( {\begin{array}{*{20}c}
{p_1 \mu _{00} ,} &  \cdots  & {,p_1 \mu _{00} }  \\
\end{array} } \right)^\prime  } \right]^2
= \left( {p_1 \mu _{00} } \right)^2.

The denominator of $\lambda _{GLS} $ is

\begin{multline*}
{\mathbf{c'\Sigma c}} = \sigma ^2 \left( {\frac{1}
{{r + 1}},\frac{1}
{{r + 1}}, \cdots ,\frac{1}
{{r + 1}}} \right)\left( {\begin{array}{*{20}c}
1 & \rho  &  \cdots  & \rho   \\
\rho  & 1 &  \ddots  &  \vdots   \\
\vdots  &  \ddots  &  \ddots  & \rho   \\
\rho  &  \cdots  & \rho  & 1  \\
\end{array} } \right)\left( {\frac{1}
{{r + 1}},\frac{1}
{{r + 1}}, \cdots ,\frac{1}
{{r + 1}}} \right)^\prime  \\
= \frac{{r\rho  + 1}}
{{r + 1}}.
\end{multline*}

The ratio of NCPs is

\frac{{\lambda _{GLS} }}
{{\lambda _{ANCOVA} }} = \frac{{\frac{{\left( {p_1 \mu _{00} } \right)^2 }}
{{\left( {\frac{{r\rho  + 1}}
{{r + 1}}} \right)}}}}
{{\frac{{\left( {p_1 \mu _{00} } \right)^2 \left( {1 - \rho } \right)^2 }}
{{ - \rho ^2  + \frac{1}
{r}\left( {1 + (r - 1)\rho } \right)}}}} = \frac{{r + 1}}
{{r(1 - \rho )}} > 1,

proving that a test statistic based on the GLS approach is more powerful than ANCOVA under CMD with a  CS covariance matrix.

§.§.§ Inefficiency of the SLANC and SLAIN tests under CMD

The vector ${\mathbf{c'}}$ for SLANC is defined by

{\mathbf{c'}} = \frac{6}
{{r(r + 1)(r + 2)}}\left( { - \rho ,\;2 - r, \cdots ,\;2j - r, \cdots ,\;r} \right),

and for SLAIN by

c_j  = \frac{{12j + 6\rho r(2j - r - 1)}}
{{r(r + 1)\left[ {\rho r(r - 1) + 2(2r + 1)} \right]}}

under CS <cit.>.  Since the vector ${\mathbf{c'}}$ are not equal to ${\mathbf{c'}}_{GLS}$, they are both inefficient.

§.§ Biasedness of the ANCOVA, SLAIN and SLANC test statistics under LDD

Under LDD, the expected value of the numerator of the ANCOVA test statistic is
\begin{multline*}
\mathbb{E}\left[ {{\mathbf{c'}}({\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 )|\,H_0 } \right] = {\mathbf{c'}}\mathbb{E}\left[ {({\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 )|\,H_0 } \right] = \\
= \left. {{\mathbf{c'}}\left( {\begin{array}{*{20}c}
{(p_1  + \frac{{p_2 p_3 }}
{\tau }t_0 )\mu _{00} ,} &  \cdots  & {,(p_1  + \frac{{p_2 p_3 }}
{\tau }t_r )\mu _{00} }  \\
\end{array} } \right)^\prime  } \right|_{H_0 :p_3  = 0}  \\
= {\mathbf{c'}}\left( {\begin{array}{*{20}c}
{p_1 \mu _{00} ,} &  \cdots  & {,p_1 \mu _{00} }  \\
\end{array} } \right)^\prime   =
\left( { - \rho ,\frac{1}
{r}, \cdots ,\frac{1}
{r}} \right)\left( {\begin{array}{*{20}c}
{p_1 \mu _{00} ,} &  \cdots  & {,p_1 \mu _{00} }  \\
\end{array} } \right)^\prime   = \\
p_1 \mu _{00} \left( {1 - \rho } \right) \ne 0.
\end{multline*}

The expected value of the numerator of the  SLANC test statistic is

\begin{multline*}
\mathbb{E}\left[ {{\mathbf{c'}}({\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 )|\,H_0 } \right] = {\mathbf{c'}}\mathbb{E}\left[ {({\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 )|\,H_0 } \right]  \\
= \left. {{\mathbf{c'}}\left( {\begin{array}{*{20}c}
{(p_1  + \frac{{p_2 p_3 }}
{\tau }t_0 )\mu _{00} ,} &  \cdots  & {,(p_1  + \frac{{p_2 p_3 }}
{\tau }t_r )\mu _{00} }  \\
\end{array} } \right)^\prime  } \right|_{H_0 :p_3  = 0} \\
= {\mathbf{c'}}\left( {\begin{array}{*{20}c}
{p_1 \mu _{00} ,} &  \cdots  & {,p_1 \mu _{00} }  \\
\end{array} } \right)^\prime   = p_1 \mu _{00} {\mathbf{c'}}\left( {1, \ldots ,1} \right)\\
= \frac{{6p_1 \mu _{00} }}
{{r(r + 1)(r + 2)}}\left( { - \rho  + \sum\limits_{j = 1}^r {\left( {2j - r} \right)} } \right) = \frac{{6p_1 \mu _{00} }}
{{r(r + 1)(r + 2)}}\left( {r - \rho } \right) \ne 0
\end{multline*}

The expected value of the numerator of the SLAIN test statistic is

\begin{multline*}
\mathbb{E}\left[ {{\mathbf{c'}}({\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 )|\,H_0 } \right] = p_1 \mu _{00} {\mathbf{c'}}\left( {\begin{array}{*{20}c}
1 &  \cdots  & 1  \\
\end{array} } \right) \\
= p_1 \mu _{00} \sum\limits_{j = 0}^r {\frac{{12j + 6\rho r(2j - r - 1)}}
{{r(r + 1)\left[ {\rho r(r - 1) + 2(2r + 1)} \right]}}} \\
= \frac{{p_1 \mu _{00} }}
{{r(r + 1)\left[ {\rho r(r - 1) + 2(2r + 1)} \right]}}\sum\limits_{j = 0}^r {12j + 6\rho r(2j - r - 1)} \\
= \frac{{6p_1 \mu _{00} (1 - \rho )}}
{{\left[ {\rho r(r - 1) + 2(2r + 1)} \right]}} \ne 0
\end{multline*}

Thus, ANCOVA, SLAIN  and SLANC are all biased under the null and therefore not valid in observational studies under LDD.

§ PROOF THAT TWO-STAGE AND GLS ARE EQUIVALENT APPROACHES UNDER CS OR RS FOR $V(T_0 ) = 0$

In the setting where all subjects are observed at the same set of time points, this appendix will proof:

(i) That the estimator of the difference of the rates of change in the two exposure groups obtained using the summary measure (two-stage) approach is algebraically equivalent to the estimator of $\gamma _3 $ obtained from fitting model (<ref>) by OLS.

(ii)That when the covariance matrix ${\mathbf{\Sigma }}_i  = {\mathbf{\Sigma }}$ has a CS or RS structure, the estimators from model (<ref>) obtained by OLS and GLS are algebraically equivalent. Given (i), this implies that the estimator from the summary measure approach is algebraically equivalent to the GLS estimator. We also show that this is not the case for DEX.

Given (i) and (ii), since the estimators from the summary measure (two-stage) approach, and GLS are the same linear combination of $\left( {{\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 } \right)$, ${\mathbf{d'}}\left( {{\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 } \right)$, once we assume a covariance structure for $Var\left[ {{\mathbf{Y}}_i |{\mathbf{X}}_i } \right] = {\mathbf{\Sigma }}_i $, the test statistic for the two methods is also equivalent and equal to

T = \frac{{{\mathbf{d'}}\left( {{\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 } \right)}}
{{\sqrt {Var\left( {{\mathbf{d'}}\left( {{\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 } \right)} \right)} }},

\begin{multline*}
Var\left( {{\mathbf{d'}}\left( {{\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 } \right)} \right) = {\mathbf{d'}}Var\left( {{\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 } \right){\mathbf{d}} \\
= {\mathbf{d'}}\left( {\frac{1}
{{Np_e }}Var\left( {{\mathbf{Y}}_{i,k_i  = 1} } \right) + \frac{1}
{{N(1 - p_e )}}Var\left( {{\mathbf{Y}}_{i,k_i  = 0} } \right)} \right){\mathbf{d}} = \frac{{{\mathbf{d'\Sigma d}}}}
{{Np_e (1 - p_e )}}.
\end{multline*}

§.§ Proof of (i)

§.§.§ Summary measure (two-stage) approach

Let ${\mathbf{Z}}_i $ be a $(r + 1) \times 2$ matrix that contains a column of ones and the column of times for participant $i$. Since all subjects are observed at the same set of time points then ${\mathbf{Z}}_i  = {\mathbf{Z}}$. Here, the summary measure is the subject-specific OLS slope associated with time from the regression of ${\mathbf{Y}}_i $ on ${\mathbf{Z}}_i  = {\mathbf{Z}}$. Let us call ${\mathbf{\hat \beta }}_i $, $i = 1, \ldots ,N$, the $(2 \times 1)$ vector containing the subject-specific intercept and slope of the regression, where ${\mathbf{\hat \beta }}_i  = \left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'Y}}_i
$. The subject-specific intercepts and slopes are averaged in each exposure group as follows,

{\mathbf{\hat \beta }}_k  = \frac{{\sum\limits_{i = 1}^N {\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}} {\mathbf{Y}}_i \,I\left\{ {k_i  = k} \right\}}}
{{n_k }} = \left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}\frac{{\sum\limits_{i = 1}^N {{\mathbf{Y}}_i I\left\{ {k_i  = k} \right\}} }}
{{n_k }} = \left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\bar Y}}_k,

where $I\left\{ {k_i  = k} \right\}$ is an indicator variable that takes the value one when $k_i  = k$ and zero otherwise; $n_k $ is the number of participants in exposure group $k$, $k = 0,1$; and ${\mathbf{\bar Y}}_k $ is the average of ${\mathbf{Y}}_i $ in group $k$. Since we are interested in the second component of ${\mathbf{\hat \beta }}_k $, the slope associated with time, we define $\bar S_k  = \left( {\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}} \right)_{(2)} {\mathbf{\bar Y}}_k $, where the subscript (2) indicates the second row of the matrix $\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}$. We are interested in the difference, which is $\left( {\bar S_1  - \bar S_0 } \right) = \left( {\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}} \right)_{(2)} \left( {{\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 } \right)$.

§.§.§ OLS approach

With the OLS approach, we fit all the data at the same time, using

\mathbb{E}\left( {Y_{ij} |X_{ij} } \right) = \gamma _0  + \gamma _1 t_{ij}  + \gamma _2 k_i  + \gamma _3 \left( {t_{ij}  \times k_i } \right),

and our interest in on $\gamma _3 $. Reparameterizing, we can fit model

\mathbb{E}\left( {Y_{ij} |X_{ij} } \right) = \gamma _0^* \left( {1 - k_i } \right) + \gamma _1^* \left( {1 - k_i } \right)t_{ij}  + \gamma _2^* k_i  + \gamma _3^* k_i t_{ij} ,

and our parameter of interest is now $\gamma _3  = \gamma _3^*  - \gamma _1^* $. The OLS estimator of the latter model can be derived as

{\mathbf{\hat \gamma }}^*  = \left( {{\mathbf{X'X}}} \right)^{ - 1} {\mathbf{X'Y}} = \left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{X}}_i } } \right)^{ - 1} \left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{Y}}_i } } \right),

where ${\mathbf{X}}_i $ is the covariate matrix for subject $i$ and can be written as ${\mathbf{X}}_i  = \left( {\begin{array}{*{20}c}
{\mathbf{Z}} & {\mathbf{0}}  \\
\end{array} } \right)$ if participant $i$ is unexposed and ${\mathbf{X}}_i  = \left( {\begin{array}{*{20}c}
{\mathbf{0}} & {\mathbf{Z}}  \\
\end{array} } \right)$ if exposed. Then,

\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{X}}_i } } \right) = \left( {\begin{array}{*{20}c}
{N(1 - p_e ){\mathbf{Z'Z}}} & {\mathbf{0}}  \\
{\mathbf{0}} & {Np_e {\mathbf{Z'Z}}}  \\
\end{array} } \right),

$$\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{X}}_i } } \right)^{ - 1}  = \frac{1}
{N}\left( {\begin{array}{*{20}c}
{{(1 - p_e )}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} } & {\mathbf{0}}  \\
{\mathbf{0}} & {\frac{1}
{{p_e }}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} }  \\
\end{array} } \right),

\begin{multline*}
\left( {\sum\limits_{i = 1}^N {{\mathbf{X'}}_i {\mathbf{Y}}_i } } \right) = \left( {\begin{array}{*{20}c}
{{\mathbf{Z'}}}  \\
{\mathbf{0}}  \\
\end{array} } \right)\left( {\sum\limits_{i = 1}^N {{\mathbf{Y}}_i I\left\{ {k_i  = 0} \right\}} } \right) + \left( {\begin{array}{*{20}c}
{\mathbf{0}}  \\
{{\mathbf{Z'}}}  \\
\end{array} } \right)\left( {\sum\limits_{i = 1}^N {{\mathbf{Y}}_i I\left\{ {k_i  = 1} \right\}} } \right) \\
= \left( {\begin{array}{*{20}c}
{N(1 - p_e ){\mathbf{Z'\bar Y}}_0 }  \\
{Np_e {\mathbf{Z'\bar Y}}_1 }  \\
\end{array} } \right),
\end{multline*}

{\mathbf{\hat \gamma }}^*  = \left( {\begin{array}{*{20}c}
{\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\bar Y}}_0 }  \\
{\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\bar Y}}_1 }  \\
\end{array} } \right).

To compute $\hat \gamma _3  = \hat \gamma _3^*  - \hat \gamma _1^* $ we need to subtract the second from the fourth component, so $\hat \gamma _3  = \left( {\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}} \right)_{(2)} \left( {{\mathbf{\bar Y}}_1  - {\mathbf{\bar Y}}_0 } \right)$ as in the two-stage approach.

§.§ Proof of (ii)

A necessary and sufficient condition for the OLS and GLS estimators to be the same is ${\mathbf{HV}} = {\mathbf{VH}}$ <cit.>, where ${\mathbf{H}}$ is the hat matrix ${\mathbf{H}} = {\mathbf{X}}({\mathbf{X'X}})^{ - 1} {\mathbf{X'}}$, ${\mathbf{X}}$ is our case the $N(r + 1) \times 4$ matrix of covariates based on model (<ref>), and ${\mathbf{V}}$ is the $N(r + 1) \times N(r + 1)$ covariance matrix of ${\mathbf{Y}}$, which is a block-diagonal matrix with the diagonal blocks equal to ${\mathbf{\Sigma }}$. As in the OLS derivation, we reparameterize the model as

\mathbb{E}\left( {Y_{ij} |X_{ij} } \right) = \gamma _0^* \left( {1 - k_i } \right) + \gamma _1^* \left( {1 - k_i } \right)t_{ij}  + \gamma _2^* k_i  + \gamma _3^* k_i t_{ij},

and for convenience we sort ${\mathbf{X}}$ sot that the first $N(1 - p_e )$ participants are unexposed and therefore have ${\mathbf{X}}_i  = \left( {\begin{array}{*{20}c}
{\mathbf{Z}} & {\mathbf{0}}  \\
\end{array} } \right)$, and the following $Np_e $ are exposed and have ${\mathbf{X}}_i  = \left( {\begin{array}{*{20}c}
{\mathbf{0}} & {\mathbf{Z}}  \\
\end{array} } \right)$. As derived in the OLS case,

({\mathbf{X'X}})^{ - 1}  = \frac{1}
{N}\left( {\begin{array}{*{20}c}
{{(1 - p_e )}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} } & {{\mathbf{0'}}}  \\
{\mathbf{0}} & {\frac{1}
{{p_e }}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} }  \\
\end{array} } \right).

Then, it can be derived that

{\mathbf{H}} = {\mathbf{X}}({\mathbf{X'X}})^{ - 1} {\mathbf{X'}} = \frac{1}
{N}\left( {\begin{array}{*{20}c}
{{\mathbf{H}}_{11} } & {{\mathbf{0'}}}  \\
{\mathbf{0}} & {{\mathbf{H}}_{22} }  \\
\end{array} } \right),

where ${\mathbf{H}}_{11} $ is a block matrix of $N(1 - p_e ) \times N(1 - p_e )$ blocks, each block being equal to $\frac{1}
{{1 - p_e }}{\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}$; and ${\mathbf{H}}_{22} $ is a block matrix with $Np_e  \times Np_e $ blocks, each block being equal to $\frac{1}
{{p_e }}{\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}$. Since ${\mathbf{V}}$ is block diagonal with the diagonal blocks equal to ${\mathbf{\Sigma }}
$, it follows that ${\mathbf{HV}}$ is going to be of the form

{\mathbf{HV}} = \frac{1}
{N}\left( {\begin{array}{*{20}c}
{\left( {{\mathbf{HV}}} \right)_{11} } & {{\mathbf{0'}}}  \\
{\mathbf{0}} & {\left( {{\mathbf{HV}}} \right)_{22} }  \\
\end{array} } \right),

where $\left( {{\mathbf{HV}}} \right)_{11} $ is a block matrix of $N(1 - p_e ) \times N(1 - p_e )$ blocks, each block being equal to $\frac{1}
{{1 - p_e }}{\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\Sigma }}$; and $\left( {{\mathbf{HV}}} \right)_{22} $ is a block matrix with $Np_e  \times Np_e $ blocks, each block being equal to $\frac{1}
{{p_e }}{\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\Sigma }}$. Similarly, we can derive that ${\mathbf{VH}}$ is of the form

{\mathbf{VH}} = \frac{1}
{N}\left( {\begin{array}{*{20}c}
{\left( {{\mathbf{VH}}} \right)_{11} } & {{\mathbf{0'}}}  \\
{\mathbf{0}} & {\left( {{\mathbf{VH}}} \right)_{22} }  \\
\end{array} } \right),

where $\left( {{\mathbf{VH}}} \right)_{11}$ is a block matrix of $N(1 - p_e ) \times N(1 - p_e )$ blocks, each block being equal to $\frac{1}
{{1 - p_e }}{\mathbf{\Sigma Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}$;  and $\left( {{\mathbf{VH}}} \right)_{22} $ is a block matrix with $Np_e  \times Np_e $ blocks, each block being equal to $\frac{1}
{{p_e }}{\mathbf{\Sigma Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}$. Clearly, then, proving that ${\mathbf{HV}} = {\mathbf{VH}}$ is equivalent to proving that ${\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\Sigma }} = {\mathbf{\Sigma Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}$.

Next, we show that the ${\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\Sigma }} = {\mathbf{\Sigma Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}$ holds for ${\mathbf{\Sigma }}$ having a CS or RS structure and therefore the OLS and GLS estimators are algebraically equivalent in those cases. We also show that the condition does not hold for DEX.

§.§.§ CS

Under CS, ${\mathbf{\Sigma }} = \sigma ^2 \left( {\rho {\mathbf{11'}} + (1 - \rho ){\mathbf{I}}} \right)$, where ${\mathbf{I}}$ is the $(r + 1) \times (r + 1)$ identity matrix and ${\mathbf{1}}$ a $(r + 1) \times 1$ vector of ones. Then,

\begin{multline*}
{\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\Sigma }} = \sigma ^2 {\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}\left( {\rho {\mathbf{11'}} + (1 - \rho ){\mathbf{I}}} \right)  \\
= \sigma ^2 \rho {\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'11'}} + \sigma ^2 (1 - \rho ){\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}.
\end{multline*}

Since ${\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}$ is a projection matrix in the subspace defined by columns of ${\mathbf{Z}}$, and the first column of ${\mathbf{Z}}$is ${\mathbf{1}}$, then ${\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'1}} = {\mathbf{1}}$ and ${\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\Sigma }} = \sigma ^2 \rho {\mathbf{11'}} + \sigma ^2 (1 - \rho ){\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}$. Now, we derive an expression for

\begin{multline*}
{\mathbf{\Sigma Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}} = \sigma ^2 \left( {\rho {\mathbf{11'}} + (1 - \rho ){\mathbf{I}}} \right){\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}} \\
= \sigma ^2 \rho {\mathbf{11'Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}} + \sigma ^2 (1 - \rho ){\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}.
\end{multline*}

For the same reasoning used above, ${\mathbf{1'Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}} = {\mathbf{1'}}$, and therefore ${\mathbf{\Sigma Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}} = \sigma ^2 \rho {\mathbf{11'}} + \sigma ^2 (1 - \rho ){\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}$, which is the same expression we derived for ${\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\Sigma }}$.

§.§.§ RS

Under RS, ${\mathbf{\Sigma }} = {\mathbf{ZDZ'}} + \sigma _{within}^2 {\mathbf{I}}$. Then,

\begin{multline*}
{\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\Sigma }} = {\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}\left( {{\mathbf{ZDZ'}} + \sigma _{within}^2 {\mathbf{I}}} \right) \\
= {\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'ZDZ'}} + \sigma _{within}^2 {\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}} = {\mathbf{ZDZ'}} + \sigma _{within}^2 {\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}.
\end{multline*}

Now, we derive an expression for

{\mathbf{\Sigma Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}} = \left( {{\mathbf{ZDZ'}} + \sigma _{within}^2 {\mathbf{I}}} \right){\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}} = {\mathbf{ZDZ'}} + \sigma _{within}^2 {\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}},

which is the same expression we derived for ${\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\Sigma }}$.

§.§.§ DEX

A counterexample is enough to show that ${\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\Sigma }} = {\mathbf{\Sigma Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}$ does not hold for DEX. With $r = 2$ then

{\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}} = \left( {\begin{array}{*{20}c}
{5/6} & {1/3} & { - 1/6}  \\
{1/3} & {1/3} & {1/3}  \\
{ - 1/6} & {1/3} & {5/6}  \\
\end{array} } \right).

If we take $\sigma ^2  = 1$, $\rho  = 0.8$ and $\theta  = 1$ (AR(1) covariance structure) then

{\mathbf{\Sigma }} = \left( {\begin{array}{*{20}c}
1 & {0.8} & {0.64}  \\
{0.8} & 1 & {0.8}  \\
{0.64} & {0.8} & 1  \\
\end{array} } \right).

\begin{multline*}
{\mathbf{Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'\Sigma }} = \left( {\begin{array}{*{20}c}
{5/6} & {1/3} & { - 1/6}  \\
{1/3} & {1/3} & {1/3}  \\
{ - 1/6} & {1/3} & {5/6}  \\
\end{array} } \right)
\left( {\begin{array}{*{20}c}
1 & {0.8} & {0.64}  \\
{0.8} & 1 & {0.8}  \\
{0.64} & {0.8} & 1  \\
\end{array} } \right) \\ % new line
= \left( {\begin{array}{*{20}c}
{0.993} & {0.866} & {0.633}  \\
{0.813} & {0.866} & {0.813}  \\
{0.633} & {0.866} & {0.993}  \\
\end{array} } \right)
\end{multline*}

\begin{multline*}
{\mathbf{\Sigma Z}}\left( {{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}} = \left( {\begin{array}{*{20}c}
1 & {0.8} & {0.64}  \\
{0.8} & 1 & {0.8}  \\
{0.64} & {0.8} & 1  \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
{5/6} & {1/3} & { - 1/6}  \\
{1/3} & {1/3} & {1/3}  \\
{ - 1/6} & {1/3} & {5/6}  \\
\end{array} } \right) \\ % new line
= \left( {\begin{array}{*{20}c}
{0.993} & {0.813} & {0.633}  \\
{0.866} & {0.866} & {0.866}  \\
{0.633} & {0.813} & {0.993}  \\
\end{array} } \right).
\end{multline*}

We can see that the the [2,1], [1,2], [3,2] and [2,3] components differ, so the condition does not hold.

§ PROOF THAT ${\MATHBF{C'\SIGMA }}_{\RM B} {\MATHBF{C}}
$ IS THE SAME FOR $R = 1$ AND $R = 2$ UNDER LDD AND $V(T_0 ) = 0$ WITH FIXED FOLLOW-UP PERIOD $\TAU $ AND EQUIDISTANT TIME POINTS.

Formula (<ref>) for the $V(t_0 ) = 0$ case and expressed as a function of $\tau $ is

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)r^2 }}
{{p_e (1 - p_e )\tau ^2 \det ({\mathbf{A}})}},

where the term $v_{jj'} $ is the $[j,j']$ component of the inverse of ${\mathbf{\Sigma }}$ and

{\mathbf{A}} = \left( {\begin{array}{*{20}c}
{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } & {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } }  \\
{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } } & {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } } }  \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
1 &  \cdots  & 1  \\
0 &  \cdots  & r  \\
\end{array} } \right){\mathbf{\Sigma }}\left( {\begin{array}{*{20}c}
1 & 0  \\
\vdots  &  \vdots   \\
1 & r  \\
\end{array} } \right).

Let us call

{\mathbf{\Sigma }}_1  = \left( {\begin{array}{*{20}c}
{\sigma _{11} } & {\sigma _{1\tau } }  \\
{\sigma _{1\tau } } & {\sigma _{\tau \tau } }  \\
\end{array} } \right)

the covariance matrix for the case $r = 1$ and

{\mathbf{\Sigma }}_2  = \left( {\begin{array}{*{20}c}
{\sigma _{11} } & {\sigma _{1,\tau /2} } & {\sigma _{1,\tau } }  \\
{\sigma _{1,\tau /2} } & {\sigma _{\tau /2,\tau /2} } & {\sigma _{\tau /2,\tau } }  \\
{\sigma _{1,\tau } } & {\sigma _{\tau /2,\tau } } & {\sigma _{\tau ,\tau } }  \\
\end{array} } \right)

for the case $r = 2$ and note that $\sigma _{11} $, $\sigma _{1,\tau } $ and $\sigma _{\tau ,\tau } $ are the same in the two matrices. Let us call ${\mathbf{A}}_1 $ the matrix ${\mathbf{A}}$ for the case $r = 1$ and ${\mathbf{A}}_2 $ for the case $r = 2$. Then, the expression ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ will be the same for $r = 1$ and $r = 2$ if and only if

\frac{{{\mathbf{A}}_1 [1,1]}}
{{\det ({\mathbf{A}}_1 )}} = \frac{{4{\mathbf{A}}_2 [1,1]}}
{{\det ({\mathbf{A}}_2 )}}.

We can now derive

{\mathbf{\Sigma }}_1^{ - 1}  = \frac{1}
{{\sigma _{11} \sigma _{\tau \tau }  - \sigma _{1\tau }^2 }}\left( {\begin{array}{*{20}c}
{\sigma _{11} } & {\sigma _{1\tau } }  \\
{\sigma _{1\tau } } & {\sigma _{\tau \tau } }  \\
\end{array} } \right)

\begin{multline*}
{\mathbf{\Sigma }}_2^{ - 1}  = \frac{1}
{{ - 2\sigma _{1,\tau } \sigma _{1,\tau /2} \sigma _{\tau /2,\tau }  + \sigma _{1,\tau }^2 \sigma _{\tau /2,\tau /2}  + \sigma _{1,\tau /2}^2 \sigma _{\tau ,\tau }  + \sigma _{11} \left( {\sigma _{\tau /2,\tau }^2  - \sigma _{\tau /2,\tau /2} \sigma _{\tau ,\tau } } \right)}}  \\
\left( {\begin{array}{*{20}c}
{\sigma _{\tau /2,\tau }^2  - \sigma _{\tau /2,\tau /2} \sigma _{\tau ,\tau } } & {} & {}  \\
{\sigma _{1,\tau /2} \sigma _{\tau ,\tau }  - \sigma _{1,\tau } \sigma _{\tau /2,\tau } } & {\sigma _{1,\tau }^2  - \sigma _{11} \sigma _{\tau ,\tau } } & {}  \\
{\sigma _{1,\tau } \sigma _{\tau /2,\tau /2}  - \sigma _{1,\tau /2} \sigma _{\tau /2,\tau } } & {\sigma _{11} \sigma _{\tau /2,\tau }  - \sigma _{1,\tau } \sigma _{1,\tau /2} } & {\sigma _{1,\tau /2}^2  - \sigma _{11} \sigma _{\tau /2,\tau /2} }  \\
\end{array} } \right)
\end{multline*}

\begin{multline*}
{\mathbf{A}}_1  = \frac{1}
{{\sigma _{11} \sigma _{\tau \tau }  - \sigma _{1\tau }^2 }}\left( {\begin{array}{*{20}c}
1 & 1  \\
0 & 1  \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
{\sigma _{11} } & {\sigma _{1\tau } }  \\
{\sigma _{1\tau } } & {\sigma _{\tau \tau } }  \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
1 & 0  \\
1 & 1  \\
\end{array} } \right) \\
= \frac{1}
{{\sigma _{11} \sigma _{\tau \tau }  - \sigma _{1\tau }^2 }}\left( {\begin{array}{*{20}c}
{\sigma _{11}  - 2\sigma _{1\tau }  + \sigma _{\tau \tau } } & {\sigma _{11}  - \sigma _{1\tau } }  \\
{\sigma _{11}  - \sigma _{1\tau } } & {\sigma _{11} }  \\
\end{array} } \right),
\end{multline*}

\det ({\mathbf{A}}_1 ) = \frac{1}
{{\sigma _{11} \sigma _{\tau \tau }  - \sigma _{1\tau }^2 }}

\frac{{\left( {\sum\limits_{j = 0}^1 {\sum\limits_{j' = 0}^1 {v_{jj'} } } } \right)}}
{{\det ({\mathbf{A}}_1 )}} = \frac{{{\mathbf{A}}_1 [1,1]}}
{{\det ({\mathbf{A}}_1 )}} = \sigma _{11}  - 2\sigma _{1\tau }  + \sigma _{\tau \tau };

\begin{multline*}
{\mathbf{A}}_2  = \frac{1}
{{ - 2\sigma _{1,\tau } \sigma _{1,\tau /2} \sigma _{\tau /2,\tau }  + \sigma _{1,\tau }^2 \sigma _{\tau /2,\tau /2}  + \sigma _{1,\tau /2}^2 \sigma _{\tau ,\tau }  + \sigma _{11} \left( {\sigma _{\tau /2,\tau }^2  - \sigma _{\tau /2,\tau /2} \sigma _{\tau ,\tau } } \right)}}  \\
\left( {\begin{array}{*{20}c}
1 & 1 & 1  \\
0 & 1 & 2  \\
\end{array} } \right) \\
\left( {\begin{array}{*{20}c}
{\sigma _{\tau /2,\tau }^2  - \sigma _{\tau /2,\tau /2} \sigma _{\tau ,\tau } } & {} & {}  \\
{\sigma _{1,\tau /2} \sigma _{\tau ,\tau }  - \sigma _{1,\tau } \sigma _{\tau /2,\tau } } & {\sigma _{1,\tau }^2  - \sigma _{11} \sigma _{\tau ,\tau } } & {}  \\
{\sigma _{1,\tau } \sigma _{\tau /2,\tau /2}  - \sigma _{1,\tau /2} \sigma _{\tau /2,\tau } } & {\sigma _{11} \sigma _{\tau /2,\tau }  - \sigma _{1,\tau } \sigma _{1,\tau /2} } & {\sigma _{1,\tau /2}^2  - \sigma _{11} \sigma _{\tau /2,\tau /2} }  \\
\end{array} } \right)
\left( {\begin{array}{*{20}c}
1 & 0  \\
1 & 1  \\
1 & 2  \\
\end{array} } \right).
\end{multline*}

It can be derived that

\begin{multline*}
\frac{{{\mathbf{A}}_2 [1,1]}}
{{\det ({\mathbf{A}}_2 )}} = \frac{1}
{{\sigma _{11}  + 2\sigma _{1,\tau }  - 4\left( {\sigma _{1,\tau /2}  + \sigma _{\tau /2,\tau }  - \sigma _{\tau /2,\tau /2} } \right) + \sigma _{\tau ,\tau } }} \\%newline
\biggl\{ - \sigma _{1,\tau }^2  - \left( {\sigma _{1,\tau /2}  - \sigma _{\tau /2,\tau } } \right)^2  + 2\sigma _{1,\tau } \left( {\sigma _{1,\tau /2}  + \sigma _{\tau /2,\tau }  - \sigma _{\tau /2,\tau /2} } \right) + \\ %newline
\left( {\sigma _{\tau /2,\tau /2}  - 2\sigma _{1,\tau /2} } \right)\sigma _{\tau ,\tau }  +   {\sigma _{11} \left( {\sigma _{\tau /2,\tau /2}  + \sigma _{\tau ,\tau }  - 2\sigma _{\tau /2,\tau } } \right)} \biggr\}
\end{multline*}

\frac{{{\mathbf{A}}_1 [1,1]}}
{{\det ({\mathbf{A}}_1 )}} = \frac{{4{\mathbf{A}}_2 [1,1]}}
{{\det ({\mathbf{A}}_2 )}}

if and only if

\begin{multline*}
\sigma _{11}  - 2\sigma _{1\tau }  + \sigma _{\tau \tau }  = \frac{4}
{{\sigma _{11}  + 2\sigma _{1,\tau }  - 4\left( {\sigma _{1,\tau /2}  + \sigma _{\tau /2,\tau }  - \sigma _{\tau /2,\tau /2} } \right) + \sigma _{\tau ,\tau } }} \\%endline
\biggl\{  - \sigma _{1,\tau }^2  - \left( {\sigma _{1,\tau /2}  - \sigma _{\tau /2,\tau } } \right)^2  + 2\sigma _{1,\tau } \left( {\sigma _{1,\tau /2}  + \sigma _{\tau /2,\tau }  - \sigma _{\tau /2,\tau /2} } \right) + \\ %endline
\left( {\sigma _{\tau /2,\tau /2}  - 2\sigma _{1,\tau /2} } \right)\sigma _{\tau ,\tau } +  {\sigma _{11} \left( {\sigma _{\tau /2,\tau /2}  + \sigma _{\tau ,\tau }  - 2\sigma _{\tau /2,\tau } } \right)} \biggr\},
\end{multline*}

which with some algebra it reduces to $\sigma _{11}  - \sigma _{\tau \tau }  = 2\left( {\sigma _{1,\tau /2}  - \sigma _{\tau /2,\tau } } \right)$. So, ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ will be the same for $r = 1$ and $r = 2$ if and only if $\sigma _{11}  - \sigma _{\tau \tau }  = 2\left( {\sigma _{1,\tau /2}  - \sigma _{\tau /2,\tau } } \right)$. We can check that for the covariance structures used in the paper, i.e. compound symmetry (CS) (section <ref>), damped exponential (DEX) (section <ref>) and random intercepts and slopes (RS) (section <ref>) this condition is met. For CS,

{\mathbf{\Sigma }}_2  = \left( {\begin{array}{*{20}c}
{\sigma _{11} } & {\sigma _{1,\tau /2} } & {\sigma _{1,\tau } }  \\
{\sigma _{1,\tau /2} } & {\sigma _{\tau /2,\tau /2} } & {\sigma _{\tau /2,\tau } }  \\
{\sigma _{1,\tau } } & {\sigma _{\tau /2,\tau } } & {\sigma _{\tau ,\tau } }  \\
\end{array} } \right) = \sigma ^2 \left( {\begin{array}{*{20}c}
1 & \rho  & \rho   \\
\rho  & 1 & \rho   \\
\rho  & \rho  & 1  \\
\end{array} } \right),

so $\sigma _{11}  - \sigma _{\tau \tau }  = \sigma ^2 \left( {1 - 1} \right) = 0
$ and $2\left( {\sigma _{1,\tau /2}  - \sigma _{\tau /2,\tau } } \right) = 2\sigma ^2 (\rho  - \rho ) = 0$ and the condition holds. For DEX,

{\mathbf{\Sigma }}_2  = \left( {\begin{array}{*{20}c}
{\sigma _{11} } & {\sigma _{1,\tau /2} } & {\sigma _{1,\tau } }  \\
{\sigma _{1,\tau /2} } & {\sigma _{\tau /2,\tau /2} } & {\sigma _{\tau /2,\tau } }  \\
{\sigma _{1,\tau } } & {\sigma _{\tau /2,\tau } } & {\sigma _{\tau ,\tau } }  \\
\end{array} } \right) = \sigma ^2 \left( {\begin{array}{*{20}c}
1 & \rho  & {\rho ^{2^\theta  } }  \\
\rho  & 1 & \rho   \\
{\rho ^{2^\theta  } } & \rho  & 1  \\
\end{array} } \right),

so $\sigma _{11}  - \sigma _{\tau \tau }  = \sigma ^2 \left( {1 - 1} \right) = 0$ and $2\left( {\sigma _{1,\tau /2}  - \sigma _{\tau /2,\tau } } \right) = 2\sigma ^2 (\rho  - \rho ) = 0$ and the condition holds. For RS,

\begin{multline*}
{\mathbf{\Sigma }}_2  = \left( {\begin{array}{*{20}c}
{\sigma _{11} } & {\sigma _{1,\tau /2} } & {\sigma _{1,\tau } }  \\
{\sigma _{1,\tau /2} } & {\sigma _{\tau /2,\tau /2} } & {\sigma _{\tau /2,\tau } }  \\
{\sigma _{1,\tau } } & {\sigma _{\tau /2,\tau } } & {\sigma _{\tau ,\tau } }  \\
\end{array} } \right) = \\
{\footnotesize \left( {\begin{array}{*{20}c}
{\sigma _{b_0 }^2  + \sigma _{within}^2 } & {} & {}  \\
{\sigma _{b_0 }^2  + \rho _{b_0 b_1 } \sigma _{b_0 } \sigma _{b_1 } } & {\sigma _{b_0 }^2  + \sigma _{b_1 }^2  + 2\rho _{b_0 b_1 } \sigma _{b_0 } \sigma _{b_1 }  + \sigma _{within}^2 } & {}  \\
{\sigma _{b_0 }^2  + 2\rho _{b_0 b_1 } \sigma _{b_0 } \sigma _{b_1 } } & {\sigma _{b_0 }^2  + 3\rho _{b_0 b_1 } \sigma _{b_0 } \sigma _{b_1 }  + 2\sigma _{b_1 }^2 } & {\sigma _{b_0 }^2  + 4\sigma _{b_1 }^2  + 4\rho _{b_0 b_1 } \sigma _{b_0 } \sigma _{b_1 }  + \sigma _{within}^2 }  \\
\end{array} } \right)},
\end{multline*}

\sigma _{11}  - \sigma _{\tau \tau }  = \sigma _{b_0 }^2  + \sigma _{within}^2  - \sigma _{b_0 }^2  - 4\sigma _{b_1 }^2  - 4\rho _{b_0 b_1 } \sigma _{b_0 } \sigma _{b_1 }  - \sigma _{within}^2  =  - 4\sigma _{b_1 }^2  - 4\rho _{b_0 b_1 } \sigma _{b_0 } \sigma _{b_1 }

\begin{multline*}
2\left( {\sigma _{1,\tau /2}  - \sigma _{\tau /2,\tau } } \right) = 2\left( {\sigma _{b_0 }^2  + \rho _{b_0 b_1 } \sigma _{b_0 } \sigma _{b_1 }  - \sigma _{b_0 }^2  - 3\rho _{b_0 b_1 } \sigma _{b_0 } \sigma _{b_1 }  - 2\sigma _{b_1 }^2 } \right) \\
=  - 4\sigma _{b_1 }^2  - 4\rho _{b_0 b_1 } \sigma _{b_0 } \sigma _{b_1 }
\end{multline*}

and the condition holds.

§ EFFECT OF $P_E $ ON $R$

We write

$$N = \frac{{\left( {{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}} \right)\,\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 }}
{{({\mathbf{c'{\rm B}}}_{H_A } )^2 }}

N = \frac{{g({\mathbf{\eta }})f(r)\,\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 }}
{{p_e (1 - p_e )({\mathbf{c'{\rm B}}}_{H_A } )^2 }}

where $g({\mathbf{\eta }})$ does not depend on $r$ or $p_e $. Then we can define $r$ implicitly as the value/s solving the equation $F(r) = 0$, where

F(r) = \frac{{g({\mathbf{\eta }})\,\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 }}
{{N({\mathbf{c'{\rm B}}}_{H_A } )^2 }} - \frac{{p_e (1 - p_e )}}

Using implicit differentiation and differentiating both sides of $F(r) = 0$ we have

\frac{{\delta F(r)}}
{{\delta p_e }} = 0,

from where we can derive $\frac{{\delta r}} {{\delta p_e }}$,

\frac{{\delta F(r)}}
{{\delta p_e }} = 0 \Leftrightarrow \frac{{(1 - 2p_e )f(r) - p_e (1 - p_e )f'(r)\frac{{\delta r}}
{{\delta p_e }}}}
{{\left[ {f(r)} \right]^2 }} = 0 \Leftrightarrow
\frac{{\delta r}}
{{\delta p_e }} = \frac{{(1 - 2p_e )f(r)}}
{{p_e (1 - p_e )f'(r)}}.

Then, to find the value of $p_e $ that minimizes $r$ we solve $\frac{{\delta r}}
{{\delta p_e }} = 0$, which results in the only root $p_e  = 0.5$. Since $(1 - 2p_e )$ is greater than zero for $p_e  < 0.5$ and smaller than zero for  $p_e  > 0.5$, $r$ has a maximum or a minimum at $p_e  = 0.5$. The sign of $\frac{{f(r)}}
{{f'(r)}}$ determines whether it is a maximum or a minimum. Since the variance ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ is always positive so it is $f(r)$, and since the variance decreases as $r$ increases, $f'(r)$ is negative. Therefore $\frac{{f(r)}} {{f'(r)}}$ is negative and $\frac{{\delta r}}
{{\delta p_e }} < 0$ for $p_e  < 0.5$ and $\frac{{\delta r}}
{{\delta p_e }} > 0$ for $p_e  > 0.5$, implying that $r$ is minimum at $p_e  = 0.5$.

§ LIMIT OF ${\MATHBF{C'\SIGMA }}_{\RM B} {\MATHBF{C}}$ WHEN $R \TO \INFTY $

§.§ CMD, CS

The inverse of a CS matrix has diagonal elements

\frac{1}
{{\sigma ^2 }}\frac{{1 + \rho (r - 2) - \rho ^2 (r - 1)}}
{{(1 - \rho )^2 \left( {1 + r\rho } \right)}}

and off-diagonal elements

\frac{1}
{{\sigma ^2 }}\frac{{ - \rho }}
{{(1 - \rho )\left( {1 + r\rho } \right)}}.

The sum of a row or a column of the inverse is

\frac{1}
{{\sigma ^2 }}\left( {\frac{{1 + \rho (r - 2) - \rho ^2 (r - 1)}}
{{(1 - \rho )^2 \left( {1 + r\rho } \right)}} - \frac{{r\rho }}
{{(1 - \rho )\left( {1 + r\rho } \right)}}} \right) = \frac{1}
{{\sigma ^2 \left( {1 + r\rho } \right)}}

and therefore

\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } }  = \frac{{r + 1}}
{{\sigma ^2 (1 + r\rho )}}.

\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } }  = \sum\limits_{j = 0}^r {j\sum\limits_{j' = 0}^r {v_{jj'} } }  = \frac{{r(r + 1)}}
{{2\sigma ^2 \left( {1 + r\rho } \right)}}

since $\sum\limits_{j' = 0}^r {v_{jj'} } $ is the sum of a row or column of the inverse. We can also derive

\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } }  = \frac{{r(r + 1)(2 + r(4 + (r - 1)\rho ))}}
{{12\sigma ^2 (1 - \rho )\left( {1 + r\rho } \right)}}.

\det ({\mathbf{A}}) = \left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } \right)\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } } } \right) - \left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } } \right)^2  = \frac{{r(r + 1)^2 (r + 2)}}
{{12\sigma ^4 (1 - \rho )\left( {1 + r\rho } \right)}}.

Plugging in all these expressions in to equation (<ref>), we have that under CMD and CS

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{\sigma ^2 (1 + r\rho )\left( {r(r + 2)(1 + r\rho )s^2  + 12(1 - \rho )V\left( {t_0 } \right)} \right)}}
{{p_e (1 - p_e )(r + 1)\left( {r(r + 2)(1 + r\rho )s^2  + 12(1 - \rho )\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)V\left( {t_0 } \right)} \right)}}.

Then, using the highest order terms of $r$ on the numerator and denominator of ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ we can derive that

\begin{multline*}
\mathop {\lim }\limits_{r \to \infty } \;{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \mathop {\lim }\limits_{r \to \infty } \;\frac{{\sigma ^2 (1 + r\rho )\left( {r(r + 2)(1 + r\rho )s^2 } \right)}}
{{p_e (1 - p_e )(r + 1)\left( {r(r + 2)(1 + r\rho )s^2 } \right)}} = \\
\mathop {\lim }\limits_{r \to \infty } \frac{{\sigma ^2 (1 + r\rho )}}
{{p_e (1 - p_e )(r + 1)}}\; = \frac{{\sigma ^2 \rho }}
{{p_e (1 - p_e )}}.
\end{multline*}

§.§ LDD, CS

Applying the results derived in Appendix <ref> to equation (<ref>), we can derive ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ as

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{12\sigma ^2 (1 - \rho ){\kern 1pt} (1 - r\rho )}}
{{N{\kern 1pt} p_e (1 - p_e )(r + 1)\left( {r(r + 2)(1 + r\rho )\,s^2  + 12(1 - \rho )\left( {1 - \rho _{\operatorname{e} ,t_0 }^2 } \right)V(t_0 )} \right)}}.

Since the denominator is a polynomial of fourth degree of $r$ while the numerator is of first degree, then $\mathop {\lim }\limits_{r \to \infty } \;\;{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = 0$.

§.§ CMD, AR(1)

§.§.§ Fixed $s$:

The AR(1) covariance matrix is given by (<ref>) with $\theta  = 1$, and its inverse is a tridiagonal matrix with the form

{\mathbf{\Sigma }}^{ - 1}  = \frac{1}
{{\left( {1 - \rho ^{2s} } \right)\sigma ^2 }}\left( {\begin{array}{*{20}c}
1 & { - \rho ^s } & 0 & 0 &  \cdots  & 0  \\
{ - \rho ^s } & {1 + \rho ^{2s} } & { - \rho ^s } & 0 & {} & 0  \\
0 & { - \rho ^s } & {1 + \rho ^{2s} } &  \ddots  &  \ddots  &  \vdots   \\
0 & 0 &  \ddots  &  \ddots  & { - \rho ^s } & 0  \\
\vdots  & {} &  \ddots  & { - \rho ^s } & {1 + \rho ^{2s} } & { - \rho ^s }  \\
0 & 0 &  \cdots  & 0 & { - \rho ^s } & 1  \\
\end{array} } \right)

<cit.>. To use equation (<ref>) we need to derive $\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } $, $\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } $ and $\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } } $. In can be easily shown that

\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } }  = \frac{{(1 + r + \rho ^s  - r\rho ^s )}}
{{\sigma ^2 {\kern 1pt} (1 + \rho ^s )}}.

\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {j \cdot v_{jj'} } }  = \frac{{r\left( {1 - \rho ^s } \right)\left( {1 + r\left( {1 - \rho ^s } \right) + \rho ^s } \right)}}
{{2\left( {1 - \rho ^{2s} } \right)\sigma ^2 }}

\left( {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } } } \right) = \frac{r}
{{6\left( {1 - \rho ^{2s} } \right)\sigma ^2 }}\left( {1 + 4\rho ^s  + \rho ^{2s}  + 3r\left( {1 - \rho ^{2s} } \right) + 2r^2 \left( {1 - \rho ^s } \right)^2 } \right).

If $V\left( {t_0 } \right) = 0$, we can use formula (<ref>) and

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{\sigma ^2 (1 + \rho ^s )}}
{{{\kern 1pt} p_e (1 - p_e )(1 + r + \rho ^s  - r\rho ^s )}}

as in Table <ref>. This formula has a polynomial of first order degree of $r$ in the denominator, and no terms involving $r$ in the numerator. Therefore,  $\mathop {\lim }\limits_{r \to \infty } \;\;{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = 0$. If $V\left( {t_0 } \right) > 0$ the formula is very long ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ and we used Mathematica <cit.> to get the formula and compute the limit, which was zero. Therefore, $\mathop {\lim }\limits_{r \to \infty } \;\;{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = 0$ also for the case of $V\left( {t_0 } \right) > 0$.

§.§ Fixed $\tau $

For the fixed $\tau $ case we need to substitute $s$ by $\tau /r$. So, for the case of $V\left( {t_0 } \right) = 0$,

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{\sigma ^2 (1 + \rho ^{\tau /r} )}}
{{{\kern 1pt} p_e (1 - p_e )(1 + r + \rho ^{\tau /r}  - r\rho ^{\tau /r} )}}.

We want to compute

\begin{multline*}
\mathop {\lim }\limits_{r \to \infty } \;\;{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \mathop {\lim }\limits_{r \to \infty } \;\frac{{\sigma ^2 (1 + \rho ^{\tau /r} )}}
{{{\kern 1pt} p_e (1 - p_e )(1 + r + \rho ^{\tau /r}  - r\rho ^{\tau /r} )}} = \\ \mathop {\lim }\limits_{r \to \infty } \;\frac{{\sigma ^2 (1 + \rho ^{\tau /r} )}}
{{{\kern 1pt} p_e (1 - p_e )\left[ {1 + \rho ^{\tau /r}  + r(1 - \rho ^{\tau /r} )} \right]}}.
\end{multline*}

By l'Hôpital's rule it can be shown that $\mathop {\lim }\limits_{r \to \infty } \;r(1 - \rho ^{\tau /r} ) =  - \tau \log \rho $, and then

\mathop {\lim }\limits_{r \to \infty } \;\;{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{2\sigma ^2 }}
{{p_e (1 - p_e )\left[ {2 - \tau \log \rho } \right]^{} }}.

If $V\left( {t_0 } \right) > 0$, we used Mathematica <cit.> to derive the limit, which in this case has a very complicated expression,

\begin{multline*}
2\sigma ^2 \left( {\left( {\tau ^3  + 12V(t_0 )\tau } \right)\left( {\log (\rho )} \right)^2  - 6\left( {\tau ^2  + 4V(t_0 )} \right)\log (\rho ) + 12\tau } \right)\\
\left[ {p_{\text{e}} (1 - p_{\text{e}} )(2 - \tau \log (\rho ))} \right]^{ - 1}  \\
\biggl\{  \left( {\tau ^3  + 12V(t_0 )\tau } \right)\left( {\log (\rho )} \right)^2  - 12V(t_0 )(\tau \log (\rho ) - 2)\rho _{e,t_0 }^2 \log (\rho ) \\
- 6\left( {\tau ^2  + 4V(t_0 )} \right)\log (\rho ) + 12\tau  \biggr\}^{-1}
\end{multline*}

§.§ LDD, AR(1)

§.§.§ Fixed $s$

Using the results from Appendix <ref> and applying formula (<ref>) for the case $V\left( {t_0 } \right) = 0$, we can derive that

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{12\sigma ^2 {\kern 1pt} (1 - \rho ^{2s} )\,\left[ {\,r\,s^2 p_e (1 - p_e )} \right]^{ - 1} }}
{{{\kern 1pt} \;\,(2 + r(r + 3) + 8\rho ^s  - 2r^2 \rho ^s  + (r - 2)(r - 1)\rho ^{2s} )}}

as is shown in Table <ref>. Since the denominator is a polynomial of second degree of $r$ while the has no terms involving $r$, then $\mathop {\lim }\limits_{r \to \infty } \;\;{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = 0
$. If $V\left( {t_0 } \right) > 0$, we used Mathematica <cit.> to derive that the limit was also zero.

§.§.§ Fixed $\tau $

For the fixed $\tau $ case we need to substitute $s$ by $\tau /r$. So, for the case of $V\left( {t_0 } \right) = 0$, we have

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{12\sigma ^2 {\kern 1pt} (1 - \rho ^{2\tau /r} )\,r\,\left[ {\,\tau ^2 p_e (1 - p_e )} \right]^{ - 1} }}
{{{\kern 1pt} \;\,(2 + r(r + 3) + 8\rho ^{\tau /r}  - 2r^2 \rho ^{\tau /r}  + (r - 2)(r - 1)\rho ^{2\tau /r} )}},

as shown in Table <ref>. This expression can be rewritten as

\frac{{12\sigma ^2 {\kern 1pt} (1 - \rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} )\,\left[ {p_e (1 - p_e )} \right]^{ - 1} }}
{{{\kern 1pt} \;\,\tau ^2 \left[ {r\left( {1 - \rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right)^2  + \left( {3\left( {1 - \rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right)} \right) + \frac{1}
{r}\left( {2 + 8\rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}}  + 2\rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right)} \right]}}.

Then, to compute $\mathop {\lim }\limits_{r \to \infty } \;\;{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ we note that the limit of the numerator is $12\sigma ^2 \mathop {\lim }\limits_{r \to \infty } \;(1 - \rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} ) = 0
$. In the denominator, the limit of last two terms is zero, and l'Hôpital's rule can be used to derive that the limit of the first term is also zero. Thus, we apply l'Hôpital's rule to derive the limit of ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}
$, where the derivative of the numerator is

\frac{{24\sigma ^2 {\kern 1pt} \rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} \tau \log \rho }}
{{r^2 }},

the derivative of the first term of the denominator is

\tau ^2 \left[ {\left( {1 - \rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right)^2  + \frac{{2\rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} \left( {1 - \rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right)\tau \log \rho }}
{r}} \right],

the derivative of the second term of the denominator is

3\tau ^2 \left[ {\frac{{2{\kern 1pt} \rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} \tau \log \rho }}
{{r^2 }}} \right],

and the derivative of the third term of the denominator is

2\tau ^2 \left[ {\frac{{ - 1 - 4\rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}}  - \rho ^{{{2\tau } \mathord{\left/
{\vphantom {{2\tau } r}} \right.
\kern-\nulldelimiterspace} r}} }}
{{r^2 }} - \frac{{2\tau \log \rho \left( {2\rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}}  + \rho ^{{{2\tau } \mathord{\left/
{\vphantom {{2\tau } r}} \right.
\kern-\nulldelimiterspace} r}} } \right)}}
{{r^3 }}} \right].

Simplifying terms,

\begin{multline*}
\mathop {\lim }\limits_{r \to \infty } \;\;{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \\
{24\sigma ^2 {\kern 1pt} \rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} \tau \log \rho \left[ {p_e (1 - p_e )} \right]^{ - 1} \tau ^{-2}} \\
\biggl\{ r^2 \left( {1 - \rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right)^2  + 2r\rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} \left( {1 - \rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right)\tau \log \rho  + 3\left[ {2\rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} \tau \log \rho } \right] \\
+ 2\left( { - 1 - 4\rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}}  - \rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right) - \frac{{4\tau \log \rho \left( {2\rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}}  + \rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right)}}
{r} \biggr\}^{-1}
\end{multline*}

Now the limit of the numerator is $24\sigma ^2 {\kern 1pt} \tau \log \rho $. In the denominator, we need to evaluate several terms. The limit of  $r^2 \left( {1 - \rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right)^2
$ can be obtained by applying l'Hôpital's rule twice and it equals $\tau ^2 \left( {\log \rho } \right)^2 $. The limit of $2r\rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} \left( {1 - \rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right)\tau \log \rho
$ is obtained by applying l'Hôpital's rule and it equals $ - 2\tau ^2 \left( {\log \rho } \right)^2 $. The limit of $3\left[ {2\rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} \tau \log \rho } \right]$ is $3\left[ {2\tau \log \rho } \right]$, the limit of $2\left( { - 1 - 4\rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}}  - \rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right)
$ is $ - 12$ and the limit of

\frac{{4\tau \log \rho \left( {2\rho ^{{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}}  + \rho ^{2{\tau  \mathord{\left/
{\vphantom {\tau  r}} \right.
\kern-\nulldelimiterspace} r}} } \right)}}

is zero. Therefore, with some algebra we can deduce that

\mathop {\lim }\limits_{r \to \infty } \;\;{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{24\sigma ^2 {\kern 1pt} \log \rho }}
{{p_e (1 - p_e )\left[ { - 12\tau  + 6\tau ^2 \log \rho  - \tau ^3 \left( {\log \rho } \right)^2 } \right]}}.

If $V\left( {t_0 } \right) > 0$, using the expression derived for fixed $s$ and substituting $s$ by $\tau /r$, we used Mathematica <cit.> to derive the limit, which has a complicated expression,

\begin{multline*}
{24\sigma ^2 \log (\rho )\left[ {p_{\text{e}} (1 - p_{\text{e}} )} \right]} \\
\biggl\{  - \left( {\tau ^3  + 12V(t_0 )\tau } \right)\left( {\log (\rho )} \right)^2  + 12V(t_0 )(\tau \log (\rho ) - 2)\rho _{e,t_0 }^2 \log (\rho ) \\
+ 6\left( {\tau ^2  + 4V(t_0 )} \right)\log (\rho ) - 12\tau  \biggr\}^{-1}
\end{multline*}

§.§ CMD, RS, $V(t_0 ) = 0$

Since for the case $V(t_0 ) > 0$ we need to use numerical methods to compute ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$, we only compute the limits of ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ for the case $V(t_0 ) = 0$. The covariance matrix of the repeated measurements is expressed as ${\mathbf{\Sigma }}_i  = {\mathbf{Z}}_i {\mathbf{DZ'}}_i  + \sigma _{within}^2 {\mathbf{I}}$, and since $V(t_0 ) = 0$ we have ${\mathbf{Z}}_i  = {\mathbf{Z}}$ and then we${\mathbf{\Sigma }}_i  = {\mathbf{\Sigma }} = {\mathbf{ZDZ'}} + \sigma _{within}^2 {\mathbf{I}}$. The matrix ${\mathbf{Z}}$ is $(r + 1) \times 2$ and contains a column of ones and the column of times ($sj,j = 0, \ldots ,r$). Note that formula (<ref>) depends on

{\mathbf{A}} = \left( {\begin{array}{*{20}c}
{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } & {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } }  \\
{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } } & {\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } } }  \\
\end{array} } \right)

only through $s^2 \det ({\mathbf{A}})$. For convenience in this proof we define the new matrix

{\mathbf{\tilde A}} = \left( {\begin{array}{*{20}c}
{\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {v_{jj'} } } } & {s\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } }  \\
{s\sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jv_{jj'} } } } & {s^2 \sum\limits_{j = 0}^r {\sum\limits_{j' = 0}^r {jj'v_{jj'} } } }  \\
\end{array} } \right)

so that ${\mathbf{\tilde A}}$ is actually ${\mathbf{Z'\Sigma }}^{ - 1} {\mathbf{Z}}$. Note that $\det \left( {{\mathbf{\tilde A}}} \right) = s^2 \det ({\mathbf{A}})
$. Then, we have that  ${\mathbf{\tilde A}} = {\mathbf{Z'\Sigma }}^{ - 1} {\mathbf{Z}} = {\mathbf{Z'}}\left( {{\mathbf{ZDZ'}} + \sigma _{within}^2 {\mathbf{I}}} \right)^{ - 1} {\mathbf{Z}}$. Using the property

({\mathbf{ABA}}' + {\mathbf{C}})^{ - 1}  = {\mathbf{C}}^{ - 1}  - {\mathbf{C}}^{ - 1} {\mathbf{A}}\left( {{\mathbf{B}}^{ - 1}  + {\mathbf{A}}'{\mathbf{C}}^{ - 1} {\mathbf{A}}} \right)^{ - 1} {\mathbf{A}}'{\mathbf{C}}^{ - 1},

which can be found in <cit.>, we have that

\begin{multline*}
\left( {{\mathbf{ZDZ'}} + \sigma _{within}^2 {\mathbf{I}}} \right)^{ - 1}  = \frac{1}
{{\sigma _{within}^2 }}{\mathbf{I}} - \frac{1}
{{\sigma _{within}^2 }}{\mathbf{IZ}}\left( {{\mathbf{D}}^{ - 1}  + {\mathbf{Z'}}\frac{1}
{{\sigma _{within}^2 }}{\mathbf{IZ}}} \right)^{ - 1} {\mathbf{Z'I}}\frac{1}
{{\sigma _{within}^2 }} \\
= \frac{1}
{{\sigma _{within}^2 }}{\mathbf{I}} - \frac{1}
{{\sigma _{within}^4 }}{\mathbf{Z}}\left( {{\mathbf{D}}^{ - 1}  + \frac{1}
{{\sigma _{within}^2 }}{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'}}.
\end{multline*}

\begin{multline*}
{\mathbf{Z'}}\left( {{\mathbf{ZDZ'}} + \sigma _{within}^2 {\mathbf{I}}} \right)^{ - 1} {\mathbf{Z}} \\
= \frac{1}
{{\sigma _{within}^2 }}{\mathbf{Z'Z}} - \frac{1}
{{\sigma _{within}^4 }}{\mathbf{Z'Z}}\left( {{\mathbf{D}}^{ - 1}  + \frac{1}
{{\sigma _{within}^2 }}{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'Z}}
\end{multline*}

and using the property

{\mathbf{A}}^{ - 1}  - {\mathbf{A}}^{ - 1} \left( {{\mathbf{A}}^{ - 1}  + {\mathbf{B}}^{ - 1} } \right)^{ - 1} {\mathbf{A}}^{ - 1}  = ({\mathbf{A}} + {\mathbf{B}})^{ - 1},

which can be found in <cit.>, we have that
\begin{multline*}
{\mathbf{\tilde A}} = {\mathbf{Z'}}\left( {{\mathbf{ZDZ'}} + \sigma _{within}^2 {\mathbf{I}}} \right)^{ - 1} {\mathbf{Z}} \\
= \frac{1}
{{\sigma _{within}^2 }}{\mathbf{Z'Z}} - \frac{1}
{{\sigma _{within}^4 }}{\mathbf{Z'Z}}\left( {{\mathbf{D}}^{ - 1}  + \frac{1}
{{\sigma _{within}^2 }}{\mathbf{Z'Z}}} \right)^{ - 1} {\mathbf{Z'Z}} = \left( {\left( {{\mathbf{Z'Z}}} \right)^{ - 1} \sigma _{within}^2  + {\mathbf{D}}} \right)^{ - 1}.
\end{multline*}

\begin{multline*}
\left( {\left( {{\mathbf{Z'Z}}} \right)^{ - 1} \sigma _{within}^2  + {\mathbf{D}}} \right)^{ - 1}  = \\
\left( {\sigma _{within}^2 \left( {\begin{array}{*{20}c}
{\frac{{2 + 4r}}
{{(r + 1)(r + 2)}}} & {\frac{{ - 6}}
{{s(r + 1)(r + 2)}}}  \\
{\frac{{ - 6}}
{{s(r + 1)(r + 2)}}} & {\frac{{12}}
{{s^2 r(r + 1)(r + 2)}}}  \\
\end{array} } \right) + \left( {\begin{array}{*{20}c}
{\sigma _{b_0 }^2 } & {\rho _{b_0 b_1 } \sigma _{b_0 } \sigma _{b_1 } }  \\
{\rho _{b_0 b_1 } \sigma _{b_0 } \sigma _{b_1 } } & {\sigma _{b_1 }^2 }  \\
\end{array} } \right)} \right)^{ - 1}
\end{multline*}

We computed this inverse using Mathematica <cit.>, and then using equation (<ref>) substituting $s^2 \det ({\mathbf{A}})
$ by $\det \left( {{\mathbf{\tilde A}}} \right)$ we derived the expression for ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$, which is

\frac{{\left( {\sigma _0^2  + \frac{{2(2r + 1)\sigma _w^2 }}
{{r^2  + 3r + 2}}} \right)\left( {\sigma _1^2  + \frac{{12\sigma _w^2 }}
{{\left( {r^3  + 3r^2  + 2r} \right)s^2 }}} \right) - \left( {\sigma _{01}  - \frac{{6\sigma _w^2 }}
{{\left( {r^2  + 3r + 2} \right)s}}} \right)^2 }}
{{(1 - p_e )p_e \left( {\sigma _1^2  + \frac{{12\sigma _w^2 }}
{{\left( {r^3  + 3r^2  + 2r} \right)s^2 }}} \right)}}.

The limit of this expression is

\frac{{\sigma _0^2 \sigma _1^2  - \sigma _{01}^2 }}
{{p_e (1 - p_e )\sigma _1^2 }}.

Equivalently, we derived the same results for the fixed $\tau $ case. This limit can be rewritten in terms of our paramterization as

\mathop {\lim }\limits_{r \to \infty } \;\;{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \frac{{\sigma _{t_0 }^2 \rho _{t_0 } \left( {1 - \rho _{01}^2 } \right)}}
{{p_e (1 - p_e )}}.

§.§ LDD, RS, $V(t_0 ) = 0$

When $V(t_0 ) = 0$, using equation (<ref>), we can derive ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ by substituting $s^2 \det ({\mathbf{A}})$ by $\det \left( {{\mathbf{\tilde A}}} \right)$ to obtain, in terms of our parameterization,

{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \left( {\frac{{12\sigma ^2 (1 - \rho _{t_0 } )}}
{{{\kern 1pt} s^2 p_e (1 - p_e )\,}}} \right)\left( {\frac{1}
{{\,r(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,s,\tilde r} }}
{{1 - \rho _{b_1 ,s,\tilde r} }}} \right)\frac{1}
{{\tilde r(\tilde r + 1)(\tilde r + 2)}}} \right),

as in Table <ref>. Then, it is easily derived that

\mathop {\lim }\limits_{r \to \infty } \;\;{\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} = \left( {\frac{{12\sigma ^2 (1 - \rho _{t_0 } )}}
{{{\kern 1pt} s^2 p_e (1 - p_e )\,}}} \right)\left( {\frac{{\rho _{b_1 ,s,\tilde r} }}
{{1 - \rho _{b_1 ,s,\tilde r} }}} \right)\frac{1}
{{\tilde r(\tilde r + 1)(\tilde r + 2)}},

and, equivalently, for the fixed $\tau $ case the limit is

\left( {\frac{{12\sigma ^2 (1 - \rho _{t_0 } )}}
{{{\kern 1pt} \tau ^2 p_e (1 - p_e )\,}}} \right)\left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{\,(\tilde r + 1)(\tilde r + 2)}}.

§ THE EFFECT OF COVARIANCE PARAMETERS ON THE MINIMUM $R$ FOR A FIXED $N$, SUBJECT TO POWER $\PI $

§.§ The effect of $\rho $ and $\rho _{t_0 } $

§.§.§ CMD, CS, $V(t_0 ) = 0$

From equation (<ref>),

r = \frac{{\beta _2^2 N\,p_e (1 - p_e ) - \left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 \sigma ^2 }}
{{\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 \sigma ^2 \rho  - \beta _2^2 N\,p_e (1 - p_e )}}.

Differentiating with respect to $\rho $, we get

\frac{{\partial r}}
{{\partial \rho }} = \frac{{\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 \sigma ^2 \left( { - \beta _2^2 N\,p_e (1 - p_e ) + \left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 \sigma ^2 } \right)}}
{{\left( {\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 \sigma ^2 \rho  - \beta _2^2 N\,p_e (1 - p_e )} \right)^2 }}.

If $\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 \sigma ^2  > \beta _2^2 N\,p_e (1 - p_e )$, then $\frac{{\partial r}} {{\partial \rho }} > 0$, so r increases as $\rho
$ increases.
If $\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 \sigma ^2  < \beta _2^2 N\,p_e (1 - p_e )$, then $\frac{{\partial r}}{{\partial \rho }} < 0$, so r decreases as $\rho $ increases.

§.§.§ LDD, CS, fixed s, $V(t_0 ) = 0$

The minimum $r$ for fixed $N$ and fixed power, $\pi $, solves

N = \frac{{12\sigma ^2 (1 - \rho )\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 {\kern 1pt} }}
{{{\kern 1pt} \gamma _3^2 p_e (1 - p_e )\;s^2 \,r\,(r + 1)(r + 2)}},

which was obtained plugging in the corresponding value of ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}} $ in Table <ref> into equation (<ref>). Defining

F\left( {r,\,\rho } \right) = \frac{{N\gamma _3^2 p_e (1 - p_e )\;s^2 \,}}
{{12\sigma ^2 \left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 }} - \frac{{(1 - \rho )}}
{{r\,(r + 1)(r + 2)}},

the equation $F\left( {r,\,\rho } \right) = 0$ implicitly defines the function $r = f(\rho )$. Using implicit differentiation and taking into account that r is a function of $\rho $, $r(\rho )$, we obtain

\frac{{\partial r}}
{{\partial \rho }} = \frac{{ - r\,\left( {r + 1} \right)\left( {r + 2} \right)}}
{{(1 - \rho )(3r^2  + 6r + 2)}}.

Since r is positive, the derivative is always negative, and r decreases as $\rho $ increases.

§.§.§ LDD, CS, fixed $\tau $, $V(t_0 ) = 0$

The minimum $r$ for fixed $N$ and fixed power, $\pi $, solves

N = \frac{{12\sigma ^2 (1 - \rho )\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 {\kern 1pt} r}}
{{{\kern 1pt} \gamma _3^2 p_e (1 - p_e )\;\tau ^2 \,\,(r + 1)(r + 2)}},

which was obtained plugging in the corresponding value of ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ in Table <ref> into equation (<ref>). Defining

F\left( {r,\,\rho } \right) = \frac{{N\gamma _3^2 p_e (1 - p_e )\;\tau ^2 \,}}
{{12\sigma ^2 \left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 }} - \frac{{(1 - \rho )r}}
{{\,(r + 1)(r + 2)}},

the equation $F\left( {r,\,\rho } \right) = 0$ implicitly defines the function $r = f(\rho )$. Using implicit differentiation and  taking into account that r is a function of $\rho $, we obtain

\frac{{\partial r}}
{{\partial \rho }} = \frac{{r\left( {r + 1} \right)\left( {r + 2} \right)}}
{{(1 - \rho )\left( { - r^2  + 2} \right)}}.

If $r \geqslant 2$, then $\frac{{\partial r}}{{\partial \rho }} < 0$. So if we are taking at least two post-baseline measures, larger values of $\rho $ lead to smaller values of r to achieve the specified power. Since $\frac{r}{{\left( {r + 1} \right)\left( {r + 2} \right)}}$ is the same for $r = 1$ and $r = 2$, it is preferable to choose $r = 1$ since fewer measurements need to be collected. Therefore, the choice between $r = 1$ and $r = 2$ is not affected by $\rho $.

§.§.§ LDD, RS, fixed s, $V(t_0 ) = 0$

The minimum $r$ for fixed $N$ and fixed power, $\pi $, solves

N = \frac{{\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 \left( {\frac{{12\sigma ^2 (1 - \rho _{t_0 } )}}
{{{\kern 1pt} s^2 \,}}} \right)\left( {\frac{1}
{{\,r(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,s,\tilde r} }}
{{1 - \rho _{b_1 ,s,\tilde r} }}} \right)\frac{1}
{{\tilde r(\tilde r + 1)(\tilde r + 2)}}} \right)}}
{{{\kern 1pt} \gamma _3^2 p_e (1 - p_e )\;}},

which was obtained by plugging in the corresponding value of ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}
$ in Table <ref> into equation (<ref>). Defining

\begin{multline*}
F\left( {r,\,\rho _{t_0 } } \right) = \\
\frac{{N\gamma _3^2 p_e (1 - p_e )s^2 }}
{{12\sigma ^2 \left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 }} - (1 - \rho _{t_0 } )\left( {\frac{1}
{{\,r(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,s,\tilde r} }}
{{1 - \rho _{b_1 ,s,\tilde r} }}} \right)\frac{1}
{{\tilde r(\tilde r + 1)(\tilde r + 2)}}} \right),
\end{multline*}

the equation $F\left( {r,\,\rho _{t_0 } } \right) = 0$ implicitly defines the function $r = f(\rho _{t_0 } )$. Using implicit differentiation and taking into account that r depends on $\rho _{t_0 } $, we obtain

\frac{{\partial r}}
{{\partial \rho _{t_0 } }} = \frac{{ - r\,\left( {r + 1} \right)\left( {r + 2} \right)\left[ {\rho _{b_1 ,s,\tilde r} r\,\left( {r + 1} \right)\left( {r + 2} \right) + \left( {1 - \rho _{b_1 ,s,\tilde r} } \right)\tilde r(\tilde r + 1)(\tilde r + 2)} \right]}}
{{\left( {1 - \rho _{b_1 ,s,\tilde r} } \right)\tilde r(\tilde r + 1)(\tilde r + 2)(1 - \rho _{t_0 } )\left( {3r^2  + 6r + 2} \right)}} < 0.

Since the derivative is always negative when $r > 0$, r decreases as $\rho _{t_0 } $ increases.

§.§.§ LDD, RS, fixed $\tau $, $V(t_0 ) = 0$

The minimum $r$ for fixed $N$ and fixed power, $\pi $, solves

N = \frac{{\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 \left( {\frac{{12\sigma ^2 (1 - \rho _{t_0 } )}}
{{{\kern 1pt} \tau ^2 \,}}} \right)\left( {\frac{r}
{{\,(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{\,(\tilde r + 1)(\tilde r + 2)}}} \right)}}
{{{\kern 1pt} \gamma _3^2 p_e (1 - p_e )\;}},

which was obtained by plugging in the corresponding value of ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ in Table <ref> into equation (<ref>). Defining
\begin{multline*}
F\left( {r,\,\rho } \right) = \\
\frac{{N\gamma _3^2 p_e (1 - p_e )\tau ^2 }}
{{12\sigma ^2 \left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 }} - (1 - \rho _{t_0 } )\left( {\frac{r}
{{\,(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{\,(\tilde r + 1)(\tilde r + 2)}}} \right),
\end{multline*}

the equation $F\left( {r,\,\rho _{t_0 } } \right) = 0$ implicitly defines the function $r = f(\rho _{t_0 } )$. Using implicit differentiation, and taking into account that r depends on $\rho _{t_0 } $, we obtain

\frac{{\partial r}}
{{\partial \rho _{t_0 } }} = \frac{{\left( {r + 1} \right)\left( {r + 2} \right)\left[ {\rho _{b_1 ,\tau ,\tilde r} \tilde r\left( {r + 1} \right)\left( {r + 2} \right) + \left( {1 - \rho _{b_1 ,\tau ,\tilde r} } \right)(\tilde r + 1)(\tilde r + 2)r} \right]}}
{{\left( {1 - \rho _{b_1 ,\tau ,\tilde r} } \right)(\tilde r + 1)(\tilde r + 2)\left( {1 - \rho _{t_0 } } \right)(2 - r^2 )}}.

If $r \geqslant 2$ then $\frac{{\partial r}} {{\partial \rho _{t_0 } }} < 0$. So if we are taking at least two post-baseline measures, larger values of $\rho _{t_0 } $ lead to smaller minimal values of r to achieve a certain power. Since $\frac{r}
{{\left( {r + 1} \right)\left( {r + 2} \right)}}$ is the same for $r = 1
$ and $r = 2$, the resulting power of both studies would be the same and it would be preferable to choose $r = 1$ since less measurements need to be collected. The choice between $r = 1$ and $r = 2$ is not affected by $\rho _{t_0 } $.

§.§ The effect of $\rho _{b_1 ,s,\tilde r} $

§.§.§ LDD, RS, fixed s, $V(t_0 ) = 0$

The minimum $r$ for fixed $N$ and fixed power, $\pi $, solves

N = \frac{{\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 \left( {\frac{{12\sigma ^2 (1 - \rho _{t_0 } )}}
{{{\kern 1pt} s^2 \,}}} \right)\left( {\frac{1}
{{\,r(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,s,\tilde r} }}
{{1 - \rho _{b_1 ,s,\tilde r} }}} \right)\frac{1}
{{\tilde r(\tilde r + 1)(\tilde r + 2)}}} \right)}}
{{{\kern 1pt} \gamma _3^2 p_e (1 - p_e )\;}},

which was obtained plugging in the corresponding value of ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ in Table <ref> into equation (<ref>). Defining

\begin{multline*}
F\left( {r,\,\rho _{b_1 ,s,\tilde r} } \right) = \\
\frac{{N\gamma _3^2 p_e (1 - p_e )s^2 }}
{{12\sigma ^2 \left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 (1 - \rho _{t_0 } )}} - \left( {\frac{1}
{{\,r(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,s,\tilde r} }}
{{1 - \rho _{b_1 ,s,\tilde r} }}} \right)\frac{1}
{{\tilde r(\tilde r + 1)(\tilde r + 2)}}} \right),
\end{multline*}

the equation $F\left( {r,\,\rho _{b_1 ,s,\tilde r} } \right) = 0$ implicitly defines the function $r = f(\rho _{b_1 ,s,\tilde r} )$. Using implicit differentiation, and taking into account that r depends on $\rho _{b_1 ,s,\tilde r} $, we obtain

\frac{{\partial r}}
{{\partial \rho _{b_1 ,s,\tilde r} }} = \frac{{r^2 (r + 1)^2 (r + 2)^2 }}
{{\tilde r(\tilde r + 1)(\tilde r + 2)(1 - \rho _{b_1 ,s,\tilde r} )^2 (3r^2  + 6r + 2)}} > 0.

Since the derivative is always positive, r increases as $\rho _{b_1 ,s,\tilde r} $ increases.

§.§.§ LDD, RS, fixed $\tau $, $V(t_0 ) = 0$

The minimum $r$ for fixed $N$ and fixed power, $\pi $, solves

N = \frac{{\left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 \left( {\frac{{12\sigma ^2 (1 - \rho _{t_0 } )}}
{{{\kern 1pt} \tau ^2 \,}}} \right)\left( {\frac{r}
{{\,(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{\,(\tilde r + 1)(\tilde r + 2)}}} \right)}}
{{{\kern 1pt} \gamma _3^2 p_e (1 - p_e )\;}},

which was obtained plugging in the corresponding value of ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ in Table <ref> into equation (<ref>). Defining

\begin{multline*}
F\left( {r,\,\rho _{b_1 ,\tau ,\tilde r} } \right) = \\
\frac{{N\gamma _3^2 p_e (1 - p_e )\tau ^2 }}
{{12\sigma ^2 \left( {z_\pi   + z_{1 - \alpha /2} } \right)^2 (1 - \rho _{t_0 } )}} - \left( {\frac{r}
{{\,(r + 1)(r + 2)}} + \left( {\frac{{\rho _{b_1 ,\tau ,\tilde r} }}
{{1 - \rho _{b_1 ,\tau ,\tilde r} }}} \right)\frac{{\tilde r}}
{{\,(\tilde r + 1)(\tilde r + 2)}}} \right),
\end{multline*}

the equation $F\left( {r,\,\rho _{b_1 ,s,\tilde r} } \right) = 0$ implicitly defines the function $r = f(\rho _{b_1 ,s,\tilde r} )$. Using implicit differentiation, and taking into account that r depends on $\rho _{b_1 ,\tau ,\tilde r} $, we obtain

\frac{\partial }
{{\partial \rho _{b_1 ,\tau ,\tilde r} }} = \frac{{\tilde r(r + 1)^2 (r + 2)^2 }}
{{\,(r^2  - 2)(\tilde r + 1)(\tilde r + 2)(1 - \rho _{b_1 ,\tau ,\tilde r} )^2 }}.

If $r \geqslant 2$, $\frac{{\partial r}}
{{\partial \rho _{b_1 ,\tau ,\tilde r} }} > 0$. So if we are taking at least two post-baseline measurements, the effect of increasing $\rho _{b_1 ,\tau ,\tilde r}
$ is to increase the minimum r needed to achieve a pre-specified power. Since $\frac{r}
{{\left( {r + 1} \right)\left( {r + 2} \right)}}$ is the same for $r = 1$ and $r = 2$, the resulting power of both studies would be the same and it is therefore preferable to choose $r = 1$ since less measurements need to be collected. The choice between $r = 1$ and $r = 2$ is not affected by $\rho _{b_1 ,\tau ,\tilde r} $.

§ CALCULATION OF THE VARIANCE UNDER RS AND $V(T_0 ) > 0$ ASSUMING $T_{0I} $ ARE NORMALLY DISTRIBUTED

We need to derive

{\mathbf{\Sigma }}_{\rm B}  = \left( {\mathbb{E}\left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } \right)} \right)^{ - 1}.

When ${\mathbf{\Sigma }}_i  = {\mathbf{\Sigma }}$ for all subjects, $\mathbb{E}\left( {{\mathbf{X'}}_i {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } \right)$ can be computed exactly. This will happen when $V(t_0 ) = 0$, and then equations (<ref>) and (<ref>) for $V(t_0 ) = 0$ provide general expression for ${\mathbf{c'\Sigma }}_{\rm B} {\mathbf{c}}$ for CMD and LDD, respectively. However, if $V(t_0 ) > 0$ then ${\mathbf{\Sigma }}_i  \ne {\mathbf{\Sigma }}$ under RS. Specifically, ${\mathbf{\Sigma }}_i $ depends on $t_{0i} $, so we have ${\mathbf{\Sigma }}(t_{0i} )$. The formula for ${\mathbf{\Sigma }}_i $ under RS is ${\mathbf{\Sigma }}_i  = {\mathbf{Z}}_i {\mathbf{DZ'}}_i  + \sigma _{within}^2 {\mathbf{I}}$, where

{\mathbf{Z'}}_i  = \left( {\begin{array}{*{20}c}
1 &  \cdots  &  \cdots  &  \cdots  & 1  \\
{t_{0i} } &  \cdots  & {t_{0i}  + js} &  \cdots  & {t_{0i}  + rs}  \\
\end{array} } \right)

{\mathbf{D}} = \left( {\begin{array}{*{20}c}
{\sigma _0^2 } & {\sigma _{01} }  \\
{\sigma _{01} } & {\sigma _1^2 }  \\
\end{array} } \right).

At this point it is convenient to introduce a new matrix

{\mathbf{W}}_i  = \left( {\begin{array}{*{20}c}
1 & 0 & {k_i } & 0  \\
0 & 1 & 0 & {k_i }  \\
\end{array} } \right).

Note that under LDD,

\begin{multline*}
{\mathbf{Z}}_i {\mathbf{W}}_i  = \left( {\begin{array}{*{20}c}
1 & {t_{0i} }  \\
\vdots  &  \vdots   \\
\vdots  & {t_{0i}  + js}  \\
\vdots  &  \vdots   \\
1 & {t_{0i}  + rs}  \\
\end{array} } \right)\left( {\begin{array}{*{20}c}
1 & 0 & {k_i } & 0  \\
0 & 1 & 0 & {k_i }  \\
\end{array} } \right) = \\
\left( {\begin{array}{*{20}c}
1 & {t_{0i} } & {k_i } & {t_{0i} k_i }  \\
\vdots  &  \vdots  &  \vdots  &  \vdots   \\
\vdots  & {t_{0i}  + js} &  \vdots  & {\left( {t_{0i}  + js} \right)k_i }  \\
\vdots  &  \vdots  &  \vdots  &  \vdots   \\
1 & {t_{0i}  + rs} & {k_i } & {\left( {t_{0i}  + rs} \right)k_i }  \\
\end{array} } \right) = {\mathbf{X}}_i.
\end{multline*}

{\mathbf{X}}_i ^\prime  {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i  = {\mathbf{W}}_i ^\prime  {\mathbf{Z}}_i ^\prime  {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{Z}}_i {\mathbf{W}}_i  = {\mathbf{W}}_i ^\prime  {\mathbf{Z}}_i ^\prime  \left( {{\mathbf{Z}}_i {\mathbf{DZ'}}_i  + \sigma _{within}^2 {\mathbf{I}}} \right)^{ - 1} {\mathbf{Z}}_i {\mathbf{W}}_i.

In Appendix <ref> we derived ,

{\mathbf{Z'}}\left( {{\mathbf{ZDZ'}} + \sigma _{within}^2 {\mathbf{I}}} \right)^{ - 1} {\mathbf{Z}} = \left( {\left( {{\mathbf{Z'Z}}} \right)^{ - 1} \sigma _{within}^2  + {\mathbf{D}}} \right)^{ - 1},

so we can deduce ${\mathbf{X}}_i ^\prime  {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i  = {\mathbf{W}}_i ^\prime  \left( {\left( {{\mathbf{Z'Z}}} \right)^{ - 1} \sigma _{within}^2  + {\mathbf{D}}} \right)^{ - 1} {\mathbf{W}}_i
$. Now,

\left( {{\mathbf{Z'}}_i {\mathbf{Z}}_i } \right) = \left( {\begin{array}{*{20}c}
{r + 1} & {\left( {r + 1} \right)t_{0i}  + \frac{{s{\kern 1pt} r(r + 1)}}
{2}}  \\
{\left( {r + 1} \right)t_{0i}  + \frac{{s{\kern 1pt} r(r + 1)}}
{2}} & {\left( {r + 1} \right)t_{0i} ^2  + st_{0i} r(r + 1) + \frac{{s^2 r(r + 1)(2r + 1)}}
{6}}  \\
\end{array} } \right),

\left( {{\mathbf{Z'}}_i {\mathbf{Z}}_i } \right)^{ - 1}  = \frac{1}
{{r(r + 1)(r + 2)s^2 }}\left( {\begin{array}{*{20}c}
{2\left( {r\left( {1 + 2r} \right)s^2  + 6r{\kern 1pt} st_{0i}  + 6t_{0i} ^2 } \right)} & { - 6\left( {r{\kern 1pt} s + 2t_{0i} } \right)}  \\
{ - 6\left( {r{\kern 1pt} s + 2t_{0i} } \right)} & {12}  \\
\end{array} } \right).

Using this result we computed

\left( {\left( {{\mathbf{Z'Z}}} \right)^{ - 1} \sigma _{within}^2  + {\mathbf{D}}} \right)^{ - 1}  = \left( {\begin{array}{*{20}c}
{a(t_{0i} )} & {c(t_{0i} )}  \\
{c(t_{0i} )} & {d(t_{0i} )}  \\
\end{array} } \right),

a(t_{0i} ) = \frac{{\left( {\frac{{12\sigma _w^2 }}
{{r(r + 1)(r + 2)s^2 }} + \sigma _1^2 } \right)}}
{{\left( {\frac{{12\sigma _w^2 }}
{{r(r + 1)(r + 2)s^2 }} + \sigma _1^2 } \right)\left( {\sigma _0^2  + \frac{{2\sigma _w^2 \left( {r(1 + 2r)s^2  + 6rst_{0i}  + 6t_{0i} ^2 } \right)}}
{{r(r + 1)(r + 2)s^2 }}} \right) - \left( {\sigma _{01}  - \frac{{6\sigma _w^2 (rs + 2t_{0i} )}}
{{r(r + 1)(r + 2)s^2 }}} \right)^2 }},

c(t_{0i} ) = \frac{{\left( { - \sigma _{01}  + \frac{{6\sigma _w^2 (rs + 2t_{0i} )}}
{{r(r + 1)(r + 2)s^2 }}} \right)}}
{{\left( {\frac{{12\sigma _w^2 }}
{{r(r + 1)(r + 2)s^2 }} + \sigma _1^2 } \right)\left( {\sigma _0^2  + \frac{{2\sigma _w^2 \left( {r(1 + 2r)s^2  + 6rst_{0i}  + 6t_{0i} ^2 } \right)}}
{{r(r + 1)(r + 2)s^2 }}} \right) - \left( {\sigma _{01}  - \frac{{6\sigma _w^2 (rs + 2t_{0i} }}
{{r(r + 1)(r + 2)s^2 }}} \right)^2 }},

d(t_{0i} ) = \frac{{\left( {\sigma _0^2  + \frac{{2\sigma _w^2 \left( {r(1 + 2r)s^2  + 6rst_{0i}  + 6t_{0i} ^2 } \right)}}
{{r(r + 1)(r + 2)s^2 }}} \right)}}
{{\left( {\frac{{12\sigma _w^2 }}
{{r(r + 1)(r + 2)s^2 }} + \sigma _1^2 } \right)\left( {\sigma _0^2  + \frac{{2\sigma _w^2 \left( {r(1 + 2r)s^2  + 6rst_{0i}  + 6t_{0i} ^2 } \right)}}
{{r(r + 1)(r + 2)s^2 }}} \right) - \left( {\sigma _{01}  - \frac{{6\sigma _w^2 (rs + 2t_{0i} }}
{{r(r + 1)(r + 2)s^2 }}} \right)^2 }}.

Pre- and post-multiplying by ${\mathbf{W}}_i $ we get

\begin{multline*}
{\mathbf{X}}_i ^\prime  {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i  = {\mathbf{W}}_i ^\prime  \left( {\left( {{\mathbf{Z'Z}}} \right)^{ - 1} \sigma _{within}^2  + {\mathbf{D}}} \right)^{ - 1} {\mathbf{W}}_i  = \\
\left( {\begin{array}{*{20}c}
{a(t_{0i} )} & {c(t_{0i} )} & {k_i a(t_{0i} )} & {k_i c(t_{0i} )}  \\
{c(t_{0i} )} & {d(t_{0i} )} & {k_i c(t_{0i} )} & {k_i d(t_{0i} )}  \\
{k_i a(t_{0i} )} & {k_i c(t_{0i} )} & {k_i a(t_{0i} )} & {k_i c(t_{0i} )}  \\
{k_i c(t_{0i} )} & {k_i d(t_{0i} )} & {k_i c(t_{0i} )} & {k_i d(t_{0i} )}  \\
\end{array} } \right).
\end{multline*}

\mathbb{E}\left[ {{\mathbf{X}}_i ^\prime  {\mathbf{\Sigma }}_i^{ - 1} {\mathbf{X}}_i } \right] = \left( {\begin{array}{*{20}c}
{\mathbb{E}\left[ {a(t_{0i} )} \right]} & {\mathbb{E}\left[ {c(t_{0i} )} \right]} & {\mathbb{E}\left[ {k_i a(t_{0i} )} \right]} & {\mathbb{E}\left[ {k_i c(t_{0i} )} \right]}  \\
{\mathbb{E}\left[ {c(t_{0i} )} \right]} & {\mathbb{E}\left[ {d(t_{0i} )} \right]} & {\mathbb{E}\left[ {k_i c(t_{0i} )} \right]} & {\mathbb{E}\left[ {k_i d(t_{0i} )} \right]}  \\
{\mathbb{E}\left[ {k_i a(t_{0i} )} \right]} & {\mathbb{E}\left[ {k_i c(t_{0i} )} \right]} & {\mathbb{E}\left[ {k_i a(t_{0i} )} \right]} & {\mathbb{E}\left[ {k_i c(t_{0i} )} \right]}  \\
{\mathbb{E}\left[ {k_i c(t_{0i} )} \right]} & {\mathbb{E}\left[ {k_i d(t_{0i} )} \right]} & {\mathbb{E}\left[ {k_i c(t_{0i} )} \right]} & {\mathbb{E}\left[ {k_i d(t_{0i} )} \right]}  \\
\end{array} } \right).

To compute the expected values in the matrix, we need to know the distribution of $t_{0i} $ and the joint distribution of $\left( {t_{0i} ,k_i } \right)$. We assume that the distribution of $t_{0i} $ has mean zero and variance $V(t_0 )$ and that $k_i $ follows a Bernoulli distribution with probability of success $p_e $. Additionally, we assume that within each exposure group, $t_{0i} $ is normally distributed with the same variance. In Appendix <ref> we deduced the means of $t_{0i} $ in each of the exposure groups as

\mathbb{E}\left( {t_0 |k = 1} \right) = \rho _{\operatorname{e} ,t_0 } \sqrt {\frac{{(1 - p_e )}}
{{p_e }}} \sqrt {V\left( {t_0 } \right)}

\mathbb{E}\left[ {t_0 |k = 1} \right] =  - \rho _{\operatorname{e} ,t_0 } \sqrt {\frac{{p_e }}
{{(1 - p_e )}}} \sqrt {V\left( {t_0 } \right)}.