# Relationship between Collider Bias and Interactions on the Log-Additive
Scale

Apostolos Gkatzionis1, Shaun R. Seaman3, Rachael A. Hughes1,2, Kate Tilling1,2

(1MRC Integrative Epidemiology Unit, University of Bristol, UK.
2Population Health Science Institute, Bristol Medical School, University of
Bristol, UK.
3MRC Biostatistics Unit, University of Cambridge, Cambridge, UK. )

###### Abstract

Collider bias occurs when conditioning on a common effect (collider) of two
variables $X,Y$. In this manuscript, we quantify the collider bias in the
estimated association between exposure $X$ and outcome $Y$ induced by
selecting on one value of a binary collider $S$ of the exposure and the
outcome. In the case of logistic regression, it is known that the magnitude of
the collider bias in the exposure-outcome regression coefficient is
proportional to the strength of interaction $\delta_{3}$ between $X$ and $Y$
in a log-additive model for the collider:
$\mathbb{P}(S=1|X,Y)=\exp\left\\{\delta_{0}+\delta_{1}X+\delta_{2}Y+\delta_{3}XY\right\\}$.
We show that this result also holds under a linear or Poisson regression model
for the exposure-outcome association. We then illustrate by simulation that
even if a log-additive model with interactions is not the true model for the
collider, the interaction term in such a model is still informative about the
magnitude of collider bias. Finally, we discuss the implications of these
findings for methods that attempt to adjust for collider bias, such as inverse
probability weighting which is often implemented without including
interactions between variables in the weighting model.

Keywords: collider bias, Berkson’s bias, log-additive model, interaction,
inverse probability weighting, ALSPAC.

##  1 Introduction

Collider bias is a common concern in epidemiological studies. When exploring
the association between an exposure $X$ and an outcome $Y$ of interest,
collider bias occurs if the analysis is conditioned on a common effect of the
exposure and outcome, or a variable that is causally downstream of the common
effect (a “child” of the collider), as illustrated in Figure 1. Numerous
examples of studies affected by collider bias can be found in the literature.
For example, collider bias has been suggested as an explanation for the
“obesity paradox”, where obesity often appears to be associated with decreased
mortality in older individuals or people suffering from chronic diseases,
despite being associated with increased mortality in the overall population
(Sperrin et al., 2016; Viallon and Dufournet, 2016).

SXYSXYR Figure 1: Causal diagram illustrating how collider bias occurs. Two
variables $X$, $Y$ that are both causes of a third variable $S$ will become
correlated when conditioning on $S$ (left) or on a variable that is causally
downstream of $S$ (right) even if they were unconditionally independent. Or,
if $X$ and $Y$ are unconditionally associated, conditioning on $S$ will change
the strength of their association.

In this paper, we focus on collider bias induced when an analysis is
restricted to a single level of a binary collider variable $S$. A number of
biases in epidemiological studies across a wide range of study designs can be
attributed to this mechanism. This includes some forms of selection bias due
to non-representative sampling (Hernán et al., 2004), survival bias, Berkson’s
hospitalization bias (Berkson, 1946) and index event bias in studies of
disease progression (Mitchell et al., 2022), among others. In all of these
cases, the collider $S$ represents selection into the study: individuals with
$S=1$ have their exposure and outcome observed, while individuals with $S=0$
do not and are hence excluded from the study.

It has long been recognized in the literature that this type of collider bias
relates to interactions between the exposure and the outcome on the log-
additive scale in their effects on the collider (Greenland, 1977; Kleinbaum et
al., 1982). Consider the following log-additive model for the collider:

$\log\mathbb{P}(S=1|X,Y)=\delta_{0}+\delta_{1}X+\delta_{2}Y+\delta_{3}XY$ (1)

The parameter $\delta_{3}$ quantifies the strength of the exposure-outcome
interaction. In analyses of binary outcomes, a number of authors have
suggested that collider bias will not affect estimates of the exposure-outcome
odds ratio when $X$ and $Y$ do not interact in their effects on $S$, i.e. when
$\delta_{3}=0$ (Greenland, 1996; Rothman et al., 2008; Greenland, 2009;
VanderWeele, 2015; Bartlett et al., 2015). An overview of the relevant
literature can be found in Jiang and Ding (2017), who also explored the
direction in which collider bias acts, while Mansournia et al. (2022) obtained
similar results and also investigated bias when conditioning on $S=0$. Despite
the interest this topic has attracted, most of the relevant papers have
focused on binary outcome variables, with some papers also restricting the
exposure to be a binary variable. One exception is Shahar and Shahar (2017),
who considered a discrete exposure and a discrete outcome, potentially with
more than two categories.

In this manuscript, we expand the literature by considering a wider range of
outcome variables, including count outcomes (Poisson regression) and
continuous outcomes (linear regression) in addition to binary ones. We also
place no conditions on the form of the exposure variable. We show that
collider bias will not affect the exposure-outcome association when the
exposure and outcome do not interact in their effects on the collider, i.e.
when $\delta_{3}=0$ in model (1). When they do interact, we show that the
magnitude of collider bias induced in the exposure-outcome regression
coefficients in linear, logistic and Poisson regression models is proportional
to $\delta_{3}$. Finally, when model (1) is misspecified, we show numerically
that for binary $X$ there is still a linear relationship between the magnitude
of the collider bias and the estimated value of $\delta_{3}$ in the
(misspecified) model (1).

The rest of this manuscript is organized as follows. In Section 2, we review
the relevant literature. We then consider regression models for a binary
outcome $Y$ (primarily logistic regression), linear regression and Poisson
regression. For each of these models, we prove that the magnitude of collider
bias is proportional to the value of the exposure-outcome interaction
$\delta_{3}$ in the log-additive model (1). In Section 3, we investigate
collider bias when the collider variable is not distributed according to the
log-additive model (1). We demonstrate numerically that when $X$ is a binary
variable, estimating $\delta_{3}$ can still provide information about the
magnitude of collider bias, even if model (1) is not the true model for $S$.
Section 4 contains an illustrative application using data from the Avon
Longitudinal Study of Parents and Children (ALSPAC) to investigate
associations between maternal traits, such as education or smoking before
pregnancy, and child sex. A discussion of our main findings and their
implications is presented in Section 5.

##  2 Collider Bias under a Log-additive Model for the Collider

###  2.1 Statement of the Problem

As in the introduction, suppose that the objective is to investigate the
association between an exposure $X$ and an outcome $Y$. We focus on the
marginal association between $X$ and $Y$. However, the results presented here
can be extended to cover the conditional association between $X$ and $Y$ given
a set of other variables, e.g. confounders, as we briefly discuss towards the
end of Section 2.2.1.

As in the previous section, we let the collider $S$ represent a binary
selection indicator. The unconditional exposure-outcome association cannot be
estimated directly using the observed data; instead, only the conditional
association given $S=1$ can be estimated. As stated in Section 1, if an
individual’s exposure and outcome values affect their likelihood of selection
into the study sample, the conditional and unconditional exposure-outcome
associations will differ. Our aim in this section is to explore the difference
between the conditional and unconditional exposure-outcome associations for a
binary collider variable distributed according to model (1).

We study collider bias separately for binary, continuous and count outcome
variables. For binary outcome variables, we examine collider bias in odds
ratios, logistic regression coefficients and risk ratios. For continuous
outcomes, we quantify collider bias in linear regression coefficients, and for
count outcome variables, we investigate collider bias in the log rate ratio
parameters of a Poisson regression model. We work under the assumption that
the collider $S$ is distributed according to the log-additive model (1);
different models for $S$ will be considered in the next section.

###  2.2 The Relationship between Collider Bias and Exposure-Outcome
Interactions

####  2.2.1 Binary Outcome - Collider Bias on the Odds Ratio Scale

Consider first the case of a binary outcome, which has received the most
attention in the literature (e.g. Jiang and Ding, 2017; Bartlett et al.,
2015). Some of these papers also restricted the exposure to be binary; here,
we do not place any assumptions on the type of the exposure variable. Our main
assumption is that the exposure and outcome affect the collider on the log-
additive scale, as in (1). Let

$\displaystyle OR_{XY}(x)$ $\displaystyle=$
$\displaystyle\frac{\mathbb{P}(Y=1|X=x+1)}{\mathbb{P}(Y=0|X=x+1)}\times\frac{\mathbb{P}(Y=0|X=x)}{\mathbb{P}(Y=1|X=x)}$
$\displaystyle OR_{XY|S=1}(x)$ $\displaystyle=$
$\displaystyle\frac{\mathbb{P}(Y=1|X=x+1,S=1)}{\mathbb{P}(Y=0|X=x+1,S=1)}\times\frac{\mathbb{P}(Y=0|X=x,S=1)}{\mathbb{P}(Y=1|X=x,S=1)}$

be the unconditional and conditional odds ratios respectively.

In Supplementary Section 1.1 (see also Jiang and Ding, 2017), we prove that

$OR_{XY|S=1}(x)=OR_{XY}(x)\;\exp\left\\{\delta_{3}\right\\}$ (2)

This shows that the magnitude of collider bias on the odds ratio scale is
fully determined by the interaction parameter $\delta_{3}$. In particular, if
$\delta_{3}=0$, we have $OR_{XY|S=1}(x)=OR_{XY}(x)$, meaning that collider
bias does not occur. It follows from equation (2) that if the logistic model

$\text{logit}\mathbb{P}(Y=1|X)=\beta_{0}+\beta_{1}X$ (3)

is correctly specified, the magnitude of collider bias in the parameter
$\beta_{1}$ is equal to $\delta_{3}$. Letting $\beta_{1}^{S}$ denote the
(population) log-odds ratio for the exposure-outcome association conditional
on $S=1$, one could write

$\beta_{1}^{S}=\beta_{1}+\delta_{3}$ (4)

Some generalizations of this result are possible. For example, equation (4)
still holds when the $X-S$ and $Y-S$ main effects in model (1) are replaced by
non-linear functions:

$\log\mathbb{P}(S=1|X,Y)=g_{1}(X)+g_{2}(Y)+\delta_{3}XY$

Finally, consider the more general case of a higher-order (in X) interaction:

$\log\mathbb{P}(S=1|X,Y)=g_{1}(X)+g_{2}(Y)+g_{3}(X)Y$

In this case, equation (2) becomes

$OR_{XY|S=1}(x)=OR_{XY}(x)\exp\left\\{g_{3}(x+1)-g_{3}(x)\right\\}$

The pattern of bias will therefore depend on the form of the function $g_{3}$.
Note that the conditional and unconditional odds ratios will be equal if and
only if $g_{3}(x+1)=g_{3}(x)$ for all $x$, i.e. $g_{3}$ is constant in $X$,
which again shows that collider bias occurs if and only if the exposure and
outcome interact in their effects on the collider.

Finally, let the variable $X$ be vector-valued; this may represent either
multiple exposures whose association with $Y$ is investigated, or a single
exposure whose association with the outcome is adjusted for the presence of
observed confounders. Under the models (1) and (3), where now
$\beta_{1},\delta_{1},\delta_{3}$ are vector-valued, one can show that
$\beta_{1}^{S}=\beta_{1}+\delta_{3}$, i.e. the bias in the regression
coefficient of an element $X_{j}$ of the vector $X$ is equal to the
interaction between $X_{j}$ and $Y$ in the collider model. Moreover,
interactions between the variables $X_{j}$ in the collider model will not
affect the bias.

####  2.2.2 Binary Outcome - Collider Bias on the Risk Ratio Scale

We now explore the magnitude of collider bias on the risk ratio scale. Once
again, we assume that the collider $S$ is distributed according to (1) and
that the outcome $Y$ is binary. Our aim is to compare the unconditional risk
ratio

$RR_{XY}(x)=\frac{\mathbb{P}(Y=1|X=x+1)}{\mathbb{P}(Y=1|X=x)}$

to the conditional risk ratio

$RR_{XY|S=1}(x)=\frac{\mathbb{P}(Y=1|X=x+1,S=1)}{\mathbb{P}(Y=1|X=x,S=1)}$

In Supplementary Section 1.2, we prove that

$RR_{XY|S=1}(x)=RR_{XY}(x)\times\frac{e^{\delta_{2}+\delta_{3}(x+1)}\;\mathbb{P}(Y=1|X=x)+e^{\delta_{3}}\;\mathbb{P}(Y=0|X=x)}{e^{\delta_{2}+\delta_{3}(x+1)}\;\mathbb{P}(Y=1|X=x+1)+\mathbb{P}(Y=0|X=x+1)}$
(5)

It is clear from this formulation that the properties we have proved for the
magnitude of collider bias on the odds ratio scale do not hold for bias on the
risk ratio scale; for example, unlike (4), expression (5) involves both the
interaction parameter $\delta_{3}$ and the outcome-collider coefficient
$\delta_{2}$. A more specific formula for the bias on the risk ratio scale can
be obtained by incorporating modelling assumptions for the exposure-outcome
relationship into (5). In practice, risk ratios are often studied under the
log-binomial regression model

$\text{log}\mathbb{P}(Y=1|X=x)=\beta_{0}+\beta_{1}x$

This implies an unconditional risk ratio of $RR_{XY}(x)=e^{\beta_{1}}$ for any
$x$, and from (5), a conditional risk ratio of

$RR_{XY|S=1}(x)=\frac{e^{\delta_{2}+\delta_{3}(x+1)}\;e^{\beta_{0}+\beta_{1}(x+1)}+e^{\delta_{3}+\beta_{1}}\;(1-e^{\beta_{0}+\beta_{1}x})}{e^{\delta_{2}+\delta_{3}(x+1)}\;e^{\beta_{0}+\beta_{1}(x+1)}+(1-e^{\beta_{0}+\beta_{1}(x+1)})}$

The difference between the unconditional and conditional risk ratios is then

$RR_{XY}(x)-RR_{XY|S=1}(x)=e^{\beta_{1}}\;\left(1-\frac{e^{\delta_{2}+\delta_{3}(x+1)}\;e^{\beta_{0}+\beta_{1}x}+e^{\delta_{3}}\;(1-e^{\beta_{0}+\beta_{1}x})}{e^{\delta_{2}+\delta_{3}(x+1)}\;e^{\beta_{0}+\beta_{1}(x+1)}+(1-e^{\beta_{0}+\beta_{1}(x+1)})}\right)$
(6)

The bias hence depends on both the interaction term $\delta_{3}$ and the
outcome-collider parameter $\delta_{2}$, and the absence of an interaction
($\delta_{3}=0$) is not enough to eliminate bias on the risk ratio scale. On
the other hand, one can easily verify that the conditional and unconditional
risk ratios are equal when $\delta_{2}=\delta_{3}=0$, i.e. when the outcome
does not associate with the collider; and likewise, there is no bias if
$\delta_{3}=0$ and $\beta_{1}=0$.

In Supplementary Section 1.2, we explore collider bias on the risk ratio scale
under a logistic regression model for the outcome and obtain results similar
to those reported here.

####  2.2.3 Continuous Outcome - Collider bias in Linear Regression
Coefficients

We now turn our attention to continuous outcome variables, and assume that the
outcome is distributed according to the linear regression model

$Y|X=\beta_{0}+\beta_{1}X+\epsilon_{Y}$

where $\epsilon_{Y}\sim N(0,\sigma^{2})$ independent of $X$. We start by
noting that $\mathbb{E}(Y|X)=\beta_{0}+\beta_{1}X$ and explore the bias in the
regression coefficient $\beta_{1}$ when conditioning on $S=1$. In
Supplementary Section 1.3, we show that the conditional expectation
$\mathbb{E}(Y|X,S=1)$ is equal to

$\mathbb{E}(Y|X,S=1)=(\beta_{0}+\delta_{2}\sigma^{2})+(\beta_{1}+\delta_{3}\sigma^{2})x$
(7)

Denoting by $\beta_{0}^{S}$ and $\beta_{1}^{S}$ the regression coefficients of
a linear regression model conditioned on $S=1$, it follows that

$\beta_{1}^{S}=\beta_{1}+\delta_{3}\sigma^{2}$ (8)

which implies that the two regression coefficients in the conditional and
unconditional exposure-outcome models will be equal if and only if
$\delta_{3}=0$. When the two coefficients differ, the magnitude of collider
bias induced is equal to the interaction term $\delta_{3}$ multiplied by the
outcome variance $\sigma^{2}$. Moreover,
$\beta_{0}^{S}=\beta_{0}+\delta_{2}\sigma^{2}$; therefore the bias in the
intercept $\beta_{0}$ is equal to the outcome-collider parameter $\delta_{2}$
multiplied by the outcome variance.

As with binary outcomes, the above derivation allows for a non-linear
exposure-collider effect,

$\log\mathbb{P}(S=1|X,Y)=g_{1}(X)+\delta_{2}Y+\delta_{3}XY$

but not for a non-linear outcome-collider effect. Finally, a more general
regression framework for the exposure-outcome relationship can be considered:

$Y|X=x=m(x;\beta)+\epsilon\;\;\;,\;\;\;\epsilon\sim N(0,\sigma^{2})$

where $m(x;\beta)=\mathbb{E}(Y|X=x)$ is a potentially non-linear function that
represents the exposure-outcome association. This yields

$\mathbb{E}(Y|X=x,S=1)=m(x;\beta)+\sigma^{2}\delta_{2}+\sigma^{2}\delta_{3}x$

In addition, as for logistic regression, our results still hold if $X$ is
vector-valued.

####  2.2.4 Count Outcome - Collider bias in Poisson Regression Coefficients

Finally, we consider the case of a count outcome variable distributed
according to the Poisson regression model

$Y|X\sim
Poisson(\lambda)\;\;\;,\;\;\;\lambda=\lambda(X)=\exp\left\\{\beta_{0}+\beta_{1}X\right\\}$
(9)

Once again, our aim is to obtain an expression for the bias in the regression
coefficient $\beta_{1}$ when the collider $S$ follows the log-additive model
(1). Our framework here has some similarities to the work of Shahar and Shahar
(2017); their paper requires that the exposure is a discrete variable but does
not place any distributional assumptions on the outcome, apart from it being
discrete.

In Supplementary Section 1.4, we show that if expression (9) holds, then
$Y|X,S=1$ is Poisson($\kappa$)-distributed, where

$\kappa=\kappa(X)=\exp\left\\{(\beta_{0}+\delta_{2})+(\beta_{1}+\delta_{3})X\right\\}$
(10)

Therefore, the relationship between the regression coefficients $\beta_{0}$
and $\beta_{1}$ in the unconditional exposure-outcome model and the
corresponding coefficients $\beta_{0}^{S}$ and $\beta_{1}^{S}$ in the
conditional model is

$\displaystyle\beta_{0}^{S}$ $\displaystyle=$
$\displaystyle\beta_{0}+\delta_{2}$ $\displaystyle\beta_{1}^{S}$
$\displaystyle=$ $\displaystyle\beta_{1}+\delta_{3}$ (11)

As in the case of logistic regression (4), this implies that the magnitude of
collider bias in the regression coefficient $\beta_{1}$ induced by
conditioning on $S=1$ is equal to $\delta_{3}$. In particular, when the
exposure and outcome do not interact in their effects on the collider in the
log-additive model (1), i.e. $\delta_{3}=0$, there is no bias.

As with binary and continuous outcome variables, a few extensions of this
result are possible, including to analyses with a non-linear exposure-collider
association: if expression (9) holds and

$\log\mathbb{P}(S=1|X,Y)=g_{1}(X)+\delta_{2}Y+g_{3}(X)Y$

then $Y|X,S=1\sim Poisson(\kappa(x))$ where
$\kappa(x)=\exp\\{\beta_{0}+\beta_{1}x+\delta_{2}+g_{3}(x)\\}$. Finally, our
results can be readily extended to Poisson regression with a vector-valued
exposure variable.

##  3 Collider Bias under Alternative Models for the Collider - An Asymptotic
Study

###  3.1 Study Design

So far we have assumed that the collider $S$ is distributed according to the
log-additive model (1). Under this assumption, we have shown that there is a
linear relationship between the magnitude of collider bias and the interaction
term $\delta_{3}$ in model (1), as seen in equations (4), (8) and (11).
However, the log-additive model (1) may be misspecified. In this section, we
conduct an asymptotic study to investigate collider bias under
misspecification of model (1). Focusing on the case of a binary exposure
variable, we demonstrate two things. First, if the true model for $S$ is not
log-additive, the exposure-outcome interaction term in that model does not
exhibit a linear relationship with the magnitude of collider bias in exposure-
outcome regression coefficients. Second, the limiting value (as the sample
size tends to infinity) of the maximum likelihood estimator (MLE) of the
interaction parameter $\delta_{3}$ obtained by fitting the log-additive model
(1) for $S$ still exhibits a linear relationship with the magnitude of
collider bias, even if this log-additive model is misspecified.

For our asymptotic study, we considered nine data generating mechanisms,
obtained by combining three different outcome models and three different
models for the selection indicator $S$. For the outcome, we used linear,
logistic and Poisson regression:

$\displaystyle(Y1)$ $\displaystyle:$
$\displaystyle\text{logit}\mathbb{P}(Y=1|X)=\beta_{0}^{Y1}+\beta_{1}^{Y1}X$
$\displaystyle(Y2)$ $\displaystyle:$ $\displaystyle
Y|X=x=\beta_{0}^{Y2}+\beta_{1}^{Y2}x+\epsilon_{Y}\;\;\;,\;\;\;\epsilon_{Y}\sim
N(0,\sigma^{2})$ $\displaystyle(Y3)$ $\displaystyle:$ $\displaystyle Y|X=x\sim
Poisson(\lambda)\;\;\;,\;\;\;\lambda=\lambda(x)=\exp\left\\{\beta_{0}^{Y3}+\beta_{1}^{Y3}x\right\\}$

For the selection indicator $S$, we considered standard logistic and probit
regression models:

$\displaystyle(S1)$ $\displaystyle:$
$\displaystyle\mathbb{P}(S=1|X,Y)\;=\;\text{expit}\left\\{\delta_{0}^{S1}+\delta_{1}^{S1}X+\delta_{2}^{S1}Y+\delta_{3}^{S1}XY\right\\}$
$\displaystyle(S2)$ $\displaystyle:$ $\displaystyle
S\;=\;\mathbf{1}_{S^{\prime}>0}\;\;\;,\;\;\;S^{\prime}\sim
N\left(\delta_{0}^{S2}+\delta_{1}^{S2}X+\delta_{2}^{S2}Y+\delta_{3}^{S2}XY,1.6^{2}\right)$

We also considered a third model, where we generated a latent, normally
distributed variable $S^{\prime}$ and then set $S=1$ for individuals for which
the latent variable took values below a lower threshold $r_{1}$ or above an
upper threshold $r_{2}$:

$(S3)\;\;\;:\;\;\;S=\mathbf{1}_{S^{\prime}>r_{1}\;\text{or}\;S^{\prime}<r_{2}}\;\;\;,\;\;\;S^{\prime}\sim
N\left(\delta_{0}^{S3}+\delta_{1}^{S3}X+\delta_{2}^{S3}Y+\delta_{3}^{S3}XY,1.6^{2}\right)$

We will refer to (S3) as a “double-threshold” model. Finally, in all nine data
generating mechanisms, the exposure values were generated from a
$\text{Bernoulli}(0.3)$ distribution.

The parameters of the three outcome models and the three data-generating
models for $S$ were specified as follows. In the outcome model, we set
$\beta_{0}^{Yj}=0$ and $\beta_{1}^{Yj}=0.2$, $j=1,2,3$. In the linear
regression model (Y2), we also set $\sigma=0.5$. In the model for $S$, the
exposure-collider and outcome-collider association parameters were set to
$\delta_{1}^{Sk}=\delta_{2}^{Sk}=0.3$, $k=1,2,3$. Finally, the residual
variance for the latent variable $S^{\prime}$ in models (S2) and (S3) was set
to $1.6$ so that the regression coefficients in these models represented a
comparable strength of association as the coefficients of the logistic
regression model (S1) (Wooldridge, 2012, , Chapter 17).

There were two parameters to be varied in our asymptotic study: the strength
of the exposure-outcome interaction $\delta_{3}^{Sk}$, and the proportion of
selected individuals, which was determined by the intercept term
$\delta_{0}^{Sk}$ in the model for $S$ (and for model (S3), by the thresholds
$r_{1},r_{2}$). We conducted two experiments varying the values of these
parameters. In the first experiment, we specified the intercept term
$\delta_{0}^{Sk}$ so that approximately $50\%$ of simulated individuals were
included in the conditional analysis ($S=1$); in model (S3), we instead set
$\delta_{0}^{S3}=0$ and set $r_{1},r_{2}$ equal to the first and third
quartile of the distribution of $S^{\prime}$. We then varied the exposure-
outcome interaction parameter $\delta_{3}^{Sk}$, letting it take the values
$\delta_{3}^{Sk}=0,\pm 0.1,\pm 0.2,\pm 0.3,\pm 0.4,\pm 0.5$. In our second
experiment, we used the same range of values for $\delta_{3}^{Sk}$ but varied
the value of the intercept $\delta_{0}^{Sk}$. We specified four values for
$\delta_{0}^{Sk}$, such that the proportion of selected individuals was
$10\%$, $30\%$, $70\%$ and $90\%$ respectively. For model (S3), we set
$\delta_{0}^{S3}=0$ and specified the proportion of selected individuals by
tuning the thresholds $r_{1},r_{2}$ instead.

For each of the nine data generating mechanisms and each set of parameter
values, we generated a single dataset of size $n=10^{7}$. We used a large
sample size to approximate an infinite sample: the values of maximum
likelihood estimators obtained using our sample will be very close to the
limiting value of these estimators. We then fitted model (Yj) using only data
on individuals with $S=1$ to calculate the estimate $\hat{\beta}_{1}^{Yj}$ of
$\beta_{1}^{Yj}$. The difference between this estimate and the true value of
$\beta_{1}^{Yj}$ is the collider bias induced in the regression coefficient of
model (Yj). This bias was plotted against the value of the interaction
parameter $\delta_{3}^{Sk}$ in the correctly specified model (Sk), $k=1,2,3$,
to assess their relationship. We then fitted the misspecified log-additive
model:

$(S0)\;\;\;:\;\;\;\mathbb{P}(S=1|X,Y)\;=\;\exp\left\\{\delta_{0}^{S0}+\delta_{1}^{S0}X+\delta_{2}^{S0}Y+\delta_{3}^{S0}XY\right\\}$

and computed the maximum likelihood estimate $\hat{\delta}_{3}^{S0}$ of the
interaction parameter $\delta_{3}^{S0}$, and then plotted this estimate
against the collider bias. The estimate $\hat{\delta}_{3}^{S0}$ was used in
our asymptotic study to approximate the limiting value
$\tilde{\delta}_{3}^{S0}$ of the maximum likelihood estimator; note that
$\tilde{\delta}_{3}^{S0}$ is the value that minimizes the Kullback-Leibler
divergence between model (S0) and the true data-generating model for $S$
(which here is model (S1), (S2) or (S3)).

Data were generated and models were fitted using R. The log-additive models
were fitted as Poisson regression models, using the glm function.

###  3.2 Results

Figure 2: Magnitude of collider bias induced in the exposure-outcome
regression coefficient by restricting the analysis to selected ($S=1$)
individuals. Data were generated for $n=10^{7}$ individuals, and the average
selection probability was $50\%$. Outcome data were generated from logistic
regression (left column), linear regression (middle column) or Poisson
regression (right column) and collider values were generated from logistic
regression (model S1, top row), probit regression (model S2, middle row) or
the “double threshold” model (model S3, bottom row). The bias is plotted
against the exposure-outcome interaction $\delta_{3}^{Sk}$ in the collider
model. Red color represents simulations with no interaction.

The results of our first asymptotic experiment are shown in Figures 2 and 3
and reported in numerical form in Supplementary Section 2. Figure 2 shows the
results of simulations where the collider $S$ was generated under a logistic
(top row of plots), probit (middle row) or “double-threshold” (bottom row)
model, while the outcome was generated from a logistic (left column), linear
(centre column) or Poisson (right column) regression model. The collider bias
is plotted against the true value of the interaction parameter
$\delta_{3}^{Sk}$ in the corresponding collider model (Sk), $k=1,2,3$. The
relationship between collider bias and the values of the interaction
parameters is not linear, with deviations from linearity being more pronounced
for Poisson regression and less so for linear regression. In addition, the
collider bias is quite small in scenarios where the exposure and outcome do
not interact in their effects on the collider ($\delta_{3}^{Sk}=0$, plotted in
red), although even here some bias still exists.

For some of the models considered here, it is possible to derive analytic
expressions for collider bias using arguments similar to those in the previous
section. As an example, in Supplementary Section 1.5 we obtain an expression
for the collider bias in the exposure-outcome regression coefficient of a
logistic regression model when the collider $S$ also follows a logistic
regression model. However, such relationships can only be derived for
relatively simple models, and the bias will generally depend on all the
parameters of the collider model, not just on the interaction term.

Figure 3: Magnitude of collider bias induced in the exposure-outcome
regression coefficient by restricting the analysis to selected ($S=1$)
individuals. Data were generated for $n=10^{7}$ individuals, and the average
selection probability was $50\%$. Outcome data were generated from logistic
regression (left column), linear regression (middle column) or Poisson
regression (right column) and collider values were generated from logistic
regression (model S1, top row), probit regression (model S2, middle row) or
the “double threshold” model (model S3, bottom row). The bias is plotted
against the estimated values of the exposure-outcome interaction parameter
$\delta_{3}^{S0}$ in a (misspecified) log-additive model for $S$. A gray
vertical line represents no interaction ($\hat{\delta}_{3}^{S0}=0$).

In Figure 3, we plot collider bias against the limiting values
$\tilde{\delta}_{3}^{S0}$ of the log-additive interaction parameter
$\delta_{3}^{S0}$, estimated by fitting the misspecified log-additive model
(S0). Note that the points in these plots are not equally spaced along the
x-axis because equally spaced interactions on the scale of models (S1)-(S3) do
not correspond to equally spaced interactions on the log-additive scale. The
relationship between collider bias and limiting values of the estimator of
$\delta_{3}^{S0}$ appears to be linear, with a slope of $1$ for logistic and
Poisson regression and $\sigma^{2}=0.25$ in the case of linear regression.
This is the same relationship suggested by our theory (equations (4), (8) and
(11)), despite the fact that the log-additive model is misspecified.

Note that the linear pattern of collider bias presented in Figure 3 only
occurs when the exposure $X$ is binary. In Supplementary Section 3, we report
results of an asymptotic study conducted with a normally distributed exposure
variable, where the relationship between bias and $\tilde{\delta}_{3}^{S0}$ is
not linear (see also Campbell et al., 2005). Unlike the theoretical results in
Section 2 of our manuscript, the distribution of the exposure can affect the
magnitude of collider bias when the collider $S$ is not generated from the
log-additive model (S0).

Figure 4: Magnitude of collider bias induced in the exposure-outcome
regression coefficient by restricting the analysis to selected ($S=1$)
individuals. Different colors represent different selection probabilities
(green: $10\%$, blue: $30\%$, purple: $50\%$, red: $70\%$, orange: $90\%$).
Outcome data were generated from logistic regression (left column), linear
regression (middle column) or Poisson regression (right column) and collider
values were generated from logistic regression (model S1, top row), probit
regression (model S2, middle row) or the “double threshold” model (model S3,
bottom row). The bias is plotted against the exposure-outcome interaction
$\delta_{3}^{Sk}$ in the collider model.

Results from our second asymptotic experiment are shown in Figures 4 and 5. In
Figure 4, we plot the magnitude of collider bias induced in exposure-outcome
regression coefficients against the true value of the exposure-outcome
interaction in models (S1)-(S3), for a range of selection probabilities:
$10\%$ (green), $30\%$ (blue), $50\%$ (purple), $70\%$ (red) and $90\%$
(orange). In Figure 5, we do the same for the limiting values
$\tilde{\delta}_{3}^{S0}$ of the interaction parameter $\delta_{3}^{P}{S0}$ in
the misspecified log-additive model (S0). As seen in Figure 4, smaller
selection probabilities resulted in more bias across all models considered.
However, a smaller proportion of selected individuals also led to a
proportional increase in $\tilde{\delta}_{3}^{S0}$ values. Hence, in Figure 5,
the relationship between bias and interactions was again linear (with a slope
of $1$ for logistic and Poisson-distributed outcomes and $\sigma^{2}$ for
normally distributed outcomes), and the magnitude of collider bias did not
depend on the proportion of selected individuals.

Figure 5: Magnitude of collider bias induced in the exposure-outcome
regression coefficient by restricting the analysis to selected ($S=1$)
individuals. Different colors represent different selection probabilities
(green: $10\%$, blue: $30\%$, purple: $50\%$, red: $70\%$, orange: $90\%$).
Outcome data were generated from logistic regression (left column), linear
regression (middle column) or Poisson regression (right column) and collider
values were generated from logistic regression (model S1, top row), probit
regression (model S2, middle row) or the “double threshold” model (model S3,
bottom row). The bias is plotted against estimated values of the exposure-
outcome interaction parameter $\delta_{3}^{S0}$ in a (misspecified) log-
additive model for $S$.

Taken together, our results suggest that for binary exposures, the linear
relationship between the magnitude of collider bias and the strength of
exposure-outcome interaction on the log-additive scale may hold true
regardless of whether the log-additive model is a realistic assumption for
$S$.

##  4 Real-Data Application

###  4.1 Data and Methods

In addition to the asymptotic study of the previous section, we performed an
analysis using data from the Avon Longitudinal Study of Parents and Children
(ALSPAC, Boyd et al., 2013; Fraser et al., 2013). ALSPAC is a longitudinal
population-based study that recruited pregnant women residing in Avon, UK,
with expected delivery dates between 1st April 1991 and 31st December 1992.
The study included $15447$ pregnancies resulting in $15658$ fetuses, $14901$
of which were alive at 1 year of age. Ethical approval for the study was
obtained from the ALSPAC Ethics and Law Committee and the Local Research
Ethics Committees. Informed consent for the use of data collected via
questionnaires and clinics was obtained from participants following the
recommendations of the ALSPAC Ethics and Law Committee at the time. The study
website (http://www.bristol.ac.uk/alspac/researchers/our-data/) contains
details on all data that is available through a fully searchable data
dictionary and variable search tool.

The aim of our analysis is to demonstrate, using real data, the relation
between collider bias and exposure-outcome interactions in a log-additive
model for selection. To do so, we investigated associations of six maternal
traits with child sex. These maternal traits included age at delivery, highest
educational qualification held, pre-pregnancy body mass index (BMI),
depression status, pre-pregnancy smoking and gestational age. Since child sex
is determined randomly at conception and unaffected by environmental
exposures, one would expect its associations with maternal traits to be null
in the absence of bias (perhaps with the exception of gestational age, Divon
et al., 2002). We obtained maternal trait and child sex data for all ALSPAC
families; these data were treated as the “complete sample” for the purposes of
our analysis. We also obtained participation data for two follow-up stages of
ALSPAC: the “Teen Focus 4” (TF4) clinic visit (age 17+) and the “It’s All
About You” (CCU) questionnaire (age 20). These two subsamples were considered
as the “selected samples” for our analysis. Selection into the two subsamples
differed by sex: in TF4, participation rates were $29.5\%$ for males and
$40.0\%$ for females, while in the CCU sample, they were $22.0\%$ for males
and $35.9\%$ for females. We explored whether these differences could bias
estimated associations between maternal traits and offspring sex in the two
selected samples compared to the complete sample.

For each of the six maternal traits, we fitted a logistic regression model
with child sex as the outcome and the maternal trait as exposure. The models
were fitted both in the complete ALSPAC sample and in the two subsamples. We
fitted a separate model for each maternal trait to mimic the previous parts of
our manuscript, where we only considered one exposure variable. In real-data
applications, it may be preferable to conduct a single joint analysis instead,
with all six maternal traits included as explanatory variables. Such an
analysis is presented in Supplementary Section 5.

We computed regression coefficient estimates from the models regressing child
sex on each maternal trait, fitted either to the full ALSPAC sample or to the
TF4/CCU subsamples. The difference between estimates in the TF4/CCU subsamples
and in the full ALSPAC sample was taken as a measure of collider bias for each
trait. We then fitted log-additive models for TF4/CCU participation, each time
using child sex and one of the maternal traits as covariates, and compared the
interaction estimates in these models to the magnitude of collider bias. In
addition, we fitted logistic regression models for TF4 and CCU participation
using child sex and one of the six maternal traits as explanatory variables
but no interaction terms. Logistic regression is often used to assess which
variables associate with study participation, or to adjust for collider bias
using inverse probability weighting. We investigated whether parameter
estimates from the logistic models could be used to quantify the magnitude of
collider bias for each maternal trait.

Note that our application here is conducted for illustrative purposes, to
demonstrate the connection between interactions and collider bias in a real
dataset. In reality, with access only to the TF4/CCU samples, we would not be
able to fit the log-additive model for selection, while with access to the
complete ALSPAC sample, there would be no need to restrict the analysis to the
TF4/CCU subsamples.

For our analyses, we excluded pregnancies that resulted in miscarriage or
early termination ($3.9\%$). We also excluded pregnancies with missing
information on maternal traits. Missingness rates for the six maternal traits
and sample sizes for our logistic regression analyses are reported in
Supplementary Section 4. Missingness in maternal data can be another source of
collider bias, and methods such as multiple imputation could be used to adjust
for it. In this illustrative application, we choose to ignore this source of
bias and focus on the bias induced by restricting to participants in the TF4
and CCU subsamples.

###  4.2 Results

Estimated associations between each maternal trait and child sex from the
respective logistic models are reported in Table 1. We report parameter
estimates, standard errors and p-values of association between each trait and
child sex, obtained either from all ALSPAC participants or only from TF4/CCU
attendants.

Mat Trait | All ALSPAC | TF4 | CCU
---|---|---|---
$\hat{\beta}_{1}$ |  s.e. $(\hat{\beta}_{1})$ | P-value | $\hat{\beta}_{1}$ |  s.e. $(\hat{\beta}_{1})$ | P-value | $\hat{\beta}_{1}$ |  s.e. $(\hat{\beta}_{1})$ | P-value
Age | 0.008 | 0.003 | 0.014 | 0.020 | 0.006 | 0.002 | 0.023 | 0.007 | 0.001
Education | -0.018 | 0.014 | 0.200 | 0.067 | 0.024 | 0.006 | 0.116 | 0.027 | $1.6\times 10^{-5}$
BMI | 0.004 | 0.005 | 0.369 | 0.010 | 0.008 | 0.245 | 0.007 | 0.009 | 0.431
Depression | -0.051 | 0.053 | 0.341 | -0.039 | 0.097 | 0.686 | -0.085 | 0.116 | 0.461
Smoking | 0.054 | 0.032 | 0.092 | -0.021 | 0.058 | 0.720 | -0.149 | 0.067 | 0.025
Gest Age | -0.033 | 0.007 | $3.3\times 10^{-6}$ | -0.059 | 0.016 | $1.4\times 10^{-4}$ | -0.066 | 0.017 | $1.4\times 10^{-4}$
Table 1: Associations of maternal traits with child sex in ALSPAC, obtained by
fitting six separate logistic regression models for child sex, each with one
maternal trait as the exposure. Estimated associations, standard errors and
p-values computed using data on either all ALSPAC participants, or only those
who attended the TF4 visit, or only those who returned the CCU questionnaire.

Mother’s age at delivery and gestational age were associated with child sex in
all three samples. The observational association between gestational age and
child sex has previously been noted in the literature (Divon et al., 2002) and
could be due to reverse causation, while the association with mother’s age in
ALSPAC was fairly weak and could be due to the missing maternal data. Mother’s
education was not associated with child sex in the full ALSPAC sample but was
seen to associate with child sex in both TF4 and CCU. Maternal smoking was
associated with child sex in the CCU sample but not in the TF4 sample or in
the full ALSPAC sample, while BMI and depression before pregnancy exhibited no
association with child sex in any of the three regression analyses. These
results suggest collider bias may be affecting the association of maternal
education and smoking with child sex. This is not unreasonable, as smoking and
education are often associated with participation in scientific studies, and
at the same time, participation rates in TF4 and CCU differed between males
and females, as mentioned earlier.

Variable | Child Sex | Maternal Trait | Interaction | Bias
---|---|---|---|---
Est | StdErr | P-value | Est | StdErr | P-value | Est | StdErr | P-value
TF4 Estimates
Age | -0.539 | 0.171 | 0.002 | 0.046 | 0.004 | $4.5\times 10^{-33}$ | 0.007 | 0.006 | 0.217 | 0.012
Education | -0.583 | 0.086 | $1.0\times 10^{-11}$ | 0.189 | 0.016 | $3.8\times 10^{-33}$ | 0.083 | 0.024 | 0.001 | 0.085
BMI | -0.399 | 0.186 | 0.032 | -0.008 | 0.005 | 0.149 | 0.005 | 0.008 | 0.562 | 0.006
Depression | -0.299 | 0.291 | 0.304 | 0.227 | 0.067 | 0.001 | 0.000 | 0.099 | 0.998 | 0.012
Smoking | -0.204 | 0.086 | 0.018 | -0.364 | 0.042 | $6.5\times 10^{-18}$ | -0.084 | 0.064 | 0.190 | -0.075
Gest Age | -0.052 | 0.519 | 0.921 | 0.029 | 0.009 | 0.001 | -0.007 | 0.013 | 0.610 | -0.026
CCU Estimates
Age | -0.767 | 0.189 | $4.8\times 10^{-5}$ | 0.048 | 0.004 | $3.4\times 10^{-33}$ | 0.009 | 0.006 | 0.176 | 0.015
Education | -0.937 | 0.096 | $1.7\times 10^{-22}$ | 0.190 | 0.016 | $3.5\times 10^{-31}$ | 0.133 | 0.027 | $5.6\times 10^{-7}$ | 0.134
BMI | -0.526 | 0.208 | 0.011 | -0.012 | 0.006 | 0.031 | 0.002 | 0.009 | 0.807 | 0.003
Depression | -0.329 | 0.341 | 0.335 | 0.371 | 0.076 | $1.1\times 10^{-6}$ | -0.050 | 0.116 | 0.665 | -0.034
Smoking | -0.196 | 0.095 | 0.040 | -0.328 | 0.043 | $3.8\times 10^{-14}$ | -0.238 | 0.072 | 0.001 | -0.203
Gest Age | -0.014 | 0.588 | 0.980 | 0.041 | 0.010 | $3.8\times 10^{-5}$ | -0.012 | 0.015 | 0.411 | -0.033
Table 2: Parameter estimates, standard errors and p-values for a log-additive
model of TF4 or CCU participation in terms of child sex, maternal traits and
interactions between child sex and maternal traits. The observed bias in
$\hat{\beta}_{1}$ estimates calculated from Table 1 is also reported for
comparison.

Table 2 contains the results of fitting log-additive models for TF4 and CCU
participation. Again, six different models were fitted, one for each maternal
trait. All models also included child sex and an interaction between the
maternal trait and child sex. We report parameter estimates, standard errors
and $95\%$ p-values for the regression parameters. For comparison, we also
report the bias observed in Table 1, computed as the difference between
$\beta_{1}$ estimates in the TF4/CCU samples and in the complete sample.

All maternal traits were associated with CCU participation, and all maternal
traits apart from BMI were associated with TF4 participation in their
respective models. However, evidence of an interaction between the maternal
traits and child sex was observed only for maternal education (in both
samples) and smoking (in the CCU sample). This was in line with our previous
analysis, in which maternal education associated with child sex among CCU or
TF4 participants, and maternal smoking did so among CCU participants. The
regression coefficient for the education-child sex interaction was estimated
to be positive in both log-additive models; this would suggest positive bias.
Indeed the TF4 and CCU regression coefficients in Table 1 were both larger
than the regression coefficients in the all-ALSPAC analysis (i.e. positive
bias). On the other hand, the smoking-child sex interaction in the CCU sample
was negative, suggesting negative bias, which was indeed the case based on
Table 1. Finally, the interaction parameter estimates were a good
approximation of the magnitude of bias caused by restricting to the TF4 or CCU
subsamples: for all six traits and for both subsamples, a $95\%$ confidence
interval for the interaction parameter in the log-additive model contained the
observed value of the bias.

Variable | Child Sex | Maternal Trait | Bias
---|---|---|---
Est | StdErr | P-value | Est | StdErr | P-value
TF4 Estimates
Age | -0.524 | 0.036 | $4.6\times 10^{-47}$ | 0.079 | 0.004 | $8.1\times 10^{-99}$ | 0.012
Education | -0.515 | 0.038 | $1.9\times 10^{-41}$ | 0.369 | 0.016 | $1.5\times 10^{-125}$ | 0.085
BMI | -0.472 | 0.039 | $1.6\times 10^{-34}$ | -0.009 | 0.005 | 0.069 | 0.006
Depression | -0.482 | 0.037 | $6.4\times 10^{-38}$ | 0.345 | 0.059 | $5.7\times 10^{-9}$ | 0.012
Smoking | -0.494 | 0.037 | $5.0\times 10^{-41}$ | -0.603 | 0.039 | $8.3\times 10^{-55}$ | -0.075
Gest Age | -0.483 | 0.036 | $1.1\times 10^{-41}$ | 0.038 | 0.008 | $9.8\times 10^{-7}$ | -0.026
CCU Estimates
Age | -0.752 | 0.038 | $2.8\times 10^{-85}$ | 0.078 | 0.004 | $5.0\times 10^{-87}$ | 0.015
Education | -0.753 | 0.040 | $4.0\times 10^{-79}$ | 0.371 | 0.016 | $2.2\times 10^{-115}$ | 0.134
BMI | -0.705 | 0.040 | $3.0\times 10^{-68}$ | -0.017 | 0.005 | 0.001 | 0.003
Depression | -0.705 | 0.039 | $5.8\times 10^{-72}$ | 0.489 | 0.066 | $1.1\times 10^{-13}$ | -0.034
Smoking | -0.722 | 0.039 | $9.0\times 10^{-78}$ | -0.594 | 0.041 | $1.3\times 10^{-47}$ | -0.203
Gest Age | -0.704 | 0.038 | $1.1\times 10^{-77}$ | 0.049 | 0.009 | $9.4\times 10^{-9}$ | -0.033
Table 3: Parameter estimates, standard errors and p-values for a logistic
regression of TF4 and CCU participation on child sex and maternal traits.

The results of fitting the logistic regression models without interactions are
given in Table 3. Again, we report parameter estimates, standard errors and
p-values for child sex and each maternal variable, as well as the bias
observed in ALSPAC. These results suggest strong associations between all six
maternal traits and participation in both samples, with the exception of BMI
in the TF4 sample. In addition, participation is also associated with child
sex, as expected. By fitting the logistic models, an applied researcher could
be led to believe that collider bias will occur when studying the associations
of maternal traits with child sex in TF4/CCU participants. However, as our
analysis in Table 1 indicates, bias is present only for maternal education and
smoking, and not for the other four maternal traits considered here. This
confirms that using a log-additive model with interactions is more informative
about collider bias than the commonly used logistic model without
interactions.

##  5 Discussion

We have shown that, in three commonly used regression models, the magnitude of
collider bias induced in the exposure-outcome association is a linear function
of the strength of interaction between the exposure and outcome in a log-
additive model for the collider. We have proved these results analytically
when the collider is truly distributed according to the log-additive model
(1), and explored them via a numerical asymptotic study and a real-data
application in cases when the collider does not follow the log-additive model.

Our results can be useful in several ways. First, modelling selection into a
study is an important task for methods that attempt to adjust for collider
bias, such as inverse probability weighting (IPW). IPW is typically
implemented using logistic regression without interactions as a weighting
model; this choice is often made for convenience, and in some applications
there is little reason to believe that the logistic model is correctly
specified (Seaman and White, 2013). Including interactions in the logistic
model can offer more flexibility and better adjustment for collider bias. In
addition, our results here raise the question of whether using a log-additive
weighting model can improve the performance of IPW compared to the standard
logistic model; this is something we will explore in future work.

Second, the fact that collider bias only depends on a single parameter in the
simple models considered here can be useful for sensitivity analyses. In some
applications, subject-specific knowledge may allow researchers to assess the
strength of exposure-outcome interactions, and hence assess whether collider
bias is likely to affect their analyses. In addition, simulation studies are
sometimes conducted as a form of sensitivity analysis to explore the impact of
collider bias in applications. These simulation studies typically work by
varying the associations of the exposure, outcome and other relevant variables
with study participation and exploring how much collider bias this induces in
analysis results. Our work suggests that it is the interactions (on the log-
additive scale) that dictate the magnitude of this bias, and therefore that
these interactions should also be varied in addition to (or instead of) the
exposure-selection and outcome-selection associations.

Third, simulation studies are also used as a tool for assessing the finite-
sample performance of novel methods. Our results may therefore be useful to
researchers working on methods to detect or adjust for collider bias. For
example, it may be desired to design a simulation where complete-case analysis
exhibits a specific degree of bias, and compare that with the performance of a
newly developed method. This can be done using a log-additive model with an
interaction, and specifying the value of the interaction parameter
accordingly.

Finally, in some applications it may be possible to fit the log-additive model
with interactions to the real data. For example, consider a study where the
outcome $Y$ is a disease progression trait. Of interest is the (unconditional)
association of an exposure $X$ with the outcome $Y$. Since the progression
outcome can only be observed in individuals who have the disease, the study
must condition on disease incidence. If the exposure $X$ also associates with
disease incidence, the study may suffer from a particular type of collider
bias called index event bias (Mitchell et al., 2022): conditioning on
incidence will induce associations between $X$ and other common causes of
incidence and progression, and hence induce bias in exposure-progression
association estimates. If the study has collected information on additional
covariates $C$ whose association with incidence is suspected, it is not
uncommon to investigate the potential for collider bias by fitting a logistic
model for incidence with covariates $X$ and $C$ and no interactions. Our
research emphasizes the importance of including interactions in this model, to
better assess the potential for collider bias.

Some extensions of our work are possible. Here, we have focused on three
simple statistical models for the exposure-outcome association, namely linear,
logistic and Poisson regression. It would be interesting to explore whether
similar results hold, for example, in survival analysis models. Moreover,
extensions of our results to studies of a causal nature are possible. In this
manuscript we have compared parameter estimates from conditional and
unconditional models. A similar comparison can be made between conditional and
unconditional causal effect estimates obtained from causal models, and the
role of interactions in the model for $S$ can be explored. This could include
instrumental variable analyses, which are known to suffer from collider bias
(Gkatzionis et al., 2020; Hughes et al., 2019).

We hope our findings will prove useful to methodologists investigating
collider bias, as well as to applied researchers attempting to adjust for the
bias in their analyses.

## Data Availability Statement

The R code used to conduct the asymptotic study of our paper is available at
the GitHub repository https://github.com/agkatzionis/Interactions-and-
collider-bias. Access to ALSPAC data for our real-data application was
obtained under application B4189; the data are available upon request to the
ALSPAC study: http://www.bristol.ac.uk/alspac/researchers/access/.

## Acknowledgments

AG and KT received funding for this project by the UK Medical Research Council
and the University of Bristol (MRC-IEU core funding, MC UU 00032/02). SRS was
funded by UKRI (Unit programme numbers MC UU 00002/10) and was supported by
the National Institute for Health Research (NIHR) Cambridge Biomedical
Research Centre (BRC-1215-20014). RAH is supported by a Sir Henry Dale
Fellowship that is jointly funded by the Wellcome Trust and the Royal Society
(grant 215408/Z/19/Z). The views expressed in this manuscript are those of the
authors and not necessarily those of PHE, the NHS, the NIHR or the Department
of Health and Social Care. For the purpose of open access, the authors have
applied a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising.

Access to ALSPAC data was obtained as part of application B4189. The UK
Medical Research Council and Wellcome (Grant ref: 217065/Z/19/Z) and the
University of Bristol provide core support for ALSPAC. A comprehensive list of
grants funding the ALSPAC study is available on the ALSPAC website
(http://www.bristol.ac.uk/alspac/external/documents/grant-
acknowledgements.pdf). GWAS data was generated by Sample Logistics and
Genotyping Facilities at Wellcome Sanger Institute and LabCorp (Laboratory
Corporation of America) using support from 23andMe. We are extremely grateful
to all the families who took part in this study, the midwives for their help
in recruiting them, and the whole ALSPAC team, which includes interviewers,
computer and laboratory technicians, clerical workers, research scientists,
volunteers, managers, receptionists and nurses.

## References

  * Bartlett et al. (2015) Bartlett, J. W., O. Harel, and J. R. Carpenter (2015, 09).  Asymptotically unbiased estimation of exposure odds ratios in complete records logistic regression.  American Journal of Epidemiology 182(8), 730–736.
  * Berkson (1946) Berkson, J. (1946).  Limitations of the application of fourfold table analysis to hospital data.  Biometrics 2(3), 47–53.
  * Boyd et al. (2013) Boyd, A., J. Golding, J. Macleod, D. A. Lawlor, A. Fraser, J. Henderson, L. Molloy, A. Ness, S. Ring, and G. Davey Smith (2013, 04).  Cohort Profile: The ‘Children of the 90s’—the index offspring of the Avon Longitudinal Study of Parents and Children.  International Journal of Epidemiology 42(1), 111–127.
  * Campbell et al. (2005) Campbell, U. B., N. M. Gatto, and S. Schwartz (2005).  Distributional interaction: Interpretational problems when using incidence odds ratios to assess interaction.  Epidemiologic Perspectives and Innovations 2(1).
  * Divon et al. (2002) Divon, M. Y., A. Ferber, H. Nisell, and M. Westgren (2002).  Male gender predisposes to prolongation of pregnancy.  General Obstetrics and Gynecology: Fetus-Placenta-Newborn 187(4), 1081–1083.
  * Fraser et al. (2013) Fraser, A., C. Macdonald-Wallis, K. Tilling, A. Boyd, J. Golding, G. Davey Smith, J. Henderson, J. Macleod, L. Molloy, A. Ness, S. Ring, S. M. Nelson, and D. A. Lawlor (2013, 04).  Cohort Profile: The Avon Longitudinal Study of Parents and Children: ALSPAC mothers cohort.  International Journal of Epidemiology 42(1), 97–110.
  * Gkatzionis et al. (2020) Gkatzionis, A., S. Burgess, D. V. Conti, and P. J. Newcombe (2020).  Bayesian variable selection with a pleiotropic loss function in Mendelian randomization.  bioRxiv.
  * Greenland (1977) Greenland, S. (1977).  Response and follow-up bias in cohort studies.  American Journal of Epidemiology 106(3), 184–187.
  * Greenland (1996) Greenland, S. (1996).  Basic methods for sensitivity analysis of biases.  International journal of epidemiology 25(6), 1107–1116.
  * Greenland (2009) Greenland, S. (2009, 09).  Bayesian perspectives for epidemiologic research: III. Bias analysis via missing-data methods.  International Journal of Epidemiology 38(6), 1662–1673.
  * Hernán et al. (2004) Hernán, M. A., S. Hernández-Díaz, and J. M. Robins (2004).  A structural approach to selection bias.  Epidemiology 15(5), 615–625.
  * Hughes et al. (2019) Hughes, R. A., N. M. Davies, G. Davey Smith, and K. Tilling (2019).  Selection bias when estimating average treatment effects using one-sample instrumental variable analysis.  Epidemiology 30(3), 350–357.
  * Jiang and Ding (2017) Jiang, Z. and P. Ding (2017).  The directions of selection bias.  Statistics & Probability Letters 125, 104–109.
  * Kleinbaum et al. (1982) Kleinbaum, D. G., L. L. Kupper, and H. Morgenstern (1982).  Epidemiologic research: principles and quantitative methods.  John Wiley and Sons, New York.
  * Mansournia et al. (2022) Mansournia, M. A., M. Nazemipour, and M. Etminan (2022, 06).  Interaction Contrasts and Collider Bias.  American Journal of Epidemiology 191(10), 1813–1819.
  * Mitchell et al. (2022) Mitchell, R. E., A. Hartley, V. M. Walker, A. Gkatzionis, J. Yarmolinsky, J. A. Bell, A. H. W. Chong, L. Paternoster, K. Tilling, and G. D. Smith (2022).  Strategies to investigate and mitigate collider bias in genetic and Mendelian randomization studies of disease progression.  medRxiv.
  * Rothman et al. (2008) Rothman, K. J., T. L. Lash, and S. Greenland (2008).  Modern Epidemiology (third ed.).  Lippincott Williams and Wilkins.
  * Seaman and White (2013) Seaman, S. R. and I. R. White (2013).  Review of inverse probability weighting for dealing with missing data.  Statistical Methods in Medical Research 22(3), 278–295.
  * Shahar and Shahar (2017) Shahar, D. J. and E. Shahar (2017).  A theorem at the core of colliding bias.  The International Journal of Biostatistics 13(1), 20160055\.
  * Sperrin et al. (2016) Sperrin, M., J. Candlish, E. Badrick, A. Renehan, and I. Buchan (2016).  Collider bias is only a partial explanation for the obesity paradox.  Epidemiology 27(4), 525–530.
  * VanderWeele (2015) VanderWeele, T. (2015).  Explanation in Causal Inference: Methods for Mediation and Interaction.  Oxford University Press, New York.
  * Viallon and Dufournet (2016) Viallon, V. and M. Dufournet (2016).  Can collider bias fully explain the obesity paradox?  arXiv:1612.06547v1.
  * Wooldridge (2012) Wooldridge, J. M. (2012).  Introductory Econometrics: A Modern Approach (Fifth ed.).  South-western Cengage Learning.


# Beck torsors, formally unramified objects, and Kähler differentials

Nicholas<EMAIL_ADDRESS>
Department of Mathematics,
University of Miami,
Coral Gables, FL

(April 2021)

###### Abstract

Let $\mathcal{C}$ be a category with pullbacks. We define a Beck torsor in
$\mathcal{C}$ as a morphism $Z\to Y$ in $\mathcal{C}$ which is a torsor for a
Beck module over $Y$. We say that an object $X$ of $\mathcal{C}$ is formally
unramified if, for every Beck torsor $Z\to Y$ in $\mathcal{C}$, the canonical
map $\text{Hom}_{\mathcal{C}}(X,Z)\to\text{Hom}_{\mathcal{C}}(X,Y)$ is
injective. If $A$ is a commutative ring with identity, then an $A$-algebra $B$
is formally unramified in the category of $A$-algebras if and only if the ring
homomorphism $A\to B$ is formally unramified. Given that $A\to B$ is formally
unramified if and only if $\Omega_{B/A}=0$, we seek a similar classification
for general formally unramified objects. We say that $\mathcal{C}$ has Kähler
differentials if, for each object $X$ of $\mathcal{C}$, the forgetful functor
$\textup{Ab}(\mathcal{C}/X)\to\mathcal{C}/X$ from the category of Beck modules
over $X$ has a left adjoint
$\Omega:\mathcal{C}/X\to\textup{Ab}(\mathcal{C}/X)$. Our main result is that
if $\mathcal{C}$ has Kähler differentials, then an object $X$ of $\mathcal{C}$
is formally unramified if and only if $\Omega_{X}$ is a zero object in
$\textup{Ab}(\mathcal{C}/X)$.

##  1 Introduction

The study of integer solutions to polynomial equations has led to the
development of rich mathematical theories, such as those used in effort to
study Fermat equations of the type $x^{n}+y^{n}=z^{n}$. However, for an
equation such as $x^{2}=0$ whose only integer solution is $x=0$, one is
compelled to explore non-integer solutions. For example, in the ring
$\mathbb{Z}[\varepsilon]/(\varepsilon^{2})$, we obtain the non-zero solution
$\varepsilon^{2}=0$. This kind of solution, known as an infinitesimal
solution, also plays a significant role in mathematics. For example, the
fundamental notions of formally unramified, formally etale, and formally
smooth morphisms in algebraic geometry are expressed in terms of lifting
criteria inspired by these infinitesimal solutions.

The goal of this paper is to initiate a program to extend the notions of
formally unramified, formally etale, and formally smooth morphisms into a
categorical setting in effort to begin developing algebraic geometry uniformly
over any algebraic theory. The notion of an algebraic theory, which we take to
be a category of algebras of a Lawvere theory [1, 2], includes more than
traditional algebraic objects. For example, there is an algebraic theory of
$C^{\infty}$-rings [3] which contains all of classical differential geometry.
Therefore, one potential application of our program would be to exhibit
immersions, local diffeomorphisms, and submersions in differential geometry as
precisely (not just analogously) the same phenomena as unramified, etale, and
smooth morphisms in algebraic geometry. It appears that Schreiber [4, 5] makes
a similar claim in the context of cohesive $\infty$-topoi, in the sense that
he relates an abstract version of these morphisms to the traditional
differential-geometric morphisms. However, the categorical interpretation of
these morphisms appears to be so fundamental that such elaborate formalism is
not needed to develop the core concepts. More speculatively, our program could
be used to explore geometric objects over algebraic theories in which geometry
is not typically formulated. For example, what is an algebraic stack over the
category of groups?

In this paper, we focus only on formally unramified morphisms and defer a
discussion of formally etale and formally smooth morphisms to a future work.
In section 2, we review the notion of a Beck module and introduce Beck torsors
which are a categorical interpretation of the classical notion of square-zero
extensions. With this categorical version of square-zero extensions at hand,
we introduce formally unramified objects in section 3. It appears that Paugam
[6] considered a similar definition of formally unramified objects in the
context of sheaves on a site, but that definition is not referred to elsewhere
in his book.

It is important to make sure that our definition of formally unramified
objects is justified to carry the name. What properties do formally unramified
objects share with formally unramified ring homomorphisms? A classical result
in commutative algebra states that a ring homomorphism $A\to B$ is formally
unramified if and only if $\Omega_{B/A}=0$, where $\Omega_{B/A}$ is the module
of Kähler differentials. Provided a category $\mathcal{C}$ is sufficiently
nice (such as an algebraic theory), it is possible to define a Beck module
$\Omega_{X}$ of Kähler differentials for each object $X$ of $\mathcal{C}$. In
section 4, we discuss this general notion of Kähler differentials and prove
that $X$ is a formally unramified object of $\mathcal{C}$ if and only if
$\Omega_{X}$ is a zero object in the category of Beck modules over $X$. This
theorem provides some justification that our definition of formally unramified
objects makes sense.

######  Conventions.

A ring is a commutative ring with identity. A ring homomorphism is an
identity-preserving ring homomorphism. We fix a ring $A$ throughout the paper.
An $A$-algebra is a ring $B$ together with a ring homomorphism $\mu_{B}:A\to
B$. If $B$ and $C$ are $A$-algebras, then a ring homomorphism $\varphi:B\to C$
is called an $A$-algebra homomorphism if $\varphi\circ\mu_{B}=\mu_{C}$. We
write $A\text{-Alg}$ for the category of $A$-algebras and $A$-algebra
homomorphisms.

##  2 Beck torsors

We want to reformulate the notion of a formally unramified ring homomorphism
in a way which can be interpreted in any category with pullbacks. We begin by
recalling the traditional definition of square-zero extensions and formally
unramified ring homomorphisms.

######  Definition 2.1.

An $A$-algebra homomorphism $\varphi:D\to C$ is called a square-zero extension
if $\varphi$ is surjective and $(\ker\varphi)^{2}=0$.

######  Definition 2.2.

Let $B$ be an $A$-algebra. We say that $\mu_{B}:A\to B$ is formally unramified
(or simply that $B$ is formally unramified) if, for every square-zero
extension $D\to C$ of $A$-algebras, the canonical map

$\text{Hom}_{A\text{-Alg}}(B,D)\to\text{Hom}_{A\text{-Alg}}(B,C)$

is injective.

The property of
$\text{Hom}_{A\text{-Alg}}(B,D)\to\text{Hom}_{A\text{-Alg}}(B,C)$ being
injective already makes sense for any category, but generalizing the notion of
square-zero extension takes some work. The fundamental observation that makes
this generalization possible is that $D\to C$ is a square-zero extension if
and only if $D\to C$ is a torsor for a Beck module over $C$. Then we note that
this kind of torsor can be interpreted in any category with pullbacks. We now
make these claims precise.

######  Definition 2.3.

Let $\mathcal{C}$ be a category with pullbacks. Let $X$ be an object of
$\mathcal{C}$. A Beck module over $X$ is a morphism $M\to X$ in $\mathcal{C}$
which is an abelian group object in the overcategory $\mathcal{C}/X$. A
morphism of Beck modules over $X$ is a morphism of abelian group objects. We
write $\textup{Ab}(\mathcal{C}/X)$ for the category of Beck modules over $X$.

We require $\mathcal{C}$ to have pullbacks because this assumption implies
that $\mathcal{C}/X$ has products. Since $\mathcal{C}/X$ automatically has a
terminal object $\text{id}_{X}:X\to X$, we therefore have sufficient structure
to interpret group objects in $\mathcal{C}/X$.

A result due to Beck [7] states that if $B$ is an $A$-algebra, then
$\textup{Ab}(A\text{-Alg}/B)$ is equivalent to the category of $B$-modules.
The main point of this equivalence is that every Beck module over $B$ is a
split square-zero extension (isomorphic to a square-zero extension of the type
$B\oplus M\to B$ for a $B$-module $M$). But what can we say about general
square-zero extensions? In a previous work [8], we showed that general square-
zero extensions are the torsors for split square-zero extensions.222We
actually showed that this is true for the category of rings rather than for
the category of $A$-algebras. However, the details of the proof for
$A$-algebras are the same provided that “ring homomorphism” is replaced
throughout with “$A$-algebra homomorphism.” We now define torsors for Beck
modules in a general category with pullbacks. This is possible because Beck
modules are group objects.

######  Definition 2.4.

Let $\mathcal{C}$ be a category with pullbacks. A Beck torsor in $\mathcal{C}$
is an effective epimorphism $Z\to Y$ in $\mathcal{C}$ together with a Beck
module $M$ over $Y$ and a group action $\tau:M\times_{Y}Z\to Z$ in
$\mathcal{C}/Y$ such that $(\tau,\pi_{Z}):M\times_{Y}Z\to Z\times_{Y}Z$ is an
isomorphism in $\mathcal{C}/Y$.

Note that a Beck module $M\to Y$ is a torsor for itself, with group action
given by the group addition $+:M\times_{Y}M\to M$. The requirement that $Z\to
Y$ is an effective epimorphism means that

$Z\times_{Y}Z\rightrightarrows Z\to Y$

is a coequalizer diagram. The reason for this condition is that an effective
epimorphism (not merely an epimorphism) is the correct analogue of a
surjection of rings. Indeed, a ring homomorphism such as
$\mathbb{Z}\to\mathbb{Q}$ is an epimorphism but not surjective. However, a
ring homomorphism is surjective if and only if it is an effective epimorphism.
Since square-zero extensions are surjective ring homomorphisms, using the
condition of effective epimorphism in the definition of Beck torsors seems
natural.

For later use, we record what the torsor property tell us on the level of Hom
sets.

######  Theorem 2.5.

Let $\mathcal{C}$ be a category with pullbacks. Let $Z\to Y$ be a Beck torsor
in $\mathcal{C}$ with Beck module $M$ and group action $M\times_{Y}Z\to Z$.
Then we have a bijection

$\text{Hom}_{\mathcal{C}}(X,M)\times_{\text{Hom}_{\mathcal{C}}(X,Y)}\text{Hom}_{\mathcal{C}}(X,Z)\approx\text{Hom}_{\mathcal{C}}(X,Z)\times_{\text{Hom}_{\mathcal{C}}(X,Y)}\text{Hom}_{\mathcal{C}}(X,Z)$

###### Proof.

Since $Z\to Y$ is a Beck torsor, we have that $M\times_{Y}Z\approx
Z\times_{Y}Z$. The theorem then follows from the universal property of
pullbacks. Indeed,

$\begin{split}\text{Hom}_{\mathcal{C}}(X,M)\times_{\text{Hom}_{\mathcal{C}}(X,Y)}\text{Hom}_{\mathcal{C}}(X,Z)&\approx\text{Hom}_{\mathcal{C}}(X,M\times_{Y}Z)\\\
&\approx\text{Hom}_{\mathcal{C}}(X,Z\times_{Y}Z)\\\
&\approx\text{Hom}_{\mathcal{C}}(X,Z)\times_{\text{Hom}_{\mathcal{C}}(X,Y)}\text{Hom}_{\mathcal{C}}(X,Z).\end{split}$

∎

##  3 Formally unramified objects

We now use Beck torsors as a categorical generalization of square-zero
extensions to define formally unramified objects.

######  Definition 3.1.

Let $\mathcal{C}$ be a category with pullbacks. Let $X$ be an object of
$\mathcal{C}$. We say that $X$ is formally unramified if, for every Beck
torsor $Z\to Y$ in $\mathcal{C}$, the canonical map
$\text{Hom}_{\mathcal{C}}(X,Z)\to\text{Hom}_{\mathcal{C}}(X,Y)$ is injective.

Given that Beck torsors in $A\text{-Alg}$ coincide with square-zero
extensions, it is clear that an $A$-algebra $B$ is a formally unramified
object of $A\text{-Alg}$ if and only if the ring homomorphism $\mu_{B}:A\to B$
is formally unramified.

We now discuss an aspect of formally unramified ring homomorphisms which is
sometimes not emphasized in the literature, but is important for this paper.
Suppose that $B$ is a formally unramified $A$-algebra. Then, for every Beck
module $C\oplus M\to C$ in $A\text{-Alg}$, the canonical map

$\text{Hom}_{A\text{-Alg}}(B,C\oplus M)\to\text{Hom}_{A\text{-Alg}}(B,C)$

is bijective. Said differently, the lifting exists and is unique with respect
to the split square-zero extensions. So what distinguishes formally unramified
ring homomorphisms from formally etale ring homomorphisms333Recall that a ring
homomorphism is formally etale if the lifting exists and is unique with
respect to all square-zero extensions. is the existence of square-zero
extensions which are not split, that is, the existence of Beck torsors which
are not Beck modules.

We now prove that this bijectivity with respect to Beck modules holds in
general for formally unramified objects. Furthermore, it is an if and only if
statement, meaning that this bijectivity fully characterizes formally
unramified objects.

######  Theorem 3.2.

Let $\mathcal{C}$ be a category with pullbacks. Let $X$ be an object of
$\mathcal{C}$. Then $X$ is formally unramified if and only if, for each Beck
module $M\to Y$ in $\mathcal{C}$, the canonical map

$\text{Hom}_{\mathcal{C}}(X,M)\to\text{Hom}_{\mathcal{C}}(X,Y)$

is bijective.

###### Proof.

First assume that $X$ is formally unramified. Let $\varphi:M\to Y$ be a Beck
module in $\mathcal{C}$. Since every Beck module is a Beck torsor, we know
that

$\text{Hom}_{\mathcal{C}}(X,M)\to\text{Hom}_{\mathcal{C}}(X,Y)$

is injective. It remains to show that this map is surjective. Let $\alpha:X\to
Y$ be a morphism in $\mathcal{C}$. Since $\varphi$ is a Beck module, $\varphi$
is an abelian group object in $\mathcal{C}/Y$. Note that $\text{id}_{Y}:Y\to
Y$ is a terminal object of $\mathcal{C}/Y$, and hence part of the abelian
group structure on $\varphi$ is a unit $u_{M}:Y\to M$ such that $\varphi\circ
u_{M}=\text{id}_{Y}$. Define $\psi:X\to M$ such that $\psi=u_{M}\circ\alpha$.
Then the image of $\psi$ under
$\text{Hom}_{\mathcal{C}}(X,M)\to\text{Hom}_{\mathcal{C}}(X,Y)$ is
$\varphi\circ\psi=\varphi\circ
u_{M}\circ\alpha=\text{id}_{Y}\circ\alpha=\alpha$. Therefore, the map is
surjective.

Now assume that, for each Beck module $M\to Y$ in $\mathcal{C}$, the canonical
map

$\text{Hom}_{\mathcal{C}}(X,M)\to\text{Hom}_{\mathcal{C}}(X,Y)$

is bijective. Let $\gamma:Z\to Y$ be a Beck torsor with Beck module $M$ and
group action $M\times_{Y}Z\to Z$. To show that $X$ is formally unramified, we
need to show that the canonical map

$\text{Hom}_{\mathcal{C}}(X,Z)\to\text{Hom}_{\mathcal{C}}(X,Y)$

is injective. Let $\alpha,\beta\in\text{Hom}_{\mathcal{C}}(X,Z)$ be such that
$\gamma\circ\alpha=\gamma\circ\beta$. Then

$(\alpha,\beta)\in\text{Hom}_{\mathcal{C}}(X,Z)\times_{\text{Hom}(X,Y)}\text{Hom}_{\mathcal{C}}(X,Z)$

Using Theorem 2.5 and the assumed bijection
$\text{Hom}_{\mathcal{C}}(X,M)\approx\text{Hom}_{\mathcal{C}}(X,Y)$, we have
that

$\begin{split}\text{Hom}_{\mathcal{C}}(X,Z)\times_{\text{Hom}(X,Y)}\text{Hom}_{\mathcal{C}}(X,Z)&\approx\text{Hom}_{\mathcal{C}}(X,M)\times_{\text{Hom}_{\mathcal{C}}(X,Y)}\text{Hom}_{\mathcal{C}}(X,Z)\\\
&\approx\text{Hom}_{\mathcal{C}}(X,Y)\times_{\text{Hom}_{\mathcal{C}}(X,Y)}\text{Hom}_{\mathcal{C}}(X,Z)\\\
&\approx\text{Hom}_{\mathcal{C}}(X,Y\times_{Y}Z)\\\
&\approx\text{Hom}_{\mathcal{C}}(X,Z),\end{split}$

where the final bijection
$\text{Hom}_{\mathcal{C}}(X,Z)\times_{\text{Hom}(X,Y)}\text{Hom}_{\mathcal{C}}(X,Z)\to\text{Hom}_{\mathcal{C}}(X,Z)$
maps $(\alpha,\beta)$ to $\beta$. Now, since $(\beta,\beta)$ also maps to
$\beta$, we have that $(\alpha,\beta)=(\beta,\beta)$ and hence $\alpha=\beta$.
∎

##  4 Kähler differentials

Finally, we discuss Kähler differentials and their relationship with formally
unramified objects. This categorical definition of Kähler differentials was
introduced by Quillen [9, 10]. We first reformulate the ordinary notion of
Kähler differentials in a way which suggests the categorical generalization.

Let $B$ be an $A$-algebra. We have a forgetful functor
$\textup{Ab}(A\text{-Alg}/B)\to A\text{-Alg}/B$. Under the identification of
$\textup{Ab}(A\text{-Alg}/B)$ with the category of $B$-modules, the left
adjoint $\Omega:A\text{-Alg}/B\to\textup{Ab}(A\text{-Alg}/B)$ is such that,
for each object $C\to B$ of $A\text{-Alg}/B$, $\Omega(C\to
B)=\Omega_{C/A}\otimes_{C}B$. In particular,
$\Omega(\text{id}_{B})=\Omega_{B/A}\otimes_{B}B\approx\Omega_{B/A}$. This
classical result motivates the following definition in general.

######  Definition 4.1.

Let $\mathcal{C}$ be a category with pullbacks. We say that $\mathcal{C}$ has
Kähler differentials if, for each object $X$ of $\mathcal{C}$, the forgetful
functor $\textup{Ab}(\mathcal{C}/X)\to\mathcal{C}/X$ has a left adjoint
$\Omega:\mathcal{C}/X\to\textup{Ab}(\mathcal{C}/X)$. We write
$\Omega_{X}=\Omega(\text{id}_{X})$ and refer to $\Omega_{X}$ as the module of
Kähler differentials of $X$.

By the adjunction of $\Omega$ and the forgetful functor, we have a bijection

$\text{Hom}_{\textup{Ab}(\mathcal{C}/X)}(\Omega_{X},M)\approx\text{Hom}_{\mathcal{C}/X}(X,M)$

for each object $X$ of $\mathcal{C}$ and each Beck module $M$ over $X$. We are
now ready to state and prove the main result of this paper.

######  Theorem 4.2.

Let $\mathcal{C}$ be a category with Kähler differentials. Let $X$ be an
object of $\mathcal{C}$. Then $X$ is formally unramified if and only if
$\Omega_{X}$ is a zero object in $\textup{Ab}(\mathcal{C}/X)$.

###### Proof.

First assume that $\Omega_{X}$ is a zero object in
$\textup{Ab}(\mathcal{C}/X)$. To show that $X$ is formally unramified, Theorem
3.2 tells us it suffices to show that, for each Beck module $M\to Y$ in
$\mathcal{C}$, the canonical map

$\text{Hom}_{\mathcal{C}}(X,M)\to\text{Hom}_{\mathcal{C}}(X,Y)$

is bijective. Let $\varphi:M\to Y$ be a Beck module in $\mathcal{C}$. Let
$\psi\in\text{Hom}_{\mathcal{C}}(X,Y)$. We want to show that there exists a
unique $\alpha\in\text{Hom}_{\mathcal{C}}(X,M)$ such that
$\varphi\circ\alpha=\psi$. Consider the pullback Beck module
$\psi^{\ast}M=X\times_{Y}M$ over $X$. Then

$\text{Hom}_{\textup{Ab}(\mathcal{C}/X)}(\Omega_{X},\psi^{\ast}M)\approx\text{Hom}_{\mathcal{C}/X}(X,\psi^{\ast}M).$

Since $\Omega_{X}$ is a zero object in $\textup{Ab}(\mathcal{C}/X)$,
$\text{Hom}_{\textup{Ab}(\mathcal{C}/X)}(\Omega_{X},\psi^{\ast}M)$ has exactly
one element and hence $\text{Hom}_{\mathcal{C}/X}(X,\psi^{\ast}M)$ has exactly
one element. The universal property of the pullback digram

${X}$${\psi^{\ast}M}$${M}$${X}$${Y}$$\scriptstyle{\alpha}$$\scriptstyle{\text{id}_{X}}$$\scriptstyle{\varphi}$$\scriptstyle{\psi}$

tells us that there exists a unique $\alpha:X\to M$ such that
$\varphi\circ\alpha=\psi\circ\text{id}_{X}=\psi$.

Now assume that $X$ is formally unramified. We want to show that $\Omega_{X}$
is a zero object of $\textup{Ab}(\mathcal{C}/X)$. Since $\text{id}_{X}:X\to X$
is a zero object, it suffices to show that $\Omega_{X}\approx\text{id}_{X}$.
Since any two initial objects are isomorphic, it suffices to show that
$\Omega_{X}$ is initial. Let $M\to X$ be a Beck module over $X$. Since $X$ is
formally unramified, Theorem 3.2 tells us that

$\begin{split}\text{Hom}_{\textup{Ab}(\mathcal{C}/X)}(\Omega_{X},M)&\approx\text{Hom}_{\mathcal{C}/X}(X,M)\\\
&\approx\text{Hom}_{\mathcal{C}/X}(X,X).\end{split}$

Since $\text{Hom}_{\mathcal{C}/X}(X,X)$ has exactly one element,
$\text{Hom}_{\textup{Ab}(\mathcal{C}/X)}(\Omega_{X},M)$ has exactly one
element and hence $\Omega_{X}$ is an initial object of
$\textup{Ab}(\mathcal{C}/X)$. ∎

##  5 Conclusion

We showed that if $\mathcal{C}$ is a category with Kähler differentials, then
an object $X$ of $\mathcal{C}$ is formally unramified if and only if
$\Omega_{X}$ is a zero object in the category of Beck modules over $X$. We did
not address the issue of which categories actually have Kähler differentials.
The good news is that any locally presentable category, such as an algebraic
theory (a category of algebras of a Lawvere theory), has Kähler differentials.
This can be shown by using the locally presentable adjoint functor theorem
[11, 12]. So, if our goal for now is to develop algebraic geometry over
algebraic theories, then we need not be concerned with the issue of existence
of Kähler differentials. However, it would be interesting in the future to
find an exhaustive classification of which categories have Kähler
differentials.

Moving forward, an attempt will soon be made to introduce and study formally
etale and formally smooth objects in a manner similar to what has been done in
the current paper. The notion of finite presentation also makes sense
categorically, meaning that we can define traditional unramified, etale, and
smooth objects (without the “formally”). Long term projects include finding
out whether immersions, local diffeomorphisms, and submersions can be
described in this language via the algebraic theory of $C^{\infty}$-rings, and
also using these categorical definitions to introduce and study analogues of
algebraic spaces and stacks over algebraic theories.

##  6 Acknowledgements

I originally had the idea for this paper while reading the mathoverflow
question [13]. Indeed, as a special case of the main theorem of this paper, we
answer this question by achieving a purely categorical proof of the classical
fact that a ring homomorphism $A\to B$ is formally unramified if and only if
$\Omega_{B/A}=0$.

## References

  * [1] Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories. Ph.D. thesis. F. William Lawvere.
  * [2] nLab page “Lawvere theory.”
  * [3] Models for Smooth Infinitesimal Analysis. Ieke Moerdijk and Gonzalo E. Reyes.
  * [4] nLab page “immersion of smooth manifolds.”
  * [5] Differential cohomology in a cohesive $\infty$-topos. Urs Schreiber.
  * [6] Towards the mathematics of quantum field theory. Frédéric Paugam.
  * [7] Triples, algebras and cohomology. Ph.D. thesis. Jonathan Mock Beck.
  * [8] First order thickenings and cotorsors. Nicholas Mertes.
  * [9] Homotopical Algebra. Daniel G. Quillen.
  * [10] Homology of commutative rings, unpublished notes. Downloadable from Wikipedia. Daniel Quillen.
  * [11] Locally Presentable and Accessible Categories. J. Adamek and J. Rosicky.
  * [12] nLab page “adjoint functor theorem.”
  * [13] “Formally unramified iff trivial Kähler differentials” using only universal properties? mathoverflow. Arrow.