# Quantification of follow-up time in oncology clinical trials with a time-to-
event endpoint: Asking the right questions

Kaspar Rufibach
Lynda Grinsted
Jiang Li
Hans Jochen Weber
Cheng Zheng
Jiangxiu Zhou Methods, Collaboration, and Outreach Group (MCO), Product
Development Data Sciences, Hoffmann-La Roche Ltd, Basel,
SwitzerlandAstraZeneca UK Ltd Cambridge, Cambridgeshire, UKBeiGene USA, Inc.,
55 Challenger Road, Ridgefield Park, NJ 07660Clinical Development and
Analytics, Novartis Pharma AG, Basel, SwitzerlandZentalis Pharmaceuticals, New
York, NY, USAStatistics and Decision Sciences, J&J, Spring House, PA, United
States

###### Abstract

For the analysis of a time-to-event endpoint in a single-arm or randomized
clinical trial it is generally perceived that interpretation of a given
estimate of the survival function, or the comparison between two groups,
hinges on some quantification of the amount of follow-up. Typically, a median
of some loosely defined quantity is reported. However, whatever median is
reported, is typically not answering the question(s) trialists actually have
in terms of follow-up quantification. In this paper, inspired by the estimand
framework, we formulate a comprehensive list of relevant scientific questions
that trialists have when reporting time-to-event data. We illustrate how these
questions should be answered, and that reference to an unclearly defined
follow-up quantity is not needed at all. In drug development, key decisions
are made based on randomized controlled trials, and we therefore also discuss
relevant scientific questions not only when looking at a time-to-event
endpoint in one group, but also for comparisons. We find that different
thinking about some of the relevant scientific questions around follow-up is
required depending on whether a proportional hazards assumption can be made or
other patterns of survival functions are anticipated, e.g. delayed separation,
crossing survival functions, or the potential for cure. We conclude the paper
with practical recommendations.

Keywords: Time-to-event; Randomized Trial; Estimand; Follow-up time.

Running head: Follow-up in time-to-event trials: Asking the right questions.

##  1 Introduction

For the analysis of a time-to-event (T2E) endpoint in a single-arm (SAT) or
randomized controlled trial (RCT) it is generally perceived that
interpretation of a given estimate of the survival function, or the comparison
between two groups, hinges on some quantification of the amount of follow-up.
This is particularly prevalent for oncology clinical trials, where T2E
endpoints, such as progression-free (PFS) and overall survival (OS), are
commonly used as primary endpoint. Positive results in such a trial typically
lead to regulatory approval and, at least in many cases, to a change of
standard clinical practice.

Often in reporting of such trials some quantification of follow-up is
provided, typically via a median of some quantity. However, as Betensky[1]
reports, still recently in about half of reported trials in her review it is
unclear what exactly this median is referring to and no interpretation beyond
the reporting of this bare figure is given. As already discussed in Shuster[2]
whatever median is reported is typically not answering the question(s)
trialists actually have in terms of follow-up quantification. While much has
been said about this broad question in the literature (see Section 2 for a
review) in this paper we would like to add the following aspects:

  * •

Following the thinking process in the ICH E9(R1) estimand addendum[3] we start
with a comprehensive list of relevant scientific questions that trialists have
when reporting T2E data and which are often "answered" with reference to some
unclearly defined quantifier of follow-up. We illustrate how instead these
questions should be answered, and that reference to an unclearly defined
"follow-up quantity" is not necessary.

  * •

The literature so far has focused on quantifying follow-up for estimation of a
survival function in one group. However, in oncology drug development key
decisions are made based on RCTs, and we discuss relevant scientific questions
in this context as well.

  * •

We define and illustrate a collection of follow-up quantifiers that are
routinely used in reporting of trials. One of these, namely $\\#7$ in Table 1,
has not been discussed in the statistical literature previously.

  * •

Finally, with the advent of new therapeutic modalities especially in (but not
restricted to) oncology, patterns of survival functions in trials emerged or
became more prevalent, e.g. delayed separation, crossing survival functions,
or the potential for cure, that may require different thinking on some of the
relevant scientific questions.

While our focus will be on oncology, the findings are applicable in general to
SATs or RCTs with a T2E endpoint in any therapeutic area.

In Section 2 we draw on the literature and our experience as drug developers
in oncology to list the questions that trialists typically would like to get
answered with the broad concept of "follow-up". We then discuss in Section 3
the various quantifiers of follow-up while Section 4 makes the above loosely
defined questions specific before we discuss two examples: one satisfying the
proportional hazards (PH) assumption in Section 5 and one with delayed
separation in Section 6. We conclude with comments on the (ir)relevance of
quantification of follow-up and general recommendations.

##  2 What questions do trialists want to answer?

Let us assume we are interested in analyzing a T2E endpoint, such as PFS or
OS, in a cohort of patients. Some patients had the event already, some
patients may be lost to-follow-up (LTFU), and some are still under
observation, i.e. remain administratively censored (= reporting event called
by the sponsor). For sake of simplicity we assume that the survival function
has been estimated using the Kaplan-Meier (KM) method, as this is by far the
most used estimation approach for such data. Under appropriate assumptions on
the censoring mechanism (e.g. random censoring) this is an unbiased estimate
of the underlying true survival function, automatically accounting for varying
follow-up times, as discussed e.g. by Pocock et al.[4]

Now, a KM estimate of a survival function is typically accompanied by some
quantification of "follow-up", where often neither the precise definition nor
the interpretation of the latter is given[1]. The first who looked into this
question was Shuster [2]. Based on interviews with oncologists he distilled
the following questions that trialists are interested in when considering the
role of median follow-up:

  1. 1.

"Maturity" of the estimated survival function.

  2. 2.

"Stability" of the estimated survival function.

  3. 3.

Time interval where KM estimate is "valid".

  4. 4.

"Quality" of follow-up.

We put those parts of these questions in inverted commas that need a
definition. In addition, the same oncologists were asked what "median follow-
up" means for them, and the group came up with four different answers:

  1. 1.

Follow-up among those who did not have the event yet.

  2. 2.

Follow-up of all patients, irrespective of event status.

  3. 3.

Time from patient trial entry to clinical cut-off date.

  4. 4.

The censoring distribution, estimated through KM based on inverting the
censoring indicator of the primary endpoint.

However, as Shuster already concludes, the median of none of these quantities
can be used to answer any of the above questions. Altman et al.[5] and
Schemper et al.[6] performed systematic literature reviews where Altman et al.
conclude that (1)"…almost half of the papers did not give any summary of
length of follow-up" and (2) "For those which did give a summary, the median
follow-up time was the most frequent value presented, although the method used
to compute it was rarely specified." The situation had not improved two
decades later: Betensky[1] repeated such a review and again found that "34/60
trials did not say what they mean by median follow-up. All that reported
anything only gave medians, none of the papers provided an interpretation."
Betensky concludes that "At the very least, the measure that is reported must
be clearly specified."

The CONSORT statement[7] also has a short paragraph on follow-up, referencing
Shuster [2] and Altman et al.[5]. They recommend for RCTs with a T2E endpoint
to always give the date when follow-up ends, in addition to minimum, maximum,
and median duration of follow-up, so basically a summary of the follow-up
distribution. However, the statement is not specific as to how to compute
follow-up.

We hope that this paper can bring some clarity and structure around relevant
scientific questions, and how they can be answered.

We will now discuss what the literature offers as definition, if at all, for
the four terms that had been quoted by Shuster. Recall that he got these as
responses from trialists on the question about how they interpret "median
follow-up".

###  2.1 Maturity

There is no uniformly accepted definition of what "maturity" means in this
context. For example, Ben-Aharon et al.[8] seem to use "more mature" simply to
mean "later snapshot" or "more data". On the other hand, Tai et al.[9] define
maturity as the proportion of patients that had experienced an event, i.e.
$d_{\text{int}}/n$ or $d_{\text{fin}}/n$, e.g. the proportion of patients who
had died. However, this sometimes appears to be confused with the information
fraction $d_{\text{int}}/d_{\text{fin}}$ at an interim analysis. As an
example, an Oncologic Drugs Advisory Committee (ODAC) meeting of the US FDA in
February 2021 assessed an RCT comparing pembrolizumab + chemotherapy vs.
placebo + chemotherapy in high-risk early-stage triple-negative breast cancer
(TNBC), the KEYNOTE-522[10] trial. In the Combined FDA and Applicant ODAC
Briefing Document[11] the trial data, based on an interim analysis for
efficacy, was considered "immature" because only 174 / 327 = 53% (for event-
free survival) and 96 / 300 = 32% (for OS) of the events planned for the final
analysis had already been observed at this interim analysis. However, note
that this trial recruited 1174 patients in total, so another way to look at
"maturity" would be to take the latter number as denominator. It all basically
boils down whether we want to assess "maturity" of estimation of the hazard
ratio (HR, then information fraction appears appropriate) or the survival
function in one or both groups. In the latter case, it is irrelevant which
portion of targeted events for a hypothesis test for the HR we have already
collected, but the proportion of patients that experienced an event matters
more.

In any case, none of these quantities, i.e. neither
$d_{\text{int}}/d_{\text{fin}}$ nor $d_{\text{int}}/n$ quantifies "follow-up"
using some statistic (e.g. median) of a cumulative distribution function of a
set of times, but rather based on proportion of events only. Also, as we
discuss below, the actual "information" available at an interim analysis only
exclusively depends on the number of events in case of PH. Things are
different in case of non-PH.

"Maturity", understood as $d_{\text{int}}/n$ or $d_{\text{fin}}/n$, needs to
be interpreted in connection with specifications of the underlying disease.
For example, if we expect that 25% of patients will be cured with the
treatment under consideration then it is not reasonable to assume that the
data from which this survival function is estimated can become more "mature"
(as measured by some fraction of events on total number of patients) than 75%.

If cure is possible in the disease under study the question of "enough follow-
up" has been given a precise meaning in [12] and put in a hypothesis test. For
a survival function $S$, let $\tau_{S}=\inf_{t\geq 0}\\{S(t)=0\\}$ (with the
infimum of the empty set equal to $\infty$) denote the upper endpoint of its
support. The null hypothesis considered in [12] is then

$\displaystyle H_{0}:\tau_{S_{C}}\ \leq\ \tau_{S_{X}}.$

If $H_{0}$ is rejected we conclude that the support of the censoring
distribution is not smaller than the one of the T2E endpoint and that we
therefore have enough follow-up to detect a potential proportion of cured
patients. Maller et al.[13] derive a general expression for the finite and
large-sample distributions of a test statistic that has been proposed in
Maller and Zhou[12]. However, practical experience first has to be gained with
this approach, e.g. in terms of how powerful the test is in various scenarios.
And the method addresses a quite specific scientific question, namely whether
the survival function under consideration corresponds to a population with a
proportion of cured individuals.

###  2.2 Stability

Similar to "maturity" the term "stability" has been loosely (and
interchangeably with the other terms) used in the literature until a formal
definition was given by Betensky: "stability" is the relative information
compared to the potential maximum information about the distribution of $X$
that is contained in the sample. Using the deterministic minimum and maximum
of the KM estimate under assumed complete follow-up of censored observations,
it quantifies the variability (not understood in a statistical sense, but
rather how much the estimate can potentially change) of our current estimate
relative to a future estimate given additional follow-up. Maximum information
is achieved if all patients had the event, i.e. if we only have complete (as
opposed to some of them being censored) observations. This can
straightforwardly be quantified using the ratio of number of events to the
number of patients. More specifically, "stability" may also refer to the
question about how much an estimate of the survival function, typically KM,
can still change until the next clinical cut-off date (CCOD) or the end of the
trial. To this end, Betensky proposes to look at the most extreme scenarios
assuming either all still censored patients will have an event on the next day
or would be censored at the last observed event date. See Figure 6 below for
an example.

However, also "stability" is used to mean different things. As an example,
Gebski et al.[14] discuss methods to quantify "stability" (in Betensky’s
sense) but they call it "maturity".

###  2.3 Validity

"Validity" is another vaguely or ambiguously defined concept. Intuitively,
"validity" may be understood as "unbiased" and therefore be a general concept
for any estimator. However, we read Definition 3 ("time interval of validity
of the KM curve") in Shuster’s letter such that his interpretation is that
this refers to how precise a given estimate of the underlying survival
function $S_{X}$ is. Again, as Shuster already emphasizes, this cannot really
be answered by giving some statistics of some kind of "follow-up", however
defined. Instead, as Betensky already discussed, the uncertainty of an
estimate can be directly quantified using pointwise confidence intervals (CI)
or confidence bands.

###  2.4 Quality

Again, it is not clear what "quality" would refer to when it comes to "median
follow-up". One could argue that "quality" refers to well-run trials, implying
that only few patients are censored for other reasons than administrative
censoring. This can be assessed by analyzing reasons for censoring, e.g. as
discussed as Question 9 in Table 3. Again, if this were what "quality"
referred to no median follow-up whatsoever can answer this question.

##  3 Quantities used in the literature to quantify "follow-up"

Following Betensky[1] let $X$ denote the time to event, $C$ the time to
censoring, $T$ the observed time, that is, $T=\min\\{X,C\\}.$ The
corresponding survival functions are $S_{X}(t)=P(X>t)$ and $S_{C}(t)=P(C>t)$.
For the analysis of the primary endpoint of the trial we are interested in the
survival function for $X$, $S_{X}$, which is typically estimated using the KM
estimator. Furthermore, denote by $T_{1},T_{2}$ the start and end of
recruitment and by $T_{3}$ the CCOD, i.e. the date which the data will be
reported up to. In the literature $T_{3}$ is typically referred to as "end of
study" [6], [1], but we prefer the term CCOD. This, because a trial can have
multiple reporting events, e.g. an efficacy interim and final analysis, where
the former must not be the "end of study". Finally, assume that either in a
SAT or RCT two CCODs are planned, both based on the primary efficacy endpoint:

  * •

An interim analysis after $d_{\text{int}}$ events.

  * •

A final analysis after $d_{\text{fin}}$ events.

Table 1 summarizes the quantities that are used in the clinical literature to
quantify "follow-up". Note that in the table a distinction is made for
censored patients between the two clinical events "administrative censoring"
(date where the patient was last confirmed event-free, e.g. at the last
available event-free tumor assessment for PFS or date last known alive for OS)
and "LTFU" or "drop-out", where a patient has left the trial prior to
experiencing the event of interest and prior to $T_{3}$.

Number |  Term |  Number in Shuster [2] |  Number in Schemper et al. [6] |  Betensky [1] |  Examples |  Patient subset |  Primary event |  Censoring: administrative |  Censoring: LTFU |  CCOD
---|---|---|---|---|---|---|---|---|---|---
1 |  Observation time regardless of censoring | (2) | 1. | $\min\\{X,C\\}$ |  Example in Betensky [1] |  | event | event | event | ignored
2 |  Observation time for those censored | (1) | 2. | $C|C<X$ |  Example in Betensky [1] | censored |  | event | event | ignored
|  |  |  |  | event | excluded
3 |  Time to censoring | (4) | 5. | $C$ |  Gallium [15] |  | censored | event | event | ignored
4 |  Time to CCOD, Potential follow-up | (3) | 3. |  |  Keynote-024 [16]*, Keynote-361[17] |  | ignored | ignored | ignored | event
5 |  Known function time |  | 4. |  |  None known. | censored |  | event | event | ignored
|  |  |  |  | event | ignored |  |  | event
6 |  Korn’s potential follow-up time |  | 6. |  |  None known. |  | see below
7 |  Potential follow-up considering events |  |  |  |  Ascend-05 [18]* | censored |  | ignored | ignored | event
|  |  |  |  | event | event |  |  | ignored
Table 1: Quantities used in the literature to define "follow-up". Explanation
of columns:
Number in Shuster [2] / Schemper et al. [6]: These are the numbers in these
papers that were given to the respective quantities. Added here for easy
identification.
Patient subset: Which patients to consider, based on their status for the
primary endpoint.
The last four columns indicate how to treat these clinical events for the
quantities of interest in each row. An entry of "1" means that this original
datapoint is set to "event", "0" to censored, or "ignored".
Censoring: administrative refers to the date where the patient was last
confirmed event-free, e.g. at the last available event-free tumor assessment
for PFS or date last known alive for OS. "CCOD" refers to the clinical cut-off
date.
* Definition of "follow-up" often not given in publicly available documents. In absence of publicly available details we know the definition either from personal communication of statisticians involved in these trials, or from comparing consecutive snapshots.

Typically, some quantile (primarily the median) of the distribution of the
quantities in Table 1 is reported in clinical publications[6]. Looking at all
the quantities and how differently they are defined it comes without a
surprise that there is a lot of heterogeneity and confusion in the medical
literature around follow-up, as exemplified in Sections 5 and 6.

Quantities #1-#6 have already been discussed by Schemper et al.[6] and we
refer to that paper for additional details.

Quantity #4 is used in the Keynote-024 trial, with primary publication Reck et
al.[16] and report of follow-up snapshot in Reck et al.[19]. We observe that
the reported difference in median follow-up is equal to the difference between
the two CCODs, which even applies to the ranges (i.e. minimum and maximum
follow-up). This is virtually impossible to happen if other quantities were
implemented than Quantity #4. Although the actual calculation of follow-up
time was not explicitly indicated in the publications, we consider it very
likely that Quantity #4 was used. Another example is Keynote-361 where
Quantity #4 is even explicitly defined is Powles et al.[17]: "…median follow-
up, defined as time from randomisation to data cutoff."

Quantity #6, Korn’s potential follow-up time[20], is a generalization of time
to censoring (Quantity #3) and estimates the probability to be under follow-up
at a given time $t$. To this end, it makes a distinction between the two
potential censoring reasons, namely LTFU and administrative censoring. While
Korn’s paper is regularly cited in the literature by papers which analyze
RCTs, the citation is typically used to refer to time to censoring (see e.g.
Blum et al.[21]). We are not aware of an actual computation of #6 in a trial.

Quantity #7 has, to the best of our knowledge, not been discussed in the
statistical literature previously but is regularly used in oncology RCTs, e.g.
in Ascend-05[18].

Another measure that has been proposed is Clarke’s $C$ [22] and variants
thereof [23]. The former is simply the ratio $\\#1/\\#7$ of the quantities in
Table 1 and the latter attenuates the Potential follow-up considering events
in the denominator, Quantity $\\#7$, to account for unobserved deaths.

While the primary reason to look at some quantification of follow-up is to
understand features of survival functions or effect estimates it is still
worthwhile mentioning that only two of the quantities in Table 1 actually
estimate a well-defined population quantity: Time to censoring #3, which
estimates $S_{C}$, the hypothetical follow-up at any time $t$ if no events
would happen. And Korn’s potential follow-up #6 which estimates the
probability to be under follow-up at $t$. All the other quantities either
condition on censoring/event status and/or depend on the CCOD and as such have
to be considered merely descriptive.

Based on our experience as statisticians in pharmaceutical drug development,
quantities #1 and #3 are those that are most often used to report "follow-up".
We have added the other quantities to the table because they had already been
discussed by Shuster[2], Schemper et al.[6], and Betensky[1], or are being
used in clinical trials.

To conclude this section we would like to share the observation that the
approach by which "follow-up" is computed is typically not explicitly
described in primary publications or trial protocols. However, these are the
only documents that are often publicly available (e.g. for the Gallium trial:
primary publication [15] and protocol in accompanying supplementary material
[24]). This makes it very difficult to find examples for each approach
outlined in Table 1.

##  4 Recommendations how to answer questions about maturity, stability,
validity, and quality

The ICH E9(R1) estimand addendum[3] defines its applicability as

> The principles outlined in this addendum are relevant whenever a treatment
> effect is estimated, or a hypothesis related to a treatment effect is
> tested, whether related to efficacy or safety.

So in theory a direct application of the ICH E9(R1) estimand addendum[3] to
the problem of quantifying follow-up does not appear indicated because we are
not estimating or testing a treatment effect. But we still found the estimand
thinking process useful for this problem, for two reasons:

  1. 1.

The emphasis on the scientific objective \- what question do we want to answer
for our primary T2E endpoint? As we describe below many implicit questions
have been "answered" using some "median follow-up", without properly
connecting the analysis to the question.

  2. 2.

The concept of Table 1 is clearly inspired by the handling of intercurrent
events when defining an estimand. Granted, not all of the clinical events
(primary endpoint event, LTFU, administrative censoring, CCOD) we consider in
Table 1 are intercurrent events in the spirit of the addendum, but the
thinking process and structure of the addendum remains applicable.

In Sections 4.1 and 4.2 we list potential questions of interest that trialists
have when they talk about "follow-up". With "milestone" we refer to a
timepoint of interest.

###  4.1 One-sample case

Having tried to give meaning to what questions trialists have for "median
follow-up" as reported by Shuster, Table 2 summarizes the questions one can
ask when considering the KM estimate of a survival function in one group.
"Precision" and "stability" had been discussed in detail in Betensky[1]
already.

Number | Term |  Question |  How to best answer
---|---|---|---
1 | Precision |  How precise is an estimate with respect to the underlying true survival function $S_{X}$? |  Pointwise CIs or confidence bands.
2 | Reliability |  How far out to extend the KM estimate, as discussed in Pocock et al. [4]. |  Pointwise CIs or confidence bands. Disregard KM estimate if less than $m$ patients remain at risk.
3 | Stability |  How much can a KM estimate possibly change in a future data snapshot? |  Consider all currently censored patients to either (1) have an event the day after censoring or (2) being censored at the latest observed event time.
4 | Information |  How much of the information necessary to achieve a targeted power for a hypothesis test, either for a milestone timepoint or the median, has been collected? |  Power depends on various quantities, see below.
Table 2: Questions trialists want to answer for a KM estimate of a survival
function in one sample, and how they are best answered.

Table 2 makes it clear that none of the quantities described by Shuster and
listed in Section 2 is needed or useful to answer Questions 1 - 4. As for
Question 2 we find that "reliability" best describes what Pocock et al.[4]
were after. They recommended "that survival plots be halted once the
proportion of patients free of an event, but still in follow-up, becomes
unduly small." and "It will often be reasonable to curtail the plot when only
around 10-20% are still in follow-up." and already mention that their view
might not be universally held. Indeed we recommend that instead of curtailing
the plot a proper quantification of uncertainty through pointwise CIs or a
confidence band for the entire estimated function should be preferred. In our
view, there is a big difference in terms of what having 20% of patients still
in follow-up in a trial with 1000 or 100 patients tells us about the
underlying true survival function $S_{X}$. The decreased uncertainty in the
larger trial is then properly reflected in the CIs or bands. Taking this even
further, a recommendation for practitioners could be to disregard any KM
estimate if less than some number of patients remain at risk. This number can
e.g. be derived from how wide a confidence interval trialists are still
willing to accept. That would make the consideration independent of the total
number of patients in the trial and would allow determination of when to start
ignoring the curve solely based on the number of patients still at risk, a
quantity that is often accompanying KM estimates as in our Figures 1 and 5.

Hypothesis tests for Question 4 have been described in the literature, e.g. in
Brookmeyer and Crowley[25] for the median (construct test from confidence
interval) and Nagashima et al.[26] for a milestone timepoint. Such tests
depend on quantities other than only the number events and we refer to these
publications as well as the discussion for restricted-mean survival time
(RMST) in Section 4.2.2 for details.

###  4.2 Two-sample case

For the two-sample case the primary question is not about estimates of
survival functions anymore but rather about the comparison of survival
functions to quantify a (precisely defined) treatment effect. Stability
remains an issue as well for the comparison of two groups, as discussed by
Betensky[1]. When it comes to Betensky’s definition of stability it may be
useful to base this assessment on the proposed upper and lower limits for the
individual KM estimates. We also found it useful to make a distinction between
the PH and non-PH scenario for precision and reliability.

####  4.2.1 Proportional hazards

We start with making the PH assumption for the underlying hazard functions and
use the HR as quantifier in the corresponding columns in Table 3. Typically,
sample size computation is performed using the formula proposed by
Schoenfeld[27]. We note that accuracy of this formula depends on

  * •

the PH assumption,

  * •

use of unweighted logrank test,

  * •

and the randomization ratio.

Alternative normal approximations of the distribution of the test statistic if
one or more of these assumptions are not met are given in Yung and Liu[28].
However, for the purpose of this paper, we make the first two assumptions and
assume a fixed randomization ratio. It is important to note that then powering
a trial exclusively based on number of events is tightly connected to using
the HR as an effect measure. Because only for the HR and assuming PH the
precision of the effect quantifier exclusively depends on the number of events
(and via a constant on the randomization ratio). This then allows to set up a
group-sequential design using the number of events as the "metric of
information", because the statistical information is defined as the inverse of
the variance of the group difference parameter, see e.g. Lan and Zucker[29].
So, as already discussed in Section 2.1, the information fraction
$d_{\text{int}}/d_{\text{fin}}$ then quantifies how much of the targeted
events that were pre-specified in the sample size computation we have already
reached, e.g. at an interim analysis. We reiterate that "information fraction"
is not to be confused with "follow-up". This directly leads to Question 7 in
Table 3.

Number |  Term | Question | How to best answer
---|---|---|---
Proportional hazards |  Non-prop. hazards |  Proportional hazards |  Non-prop. hazards
5 |  Precision |  How precise is estimate of HR? |  How precise is chosen estimate to quantify effect of interest? | Confidence interval.
6 |  Stability |  How much can an estimate of a HR change in a future data snapshot? |  How much can an estimate of the effect of interest change in a future data snapshot? |  If the PH assumption applies then an estimate of the HR will (on average) simply become more precise over time. |  Look at extreme scenarios as proposed by Betensky[1].
7 |  Information |  How much of the information necessary to achieve a targeted power for a hypothesis test for the effect of interest (HR if PH) has already been collected, if a group-sequential design is used? |  Information fraction $d_{\text{int}}/d_{\text{fin}}$. |  Not only related to the information fraction, effect measure specific.
8 |  Reliability |  Do hazard functions remain proportional? |  Not applicable. |  Standard tools to assess PH, e.g. plot nonparametric estimates of (cumulative) hazard functions over time, and ratio thereof, or hypothesis tests. |
9 |  Censoring pattern |  Is censoring distribution the same in the two arms? Are distributions of censoring reasons the same in the two arms? |  Plot nonparametric estimates of the censoring distribution $S_{C}$ per arm, potentially split by censoring reason.
Table 3: Questions trialists want to answer for the primary T2E endpoint, and
how they are best answered. For PH and NPH scenario.

As for Question 6 in Table 3 one could argue that what we are suggesting as
the best answer for PH is not really addressing stability, as it does not
consider the most extreme possibilities as Question 2 in Table 2. However, it
is important to realize that if we entertain the PH assumption over the entire
observation time then we restrict the range within which an early estimate of
the HR can vary considerably. Additional follow-up will only give us a more
precise estimate, and it is unlikely that this early HR estimate will
dramatically change. This implies that if we have a trial that stops "early"
in the sense that only a low proportion of patients already had their event,
stability is more driven by our belief in the PH assumption rather than in
what can possibly happen with more follow-up. This is especially relevant if a
trial stops at an interim analysis for efficacy. Question 9 may shed some
light on whether informative censoring might be present, especially in open-
label trials. All these aspects will be discussed further using a real trial
example in Section 5.

We are well aware that the above interpretation, exclusively relying on the PH
property to "predict" the future course of the trial, is a strong and
statistically motivated assumption. From a broader drug development
perspective (or from the perspective of a Health Authority) it is acknowledged
that after a first CCOD the stability of a potential treatment effect needs to
be assessed also based on accumulating data, not "only" on the PH assumption.
However, the point is that such stability cannot be assessed using whatever
measure of follow-up. The best way is to look at nonparametric estimates of
survival functions (i.e KM).

####  4.2.2 Non-proportional hazards

Now, if hazards are not assumed to be proportional, trialists have to quantify
the effect using a measure other than, or in addition to, the HR, depending on
the anticipated nature of the PH violation. The most prominent approaches are
either a comparison (through the difference or ratio) of survival functions at
a given milestone timepoint, e.g. one year, a comparison of medians, the use
of RMST, i.e. the area between two estimates of the survival functions, or
some weighted version of the logrank test (to which we also add max-combo
tests). See e.g. Uno et al.[30] for an introduction into the first two
measures, for max-combo tests Roychoudhury et al.[31], Yung and Liu[28] for a
discussion of statistical properties of these test statistics, and Lin et
al.[32] for a comparison of the properties of several methods. Typically, RMST
is computed based on KM estimates and is therefore truly nonparametric, i.e.
does not make an assumption about the treatment group having a hazard function
proportional to the one in the control group. However, as the name suggests,
it depends on choosing (either a priori or data-driven) a timepoint $t_{0}$
that restricts the observation time over which the area between the two
estimates is compared, so is more explicitly connected to the amount of
follow-up already. But as mentioned by A’Hern[33], the use of RMST does not
overcome at all any of the problems posed by limited patient follow-up.

For both RMST and the difference or ratio of $t$-year event-free
probabilities, which we will call milestone probabilities, precision of these
estimates is again best quantified using CIs for these quantities. For the
latter it is important, as discussed by Pocock et al.[4], to consider CIs for
the difference of the two estimated event-free probabilities, and not only
provide CIs around each individual estimated survival function at $t$.

As we have elaborated in Section 4.2.1 the PH assumption allows a group-
sequential design to be run exclusively based on number of events as the basic
measure of "information". For other measures of effect this is not necessarily
the case: information then needs to be more generally considered as the
inverse of the variance of the group difference parameter. As an example,
Hasegawa et al.[34] discuss this for RMST at a restriction timepoint $t_{0}$
and provide a formula for the information initially derived by Murray and
Tsiatis[35]. This expression for the information is much more complicated than
for the HR and depends on all of the following quantities:

  * •

total number of patients,

  * •

randomization ratio,

  * •

KM estimate of the pooled sample,

  * •

estimated censoring distribution in each arm (which can be taken as pooled if
random censoring is assumed),

  * •

observed number of events at $t_{0}$,

  * •

observed number of patients still at risk at $t_{0}$.

That for non-PH the power of a hypothesis test does not anymore exclusively
depend on the number of events, but also on the accrual and censoring pattern,
has already been discussed by Struthers and Kalbfleisch[36]. So there is no
hope that one can provide an accurate summary of "information" when running a
group-sequential design for RMST with whatever quantification of follow-up.
Also, it is very important to recognize that this implies that for non-PH
scenarios, number of events will typically not be a sufficient metric to
quantify information.

Lu and Tian[37] discuss how to plan and run a group-sequential design for RMST
while Wang et al.[38] discuss a simulation-free group-sequential design with
max-combo tests in the presence of non-proportional hazards.

A common pattern for non-PH is in immuno-oncology with delayed separation, see
e.g. Mick and Chen[39], i.e. survival functions only start to separate after a
certain time. Potentially, we may also expect or observe crossing survival
functions or that patients can be cured, see Section 2.1. Collection of
hypothesis tests for crossing survival functions are discussed in Li et
al.[40] and Dormuth et al.[41]. With all these features the first
consideration for trialists is to decide on a quantity with which they want to
quantify a potential treatment effect and perform a corresponding hypothesis
test. For all of them it is particularly important to be able to characterize
the tail behavior of survival functions out to some milestone timepoint that
is clinically relevant. One needs to ensure enough follow-up such that the
chosen hypothesis test achieves sufficient power, be it for RMST, milestone
survival, a weighted logrank, or max-combo test. We summarize the questions
and recommendations for the non-PH scenario in Table 3.

##  5 Example with proportional hazards: the Gallium trial

We will now illustrate all these quantities using two clinical trial examples,
one with approximately PH and one with non-PH.

We use the Gallium trial[15] to exemplify the situation for (initially) PH.
Gallium was a Phase 3, 1:1 RCT in first-line follicular lymphoma, comparing
Rituximab (= control) to Obinutuzumab (= experimental) using a primary
endpoint of PFS. Since administration schedule for the two drugs was different
the trial was open-label for patients and physicians, but not for the sponsor.
The trial was fully unblinded after the hypothesis test for the HR was
statistically significant within a group-sequential design at a pre-planned
interim analysis for efficacy. This interim analysis happened after 245 / 1202
(= 20.4% of patients, information fraction = 245 / 370 = 0.66) patients had
experienced a PFS event. We compare this snapshot to an update that was
reported almost four years later[42]. KM estimates of both snapshots are
provided in Figure 1. For both snapshots Obinutuzumab showed a benefit
compared to Rituximab. Key efficacy results of the trial are given in Table 4.
Figure 2 provides the empirical cumulative distribution function of accrual.
This may be useful to check any unusual patterns in recruitment to the trial.

Figure 1: PFS for Gallium. Vertical lines at milestone timepoints 36 and 60 months. | CCOD 2016-01-31 | CCOD 2019-10-31
---|---|---
HR | 0.66 | 0.76
95$\%$ CI | [0.51, 0.85] | [0.62, 0.92]
Number of events $d$ | 245 | 419
Proportion of patients with event | 20.4$\%$ | 34.9$\%$
Table 4: Key efficacy results for Gallium. Figure 2: Empirical cumulative
distribution function of accrual for Gallium.

###  5.1 Answering the scientific questions

##### Question 5: Precision.

Results for the HR are given in Table 4. Clinically, estimates of event-free
probabilities at certain milestones are also of interest, e.g. at 36 and 60
months, respectively. We provide these, including their differences and 95%
CIs in Table 5. With these everything that might be of interest in terms of
precision for these estimates is said.

As discussed in Section 4.1 we might also simply disregard KM estimates once
less than a given number remain at risk in each arm, where this number can
e.g. be informed by what confidence interval width trialists are willing to
accept. This would entail that for the KM estimates in Figure 1 there is no
reason to worry about the drops very late in the estimates for the earlier
CCOD. And indeed, with more data, i.e. at the later CCOD, these drops have
been "overwhelmed" and disappeared.

milestone | treatment arm | CCOD 2016-01-31 | CCOD 2019-10-31
---|---|---|---
| Rituximab | 0.73 [0.66, 0.80] | 0.76 [0.71, 0.79]
36 months | Obinutuzumab | 0.80 [0.73, 0.85] | 0.82 [0.79, 0.86]
| Difference Obinutuzumab - Rituximab | 0.07 [0.01, 0.12] | 0.07 [0.02, 0.12]
| Rituximab | - | 0.63 [0.58, 0.68]
60 months | Obinutuzumab | - | 0.70 [0.65, 0.75]
| Difference Obinutuzumab - Rituximab | - | 0.07 [0.02, 0.13]
Table 5: Milestone KM estimates for Gallium.

##### Question 6: Stability.

If one believes in the PH assumption then in theory additional follow-up will
not relevantly change the estimate of the HR but only make the CI more narrow.
For the first Gallium snapshot, a hypothesis test for the null hypothesis of
PH based on the scaled Schoenfeld residuals gives a $p$-value of 0.35, so
little evidence for a deviation from PH.

Townsend et al.[42] reported updated results based on 419 PFS events. The HR
estimate now was 0.76 with 95% CI [0.62, 0.92], see Table 4. So indeed, the CI
is now more narrow, as a consequence of more information. Of note, the fact
that the trial was fully unblinded and superiority of Obinutuzumab was claimed
after the first CCOD may introduce bias for the estimation of the PFS effect
of the initially assigned treatment. This e.g. through different patterns of
use of new anti-lymphoma therapy (NALT) prior to progression (in the language
of the ICH E9 estimands addendum[3] a treatment policy strategy was used for
the intercurrent event of NALT). Thus, whether the increase of the HR estimate
from 0.66 to 0.76 reflects a real change in effect potentially related to the
potential bias introduced through unblinding or is simply due to variability
in estimation cannot be decided based on the PFS data only. If we believe in
the PH assumption and assume no bias then it would be variability only.

For the updated snapshot, the test for PH gives now a $p$-value of 0.03, so
according to Pocock et al.[43] this is "some evidence" against PH.

However, no quantification of follow-up whatsoever can (1) provide information
on stability, (2) help decide the question about whether bias has been
introduced after the first CCOD and (3) whether the PH assumption holds up to
the second CCOD.

We also are of the opinion that if a PH assumption is entertained it is not
informative to look at Betensky’s stability criterion, as the extreme
scenarios proposed there do not really allow effect quantification via HR.
These criterion may be useful in the case of non-PH though, see Section 6.1.

##### Question 7: Information.

As discussed in Section 4.2.1 "information" or "information fraction"
generally refers to the inverse of the variance of the group difference
parameter in a group-sequential design. For PH this simplifies to the
proportion of events that have been collected with respect to the total number
of events needed to achieve the targeted power. In Gallium 370 events were
needed to achieve 80% power to detect a HR of 0.741 based on a two-sided
logrank test with significance level of 0.05 and an O’Brien-Fleming group-
sequential boundary for one efficacy interim analysis after about 2/3 of
information. The efficacy interim analysis was planned to happen after 248 PFS
events and actually took place after 245 events, so the information fraction
at the interim analysis was 245 / 370 = 0.66.

The second snapshot was taken to update estimates of survival functions and
the HR. Since the trial had rejected the null hypothesis of no difference
between the arms already at the interim analysis this updated analysis
happened outside of the group-sequential design and it is therefore
meaningless to talk about "information fraction" in this context.

Again, also here, these considerations are completely independent of any
quantification of follow-up whatsoever.

##### Question 8: Reliability.

This aspect has already been discussed in connection with "stability" in
Section 5.1.

##### Question 9: Comparison of censoring distributions.

Finally, Figure 3 compares censoring distributions between arms, estimated
using the reverse KM method. We conclude that the censoring pattern was
virtually identical between the two arms at both CCODs.

Figure 3: Estimates of censoring distributions $S_{C}$ for Gallium.

###  5.2 Quantification of follow-up

For completeness, and not because we think they provide any additional
information with respect to the scientific questions in Section 5.1, we
provide all the follow-up quantifiers from Table 1 in Table 6. Note that only
Quantity #6 (Korn’s potential follow-up) in Table 1 makes a distinction
between administrative censoring and LTFU. However, for the Gallium data we
did not have this information available in our dataset on a patient level, but
from the CONSORT diagram in Figure 1 in Hiddemann et al.[44] we find that only
a very small portion of patients (derived from the reasons given for
withdrawal) has to be considered LTFU. In our example, for illustrative
purposes, we therefore randomly assigned the status of "LTFU" to 5% of those
patients in the trial who were administratively censored (separately at each
CCOD). From Table 6 we then make the following observations:

  * •

Although all the quantities in Table 1 quantify some sort of follow-up there
is relevant heterogeneity among them, with values ranging from 28.8 to 36.5
months and 62.0 to 81.5 months for the two CCODs, respectively. This spread
matches the findings in the example discussed in Betensky[1] and in the
simulation study by Schemper et al.[6].

  * •

Not only are the actual values different, but also the increase from one
snapshot to the next, either quantified as difference in months (column
$\Delta$) or as relative increase (column $\Delta\%$). The former varies from
33.2 to 45 months and the latter from 115$\%$ to 128$\%$.

  * •

The difference between the two CCODs in calendar time is 45.0 months. However,
that number is not directly connected to any of the follow-up quantifiers or
their increase in Table 6, except to the increase for Time to CCOD, this is of
course exactly 45.0 months by definition as long as the accrual was finished
at the first CCOD (which was the case for Gallium). Rather, the increase in
follow-up primarily depends on the frequency of, in the case of PFS, tumor
assessments or confirmation of survival status for OS.

Number |  Quantity |  CCOD 2016-01-31 |  CCOD 2019-10-31 | $\Delta$ | $\Delta\%$
---|---|---|---|---|---
1 |  Observation time regardless of censoring |  28.8 |  62.0 | 33.2 | +115$\%$
2 |  Observation time for those event-free |  31.5 |  71.2 | 39.8 | +126$\%$
3 |  Time to censoring |  32.6 |  71.7 | 39.1 | +120$\%$
4 |  Time to CCOD |  36.5 |  81.5 | 45.0 | +123$\%$
5 |  Known function time |  32.9 |  75.0 | 42.0 | +128$\%$
6 |  Korn potential follow-up |  36.2 |  81.2 | 44.9 | +124$\%$
7 |  Potential follow-up considering events |  33.4 |  75.5 | 42.1 | +126$\%$
Table 6: Different quantifications of follow-up for Gallium, in months.

Figure 4 provides the distributions corresponding to the quantities in Table 1
from which the medians in Table 6 were computed. The figure again illustrates
the heterogeneity with which these various measures quantify follow-up. We
also see that those quantities that involve the CCOD (#4-#7) in their
computation exhibit a quite linear pattern, at different values for PFS. This
simply illustrates that they all essentially map the accrual at least for a
portion of the patients.

Figure 4: Distributions from which follow-up quantifiers are derived, for
Gallium.

##  6 Example with non-proportional hazards: delayed separation

Our second example is a trial with delayed separation. We simulated data for
this trial (using the R package rpact, [45]) making the following assumptions
for a T2E endpoint such as PFS or OS:

  * •

The base event rate is 0.012, corresponding to a median T2E of 60 months.

  * •

A piecewise exponential survival with no effect between 0 and 12 months, and a
HR of 0.65 thereafter.

  * •

In both arms, the probability that a patient is LTFU follows an exponential
distribution calibrated such that this probability amounts to 0.025 at 12
months. This corresponds to a median time-to-LTFU of 329 months.

  * •

After a ramp-up of 6 months we recruit 42 patients per month until a maximal
number of 1000 patients.

With these assumptions a CCOD after 389 events gives a power of 80.5% to
reject the null hypothesis of no effect using an unweighted logrank test and
69.7% power using the RMST difference based on KM estimates between arms, with
a data-driven restriction time $t_{0}$ of the lower of the two maximal
observed times (events and censored) in each arm. Both power values are based
on 10000 simulated trials. Figure 5 gives KM estimates of one such simulated
trial.

70 patients (7.0%) are LTFU and 541 patients (54.1%) administratively
censored.

Note that we only look at one CCOD for this example, as no additional insights
appeared to be gained from having two beyond what we have already discussed
for the first example.

Figure 5: T2E endpoint for Example 2. Vertical lines at milestone timepoints
36 and 60 months.

###  6.1 Answering the scientific questions

##### Question 5: Precision.

Compared to effect quantification in the case of proportional hazards in
Section 5.1 an effect measure different than the HR may be indicated in this
scenario. One option is the difference of RMST between arms, with an estimate
of 2.82 months between arms with 95% CI from -0.35 to 5.98. Estimated survival
probabilities at clinically meaningful milestones provided in Table 7 are also
useful to describe the effect and precision for delayed separation.

milestone | treatment arm | KM estimates and 95% CIs
---|---|---
| Control arm | 0.68 [0.62, 0.73]
36 months | Treatment arm | 0.71 [0.66, 0.76]
| Difference treatment - control | 0.04 [-0.03, 0.09]
| Control arm | 0.47 [0.30, 0.65]
60 months | Treatment arm | 0.60 [0.46, 0.73]
| Difference treatment - control | 0.13 [0.05, 0.21]
Table 7: Milestone estimates for delayed separation example.

##### Question 6: Stability.

Both, RMST and milestone probabilities, typically rely on the KM estimate and
for them to be valid effect quantifiers no further assumptions (apart from
random censoring) are either required or typically made. Without such further
assumptions it is only possible to construct rather crude bounds using the
properties of the KM estimator and the stability principle proposed by
Betensky for a KM estimate of a survival function: events will remain events
also in future snapshots, but for currently censored observations two things
can happen:

  * •

Best case scenario: every censored patient is censored at the last observed
event date.

  * •

Worst case scenario: every censored patient has event day after at censoring
date.

We exemplify this for the treatment arm in Figure 6.

Figure 6: Betensky’s stability scenarios for the treatment arm of the "delayed
separation" example.

Figure 6 can be interpreted as follows: if we had complete observations, i.e.
no censored patients, no amount of follow-up would change our KM estimate and
the lower and upper extreme scenario bound would coincide. With censored
observations the green and blue functions in the figure are the limits of
where the KM estimate can go with more follow-up. Betensky proposes to use the
RMST difference between the two extreme functions, normalized by the largest
observed event-time, as a one-number measure of stability that ranges from 0
(complete stability) to 1 (complete instability). We compute 18.0% in our
example in the treatment and 16.1% in the control arm.

In theory, we could now take the upper limit for the treatment and lower limit
for the control arm and compute the RMST difference between these two
functions. That would give the deterministically most extreme possible RMST
value that is still possible. However, we do not consider this number adding
much value to the discussion of stability.

To wrap this paragraph up, again, no quantification of follow-up whatsoever
can provide information on stability, considerations as above appear to be
more informative.

##### Question 7: Information.

The most important aspect is that for non-PH, number of events are typically
not sufficient to quantify the "information" present at a given snapshot.
Rather, information – understood as the inverse of the variance of the group
difference parameter – typically depends on many more estimated quantities. We
refer to the respective paragraph in Section 4.2.2 for a discussion of
"information" in absence of PH, applied to RMST.

##### Question 8: Reliability.

An assessment of PH is not indicated in case of non-PH.

##### Question 9: Comparison of censoring distributions.

In this simulated dataset with assumed identical recruitment and censoring
pattern in both arms censoring distributions are identical. We therefore omit
the corresponding figure.

###  6.2 Quantification of follow-up

Figure 7 again provides the distributions from which the quantities in Table 8
are derived. With values from 41.2 to 49.3 the heterogeneity in numbers is
comparable to our first example.

Figure 7: Distributions from which follow-up quantifiers are derived, for the delayed separation example. | CCOD 1
---|---
Observation time regardless of censoring | 41.2
Observation time for those event-free | 46.5
Time to censoring | 47.7
Time to CCOD | 49.3
Known function time | 48.2
Korn potential follow-up | 47.9
Potential follow-up considering events | 42.7
Table 8: Different quantifications of follow-up for delayed separation
example, in months.

##  7 Further aspects

Often, trialists insist on "enough follow-up for safety". While this is of
course an important consideration it is an openly formulated question and
again, no summary statistic of whatever follow-up quantity is helpful in
deciding whether we have "enough follow-up for safety". Instead, if collection
of safety data should inform length of follow-up, scientific questions we have
for safety aspects of the interventions in the trial need to be formulated and
estimands aligned to it defined. These then inform the data collection
strategy.

When discussing and comparing results from multiple trials follow-up is often
directly compared. Given that virtually never a definition of follow-up is
given, this bears the high risk of comparing apples to oranges. Below, we
anyway recommend to dispense with any single measure of follow-up (which would
make a concern about inaccurate cross-trial comparisons obsolete) and if
provided nevertheless, at least clearly define the considered quantities.
Ideally, trialists should focus on the questions and quantities of interest to
describe efficacy, as summarized in Tables 2-3, and compare these between
trials, not some unclearly defined measure of follow-up.

Of note, such cross-trial comparisons are often made in Health Technology
Assessments of drugs, when assessing whether the evidence base is suitable for
evidence synthesis. Comparison of median follow-up time quantities is often
part of such feasibility assessments. Again, relying on the true quantifiers
of efficacy instead of unclearly defined follow-up, or at least clearly
defining the latter, would make such assessments more robust and
scientifically valid.

A reviewer commented on the potential for Bayesian methods in this context. Of
course, wherever we estimate a clearly defined quantity we can use Bayesian
equally well as frequentist approaches.

##  8 (Ir)relevance of quantification of follow-up and recommendations

In Tables 2, 3, and 3 we made an attempt to list all questions trialists
typically have for a trial with a T2E primary endpoint, be it for one or two
arms and with potential follow-up snapshots. We also provided illustrative
examples. Remarkably, all these questions could be answered without reference
to any kind of follow-up whatsoever. So, we concur with Shuster[2] who already
wrote

> …it is recommended that fixed-term KM estimates and SEs be used to describe
> a quick numeric summary of outcome, rather than median followup. It is
> believed by this writer that many clinical investigators were unaware of the
> fact that the KM function adjusts for variable lengths of follow-up and
> provides an unbiased estimate of the true target population survival
> function, provided that competing losses are uninformative. Hence, they
> invented a term to meet a perceived need that, in fact, does not exist.

In our opinion, this conclusion still stands thirty years later. We have tried
to make the questions that trialists ask for a trial with a T2E endpoint
precise and discuss how these can be answered. No imprecisely and
heterogeneously defined single "median follow-up" number is needed or useful
to answer these questions and we therefore recommend not to report any at all.
Instead we propose the following when reporting results of a T2E endpoint in a
SAT or RCT:

  * •

Be clear on the scientific questions you want to answer.

  * •

It should be made clear to anyone involved in trials with T2E endpoint that
there is no hope that a single number, however defined, can say everything
about "follow-up", answer all the relevant questions trialists have, or allow
to compare relevant aspects of trials.

  * •

The primary goal of the trial is an assessment of efficacy, so estimate(s) of
$S_{X}$ in all treatment groups should be given.

  * •

As part of the assessment of efficacy, discuss precision, stability,
information, and potential assumptions separately for any quantity of
interest, i.e. typically estimates of survival functions and, in case of a
two-arm trial, effect measures. We have exemplified this in our examples.

  * •

As discussed in our examples, estimates of $S_{C}$ for each treatment arm are
very informative as well.

  * •

To describe accrual stating of $T_{1}$ and $T_{2}$ may be sufficient if
accrual was approximately uniform as in Figure 2, as Altman et al.[5] already
recommend. If the recruitment pattern was not uniform the CDF of the accrual
may be very informative.

  * •

The CCOD $T_{3}$ should be given.

  * •

Depending on the distribution of censoring reasons (administrative vs. LTFU)
it might also be informative to describe these. A distinction may even enter
Betensky’s stability assessment in Figure 6, as patients LTFU are known to
remain censored forever.

  * •

An option to complement, but not replace, the assessment of precision is to
add patient numbers still at risk below the KM plot as we have done in Figures
1 and 5, and as recommended by Morris et al.[46]. Again, a split into the
different censoring reasons might add additional information. One could even
consider using different colors by censoring reason for the censoring ticks in
the KM estimate. Furthermore, having these numbers may inform from which
timepoint on the reader may want to start ignoring the KM estimates, even in
absence of confidence intervals or bands.

  * •

We do not consider it necessary to provide any number that quantifies follow-
up - it is simply not clear what questions it answers that cannot be more
directly answered with other means. If trialists still want to give such a
number then at least very clearly define how it is computed, in line with the
recommendations by Altman et al.[5], later reiterated by Betensky[1]. In line
with Schemper et al.[6] we consider time-to-censoring, i.e. an estimate of
$S_{C}$, to be most informative. This also allows an assessment whether the
censoring pattern is different between arms, as in Figure 3. Reporting only
the estimated median has the usual deficiency of summarizing an entire
distribution into just one number, which likely will only give an incomplete
account of what is actually happening.

  * •

Key differences in case of non-PH are: (1) an effect quantifier other than the
HR and associated hypothesis test must be chosen, (2) because of the absence
of PH the assessment of stability of the effect estimate becomes more
important and (3) information, and as a consequence the power of any
hypothesis test, does depend on (many) more quantities than just the number of
events. Which ones exactly depends on the effect measure that is used. Valid
estimates of the censoring distributions are therefore even more relevant in
this case.

With these recommendations we hope to provide guidance to trialists what
scientific questions are relevant in connection with trial follow-up, how to
best analyze follow-up and answer these questions, and to motivate them to
finally, thirty years after Shuster’s letter, move away from single-number
quantifiers of follow-up.

##  9 Code Availability Statement

Code that implements all the methods discussed in this paper is available at
https://oncoestimand.github.io/quantFU/quantFU.html.

##  10 Data Availability Statement

Qualified researchers may request access to individual patient level data of
the Gallium trial[15] through the clinical study data request platform
https://vivli.org. Further details on Roche’s criteria for eligible studies
are available here: https://vivli.org/members/ourmembers. For further details
on Roche’s Global Policy on the Sharing of Clinical Information and how to
request access to related clinical study documents, see here:
https://www.roche.com/research_and_development/who_we_are_how_we_work/clinical_trials/our_commitment_to_data_sharing.htm.

##  11 Acknowledgments

This paper has been written within the industry working group estimands in
oncology, which is both, a European special interest group “Estimands in
oncology”, sponsored by PSI and European Federation of Statisticians in the
Pharmaceutical Industry (EFSPI) and a scientific working group of the
biopharmaceutical section of the American Statistical Association. Details are
available on www.oncoestimand.org.

We thank Keaven Anderson, Sandro Gsteiger, Tina Nielsen, and Godwin Yung, as
well as the associate editor and two reviewers for helpful comments.

## Appendix

## References

  * [1] Betensky R. A.. Measures of follow-up in time-to-event studies: Why provide them and what should they be? Clin Trials.  2015;12:403–408.
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# An elementary proof of the chromatic
Smith Fixed Point Theorem

William Balderrama<EMAIL_ADDRESS>and  Nicholas J. Kuhn
<EMAIL_ADDRESS>Department of Mathematics
University of Virginia
Charlottesville, VA 22903

###### Abstract.

A recent theorem by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel,
and N. Stapleton says that if $A$ is a finite abelian $p$–group of rank $r$,
then any finite $A$–space $X$ which is acyclic in the $n$th Morava $K$–theory
with $n\geq r$ will have its subspace $X^{A}$ of fixed points acyclic in the
$(n-r)$th Morava $K$–theory. This is a chromatic homotopy version of P. A.
Smith’s classical theorem that if $X$ is acyclic in mod $p$ homology, then so
is $X^{A}$.

The main purpose of this paper is to give an elementary proof of this new
theorem that uses minimal background, and follows, as much as possible, the
reasoning in standard proofs of the classical theorem. We also give a new
fixed point theorem for finite dimensional, but possibly infinite, $A$-CW
complexes, which suggests some open problems.

###### 2010 Mathematics Subject Classification:

Primary 55M35; Secondary 55N20, 55P42, 55P91.

##  1\. Introduction

Fixing a prime $p$ and finite group $G$, say that $G$–space $X$ is a finite
$G$–space if its $p$–localization is a retract of the $p$–localization of a
finite $G$–CW complex in the $G$–equivariant homotopy category. We let $X^{G}$
denote its subspace of fixed points.

Let $K(n)_{*}$ denote the $n$th Morava $K$–theory at the prime $p$. In
particular, $K(1)_{*}$ is a summand of complex $K$–theory with mod $p$
coefficients, and $K(0)_{*}$ is rational homology.

A key result in [6A19] can be stated as follows.

######  Theorem 1.1.

Let $A$ be a finite abelian $p$-group of rank $r$, and let $X$ be a finite
$A$–space. If $\widetilde{K}(n)_{*}(X)=0$ with $n\geq r$, then
$\widetilde{K}(n-r)_{*}(X^{A})=0$.

This is a chromatic homotopy theory analogue of the following classical
theorem of P. A. Smith [S41].

######  Theorem 1.2.

Let $P$ be a finite $p$-group, and let $X$ be a finite dimensional $P$–space.
If $\widetilde{H}_{*}(X;{\mathbb{Z}}/p)=0$, then
$\widetilde{H}_{*}(X^{P};{\mathbb{Z}}/p)=0$.

We note that Theorem 1.1 follows by iteration from the cases when
$A=C_{p^{k}}$, the cyclic $p$–group of order $p^{k}$, and Theorem 1.2 follows
from the case when $P=C_{p}$.

The main purpose of this note is to give an elementary proof of Theorem 1.1
that follows, as much as possible, the reasoning in standard proofs of the
classical theorem. All needed background material about $K(n)_{*}$ and related
theories is in papers published by 2000. We hope that our presentation will
lend some clarity about the interesting remaining problems in this area.

The classical theorem includes the statement that $X^{P}\neq\emptyset$, while
the version of Theorem 1.1 proved in [6A19] implicitly assumes that $X$ has a
point fixed by $A$, an assumption we do not need to make. Indeed, the first
steps in our proof hold when $X$ is just assumed to be finite dimensional, and
not necessarily finite, and lead to the following new fixed point theorem.

######  Theorem 1.3.

Let $A$ be a finite abelian $p$–group of rank $r$, and $X$ a finite
dimensional $A$–CW complex. If $\widetilde{K}(n)_{*}(X)=0$ with $n\geq r$,
then $X^{A}\neq\emptyset$.

We note that, if $X$ is any space, then
$\widetilde{K}(n)_{*}(X)=0\Rightarrow\widetilde{K}(r)_{*}(X)=0$ for $n\geq
r\geq 1$ [Bou99]. (This generalized Ravenel’s result [Rav84, Thm.2.11] about
finite $X$.) Thus this fixed point theorem for all $n$ is implied by the
special case when $n=r$.

In §2, we recall some needed background material, and deduce some simple
consequences. The theorems are then quickly proved in §3.

A final section has various remarks and speculations. In particular, we wonder
if some weakening of the finiteness hypothesis of Theorem 1.1 might be
possible, and we are curious about the existence of examples showing that
Theorem 1.3 is as strong as possible.

###  1.1. Acknowledgements

The second author is a PI of RTG NSF grant DMS-1839968, which has supported
the research of the first author. Some of the writing of this paper was done
while the second author was visiting the Utrecht Geometry Center, with support
from Simons Foundation Collaboration Grant 709802. We thank Neil Strickland
for helpful email about literature references, and Tom Goodwillie and Ian
Leary for their helpful answers to a Mathoverflow query (see §4.3).

##  2\. Background material and a localization result

###  2.1. Morava $E$–theory and $E_{n}^{*}(BA)$

Recall the Brown–Peterson homology theory $BP$, with coefficient ring
$BP_{\ast}={\mathbb{Z}}_{(p)}[v_{1},v_{2},\ldots]$. We work with complex
oriented theories $E$ that are $p$–local and with $p$-typical formal group
laws. The coefficient ring $E_{\ast}$ of such a theory is a
$BP_{\ast}$-algebra, and is said to be Landweber exact and $v_{n}$-periodic if
$v_{0},v_{1},\ldots$ acts as a regular sequence on $E_{\ast}$ with $v_{n}$
acting as a unit on $E_{\ast}/(v_{0},\ldots,v_{n-1})$. (Throughout, we let
$v_{0}=p$, as is standard.)

######  Lemma 2.1.

[H95a, Cor. 1.12] Let $E_{*}$ be Landweber exact and $v_{n}$–periodic. A
spectrum is $E_{*}$–acyclic if and only if it is $K(m)_{*}$–acyclic for $0\leq
m\leq n$. Thus, if a finite spectrum is $E^{*}$–acyclic then it is
$K(n)_{*}$–acyclic.

Now let $E_{n}$ be the $n$th Morava $E$–theory, as in [H95b], [HS99], or
[HKR00]. There are variants of these, so for concreteness, we will say that

$E_{n}^{*}={\mathbb{Z}}_{p}[u^{\pm 1}][[v_{1},\ldots,v_{n-1}]],$

where $u\in E_{n}^{-2}$, and with complex orientation $y\in E_{n}^{2}(BU(1))$
whose associated formal group law $F$ has $p$–series of the form

$[p](y)=py+_{F}v_{1}y^{p}+_{F}\cdots+_{F}v_{n-1}y^{p^{n-1}}+_{F}u^{p^{n}-1}y^{p^{n}}.$

In particular, $v_{n}=u^{p^{n}-1}$. This is a Landweber exact and
$v_{n}$-periodic theory, with the following additional property.

######  Lemma 2.2.

[H95b, Prop.3.6] If $f:X\rightarrow Y$ is a map of spectra, then $f$ is an
$E_{n}^{*}$–isomorphism if and only if it is a $K(n)_{*}$–isomorphism.

######  Proposition 2.3.

Let $G$ be a finite group and $X$ a $G$–space. If $\widetilde{K}(n)_{*}(X)=0$,
then the map $EG\times_{G}X\rightarrow BG$ induces an isomorphism

$E_{n}^{*}(BG)\simeq E_{n}^{*}(EG\times_{G}X).$

###### Proof.

There are implications

$\begin{split}\widetilde{K}(n)_{*}(X)=0&\Leftrightarrow K(n)_{*}(X)\simeq
K(n)_{*}(pt)\\\ &\Rightarrow K(n)_{*}(EG\times_{G}X)\simeq K(n)_{*}(BG)\\\
&\Leftrightarrow E_{n}^{*}(BG)\simeq E_{n}^{*}(EG\times_{G}X).\end{split}$

The previous lemma gives the third implication. The second implication is a
special case of a general fact: if $f:X\rightarrow Y$ is a $G$–equivariant map
between $G$–spaces that is an isomorphism in a generalized homology theory
$E_{*}$, then it induces an isomorphism on the associated Borel theory
$E_{*}(EG\times_{G}X)\xrightarrow{\sim}E_{*}(EG\times_{G}Y)$. ∎

We recall some basic calculations as in [HKR00, §5]. The complex orientation
$y\in E_{n}^{2}(BU(1))$ defines an isomorphism $E_{n}^{*}(BU(1))\simeq
E_{n}^{*}[[y]]$, and the $p^{k}$–series satisfies $[p^{k}](y)\equiv
u^{p^{nk}-1}y^{p^{nk}}\mod(v_{0},\ldots,v_{n-1},y^{p^{nk}+1})$.

The standard inclusion $i_{k}:C_{p^{k}}\hookrightarrow U(1)$ induces an
isomorphism

$E_{n}^{*}(BC_{p^{k}})\simeq E_{n}^{*}[[y_{k}]]/([p^{k}](y_{k})),$

where $y_{k}=i_{k}^{*}(y)$, and it follows from the Weierstrass preparation
theorem that $E_{n}^{*}(BC_{p^{k}})$ is a free $E_{n}^{*}$–module with basis
${1,y_{k},y_{k}^{2},\dots,y_{k}^{p^{nk}-1}}$.

Similarly, if $l<k$ and $q:C_{p^{k}}\rightarrow C_{p^{l}}$ is the standard
quotient map, then the map $q^{*}:E_{n}^{*}(BC_{p^{l}})\rightarrow
E_{n}^{*}(BC_{p^{k}})$ satisfies $q^{*}(y_{l})=[p^{k-l}](y_{k})$, and induces
an isomorphism

$E_{n}^{*}(BC_{p^{k}})\simeq
E_{n}^{*}(BC_{p^{l}})[[y_{k}]]/([p^{k-l}](y_{k})-y_{l}).$

This makes $E_{n}^{*}(BC_{p^{k}})$ into a free $E_{n}^{*}(BC_{p^{l}})$ module
of rank $p^{n(k-l)}$.

From these calculations a couple of general facts follow.

######  Lemma 2.4.

(a) If $A$ is a finite abelian $p$-group, then $E_{n}^{*}(BA)$ is a free
$E_{n}^{*}$–module of rank $|A|^{n}$, and there is a natural isomorphism

$E_{n}^{*}(BA)\otimes_{E_{n}^{*}}E_{n}^{*}(X)\simeq E_{n}^{*}(BA\times X)$

for any space $X$.

(b) If $q:A\rightarrow\bar{A}$ is an epimorphism between two finite abelian
$p$-groups, then $q^{*}:E_{n}^{*}(B\bar{A})\rightarrow E^{*}_{n}(BA)$ makes
$E^{*}_{n}(BA)$ into a free $E^{*}_{n}(B\bar{A})$–module of rank
$\lvert\operatorname{Ker}q\rvert^{n}$.

###  2.2. Localization

Let $A$ be a finite abelian $p$-group. Observe that the group $C_{p}^{\times}$
of automorphisms of $C_{p}$ acts freely on the set of nontrivial homomorphisms
$\alpha:A\rightarrow C_{p}$ via postcomposition.

If $\alpha:A\rightarrow C_{p}$ is a homomorphism, we let
$e(\alpha)=\alpha^{*}(y_{1})\in E^{2}_{n}(BA)$. This is the Euler class of the
1–dimensional complex representation defined by the composite
$A\xrightarrow{\alpha}C_{p}\hookrightarrow U(1)$. With this notation, we
define two Euler classes $e(A),\bar{e}(A)\in E_{n}^{*}(BA)$ as follows.

######  Definitions 2.5.

Let $e(A)=\prod e(\alpha)$, with the product over all nontrivial
$\alpha:A\rightarrow C_{p}$. Let $\bar{e}(A)=\prod e(\alpha)$, with the
product over one representative of each $C_{p}^{\times}$–orbit. The definition
of $\bar{e}(A)$ involves choices, but, for concreteness, we let
$\bar{e}(C_{p})=y_{1}$, i.e. we choose the identity $C_{p}\rightarrow C_{p}$.

Basic facts about these are summarized in the next lemma.

######  Lemma 2.6.

(a) If $i:A^{\prime}\hookrightarrow A$ is the inclusion of a proper subgroup,
then $i^{*}(e(A))=i^{*}(\bar{e}(A))=0$.

(b) $e(A)^{-1}E_{n}^{*}(BA)=\bar{e}(A)^{-1}E_{n}^{*}(BA)$.

###### Proof.

$e(A)$ and $\bar{e}(A)$ are products of the various $e(\alpha)$. If
$A^{\prime}$ is a proper subgroup of $A$, then at least one $\alpha$ restricts
to the trivial representation of $A^{\prime}$. As the Euler class of the
trivial representation vanishes, statement (a) follows.

To prove statement (b), we first note that, since $\bar{e}(A)$ divides $e(A)$,
inverting $e(A)$ also inverts $\bar{e}(A)$. To see the converse, we need to
show that, given a nontrivial $\alpha:A\rightarrow C_{p}$, inverting
$e(\alpha)$ also inverts $e(\beta)$ for all $\beta$ in the
$C_{p}^{\times}$–orbit of $\alpha$. Such $\beta$ are given by the composites
$A\xrightarrow{\alpha}C_{p}\xrightarrow{m}C_{p}$, with
$m\in\\{1,\dots,p-1\\}$.

Since $e(m\circ\alpha)=[m](e(\alpha))$, it suffices to show that inverting
$e(\alpha)$ also inverts $[m](e(\alpha))$ for any $m\in\\{1,\ldots,p-1\\}$.
One way to see this goes as follows. Choose $s$ such that $sm\equiv 1\mod p$.
Then $y_{1}=[s]([m](y_{1}))\in E_{n}^{*}(BC_{p})$, so that
$e(\alpha)=[s]([m](e(\alpha)))\in E_{n}^{*}(BA)$. Since $x$ always divides
$[s](x)$, we see that $[m](e(\alpha))$ divides $e(\alpha)$, and so inverting
$e(\alpha)$ also inverts $[m](e(\alpha))$. ∎

######  Proposition 2.7.

Let $A$ be a finite abelian $p$–group, and let $e$ be either $e(A)$ or
$\bar{e}(A)$. If $X$ is a finite dimensional $A$–CW complex, then the
inclusion $X^{A}\hookrightarrow X$ induces an isomorphism

$e^{-1}E_{n}^{*}(EA\times_{A}X)\xrightarrow{\sim}e^{-1}E_{n}^{*}(BA\times
X^{A}).$

###### Proof.

This is an application of classic localization theory, as in [tD87, Chapter
III], and a simple proof goes as follows. For notational simplicity, let
$F=X^{A}$. The $A$–equivariant cofibration sequence $F\rightarrow X\rightarrow
X/F$ induces a cofibration sequence

$BA\times F\rightarrow EA\times_{A}X\rightarrow EA_{+}\wedge_{A}X/F,$

so we need to show that $e^{-1}\widetilde{E}_{n}^{*}(EA_{+}\wedge_{A}X/F)=0$.
Since $X$ is finite dimensional, $X/F$ has a finite filtration by its
equivariant skeleta $(X/F)^{j}$, so it suffices to show
$e^{-1}\widetilde{E}_{n}^{*}(EA_{+}\wedge_{A}(X/F)^{j}/(X/F)^{j-1})=0$ for
each $j$. One has an equivariant equivalence of the form

$(X/F)^{j}/(X/F)^{j-1}\simeq\bigvee_{i}\Sigma^{j}(A/A_{i})_{+}$

where each $A_{i}$ is a proper subgroup of $A$. Since
$EA_{+}\wedge_{A}(A/A_{i})_{+}\simeq BA_{i+}$, we need to show that
$\displaystyle e^{-1}\prod_{i}E_{n}^{*}(BA_{i})=0$. But this follows
immediately from Lemma 2.6(a). ∎

##  3\. Proofs of the theorems

Proposition 2.3, Proposition 2.7, and Lemma 2.4 combine to prove the following
theorem, which gets us much of the way towards the proofs of Theorem 1.1 and
Theorem 1.3.

######  Theorem 3.1.

Let $A$ be a finite abelian $p$–group, and let $X$ be a finite dimensional
$A$–CW complex. If $\widetilde{K}(n)_{*}(X)=0$, then, with $e$ either $e(A)$
or $\bar{e}(A)$, the map $X^{A}\rightarrow pt$ induces an isomorphism

$e^{-1}E_{n}^{*}(BA)\xrightarrow{\sim}e^{-1}E_{n}^{*}(BA)\otimes_{E_{n}^{*}}E_{n}^{*}(X^{A}).$

Now we need to know something about $e^{-1}E_{n}^{*}(BA)$. The following
algebraic result of Mark Hovey and Hal Sadofsky will suffice to prove Theorem
1.1.

######  Proposition 3.2.

Let $e=\bar{e}(C_{p})=y\in E_{n}^{*}(BC_{p})\simeq E_{n}^{*}[[y]]/([p](y))$.
The ring $e^{-1}E_{n}^{*}(BC_{p})\simeq E_{n}^{*}((y))/([p](y))$ is Landweber
exact and $v_{n-1}$-periodic.

###### Proof.

This is given an elementary proof in [HS96, p.3583]. For completeness, and to
emphasize its elementary nature, we give a slightly different short proof.

As $E_{n}^{*}(BC_{p})$ is a free $E_{n}^{*}$-module, and regular sequences are
preserved by localization, the ring $e^{-1}E_{n}^{*}(BC_{p})$ is Landweber
exact. We must only show that it is $v_{n-1}$-periodic. Let
$I=(v_{0},v_{1},\ldots,v_{n-2})$. Then we must show that
$(e^{-1}E_{n}^{*}(BC_{p}))/I$ is nonzero and that $v_{n-1}$ is a unit in this
ring.

Recall $v_{n}=u^{p^{n}-1}$. We have $[p](y)\equiv
v_{n-1}y^{p^{n-1}}+_{F}v_{n}y^{p^{n}}\mod I$, so that

$v_{n-1}y^{p^{n-1}}\equiv[-1](v_{n}y^{p^{n}})\mod(I,[p](y))$

where $[-1](y)$ is the $(-1)$–series associated to the formal group law $F$.

When $n\geq 2$, it follows that

$\displaystyle(e^{-1}E_{n}^{*}(BC_{p}))/I$
$\displaystyle\simeq{\mathbb{Z}}/p[u^{\pm
1}][[v_{n-1}]]((y))/(v_{n-1}y^{p^{n-1}}+_{F}v_{n}y^{p^{n}})$
$\displaystyle\simeq{\mathbb{Z}}/p[u^{\pm
1}][[v_{n-1}]]((y))/(v_{n-1}-y^{-p^{n-1}}\cdot[-1](v_{n}y^{p^{n}}))$
$\displaystyle\simeq{\mathbb{Z}}/p[u^{\pm 1}]((y))$

since
$y^{-p^{n-1}}\cdot[-1](v_{n}y^{p^{n}}))=-v_{n}y^{p^{n}-p^{n-1}}\epsilon(y)$,
where $\epsilon(y)$ is a monic power series in $y$. A similar computation
applies when $n=1$, only one ends up with
$e^{-1}E_{1}^{*}(BC_{p})\simeq{\mathbb{Z}}_{p}[u^{\pm
1}]((y))/(p-y^{-1}\cdot[-1](v_{1}y^{p}))$, which will be a free
${\mathbb{Q}}_{p}[u^{\pm 1}]$–module with basis $1,y,\dots,y^{p-2}$. In either
case, this ring is visibly nonzero and contains $v_{n-1}$ as a unit as
claimed. ∎

A variant of Proposition 3.2 will suffice to prove Theorem 1.3.

######  Proposition 3.3.

Let $n\geq r$. The ring $e(C_{p}^{r})^{-1}E_{n}^{*}(BC_{p}^{r})$ is not zero.

###### Proof.

The special case when $r=n$ was analyzed in [HKR00, §6.2]: the ring
$e(C_{p}^{n})^{-1}E_{n}^{*}(BC_{p}^{n})$ is the nonzero ring called
$L_{1}(E_{*})$ there, and is in fact easily shown to be finite and faithfully
flat over $p^{-1}E_{n}^{*}$. For the general case, note that the projection
$C_{p}^{n}\rightarrow C_{p}^{r}$ onto the first $r$ coordinates induces an
algebra map from $e(C_{p}^{r})^{-1}E_{n}^{*}(BC_{p}^{r})$ to the nonzero ring
$e(C_{p}^{n})^{-1}E_{n}^{*}(BC_{p}^{n})$. ∎

Both of the propositions imply similar results for more general abelian
groups.

######  Corollary 3.4.

(a) The ring $e(C_{p^{k}})^{-1}E_{n}^{*}(BC_{p^{k}})$ is Landweber exact and
$v_{n-1}$-periodic.

(b) Let $A$ be a finite abelian $p$-group of rank $r$. Let $n\geq r$ and let
$e\in E_{n}^{*}(BA)$ be $e(A)$ or $\bar{e}(A)$. Then the ring
$e^{-1}E_{n}^{*}(BA)$ is nonzero.

###### Proof.

If $A$ has rank $r$, then any surjection $q:A\rightarrow C_{p}^{r}$ induces a
bijection
$q^{*}:\operatorname{Hom}(C_{p}^{r},C_{p})\xrightarrow{\sim}\operatorname{Hom}(A,C_{p})$.
It follows that $q^{*}(e(C_{p}^{r}))=e(A)$. Lemma 2.4(b) tells us that, via
$q^{*}$, $E_{n}^{*}(BA)$ is a finitely generated free
$E_{n}^{*}(BC_{p}^{r})$–algebra, so $e(A)^{-1}E_{n}^{*}(BA)$ will be a
finitely generated free $e(C_{p}^{r})^{-1}E_{n}^{*}(BC_{p}^{r})$–algebra. Thus
Proposition 3.2 proves statement (a) and Proposition 3.3 implies statement
(b). ∎

###  3.1. Proof of Theorem 1.3

Let $A$ have rank $r$. Theorem 3.1 tells us that if
$\widetilde{K}(n)_{*}(X)=0$, then
$e^{-1}E_{n}^{*}(BA)\xrightarrow{\sim}e^{-1}E_{n}^{*}(BA)\otimes_{E_{n}^{*}}E_{n}^{*}(X^{A})$.
Corollary 3.4(b) tells us that, if $n\geq r$, then $e^{-1}E_{n}^{*}(BA)\neq
0$. (When $r=1$, Corollary 3.4(a) already shows this.) Thus
$E_{n}^{*}(X^{A})\neq 0$, and so $X^{A}\neq\emptyset$.

###  3.2. Proof of Theorem 1.1

We first observe that if $A$ is abelian and $A^{\prime}$ is a subgroup of $A$,
then $X^{A}=(X^{A^{\prime}})^{A/A^{\prime}}$. Thus it suffices to prove
Theorem 1.1 when $A$ is cyclic, as the general case will follow by iteration.

So let $C=C_{p^{k}}$, and let $X$ be a finite $C$-space such that
$\widetilde{K}(n)_{*}(X)=0$ for some $n\geq 1$. As before, Theorem 3.1 tells
us that $X^{C}\neq\emptyset$ and then that
$e(C)^{-1}E_{n}^{*}(BC)\otimes_{E_{n}^{*}}\widetilde{E}_{n}^{*}(X^{C})=0$.

Given a finite CW complex $Y$, let
$h^{*}(Y)=e(C)^{-1}E_{n}^{*}(BC)\otimes_{E_{n}^{*}}\widetilde{E}_{n}^{*}(Y)$.
By Corollary 3.4(a), this defines a reduced cohomology theory with coefficient
ring that is Landweber exact and $v_{n-1}$–periodic. Since $h^{*}(X^{C})=0$,
Lemma 2.1 tells us that $\widetilde{K}(n-1)_{*}(X^{C})=0$.

##  4\. Further Remarks

###  4.1. Comparison with other proofs

The first thing to say is that every proof we know of Theorem 1.1 involves
inverting Euler classes at some point in the argument.

In particular, the proof of this theorem in the special case when $A=C_{p}$ by
Strickland [S10, Thm.16.9] (unpublished) or Balmer and Sanders [BS17] uses
results of [HS96], and Barthel et al. [6A19] invert Euler classes in the key
section 3 of their paper. (In all these papers, Theorem 1.1 is stated in terms
of the geometric fixed point functor $\Phi^{A}$ applied to a compact object in
$A$–spectra, but this is easily seen to be equivalent to Theorem 1.1 as stated
above, only with the added assumption that there exists a fixed point.)

Unlike those proofs, our proof here switches to $E_{n}$–cohomology, which
leads to our Theorem 3.1, and also allows for our simple deduction of Theorem
1.1 for all $A$ from the case when $A=C_{p}$ using Corollary 3.4, which itself
has an easy proof.

Stronger than we need for our purposes is the statement that, if $A$ is a
finite abelian $p$–group of rank $r$, and $n\geq r$, then
$e(A)^{-1}E_{n}^{*}(BA)$ is Landweber exact and $v_{n-r}$ periodic. In [T02],
T. Torii shows this when $A$ is elementary abelian. In [MNN19, Prop.5.28], A.
Mathew, N. Naumann, and J. Noel show this for general $A$, as an application
of Greenlees and Strickland’s analysis of the rings
$E_{n}^{*}(BA)/(v_{0},\ldots,v_{t})$ in [GS99].

Regarding our fixed point theorem, Theorem 1.3, we note that application of
the main theorem in [KL20] leads to the deduction of a generalization of
Theorem 1.3 for all finite $p$–groups $P$ [KL20, Thm.2.20], but specialized to
the finite $P$–space case. Our argument here proving Theorem 1.3 for abelian
groups is simpler, and, of course, applies to all finite dimensional complexes
with appropriate $A$–actions.

In a different direction, the second author has noted [K21] that Theorem 1.1
in the case when $n=r$ can be immediately ‘read off of’ the generalized
character theory of [HKR00], and that the theorem in the general case can be
similarly deduced from Stapleton’s more general transchromatic characters
[S13]. (This argument yields the full unbased version of Theorem 1.1.)
Constructing these characters also involves inverting appropriate classes in
the Morava $E$–theory of abelian $p$–groups (and then assembling the localized
rings into an appropriate universal ring). For the purposes of proving Theorem
1.1, our proof here uses much less analysis of $E_{n}^{*}(BA)$ than is used in
[HKR00] and [S13].

###  4.2. Generalizations to non-abelian groups

The paper [BS17] reduced the problem of understanding the topology of the
Balmer spectrum of a stable equivariant homotopy category to a problem that
can be posed as follows: given $n\geq 0$, and a subgroup $Q$ of a finite
$p$–group $P$, compute $r_{n}(P,Q)$, where $r_{n}(P,Q)$ is the minimal $r$
such that if $X$ is a finite $P$–space and $X^{Q}$ is $K(n+r)_{*}$-acyclic,
then $X^{P}$ is $K(n)_{*}$-acyclic.

As discussed in [KL20], iterated use of Theorem 1.1 shows that $r_{n}(P,Q)\leq
r(P,Q)$, where $r(P,Q)$ is the minimal $r$ such that there exists a sequence
of subgroups $Q=K_{0}\lhd K_{1}\lhd\dots\lhd K_{r}=P$ with each $K_{i-1}$
normal in $K_{i}$ and each $K_{i}/K_{i-1}$ cyclic. (In particular, $r(P,Q)$ is
the rank of $P/Q$ when $P$ is abelian.)

One might ask if this upper bound is best possible. To show this for a
particular $n$ and pair $Q<P$, one needs to find a finite $P$–space $X$ such
that

$\widetilde{K}(n+r)_{*}(X^{Q})=0\text{ but }\widetilde{K}(n)_{*}(X^{P})\neq
0,$

with $r=r(P,Q)-1$. The authors of [6A19] find such examples when $P$ is
abelian.

The main theorem of [KL20] says that the statement “for all finite $P$-spaces
$X$, if $X^{Q}$ is $K(n+r)_{*}$–acyclic then $X^{P}$ is $K(n)_{*}$-acyclic”
implies the apparently stronger statement “for all finite $P$-spaces $X$,
there is an inequality
$\dim_{K(n+r)_{*}}K(n+r)_{*}(X^{Q})\geq\dim_{K(n)_{*}}K(n)_{*}(X^{P})$.” (This
sort of conclusion is analogous to a theorem of Floyd [F52] in the classical
situation.) This leads both to interesting applications of Theorem 1.1 [KL21]
and to more families of examples showing that $r_{n}(P,Q)=r(P,Q)$ [KL20].

Some steps in our proof of Theorem 1.1 are quite formal, and can be
generalized to nonabelian groups. But the proof ultimately hinges on Corollary
3.4, which fails in the strongest possible way for nonabelian groups: one can
define an Euler class $e\in E_{n}^{*}(BG)$ for any finite group $G$, but
$e^{-1}E_{n}^{*}(BG)=0$ whenever $G$ is nonabelian. If there are pairs $Q<P$
for which $r_{n}(P,Q)$ is strictly less than $r(P,Q)$, then some clever new
ideas will be needed to prove this. [KL20] describes a number of pairs $Q<P$
for which $r_{n}(P,Q)$ is not yet known. Here we will just advertise one of
these: let $C$ be a noncentral subgroup of order 2 in the dihedral group $D$
of order 16. Is $r_{n}(D,C)$ equal to 2 or 3? (We note that $r(D,C)=3$ and
$r_{n}(D,\\{e\\})=r(D,\\{e\\})=2$.)

###  4.3. Questions about finite dimensional complexes

An obvious question is whether Theorem 1.1 might hold for finite dimensional
complexes that aren’t necessarily finite. Finiteness appears in our proof to
ensure that $e^{-1}E_{n}^{*}(BC)\otimes_{E_{n}^{*}}E_{n}^{*}(X^{C})=0\implies
K(n-1)_{*}(X^{C})=0$. Still one can ask:

######  Question 4.1.

Let $A$ be a finite abelian $p$-group of rank $r$, and let $X$ be a finite
dimensional $A$–CW complex. If $\widetilde{K}(n)_{*}(X)=0$ with $n\geq r$,
must $\widetilde{K}(n-r)_{*}(X^{A})=0$?

The first thing to say is that the answer is no, if $n=r$. If
$M({\mathbb{Q}})$ is the mapping telescope of
$S^{1}\xrightarrow{2}S^{1}\xrightarrow{3}S^{1}\xrightarrow{4}\dots$, then
$\widetilde{H}_{*}(M({\mathbb{Q}});{\mathbb{Z}})={\mathbb{Q}}$, concentrated
in dimension 1. Thus if we let an abelian $p$–group $A$ act trivially on
$M({\mathbb{Q}})$, then $\widetilde{K}(n)_{*}(M({\mathbb{Q}}))=0$ for all
$n>0$ (and all $p$), but $\widetilde{K}(0)_{*}(M({\mathbb{Q}})^{A})\neq 0$.
But this is really a non-equivariant example, and Bousfield’s result in
[Bou99], that if $X$ is a space then $\widetilde{K}(n)_{*}(X)=0$ implies
$\widetilde{K}(n-r)_{*}(X)=0$ for $n-r\geq 1$, makes it plausible that the
question could have a positive answer for $n-r\geq 1$. We note that [KL20]
shows that such a chromatic Smith theorem would imply that the analogous
chromatic Floyd theorem would hold, and, furthermore, the fixed point theorem
[KL20, Thm.2.20] would generalize to finite dimensional complexes with an
action of a finite $p$–group.

One might also wonder if our fixed point theorem Theorem 1.3 is best possible.

######  Question 4.2.

For all $r\geq 1$, does there exist a finite dimensional $C_{p}^{r}$–space $X$
that is $K(r-1)_{*}$–acyclic, and has no fixed points?

The answer is yes, when $r=1$. The second author asked on Mathoverflow if
$C_{p}$ could act on a rationally trivial finite dimensional complex without
fixed points. As answers, Tom Goodwillie [G22] described a free action of
$C_{2}$ on a rationally acyclic $2$–dimensional complex, and Ian Leary pointed
to a paper of his that constructs fixed point free actions of any finite group
on a $3$–dimensional rationally acyclic complex [L05, Thm.13.1].

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