Confounding is defined as the presence of common causes of the exposure and the outcome, and the common causes are often referred to as confounders. The consequence of confounding is that the exposure–outcome association becomes biased, as a measure of the causal exposure effect. As such, confounding is one of the most ubiquitous threats to the validity of observational studies. Virtually every observational study must tackle this threat in the design phase, by matching or other selection processes, or in the statistical analysis, by controlling for measured confounders.

A convenient way of dealing with confounding is the sibling comparison design, where the outcome in exposed individuals is compared with the outcome in their unexposed siblings. Since siblings are “matched by nature” on all factors that are shared (constant) within the family, a potentially large set of confounders are thereby eliminated by design. A prominent special case is the co-twin control design, which, if restricted to monozygotic twins, eliminates all confounding from genetic factors because monozygotic twins are genetically identical.

The standard analysis of sibling comparison designs assumes that the exposure and outcome of an individual do not affect the exposure and outcome of his/her siblings, sometimes referred to as an absence of sibling carryover or contagion effects. Unfortunately, there are many situations where carryover effects are likely to be present. For instance, when the exposure is “being delivered by Cesarean section” there is likely to be exposure-to-exposure carryover because the risk of being delivered by Cesarean section is greatly increased in a delivery following a prior Cesarean. When the outcome is antisocial or criminal behavior there may be outcome-to- outcome carryover because children and adolescents may imitate their older siblings. Whether, and how much, such sibling carryover effects would bias estimates from sibling comparison designs has, to our knowledge, not been explored previously.

The aim of this article is to explore the consequences of carryover effects for sibling comparison designs. We will show, using causal diagrams,^{1}^{,}^{2} when and why carryover effects lead to bias, and we will investigate the sign and magnitude of this bias under various scenarios. The article is organized as follows. In section “The standard model and the conditional maximum likelihood estimator,” we review the model commonly used in sibling comparison designs, and the standard conditional maximum likelihood estimator of the parameters of this model. We explain why the conditional maximum likelihood estimator is often biased in the presence of carryover effects. In section “Carryover effects,” we investigate how the conditional maximum likelihood estimator behaves under various types of carryover effects. Finally, in section “Testing for the presence of carryover effects,” we discuss several methods that have been proposed in earlier literature, to test for the presence of carryover effects. We will, for simplicity, focus on studies with only two siblings throughout, and because our focus is on bias rather than statistical efficiency, we will not discuss the estimators’ precision.

## THE STANDARD MODEL AND THE CONDITIONAL MAXIMUM LIKELIHOOD ESTIMATOR

The causal diagram in Figure A illustrates the sibling comparison design for a family with two siblings. Here, *X _{ij}* and

*Y*are the exposure and outcome of interest, respectively, for sibling

_{ij}*j*in family

*i*. The arrow from

*X*to

_{ij}*Y*represents a causal effect of the exposure on the outcome; we assume that this effect is the target of the study. The variable

_{ij}*U*represents those confounders for

_{i}*X*and

_{ij}*Y*that are shared (constant) within the family. The shared confounders are typically partly or fully unmeasured. Often, data are collected on some confounders that are nonshared (i.e., that may vary within the family). The variable

_{ij}*C*represents measured nonshared confounders for

_{ij}*X*and

_{ij}*Y*. The arrows from

_{ij}*U*to

_{i}*C*

_{i1}and

*C*

_{i2}allow for the shared confounders to have an effect on the nonshared confounders. For ease of exposition, we leave out confounders that are both unmeasured and nonshared. In reality, such confounders are almost always present, to some extent, and may lead to confounding bias on top of the carryover effect bias that we consider in this article.

To analyze data from a sibling comparison design it is common to use the model

where

is the conditional mean of *Y _{ij}*, given

, and

is an appropriate link function. We will restrict attention to the two most common link functions, which are the identity link and the logit link. These are appropriate for continuous and binary outcomes, respectively. In model (1), the effect of *U _{i}* on

*Y*is absorbed into the family-specific intercept α

_{ij}*, and the effect of*

_{i}*X*on

_{ij}*Y*is modeled by the parameter

_{ij}*β*. Like all statistical models, the model in (1) may be incorrectly specified. We show below that the model in (1) is robust against a certain type of misspecification, which may occur when there are carryover effects.

A naive way to fit the model (1) is to treat the family-specific intercept as a categorical variable with one level per family, and use standard maximum likelihood estimation to obtain parameter estimates. This approach can be very time consuming when there are many families in the dataset, and thus many α* _{i}*’s to estimate. More severely, when

is the logit link, the maximum likelihood estimator of (*β,δ*) is biased. This is because standard maximum likelihood theory assumes that the number of parameters in the model is limited and does not increase with sample size.^{3} A more feasible approach is to use the conditional maximum likelihood method. This method “eliminates” the family-specific intercept α_{i} by conditioning on the sufficient statistic *Y*_{i1}+*Y*_{i2} for each family, so that only *β* and *δ* are left to be estimated.^{4} When

is the logit link the conditional maximum likelihood method is synonymous with conditional logistic regression. We let

and

denote the conditional maximum likelihood estimator of *β* when

is the identity link and logit link, respectively. The conditional maximum likelihood estimator is implemented in R with the package drgee, and in Stata with the commands xtreg and xtlogit for linear and logistic models, respectively. In SAS, the conditional maximum likelihood estimator is implemented with PROC LOGISTIC for logistic models. There is, to the best of our knowledge, no implementation of the conditional maximum likelihood estimator in SAS for linear models. However, by centering all variables around their family-means, PROC GLM can be “tricked” to carry out the conditional maximum likelihood estimation for linear models.^{5}

Yet another approach is to use mixed modeling, that is, to assume a distribution (e.g., a normal distribution) for the intercept α* _{i}*, and estimate (

*β,δ*) by integrating out α

*from the likelihood. However, this approach is problematic as it assumes that α*

_{i}*, and then in effect also*

_{i}*U*, is independent of

_{i}*X*and

_{ij}*C*. Because a confounder has to be associated with the exposure, the mixed model approach thus rules out shared confounding a priori. As a consequence, it fails to control for shared confounding whenever shared confounding is present.

_{ij}^{6}This problem can be solved by centering all variables around their family-means; the resulting model has been referred to as the “hybrid model”

^{6}or the “between-within model.”

^{7}The between-within model and conditional maximum likelihood method often give identical or very similar estimates.

^{4}In this article, we focus on the conditional maximum likelihood method.

The conditional maximum likelihood estimator is unbiased under certain independence assumptions. To facilitate later discussions, it is useful to formulate these assumptions for the general model

in which **Z*** _{ij}* is a vector of covariates and

*θ*is a parameter vector of the same dimension as

**Z**

*. Model (1) is a special case of model (2), with*

_{ij}**Z**

*= (*

_{ij}*X*,

_{ij}*C*) and

_{ij}*θ*= (

*β,δ*). When

is the identity link, the conditional maximum likelihood estimator of *θ* is unbiased if the outcome for one sibling is conditionally independent of the covariates for the other sibling in the same family, given the sibling’s own covariates and the shared confounders^{8}^{,}^{9}:

When

is the logit link, the conditional maximum likelihood estimator of *θ* is unbiased if assumption (3) holds, and the outcome for one sibling is conditionally independent of the outcome for the other sibling, given the two siblings covariates and the shared confounders^{8}^{,}^{9}:

Assumptions (3) and (4) both hold in the causal diagram of Figure A. This can be seen by noting that **Z**_{i1} = (*X*_{i1},*C*_{i1}) and *U _{i}* block all paths between

*Y*

_{i1}and

**Z**

_{i2}= (

*X*

_{i2},

*C*

_{i2}), and that

**Z**

_{i2}and

*U*block all paths between

_{i}*Y*

_{i2}and

**Z**

_{i1}, so that assumption (3) holds. Similarly,

**Z**

_{i1},

**Z**

_{i2}, and

*U*block all paths between

_{i}*Y*

_{i1}and

*Y*

_{i2}, so that assumption (4) holds.

The dependence of the conditional maximum likelihood estimator on assumptions (3) and (4) is often not explicitly stated in standard epidemiologic textbooks. The reason for this may be that the conditional maximum likelihood estimator is usually proposed for matched studies where these assumptions hold by design (e.g., matched case–control studies). However, when the study participants are matched “by nature,” like siblings, there is no guarantee that the assumptions hold. In particular, in sibling comparison designs, the assumptions would often be violated when there are carryover effects. Thus, a consequence of carryover effects in sibling comparison designs is that the conditional maximum likelihood estimator of *β* may be biased, due to violations of assumptions (3) and (4).

In the next section, we discuss various types of carryover effects. We discuss whether they violate assumptions (3) and (4), and whether it is possible to say anything about the direction and magnitude of the induced bias. We also discuss if the bias can be avoided by alternative analytic approaches.

We end this section with a technical note. When *Y _{ij}* is binary, the conditional distribution of

*Y*, given (

_{ij}*X*,

_{ij}*C*,

_{ij}*U*), is completely specified by the mean model in (1). When

_{i}*Y*is continuous and

_{ij}= identity, the conditional maximum likelihood estimator can be derived by additionally assuming that *Y _{ij}* has a normal distribution with constant variance, conditional on (

*X*,

_{ij}*C*,

_{ij}*U*).

_{i}^{4}However, the conditional maximum likelihood estimator can also be derived as a semiparametric estimator from the mean model in (1) alone,

^{8}and thus it does not require these additional distributional assumptions.

## CARRYOVER EFFECTS

### Carryover from *X*_{i1} to *X*_{i2}

The most benign type of carryover effect is when the exposure for the first sibling affects the exposure for the second sibling, as in the causal diagram of Figure B. For instance, when the exposure is “being delivered by Cesarean section” there is likely to be exposure-to-exposure carryover, because the risk of being delivered by Cesarean section is greatly increased in a delivery following a prior Cesarean. The asymmetric relationship in Figure B would perhaps be considered unrealistic if the two siblings were born at the same time (i.e., if they were twins), but may otherwise be more realistic. This type of carryover effect does not lead to bias for the conditional maximum likelihood estimator, because it does not violate assumption (3) or (4). This can be seen by noting that **Z**_{i1} = (*X*_{i1},*C*_{i1}) and *U _{i}* block all paths between

*Y*

_{i1}and

**Z**

_{i2}= (

*X*

_{i1},

*C*

_{i1}), and that

**Z**

_{i2}and

*U*block all paths between

_{i}*Y*

_{i1}and

**Z**

_{i1}, so that assumption (3) holds. Similarly,

**Z**

_{i1},

**Z**

_{i2}, and

*U*block all paths between

_{i}*Y*

_{i1}and

*Y*

_{i2}, so that assumption (4) holds.

### Carryover from *X*_{i1} to *Y*_{i2}

A more harmful type of carryover effect is when the exposure for the first sibling affects the outcome for the second sibling, as in the causal diagram of Figure C. For instance, when the exposure is smoking during pregnancy and the outcome is fetal malformations, there could be exposure-to-outcome carryover if smoking has long-term effects, so that the risk of malformations depends on smoking history. This type of carryover effect can be expected to lead to bias for the conditional maximum likelihood estimator, since it violates assumption (3). In particular, even if we condition on (control for) **Z**_{i2} = (*X*_{i2},*C*_{i2}) and *U _{i}*, we would observe an association between

*Y*

_{i2}and

*X*

_{i1}, due to the direct path from

*X*

_{i1}to

*Y*

_{i2}.

The degree of bias depends on the data generating model and on the strength of the carryover effect. Suppose that the data generating model is given by

Suppose further that the conditional distribution for (*X*_{i1},*C*_{i1},*X*_{i2},*C*_{i2}), given *U _{i}*, is symmetric:

Here, x_{1} and x_{2} are arbitrary and generally nonequal values, similar for c_{1} and c_{2}. The symmetry condition in (6) would typically hold if there are no “birth order effects” on *X* and *C*. We note that this symmetry condition has testable implications. For instance, if we observe that exposure distribution among the first born siblings is different from the exposure distribution among the second born siblings, then we may conclude that the symmetry condition is violated. Under (6) it can then be shown (see eAppendix A; https://links.lww.com/EDE/B97) that

converges to *β*—*γ*/2, meaning that

is biased by a term —*γ*/2, where *γ* is the strength of carryover effect from *X*_{i1} to *Y*_{i2}. In many cases it would be reasonable to assume that the carryover effect *γ* has the same sign as the true exposure effect *β*, but smaller magnitude. Under this assumption,

is conservative (i.e., asymptotically closer to 0 than *β*), and provides a valid test of the null hypothesis that *β* = 0 (i.e., asymptotically equal to 0 if *β* = 0). These bias results do not generally hold when

is the logit link; we give an example in eAppendix A (https://links.lww.com/EDE/B97), where

does not converge to *β*—*γ*/2.

The reason for including the parameter *ψ* in the data generating model (5) is to avoid making the assumption that

This assumption would essentially say that the “effect” on the outcome of having a prior sibling with *X*_{i1} = 0 is the same as the “effect” on the outcome of having no prior sibling at all. In most cases there is no a priori reason that this assumption would hold.

The data generating model (5) does not imply the working model (1), even when

is the identity link. Thus, one may wonder if the bias that arises when fitting the working model (1) is due to model misspecification, rather than to the violation of assumption (3). However, it can be shown (see eAppendix B; https://links.lww.com/EDE/B97) that the working model with

= identity is robust against this type of model misspecification, under the symmetry condition (6). Thus, the bias of

is entirely due to the violation of assumption (3).

When the data generating model is given by (5), we can avoid bias completely by simply fitting this data generating model to the data, using the conditional maximum likelihood method. In practice this may be accomplished by defining the variables *W _{ij}* =

*I*(

*j*= 2) and

*V*=

_{ij}*W*

_{ij}*X*

_{i1}, where

is the indicator function, and fitting the model

Model (7) is equivalent to model (2), with **Z**_{ij} = (*X _{ij}*,

*W*,

_{ij}*V*,

_{ij}*C*) and

_{ij}*θ*= (

*β,ψ,γ,δ*). With

**Z**

_{ij}defined in this way it is easy to show that assumptions (3) and (4) hold under the causal diagram in Figure C, and thus the conditional maximum likelihood estimator of (

*β,ψ,γ*) is unbiased. This is true regardless of whether

is the identity link or logit link.

### Carryover from *X*_{i1} to *Y*_{i2} and from *X*_{i2} to *Y*_{i1}

We now consider the symmetric situation where the exposure for one sibling affects the outcome for the other sibling, and vice versa, as in the causal diagram of Figure D. For instance, when the exposure is attention deficit hyperactivity disorder and the outcome is abusive or criminal behavior there may be “cross-wise” exposure-to-outcome carryover in twins, because severe attention deficit hyperactivity disorder in one twin may have a negative influence on the familial environment, which in turn could increase the risk of criminality in the other twin, and vice versa. This type of carryover effect can be expected to lead to bias for the conditional maximum likelihood estimator, because it violates assumption (3). Suppose that the data generating model is given by

It can then be shown (see eAppendix A; https://links.lww.com/EDE/B97) that both

and

converge to *β*—*γ*, regardless of whether the symmetry condition in (6) holds or not. If *γ* has the same sign as *β*, but smaller magnitude, then both

and

are conservative and provide valid tests of the null hypothesis that *β* = 0.

We note that the carryover effect from

to *Y _{ij}* does not invalidate the working model (1) per se. For instance, under the data generating model (8) with

= identity, we have that

so that the working model (1) is correct. In the second-to-last equality, we have used the fact that *X _{ij}* and (

*X*,

_{ij}*C*) are conditionally independent in Figure D, given

_{ij}*U*, and in the last equality we have defined

_{i}. This highlights that the bias that occurs when fitting the working model (1) is due to the violation of assumption (3), and not due to model misspecification.

A natural question is whether the bias can be avoided by the same strategy as when there is carryover from *X*_{i1} to *Y*_{i2}, that is, by fitting the data generating model (8) with the conditional maximum likelihood method. Unfortunately it can be shown (see eAppendix C; https://links.lww.com/EDE/B97) that this strategy does not work. This is because the data generating model (8) suffers from an identifiability problem, which makes it impossible to disentangle *β* and *γ*.

### Carryover from *Y*_{i1} to *Y*_{i2}

So far we have considered carryover effects from the exposure, but there could also be carryover effects from the outcome. The causal diagram in Figure E illustrates a scenario where the outcome for the first sibling affects the outcome for the other sibling. For instance, when the outcome is antisocial or criminal behavior there may be outcome-to-outcome carryover, because children and adolescents may imitate their older siblings. This type of carryover effect can be expected to lead to bias for the conditional maximum likelihood estimator, because it violates both assumption (3) and assumption (4). Suppose that the data generating model is given by

Suppose further that the symmetry condition in (6) holds. In this case, it can be shown (see eAppendix A; https://links.lww.com/EDE/B97) that

converges to *β* = (1—*γ*/2). Thus, even if *γ* has the same sign as *β* and is smaller in magnitude than *β*,

is not generally conservative and may asymptotically have a different sign than *β*. However, if *β* = 0, then

is asymptotically equal to 0 as well. Thus,

provides a valid test of the null hypothesis that *β* = 0, even if *γ* does not have the same sign as *β* and is not smaller in magnitude than *β*. These bias results do not generally hold when

is the logit link; we give an example in eAppendix A(https://links.lww.com/EDE/B97) where

does not converge to *β* = (1—*γ*/2). However, if there are no nonshared confounders (*C _{ij}* is the empty set) and

*β*= 0, then

converges to 0 (see eAppendix A; https://links.lww.com/EDE/B97). Thus, in the absence of nonshared confounders

provides a valid test of the null hypothesis, like

.

As in section “Carryover from *X*_{i1} to *Y*_{i2},” the data generating model does not imply the working model. But also here it can be shown (see eAppendix B; https://links.lww.com/EDE/B97) that with

= identity, the latter working model is robust against this type of model misspecification so that the bias of

is entirely due to violation of assumption (3).

The bias cannot be avoided by fitting the data generating model (9). To see this, define *W _{ij}* =

*I*(

*j*= 2) and

*V*=

_{ij}*W*

_{ij}*Y*

_{i2}, and note that model (7) is a special case of model, (2) with

**Z**

_{ij}= (

*X*,

_{ij}*W*,

_{ij}*V*,

_{ij}*C*) and

_{ij}*θ*= (

*β,ψ,γ,δ*). The conditional maximum likelihood estimator of

*θ*is unbiased if assumptions (3) and (4) hold. For the first sibling (

*j*= 1) assumption (3) states that

*Y*

_{i1}should be conditionally independent of

**Z**

_{i2}= (

*X*

_{i2},1,

*Y*

_{i1},

*C*

_{i2}), given

**Z**

_{i1}= (

*X*

_{i1},0,0,

*C*

_{i1}) and

*U*. This is clearly not true, because

_{i}*Y*

_{i1}can never be independent of

*Y*

_{i1}. Thus, assumption (3) is violated when

**Z**

_{ij}is defined in this way, so that the conditional maximum likelihood estimator of (

*β,ψ,γ,δ*) is biased.

### Carryover from *Y*_{i1} to *X*_{i2}

The last case we consider is when the outcome for the first sibling affects the exposure for the second sibling, in Figure F. For instance, a fetal malformation (outcome) in the first pregnancy may alert the mother to the potential consequences of smoking during pregnancy (exposure), thus reducing her willingness to smoke in the second pregnancy. This type of carryover effect can be expected to lead to bias for the conditional maximum likelihood estimator, because it violates assumption (3). The bias has a complicated expression, even under relatively simple data generating models. For instance, suppose that there are no nonshared confounders and that the data generating model is given by

where

,

,

,

, and

are independent, with mean 0 and variance 1. It can then be shown (see eAppendix A; https://links.lww.com/EDE/B97) that

converges to

Clearly, there is no guarantee that conditional maximum likelihood estimator is conservative even under this simple data generating model, or that it provides a valid test of the null hypothesis. In eAppendix A (https://links.lww.com/EDE/B97), we suggest an alternative method test the null hypothesis.

## TESTING FOR THE PRESENCE OF CARRYOVER EFFECTS

Given that carryover effects can bias the results of sibling comparisons, it is important to determine, if possible, whether carryover effects are present. Occasionally, it may be possible to rule out some types of carryover effects based on temporality. For instance, the asymmetric carryover effects in Figure B, C, E, F may be considered unlikely when the siblings are twins. Some types of carryover effects may also be considered likely or unlikely based on subject matter (e.g., biological) knowledge about the exposure and outcome of interest.

To some extent, the observed data can also be informative about the presence or absence of carryover effects. Meyer et al.^{10} used a sibling comparison design to study the effect of maternal smoking during pregnancy on the risk of oral cleft in the offspring. To test for carryover from oral cleft in the first sibling (*Y*_{i1}) to smoking during pregnancy in the second pregnancy (*X*_{i2}) they used ordinary logistic regression to examine the association between *Y*_{i1} and *X*_{i2}, while controlling for *X*_{i1}.They obtained an odds ratio equal to 1.0, with 95% confidence interval equal to (0.8, 1.3), and concluded that a carryover effect from *Y*_{i1} to *X*_{i2} is probably absent. The rationale for this “cross-wise association test” can be understood from the causal diagram in Figure F; if the arrow from *Y*_{i1} to *X*_{i2} was present, then we would also expect *Y*_{i1} and *X*_{i2} to be statistically associated (regardless of whether we control for *X*_{i1}). Thus, the absence of association between *Y*_{i1} and *X*_{i2} indicates absence of carryover effect from *Y*_{i1} to *X*_{i2}. We note that the implication does not go the other way around; had a statistical association between *Y*_{i1} and *X*_{i2} been present, then this could very well have been entirely due to familial confounding through the path

. Thus, this cross-wise association test can be used to “rule out” carryover effects, but cannot be used to “rule in” carryover effects. We also note that an absence of association between *Y*_{i1} and *X*_{i2} could in principle be explained by “cancelling out” of effects, rather than absence of carryover effects. For instance, if the association through

has the same magnitude as the association through

, but opposite sign, then we may observe that *Y*_{i1} and *X*_{i2} are independent, even though carryover is present. In practice although, we may not consider such perfect cancelation as likely.

D’Onofrio et al.^{11} used a sibling comparison design to study the effect of preterm birth, coded as 0/1, on mortality. To test for carryover from preterm birth in the first sibling (*X*_{i1}) on mortality in the second sibling (*Y*_{i2}) they divided the exposure-discordant sib-pairs (i.e., those pairs with *X*_{i1} ≠ *X*_{i2}) into two subgroups; those where the first sibling was preterm (*X*_{i1} = 1,*X*_{i2} = 0) and those where the second sibling was preterm (*X*_{i1} = 0,*X*_{i2} = 1). They then used conditional logistic regression to analyze these subgroups separately. They obtained similar results for the two subgroups, and concluded that carryover effects are probably absent. The rationale for this “bidirectional analysis” can be understood from the causal diagram in Figure C; the presence of carryover effect in this causal diagram implies an asymmetric relation between the two siblings, and would therefore lead to different results in the two subgroups. For instance, suppose that there is carryover from *X*_{i1} to *Y*_{i2}, and that the data generating model is given by (5), with

= identity. It can then be shown (see eAppendix D; https://links.lww.com/EDE/B97) that

converges to *β*—*β*—*ψ* in the subgroup defined by *X*_{i1} = 1,*X*_{i2} = 0, and that

converges to *β*+*ψ* in the subgroup defined by *X*_{i1} = 1,*X*_{i2} = 0. Thus, similar results in the two subgroups indicate absence of asymmetric carryover effects. We note although that symmetric carryover effects, as in Figure D, would not lead to different results in the subgroups. Thus, the bidirectional analysis can be used to determine the presence of asymmetric carryover effects, as in Figure C, E, F, but cannot be used to determine the presence of symmetric carryover effects. It is also unclear whether the bidirectional design has a natural extension to nonbinary exposures.

## DISCUSSION

In this article, we have considered various types of carryover effects in sibling comparison designs, and shown why and how they may lead to bias when estimating the exposure effect. For several cases, we have derived analytic expressions for the bias terms.

Ideally, one could use these analytic expressions to correct the estimated exposure effect for the amount of bias. In practice this would require knowledge about the type and strength of carryover effect. For a given type of carryover effect, one could perform a sensitivity analysis where the strength of carryover effect is varied over a range of plausible values, producing a range of plausible values for the exposure effect. In many cases, this quantitative bias correction may not be possible, due to, for instance, lack of knowledge about the strength of carryover effect. In these cases, our results may still give important qualitative insights into the resulting bias. For instance, when there is symmetric carryover from both exposures to both outcomes (section “Carryover from *X*_{i1} to *Y*_{i2} and from *X*_{i2} to *Y*_{i1}”), we have argued that the estimated exposure effect is likely to be conservative, and that a test of the null hypothesis is still likely to be valid.

We have restricted attention to sibling pairs, and to one type of carryover effect at a time. In real studies, data may often consist of families of varying size, and several types of carryover effects may be present simultaneously. However, if a majority of all families have two children and one type of carryover effect is dominating, then our results may still hold as an approximation.

We have not explicitly considered carryover effects from or to the nonshared confounders *C*_{i1} and *C*_{i2}. However, in the eAppendix (https://links.lww.com/EDE/B97), we have derived bias terms for the conditional maximum likelihood estimators

and

under the assumed model in (1), for each of the causal diagrams in Figure B–E. By symmetry arguments, it is easy to show that the biases of

and

in the presence of carryover to/from *X _{ij}* are identical to the biases of

and

in the presence of “analogous” carryover from/to *C _{ij}*. For instance,

is unbiased when there is carryover from *X*_{i1} to *Y*_{i2} (see the eAppendix; https://links.lww.com/EDE/B97), so

is unbiased when there is carryover from *C*_{i1} to *Y*_{i2}. In principle, one can allow for simultaneous carryover from/to *X _{ij}* and

*C*, but this makes the bias calculations more difficult.

_{ij}