Estimation of regression models for the mean of repeated outcomes under nonignorable nonmonotone nonresponse

Summary

We propose a new class of models for making inference about the mean of a vector of repeated outcomes when the outcome vector is incompletely observed in some study units and missingness is nonmonotone. Each model in our class is indexed by a set of unidentified selection bias functions which quantify the residual association of the outcome at each occasion t and the probability that this outcome is missing after adjusting for variables observed prior to time t and for the past nonresponse pattern. In particular, selection bias functions equal to zero encode the investigator’s a priori belief that nonresponse of the next outcome does not depend on that outcome after adjusting for the observed past. We call this assumption sequential explainability. Since each model in our class is nonparametric, it fits the data perfectly well. As such, our models are ideal for conducting sensitivity analyses aimed at evaluating the impact that different degrees of departure from sequential explainability have on inference about the marginal means of interest. Although the marginal means are identified under each of our models, their estimation is not feasible in practice because it requires the auxiliary estimation of conditional expectations and probabilities given high-dimensional variables. We henceforth discuss estimation of the marginal means under each model in our class assuming, additionally, that at each occasion either one of following two models holds: a parametric model for the conditional probability of nonresponse given current outcomes and past recorded data, or a parametric model for the conditional mean of the outcome on the nonrespondents given the past recorded data. We call the resulting procedure 2^T-multiply robust as it protects at each of the T time points against misspecification of one of these two working models, although not against simultaneous misspecification of both. We extend our proposed class of models and estimators to incorporate data configurations which include baseline covariates and a parametric model for the conditional mean of the vector of repeated outcomes given the baseline covariates.

1 Introduction

Consider a follow-up study whose design prescribes measurements of an outcome of interest to be taken on n independent subjects at fixed time-points. The goal of the study is to make inference about the mean outcome vector possibly as a function of baseline covariates and time. The intended vector of outcomes is often not completely recorded because some subjects miss some study cycles. When, as usual, the mechanism leading to these outcomes being missing is unknown to the investigator, the expected outcome at each time-point is not identified from the observed data. Inference must then rely on unverifiable assumptions about the missing data.

The problem of missing data in follow-up studies has received much attention in the statistical literature, but most emphasis has been given to settings where the missing pattern is monotone, in which no subject returns to subsequent study cycles after missing previous cycles. Robins et al. (1999) described a model for monotone missing data patterns which requires the a priori specification of a selection bias parameter that encodes the residual association between the outcome vector and missingness at each occasion after adjusting for past recorded data. They showed that, regardless of the value of the selection bias parameter, the model is nonparametric (just) identified as it imposes no restriction on the observed data distribution and yet identifies the mean of the repeated outcomes. Since all values of the selection bias parameter determine the same model for the observed data distribution, the selection bias parameter is not identified. Robins et al. (1999) therefore recommended conducting inference about the mean of the outcome vector by repeating the estimation under different plausible values for the selection bias parameter as a form of sensitivity analysis. The goal of this paper is to extend the work of Robins et al. (1999) to the case in which the outcome vector is incompletely observed in some study units and missingness is nonmonotone.

Our interest in nonparametric identified models is motivated by the fact that other models fail to distinguish (i) the nonidentifiable, i.e. untestable, restrictions on the missing data process necessary to identify the full-data parameter of interest from (ii) additional identifiable restrictions that serve to increase the efficiency of estimation. The distinction between (i) and (ii) is not only conceptually important but can also be practically important. For example, when one has available a nonparametric identified model, one can first fit the model to the data. If the resulting uncertainty concerning the functionals of interest, such as the marginal means of the outcomes, is too large to be of substantive use, as measured for instance by the volume of a 95% confidence region, then, as in any inferential problem, to reduce uncertainty one can choose between fitting a nested submodel and refitting the same model after collecting data on additional subjects, where possible, to increase the sample size. Clearly, when logistically and financially feasible, the second option is to be preferred. Thus, fitting a model that is not nonparametrically identified is tantamount to supplementing additional modelling restrictions for the unavailable additional data. Furthermore, follow-up studies routinely collect high-dimensional data and models that are not nonparametrically identified require assumptions to be imposed on the mechanism generating these high-dimensional data. However, specification of realistic models is difficult, if not impossible. Nonparametric identified models meet the challenge posed by high-dimensional data because they only make assumptions about the missing data mechanism, thereby reducing the possibility of model misspecification.

Several available methods for the analysis of nonmonotone missing data assume that the data are missing at random (Laird & Ware, 1982; Shah et al., 1997; Andersson & Perlman, 2001; Fairclough et al., 1998; Little & Rubin, 1987; Troxel, Fairclough, Curran & Hahn, 1998). Although the missing at random assumption enables a fairly straightforward likelihood-based analysis without needing to model the missing process (Little & Rubin, 1987), we will argue in §3.1 that this assumption is rarely realistic for nonmonotone missing data. A recent model discussed by Lin et al. (2003) and van der Laan & Robins (2003, Ch. 6) relies on a more plausible assumption about the missingness process which nonetheless assumes no selection on unobservables for the marginal distribution of the responses. In §3.4 we show that it can be viewed as a special case of the model presented in this paper, in which selection bias is absent.

Several proposals also exist for nonmonotone missing data where selection depends on unobservables. With the exception of the selection bias permutation missingness model of Robins et al. (1999), none of the available models is nonparametric identified. The currently available models rely on parametric assumptions for both the full data and the missing data mechanisms (Deltour et al., 1999; Albert, 2000; Ibrahim et al., 2001; Fairclough et al., 1998; Troxel, Fairclough, Curran & Hahn, 1998; Troxel, Lipsitz & Harrington, 1998) or on parametric assumptions for just the missing data process (Rotnitzky et al., 1998; Robins et al., 1995). The selection bias permutation missingness model generalizes the permutation missingness model of Robins (1997) and the sequential coarsening model of Gill & Robins (1997). This model differs from but is related to the nonparametric identified model we propose in the current paper. The two models are compared in §3.5.

2 The formal setting

Consider a longitudinal study design that calls for measurements on a vector of variables L_it to be recorded at study cycles t = 0, …, T, for the ith of n independent subjects. The vector L_it, t = 1, …, T, includes an outcome of interest Y_it as well as other variables V_it recorded for secondary analyses. The vector L_i0 may include, in addition to Y_i0 and V_i0, a baseline covariate vector X_i of interest.

Suppose that L_i = (L_i0, …, L_iT) is not always fully recorded because some subjects miss some study cycles. In particular, for each t, L_it is either completely observed or completely missing and L_i0 is always observed. Thus, for each t, the observed data for subject i is the vector O_it = (R_it,c (R_it,L_it)), where R_it is a response indicator which equals 1 if L_it is observed and is 0 otherwise, and where, for any random vector W, c (1, W) = W and c (0, W) is set to zero by convention. Under our setting, the observed data O_i = (L_i0, O_i1, …, O_iT), i = 1, …, n, can be regarded as n independent realizations of the random vector O = (O₀, O₁, …, O_T). Here and throughout O₀ denotes L₀. Furthermore, for any vector Z = (Z₀, …, Z_T), Z̅_t denotes the history (Z₀, …, Z_t) up to and including cycle t and Z_(t) denotes the vector (Z_t, …, Z_T). Throughout we assume for each t that Y_t is either a continuous or discrete random variable; for any random vector W, f (Y_t|W) denotes a fixed version of the conditional density of Y_t given W with respect to either Lebesgue measure or a counting measure and pr(R_t = 1|W) denotes a fixed version of the conditional probability that R_t equals 1 given W.

We assume that the nonresponse patterns are nonmonotone so that R_t = 0 does not imply that R_t+1 = 0. We additionally assume that no recorded past O̅_t−1 and no current outcome Y_t can prevent the possibility of returning to study cycle t; that is,

$$\text{pr}(R_t=1|{\mathrm{O̅}}_{t-1},Y_t)>0\text{ with probability }1.$$ (1)

Note that (1) would not hold in a study design where patients are withdrawn when they miss, say, four consecutive visits (Zeuzem et al., 2000) or when all individuals with extreme values of Y_t are so physically impaired that clinic attendance at visit t is impossible. The methods we propose here are not applicable in such cases.

Condition (1) was also assumed in Lin et al. (2003) and van der Laan & Robins (2003, Ch. 6) except that visits could be in continuous rather than in discrete times. These authors obtained identifiability by imposing the additional assumption of sequential explainability,

$$\text{pr}(R_t=1|{\mathrm{O̅}}_{t-1},Y_t)=\text{pr}(R_t=1|{\mathrm{O̅}}_{t-1}),$$ (2)

which we discuss in detail in §3.4. A conflict as to the number and type of variables to include in the components V_t of the full data vector L_t at each cycle t arises when one wishes to impose both assumptions (1) and (2): to make sequential explainability (2) plausible one would generally wish to choose V_t−1 to be high-dimensional; however, the positivity assumption (1) may be unrealistic for high-dimensional V_t−1, as V_t−1 may then well include covariates, such as for example the subject’s state of consciousness, certain values of which, e.g. being unconscious, preclude the possibility of being observed at occasion t. Since it will often be unrealistic to impose both (2) and (1), we propose to relax the assumption of sequential explainability: we will describe methods for conducting inference about the marginal mean E (Y_t), t = 1, …, T, and the conditional mean given baseline covariates E (Y_t|X), t = 1, …, T, when (1) holds but (2) may fail.

3 Identifying assumptions

3.1 Preamble

Unless R_t = 1 with probability 1, the observed data O identify neither the distribution of Y_t nor its conditional distribution given X because, as Theorem 1 below implies, many distinct conditional laws of Y_t given O̅_t−1 are compatible with the observed data law. To identify these distributions we must make unverifiable assumptions. We now review one popular such assumption, that the data are missing at random, and argue that it represents processes that are unlikely to generate nonmonotone missing data patterns in longitudinal studies. We then propose a class of unverifiable assumptions that are naturally suited to conducting sensitivity analysis of the investigator’s a priori belief about the process that generates the intermittent nonresponse in a follow-up study. In subsequent sections we discuss inference under any such assumption.

3.2 Missing at random

Robins & Rotnitzky (1992) and Gill et al. (1997) showed that the distribution of the full data vector L is identified if the data are missing at random, provided that there is a positive conditional probability of observing the full data, i.e. pr(R̅_T = 1*|L) > 0 with probability 1, where 1* denotes the T × 1 vector of ones. The missing at random assumption states that

$$\text{pr}({\mathrm{R̅}}_T={\mathrm{r̅}}_T|L)=\text{pr}({\mathrm{R̅}}_T={\mathrm{r̅}}_T|L_{({\mathrm{r̅}}_T)}),$$ (3)

where L_(r̅T) denotes the observed part of L when R̅_T = r̅_T.

Under any model in which pr(R̅_T|L) and f (L) are variation independent and the missing at random condition (3) is imposed, the likelihood factorizes into a part that depends on f (L) and another that depends on pr(R̅_T|L). Any method that obeys the likelihood principle then yields the same inference whether pr(R̅_T|L) is fully known, unknown or known to follow a model. As a result of this, many authors have proposed analyzing nonmonotone missing data using likelihood-based methods under models that assume missing at random. However, convenient as the missing at random assumption may be, the assumption should only be adopted if it is believed plausible. Following Robins & Gill (1997) we will now argue that missing at random mechanisms that could plausibly generate the observed data in follow-up studies with nonmonotone nonresponse are quite restrictive and would rarely be plausible.

Robins & Gill (1997) showed that the set of missingness probabilities pr(R̅_T|L) that satisfy missing at random can be divided into two disjoint subsets. The first set contains processes in which the observed data are generated as follows. The variable L₀ is always observed. Then, with probability p₀₀ possibly depending on L₀, no further variable is observed. Otherwise one selects which of L₁, …, L_T to observe next by flipping a T-sided coin with probabilities p₀₁, …, p_0T that may depend on L₀. One then observes nothing else with probability p₁₀ that may depend on L₀ and the most recently observed L_t. Otherwise one selects which of the T − 1 still unobserved L_t’s to observe next with probabilities p₁₁, …, p_1(T−1) that may depend on the already observed L_t’s, and so on. The second subset contains all remaining missingness processes that satisfy (3). Robins & Gill (1997) showed that the second subset is not empty. They also showed that to generate the observed data O according to a missingness mechanism in the second set it is required, in the course of the data-generation procedure, to use information about the components of L that are not in L_(R̅T), and thus are ultimately missing, in a subtle and often highly contrived manner to ensure that missing at random holds. In agreement with the discussion in Robins & Gill (1997), we believe that such missing at random missingness mechanisms are often unrealistic. Consequently, the most reasonable missing at random processes with nonmonotone data are in the first set. Moreover, of the processes in the first set, the only plausible choices for longitudinal data are those in which, with conditional probability equal to 1, the next variable to be observed comes later in time than any variable already observed, as decisions today cannot affect attendance in the past. However, even these are rather unlikely processes when missingness is nonmonotone because they effectively imply for example that, if a patient chooses today to miss his next two visits and then to return, he will not reassess this decision based on evolving time-dependent covariates associated with the response. This is unlikely, as often the decision to miss a given study cycle is influenced by aspects of the subject’s health and psychological status that evolved during earlier missed study cycles.

3.3 Occasion-specific tilted models

Part (ii) of Theorem 1 below establishes that, when (1) holds for a fixed t, t = 1, …, T, the distributions f (Y_t|X) and f (Y_t) are identified under the following Assumption 1 which postulates that, among subjects with a given observed past O̅_t−1, the distribution of Y_t in the nonresponders at cycle t is equal to the distribution of Y_t in the responders at cycle t tilted by a known function.

Assumption 1. If

$$\text{pr}(R_t=0|{\mathrm{O̅}}_{t-1})>0,$$ (4)

then

$$f(Y_t|{\mathrm{O̅}}_{t-1},R_t=0)=f(Y_t|{\mathrm{O̅}}_{t-1},R_t=1)\frac{\text{exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}}{E[\text{exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}|R_t=1,{\mathrm{O̅}}_{t-1}]}$$ (5)

for some user-specified, i.e. known, function q_t(O̅_t−1, Y_t).

For ease of reference in the forthcoming discussion, we use 𝒜_t (q) (𝒜(q)) to denote the model for the full data (L,R̅_T) defined by (1) and Assumption 1 for a fixed t, for all t = 1, …, T. Note that 𝒜(q) is the intersection of models 𝒜_t (q), t = 1, …, T.

By Bayes rule, Assumption 1 is equivalent to

$$\text{pr}(R_t=0|{\mathrm{O̅}}_{t-1},Y_t)=\text{expit}\{h_t({\mathrm{O̅}}_{t-1})+q_t({\mathrm{O̅}}_{t-1},Y_t)\}$$ (6)

whenever f(Y_t|O̅_t−1, R_t = 1) > 0,, where expit(․) = exp(․)/{1 + exp(․)} and

$$\text{exp}\{h_t({\mathrm{O̅}}_{t-1})\}=\frac{\text{pr}(R_t=0|{\mathrm{O̅}}_{t-1})}{\text{pr}(R_t=1|{\mathrm{O̅}}_{t-1})E[\text{exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}|R_t=1,{\mathrm{O̅}}_{t-1}]}.$$ (7)

From (6), we interpret each function q_t(O̅_t−1, Y_t) as quantifying, on the logistic scale, the magnitude of the residual association between the missingness probability at cycle t and the possibly missing outcome Y_t, after adjustment for the observed past O̅_t−1. Thus, model 𝒜(q) encodes the investigator’s a priori belief of the degree to which, for each t the decision to return to study cycle t is influenced by prognostic factors for Y_t other than those included in the observed past, O̅_t−1. For example, the choice q_t(O̅_t−1, Y_t) = (1 − R_t−1) λY_t encodes the belief that, for those that did not miss the prior cycle t − 1, the recorded variables L_t−1 at the prior cycle together with the observed past O̅_t−2 prior to cycle t − 1 are sufficient to explain missingness at cycle t, but, for those that missed cycle t − 1, the observed past O̅_t−2 is not sufficient to explain missingness at cycle t. The choice q_t(O̅_t−1, Y_t) = {(1 − R_t−1) λ₁ + λ₂} Y_t additionally allows a residual dependence on the current outcome. In line with the terminology used in some of the missing data literature, we call q_t(․) a selection bias function (Scharfstein et al., 1999).

Part (i) of Theorem 1 below establishes that model 𝒜(q), and therefore 𝒜_t (q) for each t, places no restriction on the observed data law beyond the restriction that pr(R_t = 1|O̅_t−1) > 0 for all t. As such, each choice of selection bias functions q_t(O̅_t−1, Y_t), t = 1, …, T, fits the data perfectly and cannot be rejected by any statistical test. Since there will never be any evidence from the data that can help determine the functions q_t(O̅_t−1, Y_t), the analyst should be reluctant to analyze the data solely under one choice of functions q_t(O̅_t−1, Y_t). Instead, he should pose a range of plausible selection bias functions and, as a form of sensitivity analysis to his prior beliefs about the missingness mechanism, repeat the analysis under each choice of q_t(O̅_t−1, Y_t). This raises the question of how to choose the selection bias functions in practice. We suggest that one chooses, as in the example above, a collection of simple selection bias functions indexed by one or two parameters that are to be varied in a sensitivity analysis. Ideally, the parameterization should satisfy the following properties: it is easily interpretable so that a plausible parameter range can be specified by subject matter experts; values of the parameters equal to zero correspond to the assumption of no selection bias for the outcomes; and nonparametric bounds are attained when the parameters go to ±∞.

In the following theorem, proved in the Appendix, and throughout, if (4) holds we define

$$m_t\left({\mathrm{O̅}}_{t-1}\right)\equiv E\left[Y_t\text{exp}\right\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}|R_t=1,{\mathrm{O̅}}_{t-1}]/E[\text{exp}\left\{q_t\right({\mathrm{O̅}}_{t-1},Y_t\left)\right\}|R_t=1,{\mathrm{O̅}}_{t-1}].$$

Furthermore, we define π_t(O̅_t−1, Y_t) ≡ 1 if (4) does not hold and

$${\pi }_t({\mathrm{O̅}}_{t-1},Y_t)\equiv {[1+\text{exp}\{h_t({\mathrm{O̅}}_{t-1})+q_t({\mathrm{O̅}}_{t-1},Y_t)\left\}\right]}^{-1}$$

otherwise, where h_t(O̅_t−1) satisfies (7).

Theorem 1. (i) Model 𝒜(q) determines a model for the law of the observed data whose only restriction is pr(R_t = 1|O̅_t−1) > 0 with probability 1, for t = 1, …, T.

(ii) The conditional density f(Y_t|O̅_t−1) is identified under model 𝒜_t (q), and therefore under 𝒜(q). Furthermore, under models 𝒜_t (q) and 𝒜(q), E (Y_t) is equal to the following functional of the observed data distribution:

$$E(Y_t)=E\left\{\frac{R_t{Y}_t}{{\pi }_t({\mathrm{O̅}}_{t-1},Y_t)}\right\}$$ (8)

$$=E\{R_t{Y}_t+(1-R_t)m_t({\mathrm{O̅}}_{t-1})\}.$$ (9)

Part (ii) of the Theorem implies the identifiability of the marginal density of Y_t and its conditional distribution given X under model 𝒜_t (q) and therefore also under model 𝒜(q). However, the theorem says nothing about the identifiability of the dependence among outcomes at different occasions. This is so because this dependence is generally not identified under model 𝒜(q). In particular, under model 𝒜(q) the correlations among the repeated outcomes are not identified for any choice of selection-bias function.

3.4 Sequential explainability

The choice q_t = 0 postulates the conditional independence of Y_t and R_t given O̅_t−1. This assumption would hold if the observed past variables O̅_t−1 included all the predictors of Y_t that explain missingness at cycle t. We therefore refer to the assumption that q_t = 0 for all t as the assumption of sequential explainability. Lin et al. (2003) and van der Laan & Robins (2003) consider such processes except that visits occur in continuous time.

The assumption that q_t = 0 is less restrictive than the missing at random condition

$$\text{pr}(R_t=1|{\mathrm{R̅}}_{t-1},L)=\text{pr}(R_t=1|{\mathrm{O̅}}_{t-1})$$ (10)

for t = 1, …, T, because (10) implies (6) with q_t = 0 but the opposite is false. For example, (10) postulates the conditional independence given O̅_t−1 of R_t with the current components L_t, the future components L_(t+1) and the components of L̅_t−1 corresponding to missed cycles. However, (6) says nothing about the dependence of R_t on future components L_(t+1) and the components of L̅_t−1 corresponding to missed cycles. In fact, while assumption (10) imposes restrictions on the observed data distribution and hence is a testable assumption, this is not true for (6) by part (i) of Theorem 1. Note also that, under model 𝒜(q) with q_t = 0 for all t, nonresponse is nonignorable for inference about f (Y_t), t = 1, …, T, in the sense that likelihood-based methods do not result in the same inference if pr(R_t = 1|O̅_t−1) is known, unknown or known to follow a model. This is because the likelihood of the observed data at each cycle t does not factorize into a part that depends on pr(R_t = 1|O̅_t−1) and another that depends on f (Y_t).

3.5 Relationship between model 𝒜(q) and the selection bias permutation missingness model

Members of the class of selection bias permutation missingness models of Robins et al. (1999) are indexed by permutations of the visit subscripts 1, …, T. One such model is particularly appropriate for longitudinal studies and is defined by the assumptions that, for each t,

$$\text{pr}(R_t=1|{\mathrm{O̅}}_{t-1},Y_t,Y_{(t+1)})>0\text{ with probability }1,$$

$$f(Y_t|{\mathrm{O̅}}_{t-1},Y_{(t+1)},R_t=0)=\frac{f(Y_t|{\mathrm{O̅}}_{t-1},Y_{(t+1)},R_t=1)\text{ exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t,Y_{(t+1)})\}}{E[\text{exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t,Y_{(t+1)})\}|R_t=1,{\mathrm{O̅}}_{t-1},Y_{(t+1)}]},$$

with q_t(O̅_t−1, Y_t, Y_(t+1)) known. It differs from model 𝒜(q) only in that the future outcome history Y_(t+1) is added to each conditioning event and to the function q_t(O̅_t−1, Y_t, Y_(t+1)). Robins et al. (1999) showed that for each q_t, t = 1, …, T, this model places no restriction on the distribution of the observed data and identifies the joint distribution of Y̅_T = (Y₁, ․…, Y_T). Thus this model can be used instead of model 𝒜(q) when the substantive question at issue depends on the joint law of Y̅_T rather than simply on the marginals Y_t of Y̅_T. However, the ability to make inferences about the joint law comes at a price as it is more difficult to model f(Y_t|O̅_t−1, Y_(t+1), R_t = 0) than f(Y_t|O̅_t−1, R_t = 0) both because the conditioning event (O̅_t−1, Y_(t+1), R_t = 0) is of greater dimension than the event (O̅_t−1, R_t = 0) and because it is less natural to model the law of Y_t given the observed past O̅_t−1 and the, possibly unobserved, future Y_(t+1) than to model the law of Y_t given only the observed past O̅_t−1.

4 Estimation of the unconditional occasion-specific outcome means

4.1 Nonparametric inference under model 𝒜(q)

We now discuss inference about the marginal means ${\beta }_t^{*}\equiv E(Y_t)$ under models that assume (6) for user-specified functions q_t(O̅_t−1, Y_t), t = 1, …, T.

Models 𝒜(q) and 𝒜_t (q) define the same model, i.e. the nonparametric model, for the observed data distribution. Furthermore, as established in part (ii) of Theorem 1, under either model, ${\beta }_t^{*}$ is the same, unique, functional of the observed data law. Thus, inference about ${\beta }_t^{*}$ under either model is identical. In particular, the nonparametric maximum likelihood estimator β̂_NPML,t of ${\beta }_t^{*}$ under either model is obtained by calculating the expressions in the right-hand side of (8) or (9) under the empirical distribution of O, i.e.

$${\mathrm{\beta ̂}}_{\text{NPML},t}=E_n\left(R_t{Y}_t\right[1+\text{exp}\left\{ĥ{({\mathrm{O̅}}_{t-1})}_{\text{NPML},t}\right\}\text{exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}\left]\right)=E_n\{R_t{Y}_t+(1-R_t){\mathrm{m̂}}_{\text{NPML},t}({\mathrm{O̅}}_{t-1}\left)\right\},$$

where ĥ(O̅_t−1)_NPML,t = −∞ if E_n(R_t|O̅_t−1) = 1 and otherwise

$$ĥ{({\mathrm{O̅}}_{t-1})}_{\text{NPML},t}=\text{log }\left\{\frac{1-E_n(R_t|({\mathrm{O̅}}_{t-1})}{E_n[R_t\text{ exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}|{\mathrm{O̅}}_{t-1}]}\right\},$$

$${\mathrm{m̂}}_{\text{NPML},t}({\mathrm{O̅}}_{t-1})=\frac{E_n[Y_t\text{exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}|R_t=1,{\mathrm{O̅}}_{t-1}]}{E_n[\text{exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}|R_t=1,{\mathrm{O̅}}_{t-1}]},$$

and where for random variables W and Z, $E_n(W|Z)={\displaystyle {\sum }_{i=1}^n}{W}_{i}I(Z_i=Z)/{\displaystyle {\sum }_{i=1}^n}I(Z_i=Z)\text{ and }{E}_n(W)=n^{-1}{\displaystyle {\sum }_{i=1}^n}{W}_i$.

Unfortunately, unless T is small and L_t is discrete with few levels, with the sample sizes found in practice the data available for estimating the required conditional expectations will be sparse and consequently the estimator β̂_NPML,t will be undefined. One could assume that the required conditional expectations are smooth in O̅_t−1 and use multivariate smoothing techniques to estimate them. However, when O̅_t−1 is high-dimensional, they would not be well estimated with moderate sample sizes because no two units would have values of O̅_t−1 close enough to allow the borrowing of information needed for smoothing. Thus, in practice, because of the curse of dimensionality, we are forced to place more stringent dimension-reducing modelling restrictions on the law of the observed data.

Two dimension-reducing strategies are suggested by expressions (8) and (9) for ${\beta }_t^{*}$. The first strategy is to assume that the function h_t(O̅_t−1) follows a parametric model,

$$h_t({\mathrm{O̅}}_{t-1})=h_t({\mathrm{O̅}}_{t-1};{\alpha }_t^{*}),$$ (11)

where h_t(O̅_t−1; α_t) is a known function smooth in α_t, and ${\alpha }_t^{*}$ is an unknown p_t,h × 1 parameter vector. The second strategy is to assume that m_t (O̅_t−1) follows a parametric model,

$$m_t({\mathrm{O̅}}_{t-1})=m_t({\mathrm{O̅}}_{t-1};{\theta }_t^{*}),$$ (12)

where m_t (O̅_t−1; θ_t) is a known function, smooth in θ_t, and ${\theta }_t^{*}$ is an unknown p_t,m × 1 parameter vector.

We use ℬ_t (q) (ℬ (q)) to denote the model for the full data (L,R̅_T) defined by the assumptions of model 𝒜_t (q) and the additional restrictions (4) and (11) specified just for a fixed t (for all t). Likewise, we use 𝒞_t (q) (𝒞 (q)) to denote the model for the full data (L,R̅_T) defined by the assumptions of model 𝒜_t (q) and the additional restriction (12) specified just for a fixed t, for all t.

Models (11) and (12) are not in themselves of scientific interest. However, in practice we are forced to impose one of the two models. Estimation of ${\beta }_t^{*}$ under model ℬ_t (q) is not entirely satisfactory because the resulting estimators of ${\beta }_t^{*}$ can be biased if model (11) is incorrect, and estimation under 𝒞_t (q) suffers from the same limitation if model (12) is incorrect. Luckily there is an alternative strategy for estimation of ${\beta }_t^{*}$. This consists of computing an estimator that is consistent and asymptotically normal in the union model ℬ_t (q) ∪ 𝒞_t (q), i.e. an estimator of ${\beta }_t^{*}$ that is consistent and asymptotically normal so long as one of the models ℬ_t (q) or 𝒞_t (q), but not necessarily both, is correctly specified. Following Robins (2000) and Robins & Rotnitzky (2001) we call such an estimator a doubly robust estimator in the union model ℬ_t (q) ∪ 𝒞_t (q) as it can protect against misspecification of either (11) or (12), although not against simultaneous misspecification of both. The following definition introduces a generalization of double robustness.

Definition. Given a collection {ℳ_u; u ∈ 𝕌} of models for a law F indexed by the elements of a finite set 𝕌 with K elements, we say that an estimator λ̂ of a parameter λ ≡ λ (F) is a K-multiply robust estimator in the union model ∪_u∈𝕌ℳ_u if it is a consistent and asymptotically normal estimator of λ when one of the models ℳ_u, u ∈ 𝕌, but not necessarily more than one of them, holds.

4.2 Doubly and 2^T-multiply robust estimation

In this subsection we propose a doubly robust estimator of ${\beta }_t^{*}$ in the union model ℬ_t (q) ∪ 𝒞_t (q) and a multiply robust estimator of β* in the union model ∪_u∈𝕌ℳ_u (q). Throughout, 𝕌 denotes the collection of T ×1 vectors u whose components are either 0 or 1. For each such vector u, ℳ_u = {∩_t:ut=0ℬ_t (q)} ∩ {∩_t:ut=1𝒞_t (q)}. Thus, a 2^T-multiply-robust estimator of β* is consistent and asymptotically normal for ${\beta }^{*}=({\beta }_1^{*},\dots ,{\beta }_T^{*})$ so long as at each t one of the models ℬ_t(q) or 𝒞_t(q), but not necessarily both, is correctly specified.

To construct the doubly and 2^T-multiply robust estimators we reason as follows. Suppose that we have specified working models (11) and (12). For any constant column vector d_t and conformable column vector functions ϕ_t (O̅_t−1) and ψ_t (O̅_t−1), define

$$H_t(d_t,{\varphi }_t,{\beta }_t,{\alpha }_t)=\frac{d_t{R}_t{\epsilon }_t({\beta }_t)}{{\pi }_t({\mathrm{O̅}}_{t-1},Y_t;{\alpha }_t)}+\{1-\frac{R_t}{{\pi }_t({\mathrm{O̅}}_{t-1},Y_t;{\alpha }_t)}\}{\varphi }_t({\mathrm{O̅}}_{t-1})$$ (13)

$$M_t(d_t,{\psi }_t,{\beta }_t,{\theta }_t)=d_t{R}_t{\epsilon }_t({\beta }_t)+d_t(1-R_t)\{m_t({\mathrm{O̅}}_{t-1};{\theta }_t)-{\beta }_t\}+R_t{\psi }_t({\mathrm{O̅}}_{t-1})\text{ exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}\{Y_t-m_t({\mathrm{O̅}}_{t-1};{\theta }_t)\},$$ (14)

where ε_t(β_t) = Y_t − β_t and π_t(O̅_t−1, Y_t; α_t) ≡ [1 + exp {h_t(O̅_t−1; α_t) + q_t(O̅_t−1, Y_t)}]⁻¹. In the Appendix we show that, provided we choose ϕ_t(O̅_t−1) in (13) and ψ_t(O̅_t−1) in (14) to have the specific functional forms given by

$${\varphi }_{{\theta }_t,{\beta }_t,t}({\mathrm{O̅}}_{t-1})=m_t({\mathrm{O̅}}_{t-1};{\theta }_t)-{\beta }_t,{\psi }_{{\alpha }_t,t}({\mathrm{O̅}}_{t-1})=\text{exp }\{h_t({\mathrm{O̅}}_{t-1};{\alpha }_t)\},$$ (15)

for any fixed θ_t, α_t and β_t, H_t(1, ϕ_θt,βt,t, β_t, α_t) and M_t(1, ψ_αt,t, β_t, θ_t) are identical, where 1 is a scalar constant function equal to 1. We therefore write, for short,

$$Q_t({\beta }_t,{\theta }_t,{\alpha }_t)\equiv H_t(1,{\varphi }_{{\theta }_t,{\beta }_t,t},{\beta }_t,{\alpha }_t)=M_t(1,{\psi }_{{\alpha }_t,t},{\beta }_t,{\theta }_t).$$

In the Appendix we also show that if model (11) holds then, for any θ_t and regardless of whether or not model (12) holds, $H_t(1,{\varphi }_{{\theta }_t,{\beta }_t^{*},t},{\beta }_t^{*},{\alpha }_t^{*})$ has mean zero and therefore so does $Q_t({\beta }_t^{*},{\theta }_t,{\alpha }_t^{*})$. Furthermore, if model (12) holds then, for any α_t and regardless of whether or not model (11) holds, $M_t(1,{\psi }_{{\alpha }_t,t},{\beta }_t^{*},{\theta }_t^{*})$ has mean zero and therefore so does $Q_t({\beta }_t^{*},{\theta }_t^{*},{\alpha }_t)$. These results suggest that we can construct a doubly robust estimator β̂_t of ${\beta }_t^{*}$, throughout also denoted by β̂_t (ψ_t, ϕ_t), in model ℬ_t (q) ∪ 𝒞_t (q) by solving the scalar estimating equation

$$E_n\left\{Q_t\right({\beta }_t,{\mathrm{\theta ̂}}_t,{\mathrm{\alpha ̂}}_t\left)\right\}=0,$$ (16)

where θ̂_t and α̂_t solve

$$E_n\{M_t(0,{\psi }_t,0,{\theta }_t)\}=0,E_n\{H_t(0,{\varphi }_t,0,{\alpha }_t)\}=0,$$

using arbitrary p_t,m × 1 and p_t,h × 1 functions ψ_t(O̅_t−1) and ϕ_t(O̅_t−1) respectively. Theorem 2 below establishes that β̂_t is doubly robust in model ℬ_t (q) ∪ 𝒞_t (q) and that β̂ = (β̂₁, …, β̂_T)′, throughout also denoted by β̂ (ψ, ϕ), is 2^T-multiply robust in model ∪_u∈𝕌ℳ_u (q).

To state the asymptotic properties of β̂_t and β̂ in Theorem 2, we define

$$U_t({\beta }_t,{\theta }_t,{\alpha }_t)=Q_t({\beta }_t,{\theta }_t,{\alpha }_t)-I_{{\alpha }_t,Q_t}({\beta }_t,{\theta }_t,{\alpha }_t)I_{{\alpha }_t,H_t}^{-1}({\alpha }_t)H_t(0,{\varphi }_t,0,{\alpha }_t)-I_{{\theta }_t,Q_t}({\beta }_t,{\theta }_t,{\alpha }_t)I_{{\theta }_t,M_t}^{-1}({\theta }_t)M_t(0,{\psi }_t,0,{\theta }_t)$$

$$Q(\beta ,\theta ,\alpha )=(Q_1({\beta }_1,{\theta }_1,{\alpha }_1),\dots ,Q_T({\beta }_T,{\theta }_T,{\alpha }_T))\prime$$

$$U(\beta ,\theta ,\alpha )=(U_1({\beta }_1,{\theta }_1,{\alpha }_1),\dots ,U_T({\beta }_T,{\theta }_T,{\alpha }_T))\prime ,$$

where for any random vector function K (τ, η) of a parameter (τ, η), I_τ,K (τ, η) denotes E {∂K (τ, η)/∂τ} and ∂K (τ, η)/∂τ is a derivative matrix with (i, j) entry equal to ∂K_i (τ, η)/∂τ_j. In what follows and throughout, we use a hat to indicate expectations calculated under the empirical distribution of the observed data and evaluation at (β_t, θ_t, α_t) = (β̂_t, θ̂_t, α̂_t); for example,

$${Û}_t=Q_t({\mathrm{\beta ̂}}_t,{\mathrm{\theta ̂}}_t,{\mathrm{\alpha ̂}}_t)-{Î}_{{\alpha }_t,Q_t}{Î}_{{\alpha }_t,H_t}^{-1}{H}_t(0,{\varphi }_t,0,{\mathrm{\alpha ̂}}_t)-{Î}_{{\theta }_t,Q_t}{Î}_{{\theta }_t,M_t}^{-1}{M}_t(0,{\psi }_t,0,{\mathrm{\theta ̂}}_t)$$

and Û =(Û₁, …, Û_T)′, where Î_αt,Qt = E_n{∂Q_t (β̂_t, θ̂_t, α_t)/∂α_t|_αt=α̂t}, and so on.

Parts (i) and (iii) of Theorem 2, proved in the Appendix, state the asymptotic distribution of β̂_t and β̂ under models ℬ_t (q) ∪ 𝒞_t (q) and ∪_u∈𝕌ℳ_u (q), respectively. Parts (ii) and (iv) state that the asymptotic variance of β̂_t and β̂ under these models remains the same regardless of the choice of the functions ψ_t and ϕ_t used to compute the estimators θ̂_t and α̂_t of ${\theta }_t^{*}\text{ and }{\alpha }_t^{*}$, when in fact the true data-generating process satisfies models ℬ_t (q) ∩ 𝒞_t (q) and ${\cap }_{t=1}^T\{{ℬ}_t(q)\cap {𝒞}_t(q)\}$, respectively. In practice, the choice of functions ψ_t and ϕ_t should therefore have little impact on the efficiency of β̂_t and β̂ when the models ℬ_t (q) and 𝒞_t (q), t = 1, …, T, cannot be rejected using efficient goodness-of-fit tests. In what follows, for any matrix A, A^⊗2 denotes AA′.

Theorem 2. Suppose that the regularity conditions stated in the Appendix hold.

(i) Under model ℬ_t (q) ∪ 𝒞_t (q), $\surd n\left({\mathrm{\beta ̂}}_t-{\beta }_t^{*}\right)\to N(0,{\mathrm{\Gamma }}_t)$ in distribution, where

$${\mathrm{\Gamma }}_t=E\left[{\{I_{{\beta }_t,Q_t}^{-1}({\beta }_t^{*},{\theta }_t^0,{\alpha }_t^0)U_t({\beta }_t^{*},{\theta }_t^0,{\alpha }_t^0)\}}^{\otimes 2}\right]$$

and ${\theta }_t^0\text{ and }{\alpha }_t^0$ are the probability limits of θ̂_t and α̂_t. The matrix Γ_t can be consistently estimated with

$${\mathrm{\Gamma ̂}}_t={Î}_{{\beta }_t,Q_t}^{-1}{E}_n\left({Û}_t^{\otimes 2}\right){Î}_{{\beta }_t,Q_t}^{{-1}_{\prime }}.$$

(ii) Let $({\psi }_t^{(1)},{\varphi }_t^{(1)})\mathit{\text{ and }}({\psi }_t^{(2)},{\varphi }_t^{(2)})$ be two pairs of distinct p_t,m × 1 and p_t,h × 1 functions (ψ_t, ϕ_t) of O̅_t−1. Then, under the intersection model ℬ_t (q) ∩ 𝒞_t (q), $\surd n\left\{{\mathrm{\beta ̂}}_t\right({\psi }_t^{(1)},{\varphi }_t^{(1)})-{\mathrm{\beta ̂}}_t({\psi }_t^{(2)},{\varphi }_t^{(2)}\left)\right\}=o_p(1)$.

(iii) Under model ∪_u∈𝕌ℳ_u (q), √n (β̂ − β*) → N (0, Γ) in distribution, where

$$\mathrm{\Gamma }=E\left[{\{I_{\beta ,Q}^{-1}({\beta }^{*},{\theta }^0,{\alpha }^0)U({\beta }^{*},{\theta }^0,{\alpha }^0)\}}^{\otimes 2}\right]$$

and θ⁰ and α⁰ are the probability limits of θ̂ = (θ̂₁, …, θ̂_T)′ and α̂ = (α̂₁, …, α̂_T)′. The matrix Γ can be consistently estimated with $\mathrm{\Gamma ̂}={Î}_{\beta ,Q}^{-1}{E}_n\left({Û}^{\otimes 2}\right){Î}_{\beta ,Q}^{{-1}_{\prime }}$.

(iv) Let (ψ⁽¹⁾, ϕ⁽¹⁾) and (ψ⁽²⁾, ϕ⁽²⁾) be two distinct sets of functions {(ψ_t (O̅_t−1), ϕ_t (O̅_t−1)), t = 1, …, T}. Then, under the intersection model ${\cap }_{t=1}^T\{{ℬ}_t(q)\cap {𝒞}_t(q)\},\surd n\left\{\mathrm{\beta ̂}\right({\psi }^{(1)},{\varphi }^{(1)})-\mathrm{\beta ̂}({\psi }^{(2)},{\varphi }^{(2)}\left)\right\}=o_p(1)$.

It is possible to show that, at the intersection model ℬ_t (q) ∩ 𝒞_t (q), every doubly robust estimator β̂_t (ψ_t, ϕ_t) has asymptotic variance that attains neither the semiparametric variance bound for estimation of ${\beta }_t^{*}$ under model 𝒞_t (q) nor, except when q_t = 0, the semiparametric variance bound for estimation of β_t under model ℬ_t (q). In our opinion, the hope to control bias is more important than efficiency concerns, and we therefore recommend using doubly or 2^T-multiply robust estimators of ${\beta }_t^{*}$ and β*, respectively.

So far, we have not allowed the parameters α_t in model (11) and θ_t in model (12), respectively, to be shared across occasions. When we are faced with sample sizes that are not large enough to yield well-behaved estimators of ${\alpha }_t^{*}\text{ and }{\theta }_t^{*}$ at each occasion t, two dimension-reducing strategies can be envisaged. The first strategy is to reduce further the dimension of models (11) and (12) at each t. The second strategy is to allow parameters α_t and θ_t, respectively, to be shared across occasions and to compute an estimator of β* that is consistent and asymptotically normal so long as at least one of ${\cap }_{t=1}^T{ℬ}_t(q)\text{ or }{\cap }_{t=1}^T{𝒞}_t(q)$ holds. Denote by α and θ the resulting p_h × 1 and p_m × 1 parameter vectors indexing models (11) and (12) respectively, for all occasions t. Then such an estimator β̃ of β* can be obtained by solving estimating equations (16) at each occasion t, in which θ̂_t ≡ θ̂ and α̂_t ≡ α̂ now solve E_n {M(ψ, θ)} = 0 with $M(\psi ,\theta )={\displaystyle {\sum }_{t=1}^T}{M}_t(0,{\psi }_t,0,\theta )$ and E_n {H(ϕ, α)} = 0 with $H(\varphi ,\alpha )={\displaystyle {\sum }_{t=1}^T}{H}_t(0,{\varphi }_t,0,\alpha )$, where ψ and ϕ are vectors of p_m × 1 functions ψ_t(O̅_t−1) and p_h × 1 functions ϕ_t(O̅_t−1), t = 1, …, T, respectively. Parts (iii) and (iv) of Theorem 2 continue to hold for the resulting estimator β̃ if we replace model ∪_u∈𝕌ℳ_u (q) by $\{{\cap }_{t=1}^T{ℬ}_t(q)\}\cup \{{\cap }_{t=1}^T{𝒞}_t(q)\}$, β̂ by β̃ and U (β, θ, α) by

$$Q(\beta ,\theta ,\alpha )-I_{\alpha ,Q}(\beta ,\theta ,\alpha )I_{\alpha ,H}^{-1}(\alpha )H(\varphi ,\alpha )-I_{\theta ,Q}(\beta ,\theta ,\alpha )I_{\theta ,M}^{-1}(\theta )M(\psi ,\theta ).$$

5 Estimation of the occasion-specific conditional outcome means given baseline covariates

Suppose now that we are interested in inference about a parameter, which we denote again by β*, indexing a regression model for the conditional mean of Y_t, t = 1, …, T, given baseline covariates X; that is, for t = 1, …, T,

$$E(Y_t|X)=g_t(X;{\beta }^{*}),$$ (17)

where g_t(X; β) is a known function that is smooth in β and β* ∈ Θ ⊆ IR^r is unknown. Denote by 𝒜* (q) the model for (L,R̅_T) defined by the restrictions of model 𝒜(q) and the additional restriction (17) for all t = 1, …, T.

Part (i) of Theorem 1 is no longer true if model 𝒜(q) is replaced with model 𝒜* (q), i.e. 𝒜* (q) does not determine a nonparametric model for the observed data law. Hence, in principle, under (17) the postulated functions q_t may sometimes be subject to an empirical test. Moreover, β* and q_t, t = 1, …, T, may be jointly identified under (17). However, there would generally be very limited independent information about β* and q_t, t = 1, …, T, and therefore their joint estimation would require very large sample sizes. In fact, it follows from Proposition B1, Part 6, of Rotnitzky et al. (1998) that, when pr (R₁ = … = R_T = 0|L) > σ > 0 with probability 1 and both h_t and q_t, t = 1, …, T, in (6) are unknown, β* cannot be estimated at rate√n. Thus, we continue to recommend that one regard the functions q_t, t = 1, …, T, as fixed and known when estimating β* and then vary these functions in a sensitivity analysis.

As was the case for estimation of the marginal means E (Y_t), unless T is small and L_t is discrete with few levels, inference about β* requires placing dimension-reducing assumptions on either h_t or m_t, in addition to the restrictions of model 𝒜* (q). We therefore consider, for each t = 1, …, T, models ${ℬ}_t^{*}(q)\text{ and }{𝒞}_t^{*}(q)$ defined like models ℬ_t (q) and 𝒞_t (q) respectively but with the additional restriction (17). Furthermore, we let ${ℳ}_u^{*}(q)$ be defined like ℳ_u (q) in §4.2 but with ${ℬ}_t^{*}(q)\text{ and }{𝒞}_t^{*}(q)$ instead of ℬ_t (q) and 𝒞_t (q) so that the union model ${\cup }_{u\in 𝕌}{ℳ}_u^{*}(q)$ stands for the model in which at each t either ${ℬ}_t^{*}(q)\text{ or }{𝒞}_t^{*}(q)$ holds, but not necessarily both. In this section we consider estimation of β* under the union model ${\cup }_{u\in 𝕌}{ℳ}_u^{*}(q)$.

Although, as shown in the Appendix, the restrictions defining 𝒞_t (q) for each fixed t, and indeed simultaneously for all t, are guaranteed to be compatible, the same is not true for ${𝒞}_t^{*}(q)$. To be specific, for each t, given a function q_t the function m_t (O̅_t−1) and the conditional mean function E (Y_t|X) are not variation independent; that is, fixing one restricts the range of possible functions for the other. Thus, it may happen that there exists no joint distribution of (L,R̅_T) of which the marginal of O is the observed data distribution and that satisfies simultaneously (5), (12) and (17). Furthermore, even if such incompatibility is not present, it may still happen that the parameter space for β* under ${𝒞}_t^{*}(q)$ is much smaller than that under 𝒜* (q). This is clearly undesirable because any reasonable dimension-reducing strategy should not, a priori, eliminate values of β* regarded plausible under the model of scientific interest. The following simple example illustrates these points.

Example. Suppose that T = 1, L₀ = X and Y₁ is binary. Suppose that in (17) we assume that logit pr (Y₁ = 1|X) = β₀ + β₁X, q₁ (Y₁, X, V) = λY₁ with λ > 0 and logit pr (Y₁ = 1|R₁ = 0, X) = θ₀ + θ₁X. Under this model logit pr (Y₁ = 1|R₁ = 1, X) = λ + θ₀ + θ₁X and hence pr (Y₁ = 1|R₁ = 1, X) > pr (Y₁ = 1|R₁ = 0, X). Therefore, since

$$\text{pr}(R_1=1|X)=\frac{\text{pr}(Y_1=1|X)-\text{pr}(Y_1=1|R_1=0,X)}{\text{pr}(Y_1=1|R_1=1,X)-\text{pr}(Y_1=1|R_1=0,X)},$$

it must be that pr (Y₁ = 1|R₁ = 0, X) < pr (Y₁ = 1|X) < pr (Y₁ = 1|R₁ = 1, X). This implies that logit pr (Y₁ = 1|X) = λ* + θ₀ + θ₁X for some 0 < λ* < λ. In particular, β₁ = θ₁. It follows that the parameter space for β may be more restricted once we impose the restrictions on pr (Y₁ = 1|R₁ = 0, X). For example, if the model for pr (Y₁ = 1|R₁ = 0, X) restricts θ₁ to lie in a strict subset of the real line and the model for pr (Y₁ = 1|X) leaves β₁ unrestricted, then, once the model for pr (Y₁ = 1|R₁ = 0, X) is imposed, the parameter space for β₁ is reduced to that for θ₁. The models would even become incompatible if a probit regression were considered for pr (Y₁ = 1|X) and a logistic regression for pr (Y₁ = 1|R₁ = 0, X).

In the Appendix we show that ${ℬ}_t^{*}(q)$, and indeed ${\cap }_{t=1}^T{ℬ}_t^{*}(q)$, impose restrictions that are always compatible.

When model ${𝒞}_t^{*}(q)$ is incompatible with model (17) then an estimator of β* that is consistent under the union model ${ℬ}_t^{*}(q)\cup {𝒞}_t^{*}(q)$ actually converges in probability to β* only if the working model ${ℬ}_t^{*}(q)$ holds and hence the estimator is not really doubly robust. We do not regard this theoretical difficulty to be of concern in practice since it is ameliorated if one postulates a richly parameterized model (12). To see this note that, when no restriction is placed on m_t (O̅_t−1), model ${𝒞}_t^{*}(q)$ becomes model ${𝒜}_t^{*}(q)$ defined like 𝒜_t (q) but with the additional restriction (17). As shown in the Appendix, the restrictions defining model ${𝒜}_t^{*}(q)$ are always compatible. Consequently, if (17) is correctly specified then a flexible model for m_t (O̅_t−1) should result in a nearly correctly specified model ${𝒞}_t^{*}(q)$. In order to highlight the possibility of model incompatibility, we refer to estimators that are consistent and asymptotically normal under model ${\cup }_{u\in 𝕌}{ℳ}_u^{*}(q)$ as generalized 2^T-multiply robust estimators.

In agreement with the discussion in Robins & Rotnitzky (2001), we recommend estimating β* with generalized 2^T-multiply robust estimators because such estimators are expected to have small asymptotic bias if, at each t = 1, …, T, at leats one of the models ${ℬ}_t^{*}(q)\text{ or }{𝒞}_t^{*}(q)$ is approximately correct.

We construct generalized 2^T-multiply robust estimators of β* in model ${\cup }_{u\in 𝕌}{ℳ}_u^{*}(q)$ as follows. Redefine H_t(d_t, ϕ_t, β_t, α_t), M_t(d_t, ψ_t, β_t, θ_t) and ϕ_θt,βt,t(O̅_t−1) as in (13), (14) and (15) but with ε_t(β_t) replaced by ε_t(β) = Y_t − g_t(X; β), m_t (O̅_t−1; θ_t) − β_t by m_t (O̅_t−1; θ_t) − g_t (X; β) and with d_t= d_t(X) an arbitrary conformable vector function of X. With these redefinitions, it is true that H_t(d_t, d_tϕ_θt,β,t, β, α_t) = M_t(d_t, d_tψ_αt,t, β, θ_t) for any arbitrary r × 1 vector function d_t(X). We therefore write

$$Q_t(d_t,\beta ,{\theta }_t,{\alpha }_t)\equiv H_t(d_t,d_t{\varphi }_{{\theta }_t,\beta ,t},\beta ,{\alpha }_t)=M_t(d_t,d_t{\psi }_{{\alpha }_t,t,}\beta ,{\theta }_t).$$

Similarly to §4.2, we construct generalized 2^T-multiply robust estimators β̂ of β*, also denoted throughout by β̂ (d, d_θ, ψ, d_α, ϕ), in model ${\cup }_{u\in 𝕌}{ℳ}_u^{*}(q)$ by solving an r × 1 estimating equation of the form

$$E_n\left\{Q\right(d,\beta ,\mathrm{\theta ̃},\mathrm{\alpha ̃}\left)\right\}=0,$$ (18)

where $Q(d,\beta ,\theta ,\alpha )={\displaystyle {\sum }_{t=1}^T}{Q}_t(d_t,\beta ,{\theta }_t,{\alpha }_t)$, d (X) = (d₁ (X), …, d_T (X)) for arbitrary r × 1 vector functions d_t (X), t = 1, …, T, and where θ̃ = (θ̃₁, …, θ̃_T)′ and α̃ = (α̃₁, …, α̃_T)′ solve

$$E_n\{M_t(d_{t\theta },{\psi }_t,\beta ,{\theta }_t)\}=0,E_n\{H_t(d_{t\alpha },{\varphi }_t,\beta ,{\alpha }_t)\}=0$$

respectively, for t = 1, …, T, using arbitrary collections of functions,

$$d_{\theta }=\{d_{t\theta }(X):d_{t\theta }(·)\text{ is }\mathrm{a}{p}_{t,m}\times 1\text{ vector value map},t=1,\dots ,T\},$$

$$\psi =\{{\psi }_t({\mathrm{O̅}}_{t-1}):{\psi }_t(·)\text{ is }\mathrm{a}{p}_{t,m}\times 1\text{ vector value map},t=1,\dots ,T\},$$

$$d_{\alpha }=\{d_{t\alpha }(X):d_{t\alpha }(·)\text{ is }\mathrm{a}{p}_{t,h}\times 1\text{ vector value map},t=1,\dots ,T\},$$

$$\varphi =\{{\varphi }_t({\mathrm{O̅}}_{t-1}):{\varphi }_t(·)\text{ is }\mathrm{a}{p}_{t,h}\times 1\text{ vector value map},t=1,\dots ,T\}.$$

Parts (iii) and (iv) of Theorem 2 remain valid if we replace ∪_u∈𝕌ℳ_u (q) by ${\cup }_{u\in 𝕌}{ℳ}_u^{*}(q)$, β̂ (ψ^(j), ϕ^(j)) by $\mathrm{\beta ̂}(d,d_{\theta }^{(j)},{\psi }^{(j)},d_{\alpha }^{(j)},{\varphi }^{(j)})$, j = 1, 2, Q_t (β, θ_t, α_t) by Q_t (d_t, β, θ_t, α_t) and we use the redefinition $U(\beta ,\theta ,\alpha )={\displaystyle {\sum }_{t=1}^T}{U}_t(\beta ,{\theta }_t,{\alpha }_t)$, where

$$U_t(\beta ,{\theta }_t,{\alpha }_t)=Q_t(d_t,\beta ,{\theta }_t,{\alpha }_t)-I_{{\alpha }_t,Q_t}(\beta ,{\theta }_t,{\alpha }_t)I_{{\alpha }_t,H_t}^{-1}({\alpha }_t)H_t(d_{t\alpha },{\varphi }_t,\beta ,{\alpha }_t)-I_{{\theta }_t,Q_t}(\beta ,{\theta }_t,{\alpha }_t)I_{{\theta }_t,M_t}^{-1}({\theta }_t)M_t(d_{t\theta },{\psi }_t,\beta ,{\theta }_t),$$

with d_t = d_t(X), d_tα = d_tα(X) and d_tθ = d_tθ(X).

Theorem 3 below, proved in the Appendix, provides the optimal r × T matrix function d_opt (X) = (d_1,opt (X), …, d_T,opt (X)) in the sense that, among all estimators β̂ (d, d_θ, ψ, d_α, ϕ) that solve (18) using an arbitrary r × T matrix functions d (X) and fixed collections of functions d_θ, ψ, d_α and ϕ, the estimator with the smallest asymptotic variance is the one that uses d = d_opt. In particular, since under laws in the intersection model ${\cap }_{t=1}^T\{{ℬ}_t^{*}(q)\cap {𝒞}_t^{*}(q)\}$ the limiting distribution of β̂ (d, d_θ, ψ, d_α, ϕ) does not depend on the choice of d_θ, ψ, d_α and ϕ, we conclude that the estimator that solves (18) using d = d_opt has the smallest asymptotic variance among all estimators β̂ (d, d_θ, ψ, d_α, ϕ) under any law in ${\cap }_{t=1}^T\{{ℬ}_t^{*}(q)\cap {𝒞}_t^{*}(q)\}$. In the following theorem, Γ (d, d_θ, ψ, d_α, ϕ) denotes the variance of the limiting normal distribution of √n {β̂ (d, d_θ, ψ, d_α, ϕ) − β*} under model ${\cup }_{u\in 𝕌}{ℳ}_u^{*}(q)\text{ and }{U}^{*}(\beta ,\theta ,\alpha )\equiv (U_1^{*}(\beta ,{\theta }_1,{\alpha }_1),\dots ,U_T^{*}(\beta ,{\theta }_T,{\alpha }_T))\prime$, where each $U_t^{*}(\beta ,{\theta }_t,{\alpha }_t)$ is defined as U_t (β, θ_t, α_t) but with the r × 1 function d_t(X) replaced by the constant real valued function d_t (X)≡1. Also, θ⁰ and α⁰ denote the probability limits of θ̃ and α̃. In addition, for any pair of conformable square matrices A and B, A ≥ B indicates that A − B is positive semidefinite.

Theorem 3. For every fixed collection of functions d_θ, ψ, d_α and ϕ, we have that Γ (d_opt, dθ, ψ, d_α, ϕ) ≤ Γ (d, d_θ, ψ, d_α, ϕ), where

$$d_{\mathit{\text{opt}}}(X)=E\left\{{\frac{\partial U^{*}(\beta ,{\theta }^0,{\alpha }^0)}{\partial \beta }|}_{\beta ={\beta }^{*}}|X\right\}{[\text{var}\{U^{*}({\beta }^{*},{\theta }^0,{\alpha }^0)|X\}]}^{-1}.$$

6 Simulation study

We conducted two simulation experiments. The first compares the behaviour in finite samples of the 2^T-multiply robust estimators of marginal means with competitors that are not 2^T-multiply robust, and the second evaluates the behaviour of generalized 2^T-multiply robust estimators of parameters of regression models for the marginal means. Each experiment was based on 1000 replications of random samples of size 500 generated as follows. In both experiments, for t > 1, L_t comprised just the outcome Y_t. In the first experiment L₀ was standard normal and, for t = 1, …, 4, given (L̅_t−1, R̅_t−1), R_t was generated from

$$\text{pr}(R_t=0|{\mathrm{R̅}}_{t-1},{\mathrm{L̅}}_{t-1})=\text{expit}[-2.25+0.25t+0.5\{1+I(t=1)\}{L}_0+2(1-R_{t-1})+2R_{t-1}{\epsilon }_{t-1}],$$

and then L_t was generated as L_t = 2t + 3L₀ + R_t−1 + 2R_t−1ε_t−1 − γR_t + ε_t for the choices γ = 0 and γ = −0.5, where ε₁, …, ε₄ are four independent standard normal variates. It is easy to check that the law of our simulated data satisfies the restrictions of ℬ (q) and 𝒞 (q) with q_t(O̅_t−1, Y_t) = γY_t, h_t(O̅_t−1; α_t) = α_t0 + α_t1L₀ + I(t > 1)(α_t2R_t−1 + α_t3R_t−1Y_t−1) and m_t(O̅_t−1; θ_t) = θ_t0 + θ_t1L₀ + I(t > 1)(θ_t2R_t−1 + θ_t3R_t−1Y_t−1) for specific vector values α_t and θ_t.

In the second experiment, given a standard normal variate X, we generated a 4 × 1 multivariate normal vector (Y₁, …, Y₄) with E (Y_t|X) = β₁t + β₂X, β₁ = 2, β₂ = 3, var(Y_t|X) = 5 and cov(Y_t, Y_t′|X) = 4, t ≠ t′. Then we generated R̅₄ given Y̅₄ iteratively for t = 1, …, 4 from

$$\text{pr }(R_t=0|{\mathrm{R̅}}_{t-1},{\mathrm{L̅}}_4)=\text{expit}[-2.25+0.25t+0.5\{1+I(t=1)\}X+\{2(1-R_{t-1})+2R_{t-1}{\epsilon }_{t-1}\}I(t>1)+\gamma {\epsilon }_t],$$

where ε_t = Y_t − β₁t − β₂X and γ = 0 or γ = −0.5. Under our data-generating process, model ℬ (q) holds for q_t(O̅_t−1, Y_t) = γY_t and h_t(O̅_t−1; α_t) = α_t0 + α_t1X + I(t > 1)(α_t2R_t−1 + α_t3R_t−1Y_t−1) for a specific value of α_t. Since the functional form of m_t(O̅_t−1) is complicated we have considered an approximate working model for it, given by m_t(O̅_t−1; θ) = θ_t0 + θ_t1X + I(t > 1) {θ_t2R_t−1 + θ_t3R_t−1X + θ_t4R_t−1Y_t−1}, and thus computed a generalized 2^T-multiply robust estimator of (β₁, β₂) such that, at each t, model 𝒞_t (q) assumes that m_t (O̅_t−1) = m_t(O̅_t−1; θ) for some θ.

Under the data-generating process of the first experiment, the probability of not missing the outcome is 84.4, 69.0, 60.5 and 53.7 at the four occasions, respectively, for both γ = 0 and γ = −0.5. In the second experiment, these probabilities are 84.4, 60.1, 61.5 and 56.5 for γ = 0. The values when γ = −0.5 are similar.

Table 1 summarizes the results for estimation of β₃ = E (Y₃) and β₄ = E (Y₄) in the first experiment for the following methods: inverse probability weighted estimators, i.e. those solving E_n {H_t(1, 0, β_t, α̂_t)} = 0, labelled ‘IPW’; conditional mean imputation estimators solving E_n {M_t(1, 0, β_t, θ̂_t)} = 0, labelled ‘CM’; and 2^T-multiply robust estimators, labelled ‘MR’. The estimators were computed under various conditions: correctly specified working models h_t(O̅_t−1; α_t) and m_t(O̅_t−1; θ_t) as defined above for all t, labelled ‘None’; models m₃(O̅₂; θ₃) and h₄(O̅₃; α₄) that incorrectly set a priori to zero the coefficients multiplying the term R_t−1Y_t−1, labelled ‘𝒞₃&ℬ₄’; and models h_t(O̅_t−1; α_t) and m_t(O̅_t−1; θ_t) that for all t incorrectly set a priori to zero the coefficients multiplying the term R_t−1Y_t−1, labelled ‘All’.

Table 1.

Results from the first simulation study. Bias is 100 times the Monte Carlo mean minus true parameter value, SD is 100 times the Monte Carlo standard deviation, Cov is the Monte Carlo coverage probability times 100.

			IPW			CM			MR
qt	Estimand	Misspec.	Bias	SD	Cov	Bias	SD	Cov	Bias	SD	Cov
0	β3 = 6.69	None	−1.6	27.7	92.6	0.1	16.4	95.1	−0.1	16.8	94.3
		C3 & B4	−1.6	27.7	92.6	−48.8	16.2	14.4	−0.6	30.7	93.8
		All	−52.2	17.7	15.8	−48.8	16.4	14.4	−48.9	16.2	15.4
	β4 = 8.61	None	−1.3	28.5	91.5	0.2	16.4	95.4	0.3	16.8	95.5
		C3 & B4	−52.1	18.4	18.7	0.2	16.4	95.4	0.3	16.4	95.4
		All	−52.1	18.4	18.7	−47.6	16.4	18.3	−46.7	16.2	19.4
−0.5Yt	β3 = 6.99	None	−0.4	28.5	93.0	−2.3	22.0	89.5	1.0	17.9	95.0
		C3 & B4	−0..4	28.5	93.0	−86.2	29.5	11.6	0.7	44.1	92.9
		All	−82.1	19.4	1.0	−86.2	29.5	11.6	−77.3	17.9	0.9
	β4 = 8.87	None	−0.7	27.6	92.2	−4.0	23.9	94.1	1.2	17.3	94.9
		C3 & B4	−81.3	20.2	2.2	−4.0	23.9	94.1	−1.0	18.8	94.1
		All	−81.3	20.2	2.2	−86.9	34.7	15.2	−74.6	18.3	1.4

IPW, inverse probability weighted estimator; CM, conditional mean imputation estimator; MR, 2^T-multiply robust estimator; None, no model misspecification; C₃ & B₄, misspecification of models C₃ and B₄; All, misspecification of C_t and B_t for each t.

The results for the 2^T-multiply robust estimators in the first simulation study are as predicted by the theory: they are nearly unbiased and the Wald confidence intervals centred at them cover roughly at the nominal 95% level when, at each occasion, none or one of the working models, but not both, is incorrect. In contrast, and also as expected, the inverse probability weighted estimators of β₃ are nearly unbiased but those of β₄ are not unbiased when the model for h₃(O̅₂) is correctly specified but that of h₄(O̅₃) is incorrectly specified. The reverse occurs for the conditional mean imputation estimators. No estimator is unbiased when all working models are misspecified. In addition, as predicted by theory, when q_t = 0, the 2^T-multiply robust estimator is more efficient than the inverse probability weighted estimator when all working models are correct, but is less efficient than the conditional mean imputation estimator. Interestingly, the 2^T-multiply robust estimators of β₃ and β₄ are also more efficient than the corresponding inverse probability weighted estimators when q_t = −0.5Y_t, and both working models are correct even though this cannot be deduced from the theory. Note also that the 2^T-multiply robust estimator of β₃ is less efficient than the inverse probability weighted estimator when ℬ₃ is correct but 𝒞₃ is incorrect.

Table 2 summarizes the results from the second simulation for the generalized 2^T-multiply robust estimators of β₁ and β₂ solving the equations (18) that use, instead of d(X), the vector function of X and β,

$$d^{*}(X;\beta )=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill X\hfill & \hfill X\hfill & \hfill X\hfill & \hfill X\hfill \end{array}\right)E_n\left\{Q^{*}{(\beta ,\mathrm{\theta ̃},\mathrm{\alpha ̃})}^{\otimes 2}\right\},$$

where Q* (β, θ, α) = (Q₁ (1, β, θ₁, α₁), …, Q_T (1, β, θ_T, α_T))′. It can be shown that using d* (X; β) in (18) results in generalized 2^T-multiply robust estimators that, under the union model ${\cup }_{u\in 𝕌}{ℳ}_u^{*}(q)$, are asymptotically equivalent to those that solve (18) with d (X) = d* (X; β*). The estimators were computed under various conditions: the correct working model h_t(O̅_t−1; α_t) and the approximately correct model m_t(O̅_t−1; θ), labelled ‘None’; the same models as in the previous case except that the coefficients corresponding to R_t−1Y_t−1 of h₂(O̅₁; α₂), h₄(O̅₃; α₄) and m₃(O̅₂; θ₃) were set to 0, labelled ‘ℬ₂, 𝒞₃ and ℬ₄’; and the same models as in the first case but with the coefficients of R_t−1Y_t−1 equal to 0 for all t and for all models, labelled ‘All’. The results confirm that the generalized 2^T-multiply robust estimators perform well even if the model for m_t(O̅_t−1) is incorrectly specified provided, as in our simulations, the model is richly parameterized.

Table 2.

Results from the second simulation study. Bias is 100 times the Monte Carlo mean minus true parameter value, SD is 100 times the Monte Carlo standard deviation, Cov is the Monte Carlo coverage probability times 100.

qt	Estimand	Misspec.	Bias	SD	Cov
0	β1 = 3	None	0.08	10.60	94.6
		B2,C3 & B4	0.12	10.30	94.0
		All	1.74	9.58	95.7
	β2 = 2	None	0.007	4.25	95.3
		B2,C3 & B4	−0.66	3.68	95.9
		All	−10.7	3.87	17.2
−0.5Yt	β1 = 3	None	1.70	12.00	95.7
		B2,C3 & B4	1.36	12.20	93.1
		All	3.37	11.30	91.2
	β2 = 2	None	1.31	6.43	96.7
		B2,C3 & B4	−4.21	7.69	88.1
		All	−24.8	9.10	5.71

None, no model misspecification; B₂,C₃ & B₄, misspecification of models B₂,C₃ and B₄; All, misspecification of C_t and B_t for each t.

Appendix

Proofs

Proof of Theorem 1. Every observed data distribution is defined by a given collection of conditional densities and probabilities, {f(L_t|O̅_t−1, R_t = 1), pr(R_t|O̅_t−1) : t = 1, …, T}. Thus, to show that 𝒜(q) is a model for the observed data law restricted only by pr(R_t = 1|O̅_t−1) > 0 for t = 1, …, T, it suffices to show the following: (a) given {f(L_t|O̅_t−1, R_t = 1), pr(R_t|O̅_t−1) : pr(R_t = 1|O̅_t−1) > 0, t = 1, …, T}, there exists a distribution f* (L̅_T,R̅_T) satisfying the restrictions of 𝒜(q) and such that, for t = 1, …, T,

$$f(L_t|{\mathrm{O̅}}_{t-1},R_t=1)=f^{*}(L_t|{\mathrm{O̅}}_{t-1},R_t=1),\text{ pr}(R_t=1|{\mathrm{O̅}}_{t-1})={\text{pr}}^{*}(R_t=1|{\mathrm{O̅}}_{t-1});$$ (A1)

and (b) pr*(R_t = 1|O̅_t−1) > 0 for every joint distribution f* (L̅_T,R̅_T) that satisfies the restrictions of model 𝒜(q).

We prove (a) by constructing a joint distribution f* (L̅_T,R̅_T) that satisfies (A1) iteratively as follows. For t = 1, R_t−1 is nil and f*(L̅_t−1) = f (L₀). For each t = 1, …, T, we define f*(L_t,R_t|L̅_t−1, R̅_t−1) equal to f*(L_t,R_t|O̅_t−1), where f*(L_t,R_t|O̅_t−1) is defined by f*(L_t|O̅_t−1,R_t = 1) = f(L_t|O̅_t−1,R_t = 1) and pr*(R_t = 1|O̅_t−1) = pr(R_t = 1|O̅_t−1), and when (4) holds f*(L_t\Y_t|Y_t,O̅_t−1,R_t = 0) is equal to an arbitrary law and f*(Y_t|O̅_t−1,R_t = 0) is equal to the right-hand side of (5). By construction, f*(L̅_T,R̅_T) satisfies (1) and Assumption 1 for t = 1, …, T, and thus is in model 𝒜(q), and additionally satisfies (A1), thus proving (a). Part (b) holds because it is implied by (1). This concludes the proof of part (i). To show part (ii), note that, under model 𝒜_t(q),

$$f(Y_t|{\mathrm{O̅}}_{t-1})=f(Y_t|{\mathrm{O̅}}_{t-1},R_t=1)(\text{pr}(R_t=1|{\mathrm{O̅}}_{t-1})+\frac{\text{pr}(R_t=0|{\mathrm{O̅}}_{t-1})\text{ exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}}{E[\text{exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}|{\mathrm{O̅}}_{t-1},R_t=1]}),$$

where q_t(O̅_t−1, Y_t) is defined arbitrarily if pr(R_t = 0|O̅_t−1) = 0. Thus, f (Y_t|O̅_t−1) is identified under 𝒜_t(q) because the right-hand side of the last display is determined by the observed data law. To show that (8) holds, we use expression (7) and note that, under models 𝒜_t(q) and 𝒜(q), the right-hand side of (8) equals

$$E\left\{R_t{Y}_t\right(1+\frac{\text{pr}(R_t=0|{\mathrm{O̅}}_{t-1})\text{ exp}\{q_t(Y_t,{\mathrm{O̅}}_{t-1})\}}{\text{pr}(R_t=1|{\mathrm{O̅}}_{t-1})E[\text{exp}\{q_t(Y_t,{\mathrm{O̅}}_{t-1})\}|R_t=1,{\mathrm{O̅}}_{t-1})]}\left)\right\}=E(\text{pr}(R_t=1|{\mathrm{O̅}}_{t-1})E(Y_t|R_t=1,{\mathrm{O̅}}_{t-1})+\text{pr}(R_t=0|{\mathrm{O̅}}_{t-1})\frac{E[Y_t\text{ exp}\{q_t(Y_t,{\mathrm{O̅}}_{t-1})\}|R_t=1,{\mathrm{O̅}}_{t-1}]}{E[\text{exp}\{q_t(Y_t,{\mathrm{O̅}}_{t-1})\}|R_t=1,{\mathrm{O̅}}_{t-1})]}).$$

From

$$\frac{E[Y_t\text{ exp}\{q_t(Y_t,{\mathrm{O̅}}_{t-1})\}|R_t=1,{\mathrm{O̅}}_{t-1}]}{E[\text{exp}\{q_t(Y_t,{\mathrm{O̅}}_{t-1})\}|R_t=1,{\mathrm{O̅}}_{t-1})]}=E(Y_t|R_t=0,{\mathrm{O̅}}_{t-1})$$

under model 𝒜_t(q), it follows that the right-hand side is equal to E(Y_t). The proof of (9) is now immediate.

Proof that the restrictions imposed by models 𝒜*(q), ℬ*(q), ℬ(q), 𝒞(q) and ${\cap }_{t=1}^T\{{ℬ}_t(q)\cap {𝒞}_t(q)\}$ are compatible. To show that the restrictions of ℬ*(q) are compatible we will exhibit a joint law f* (L̅_T,R̅_T) satisfying the restrictions defining ℬ*(q). We construct such a law recursively as follows. We define f*(L₀) as an arbitrary law and set R₀ as nil. Then, having defined f*(L̅_t−1,R̅_t−1) for t = 1, …, T, we define f*(L_t,R_t|L̅_t−1,R̅_t−1) as follows. The density f* (Y_t|L̅_t−1,R̅_t−1) satisfies ${\mathrm{L̅}}_{t-1}^{-}={\mathrm{L̅}}_{t-1}\backslash X$ and the integral is taken with respect to the counting dominating measure for R̅_t−1 and the adequate dominating measures for ${\mathrm{L̅}}_{t-1}^{-}$ and Y_t. This ensures that (17) holds. Next, we define f*(L_t \ Y_t|Y_t,R_t,O̅_t−1) as an arbitrary law. Finally, for a given fixed function $h_t({\mathrm{O̅}}_{t-1};{\alpha }_t^{*})$, we set ${\text{pr}}^{*}(R_t=1|Y_t,{\mathrm{L̅}}_{t-1},{\mathrm{R̅}}_{t-1})={\text{pr}}^{*}(R_t=1|Y_t,{\mathrm{O̅}}_{t-1})={[1+\text{exp}\{h_t({\mathrm{O̅}}_{t-1};{\alpha }_t^{*})+q_t({\mathrm{O̅}}_{t-1},Y_t)\}]}^{-1}$, which ensures that (5) and (11) hold. Thus, by construction, f* (L̅_T,R̅_T) satisfies the restrictions defining ℬ*(q). Since ℬ*(q) is more restrictive than ℬ(q) and 𝒜*(q), this implies that the restrictions of ℬ(q) and 𝒜*(q) are compatible.

We next recursively construct a law f* (L̅_T,R̅_T) that satisfies the restrictions imposed by the intersection model ${\cap }_{t=1}^T\{{ℬ}_t(q)\cap {𝒞}_t(q)\}$. We define f*(L₀) as an arbitrary law and set R₀ as nil. Then, having defined f*(L̅_t−1,R̅_t−1) for t = 1, …, T, we define f*(L_t,R_t|L̅_t−1,R̅_t−1) as follows. Given fixed functions $h_t({\mathrm{O̅}}_{t-1};{\alpha }_t^{*})\text{ and }{m}_t({\mathrm{O̅}}_{t-1};{\theta }_t^{*})$, for each t = 1, …, T, we define f* (Y_t|L̅_t−1,R̅_t−1,R_t = 0) = f*(Y_t|R_t = 0,O̅_t−1), where f*(Y_t|R_t = 0,O̅_t−1) is any law that satisfies $\int Y_t{f}^{*}(Y_t|{\mathrm{O̅}}_{t-1},R_t=0){\mathit{\text{dY}}}_t=m_t({\mathrm{O̅}}_{t-1};{\theta }_t^{*})$. Next, we define pr*(R_t = 1|L̅_t−1,R̅_t−1) = pr*(R_t = 1|O̅_t−1) by the identity

$${\displaystyle \int }\text{exp}\{-h_t({\mathrm{O̅}}_{t-1};{\alpha }_t^{*})-q_t({\mathrm{O̅}}_{t-1},Y_t)\}{f}^{*}(Y_t|{\mathrm{O̅}}_{t-1},R_t=0){\mathit{\text{dY}}}_t=\frac{{\text{pr}}^{*}(R_t=1|{\mathrm{O̅}}_{t-1})}{{\text{pr}}^{*}(R_t=0|{\mathrm{O̅}}_{t-1})},$$

and we define f* (Y_t|L̅_t−1,R̅_t−1,R_t = 1) = f*(Y_t|O̅_t−1,R_t = 1), where

$$f^{*}(Y_t|{\mathrm{O̅}}_{t-1},R_t=1)=\frac{{\text{pr}}^{*}(R_t=0|{\mathrm{O̅}}_{t-1})}{{\text{pr}}^{*}(R_t=1|{\mathrm{O̅}}_{t-1})}\frac{f^{*}(Y_t|{\mathrm{O̅}}_{t-1},R_t=0)}{\text{exp}\{h_t|{\mathrm{O̅}}_{t-1};{\alpha }_t^{*}+q_t({\mathrm{O̅}}_{t-1},Y_t)\}}.$$

Finally, we choose f*(L_t \ Y_t|Y_t,L̅_t−1,R̅_t−1) to be an arbitrary law. By construction, f* (L̅_T,R̅_T) satisfies for each t, (12), and hence model 𝒞_t(q), as well as (5) and (11), and hence model ℬ_t(q). This is seen because, for the chosen law,

$$\text{exp}\{h_t({\mathrm{O̅}}_{t-1};{\alpha }_t^{*}+q_t({\mathrm{O̅}}_{t-1},Y_t)\}=\frac{f^{*}(Y_t,R_t=0|{\mathrm{O̅}}_{t-1})}{f^{*}(Y_t,R_t=1|{\mathrm{O̅}}_{t-1})}=\frac{{\text{pr}}^{*}(R_t=0|Y_t,{\mathrm{O̅}}_{t-1})}{{\text{pr}}^{*}(R_t=1|Y_t,{\mathrm{O̅}}_{t-1})}.$$

We conclude that f* (L̅_T,R̅_T) satisfies the restrictions of model ${\cap }_{t=1}^T\{{ℬ}_t(q)\cap {𝒞}_t(q)\}$ and therefore also those of 𝒞(q).

Proof that H_t(1, ϕ_θt,βt,t, β_t, α_t)=M_t(1, ψ_αt,t, β_t, θ_t). We have

$$\frac{R_t{\epsilon }_t({\beta }_t)}{{\pi }_t({\mathrm{O̅}}_{t-1},Y_t;{\alpha }_t)}+(1-\frac{R_t}{{\pi }_t({\mathrm{O̅}}_{t-1},Y_t;{\alpha }_t)})\{m_t({\mathrm{O̅}}_{t-1};{\theta }_t)-{\beta }_t\}=(1-\frac{R_t}{{\pi }_t({\mathrm{O̅}}_{t-1},Y_t;{\alpha }_t)})\{m_t({\mathrm{O̅}}_{t-1};{\theta }_t)-Y_t\}+Y_t-{\beta }_t=(1-R_t-\frac{R_t\{1-{\pi }_t({\mathrm{O̅}}_{t-1},Y_t;{\alpha }_t)\}}{{\pi }_t({\mathrm{O̅}}_{t-1},Y_t;{\alpha }_t)})\{m_t({\mathrm{O̅}}_{t-1};{\theta }_t)-Y_t\}+R_t{\epsilon }_t({\beta }_t)+(1-R_t)(Y_t-{\beta }_t)=R_t{\epsilon }_t({\beta }_t)+(1-R_t)\{m_t({\mathrm{O̅}}_{t-1};{\theta }_t)-{\beta }_t\}+R_t{\psi }_{{\alpha }_t,t}({\mathrm{O̅}}_{t-1})\text{ exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}\{Y_t-m_t({\mathrm{O̅}}_{t-1};{\theta }_t)\}.$$

Now, suppose that (11) and (4) hold. Then

$$E\left\{\frac{R_t{\epsilon }_t({\beta }_t^{*})}{{\pi }_t({\mathrm{O̅}}_{t-1},Y_t;{\alpha }_t^{*})}\right\}=E\{{\epsilon }_t({\beta }_t^{*})\}=0$$

and

$$E\left[\right\{1-\frac{R_t}{{\pi }_t({\mathrm{O̅}}_{t-1},Y_t;{\alpha }_t^{*})}\}\{m_t({\mathrm{O̅}}_{t-1};{\theta }_t)-{\beta }_t^{*}\}]=E\left[E\right[\{1-\frac{R_t}{{\pi }_t({\mathrm{O̅}}_{t-1},Y_t;{\alpha }_t^{*})}\}|{\mathrm{O̅}}_{t-1},Y_t]\{m_t({\mathrm{O̅}}_{t-1};{\theta }_t^{*})-{\beta }_t^{*}\}]=0$$

because

$$E\left\{\right(1-\frac{R_t}{{\pi }_t({\mathrm{O̅}}_{t-1},Y_t;{\alpha }_t^{*})})|{\mathrm{O̅}}_{t-1},Y_t\}=0.$$

Thus, under (11) and (4), $E\{H_t(1,{\varphi }_{{\theta }_t,{\beta }_t^{*},t,{\beta }_t^{*},{\alpha }_t^{*}})\}=0$ for any θ_t. Next, suppose (12) holds. Then $(1-R_t)m_t({\mathrm{O̅}}_{t-1};{\theta }_t^{*})=(1-R_t)E(Y_t|{\mathrm{O̅}}_{t-1},R_t=0)$, where E (Y_t|O̅_t−1,R_t = 0) is defined arbitrarily if (4) does not hold, and hence $E[R_t{\epsilon }_t({\beta }_t^{*})+(1-R_t)\{m_t({\mathrm{O̅}}_{t-1};{\theta }_t)-{\beta }_t^{*}\left\}\right]=E\left\{{\epsilon }_t\right({\beta }_t^{*}\left)\right\}=0$. Also,

$$E[R_t{\psi }_{{\alpha }_t,t}({\mathrm{O̅}}_{t-1})\text{exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}\{Y_t-m_t({\mathrm{O̅}}_{t-1};{\theta }_t^{*})\}]=E[{\psi }_{{\alpha }_t,t}({\mathrm{O̅}}_{t-1})E[\text{exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}\{Y_t-m_t({\mathrm{O̅}}_{t-1};{\theta }_t^{*})\}|{\mathrm{O̅}}_{t-1},R_t=1]\text{ pr }(R_t=1|{\mathrm{O̅}}_{t-1})]=0$$

because $E[\text{exp}\{q_t({\mathrm{O̅}}_{t-1},Y_t)\}\{Y_t-m_t({\mathrm{O̅}}_{t-1};{\theta }_t^{*})\}|{\mathrm{O̅}}_{t-1},R_t=1]=0$. Thus, under model (12), $E\{M_t(1,{\psi }_{{\alpha }_t,t},{\beta }_t^{*},{\theta }_t^{*})\}=0$ for any α_t.

Proof of Theorem 2. To prove part (i) of the theorem, we assume that the regularity conditions 1–9 of Appendix B of Robins et al. (1994) hold with U_t(β_t, θ_t, α_t) and $({\beta }_t^{*},{\theta }_t^{*},{\alpha }_t^{*})$ replacing their H_i(γ) and γ₀ respectively, and their regularity condition 3 being replaced by the assumption that pr (R_t = 1|O̅_t−1, Y_t; α_t) > σ > 0 with probabaility 1 for some σ and arbitrary α_t in the parameter space. By standard Taylor expansion arguments we have that $\surd n({\mathrm{\theta ̂}}_t-{\theta }_t^0)=-I_{{\theta }_t,M_t}^{-1}({\theta }_t^0)n^{-1/2}{\displaystyle {\sum }_i}{M}_{\mathit{\text{it}}}(0,{\psi }_t,0,{\theta }_t^0)+o_p(1)\text{ and }\surd n({\mathrm{\alpha ̂}}_t-{\alpha }_t^0)=-I_{{\alpha }_t,H_t}^{-1}({\alpha }_t^0)n^{-1/2}{\displaystyle {\sum }_i}{H}_{\mathit{\text{it}}}(0,{\varphi }_t,0,{\alpha }_t^0)+o_p(1)$, where o_p(1) denotes a random variable converging to 0 in probability. Furthermore, because $E\{Q_t({\beta }_t^{*},{\theta }_t^0,{\alpha }_t^0)\}=0$ under model ℬ_t(q) ∪ 𝒞_t(q), another Taylor expansion gives

$$0=n^{-1/2}{\displaystyle \sum _i}{Q}_{\mathit{\text{it}}}({\beta }_t^{*},{\theta }_t^0,{\alpha }_t^0)+I_{{\beta }_t,Q_t}({\beta }_t^{*},{\theta }_t^0,{\alpha }_t^0)\surd n({\mathrm{\beta ̂}}_t-{\beta }_t^{*})-I_{{\alpha }_t,Q_t}({\beta }_t^{*},{\theta }_t^0,{\alpha }_t^0)I_{{\alpha }_t,H_t}^{-1}({\alpha }_t^0)\times n^{-1/2}{\displaystyle \sum _i}{H}_{\mathit{\text{it}}}(0,{\varphi }_t,0,{\alpha }_t^0)-I_{{\theta }_t,Q_t}({\beta }_t^{*},{\theta }_t^0,{\alpha }_t^0)I_{{\theta }_t,M_t}^{-1}({\theta }_t^0)n^{-1/2}{\displaystyle \sum _i}{M}_{\mathit{\text{it}}}(0,{\psi }_t,0,{\theta }_t^0)+o_p(1).$$

When, as in regularity condition 6 of Robins et al. (1994), $I_{{\beta }_t,Q_t}({\beta }_t^{*},{\theta }_t^{*},{\alpha }_t^{*})$ is nonsingular, this is equivalent to

$$\surd n({\mathrm{\beta ̂}}_t-{\beta }_t^{*})=I_{{\beta }_t,Q_t}^{-1}({\beta }_t^{*},{\theta }_t^0,{\alpha }_t^0)n^{-1/2}{\displaystyle \sum _i}{U}_{\mathit{\text{it}}}({\beta }_t^{*},{\theta }_t^0,{\alpha }_t^0)+o_p(1).$$

The asymptotic distribution of $\surd n({\mathrm{\beta ̂}}_t-{\beta }_t^{*})$ under model ℬ_t(q) ∪ 𝒞_t(q) follows from the previous equation by Slutsky’s Theorem and the Central Limit Theorem. The consistency of the variance estimator follows from the Law of Large Numbers. This proves part (i). Since α_t and θ_t, for t = 1, …, T, are variation independent parameters, it also follows that $(I_{{\beta }_1,Q_1}^{-1}({\beta }_1^{*},{\theta }_1^0,{\alpha }_1^0)U_1({\beta }_1^{*},{\theta }_1^0,{\alpha }_1^0),\dots ,(I_{{\beta }_T,Q_T}^{-1}({\beta }_T^{*},{\theta }_T^0,{\alpha }_T^0)U_T({\beta }_T^{*},{\theta }_T^0,{\alpha }_T^0))\prime$ is the influence function corresponding to the 2^T-multiply robust CAN estimator for β* under model ${\cap }_{t=1}^T\{{ℬ}_t(q)\cup {𝒞}_t(q)\}$. This proves part (iii).

At the intersection model ℬ_t (q) ∩ 𝒞_t (q), $I_{{\alpha }_t,Q_t}({\beta }^{*},{\theta }_t^0,{\alpha }_t^0)=I_{{\theta }_t,Q_t}({\beta }^{*},{\theta }_t^0,{\alpha }_t^0)=0$ and hence $U_{\mathit{\text{it}}}({\beta }_t^{*},{\theta }_t^0,{\alpha }_t^0)=Q_{\mathit{\text{it}}}({\beta }_t^{*},{\theta }_t^0,{\alpha }_t^0)$. It follows that the estimators ${\mathrm{\beta ̂}}_t({\psi }_t^{(1)},{\varphi }_t^{(1)})\text{ and }{\mathrm{\beta ̂}}_t({\psi }_t^{(2)},{\varphi }_t^{(2)})$ have the same influence functions at the intersection model ℬ_t (q) ∩ 𝒞_t (q). This proves part (ii). Part (iv) is similarly proved.

Proof of Theorem 3. By definition, $U(\beta ,{\theta }^0,{\alpha }_t^0)=d(X)U^{*}(\beta ,{\theta }_t^0,{\alpha }_t^0)\prime$ and, by analogous arguments to Theorem 2 for estimators β̂ (d) ≡ β̂ (d, d_θ, ψ, d_α, ϕ), the variance matrix of the limiting distribution of √n {β̂ (d) − β*} is equal to Γ (d) = Ψ (d) Ω (d) Ψ (d)′, where

$$\mathrm{\Psi }(d)=E{\left({d(X)\frac{\partial }{\partial \beta \prime }{U}^{*}(\beta ,{\theta }_t^0,{\alpha }_t^0)\prime |}_{\beta ={\beta }^{*}}\right)}^{-1},\mathrm{\Omega }(d)=E\left\{{(d(X)U^{*}({\beta }^{*},{\theta }_t^0,{\alpha }_t^0)\prime )}^{\otimes 2}\right\}$$

That Γ (d_opt) ≤ Γ (d) follows after applying the Cauchy-Schwarz inequality.