Nonresponse Models Research Articles

Multiple imputation of missing blood pressure covariates in survival analysis.

This paper studies a non-response problem in survival analysis where the occurrence of missing data in the risk factor is related to mortality. In a study to determine the influence of blood pressure on survival in the very old (85+ years), blood pressure measurements are missing in about 12.5 per cent of the sample. The available data suggest that the process that created the missing data depends jointly on survival and the unknown blood pressure, thereby distorting the relation of interest. Multiple imputation is used to impute missing blood pressure and then analyse the data under a variety of non-response models. One special modelling problem is treated in detail; the construction of a predictive model for drawing imputations if the number of variables is large. Risk estimates for these data appear robust to even large departures from the simplest non-response model, and are similar to those derived under deletion of the incomplete records.

On identifiability in models for incomplete binary data

In recent work by various authors, models for categorical response data subject to non-ignorable non-response have been considered. While the emphasis of those papers has been largely on the computational aspects of fitting those models, it has also been observed that such models present certain analytical difficulties. In particular, it can happen that the parameters of the model are not identifiable. In this note, the issue of identifiability is examined. A simple non-ignorable non-response model is considered and necessary and sufficient conditions for identifiability are given. An example exhibiting a type of non-identifiability not previously discussed is presented.

Semiparametric Regression for Repeated Outcomes with Nonignorable Nonresponse

Abstract We consider inference about the parameter β* indexing the conditional mean of a vector of correlated outcomes given a vector of explanatory variables when some of the outcomes are missing in a subsample of the study and the probability of response depends on both observed and unobserved data values; that is, nonresponse is nonignorable. We propose a class of augmented inverse probability of response weighted estimators that are consistent and asymptotically normal (CAN) for estimating β* when the response probabilities can be parametrically modeled and a CAN estimator exists. The proposed estimators do not require full specification of a parametric likelihood, and their computation does not require numerical integration. Our estimators can be viewed as an extension of generalized estimating equation estimators that allows for nonignorable nonresponse. We show that our class essentially consists of all CAN estimators of β*. We also show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model. When the model for nonresponse is richly parameterized, joint estimation of the regression parameter β* and the nonresponse model parameter τ* which encodes the magnitude of nonignorable selection bias, may be difficult or impossible. Therefore we propose regarding the selection bias parameter τ* as known, rather than estimating it from the data. We then perform a sensitivity analysis that examines how inference concerning the regression parameter β* changes as we vary τ* over a range of plausible values. We apply our approach to the analysis of ACTG Trial 002, an AIDS clinical trial.

Semiparametric Regression for Repeated Outcomes with Nonignorable Nonresponse

We consider inference about the parameter β* indexing the conditional mean of a vector of correlated outcomes given a vector of explanatory variables when some of the outcomes are missing in a subsample of the study and the probability of response depends on both observed and unobserved data values; that is, nonresponse is nonignorable. We propose a class of augmented inverse probability of response weighted estimators that are consistent and asymptotically normal (CAN) for estimating β* when the response probabilities can be parametrically modeled and a CAN estimator exists. The proposed estimators do not require full specification of a parametric likelihood, and their computation does not require numerical integration. Our estimators can be viewed as an extension of generalized estimating equation estimators that allows for nonignorable nonresponse. We show that our class essentially consists of all CAN estimators of β*. We also show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model. When the model for nonresponse is richly parameterized, joint estimation of the regression parameter β* and the nonresponse model parameter τ* which encodes the magnitude of nonignorable selection bias, may be difficult or impossible. Therefore we propose regarding the selection bias parameter τ* as known, rather than estimating it from the data. We then perform a sensitivity analysis that examines how inference concerning the regression parameter β* changes as we vary τ* over a range of plausible values. We apply our approach to the analysis of ACTG Trial 002, an AIDS clinical trial.

Model-based Inference for Categorical Survey Data Subject to Non-ignorable Non-response

Summary We consider non-response models for a single categorical response with categorical covariates whose values are always observed. We present Bayesian methods for ignorable models and a particular non-ignorable model, and we argue that standard methods of model comparison are inappropriate for comparing ignorable and non-ignorable models. Uncertainty about ignorability of non-response is incorporated by introducing parameters describing the extent of non-ignorability into a pattern mixture specification and integrating over the prior uncertainty associated with these parameters. Our approach is illustrated using polling data from the 1992 British general election panel survey. We suggest sample size adjustments for surveys when non-ignorable non-response is expected.

Weighted Estimating Equations with Nonignorably Missing Response Data

We propose weighted estimating equations for data with nonignorable nonresponse in order to reduce the bias that can occur with a complete case analysis. A survey concerning medical practice guidelines, malpractice litigation, and settlement provides the framework. The survey was sent to recipients in two waves: those who responded on the first or second wave are used to estimate a nonignorable nonresponse model, while the fraction of recipients who never responded is used to allow the percentage of missing data to change with each wave. We use the structure of the GEE of Liang and Zeger (1986, Biometrika 73, 13-22), adding weights equal to the inverse probability of being observed. We present simulations demonstrating the bias that can occur with an unweighted analysis and use the survey data to illustrate the methods.

Multivariate Logistic Models for Incomplete Binary Responses

Abstract In this article we describe a likelihood-based regression model appropriate for analyzing incomplete multivariate binary responses. We focus on “marginal models”; that is, models where the marginal mean or expectation of the binary response is related to a set of covariates. The association between the binary responses is modeled in terms of conditional log odds ratios. When the nonresponse mechanism is ignorable, it is not necessary to specify a nonresponse model, and valid inferences can be obtained provided that the likelihood for the responses has been correctly specified. But when the nonresponse mechanism is nonignorable, valid inferences can only be obtained by incorporating a model for nonresponse. An unresolved issue with nonignorable models concerns the identifiability of the parameters. So far, no general and practically useful necessary and sufficient conditions for identifiability are available. Here we suggest some simple procedures for examining the identifiability status of nonignorable models when the response variable is discrete. Finally, we present results for an analysis of multiple informant data from the New Haven Child Survey and the Eastern Connecticut Child Survey.

Multivariate Logistic Models for Incomplete Binary Responses

Abstract In this article we describe a likelihood-based regression model appropriate for analyzing incomplete multivariate binary responses. We focus on “marginal models”; that is, models where the marginal mean or expectation of the binary response is related to a set of covariates. The association between the binary responses is modeled in terms of conditional log odds ratios. When the nonresponse mechanism is ignorable, it is not necessary to specify a nonresponse model, and valid inferences can be obtained provided that the likelihood for the responses has been correctly specified. But when the nonresponse mechanism is nonignorable, valid inferences can only be obtained by incorporating a model for nonresponse. An unresolved issue with nonignorable models concerns the identifiability of the parameters. So far, no general and practically useful necessary and sufficient conditions for identifiability are available. Here we suggest some simple procedures for examining the identifiability status of nonign...

Nonresponse and Urban Travel Models

Considerable attention has been paid to the presence of nonresponse in large-scale travel surveys on the basis of which urban travel demand models are developed. It has been shown that the effect of nonresponse can be reduced by careful model building, with categorical trip generation models as an example. The same philosophy is extended to logit mode split models and exponential gravity models to show that the usual levels of nonresponse that one encounters in urban travel surveys have virtually no adverse effects on the parameter estimates of these models if the model has been specified correctly. Some simulation results are also presented to show the behavior of logit and exponential gravity model parameter estimates under conditions on nonresponse.

Nonresponse and Urban Travel Models

Considerable attention has been paid to the presence of nonresponse in large-scale travel surveys on the basis of which urban travel demand models are developed. It has been shown that the effect of nonresponse can be reduced by careful model building, with categorical trip generation models as an example. The same philosophy is extended to logit mode split models and exponential gravity models to show that the usual levels of nonresponse that one encounters in urban travel surveys have virtually no adverse effects on the parameter estimates of these models if the model has been specified correctly. Some simulation results are also presented to show the behavior of logit and exponential gravity model parameter estimates under conditions on nonresponse.

Causal Nonresponse Models for Repeated Categorical Measurements

This paper uses causal models for nonresponse (Fay, 1986, Journal of the American Statistical Association 81, 354-365) to extend the conditional likelihood procedure for repeated categorical outcome variables to allow for nonrandomly missing data. The extension based on causal models is similar to the log-linear approach presented by Conaway (1992, Journal of the American Statistical Association, 87, 817-824), but has the advantage that the parameters are directly interpretable in terms of the distribution of the outcome variables. As with log-linear model approach, all of the computations can be done with standard statistical software. The methods are first described in terms of a simple example with three binary responses with no covariates and then are applied to a more complicated example. A simulation study evaluates the properties of the estimates based on the proposed method.

Log-Linear Models for Survey Data with Non-Ignorable Non-Response

SUMMARY We demonstrate the feasibility of constructing, interpreting and fitting computable log-linear models to categorical survey data with arbitrary non-nested patterns of non-ignorable non-response. Under our approach, the non-response probability for each cell of the classification defined by the categories of interest is modelled separately from the classification probability itself, and we adopt a model formulation which allows the non-response model to depend on scores, discrete covariates, continuous covariates or a mixture of types of covariate. We obtain explicit expressions for the score and information functions generated by the observed data which allow us to compute approximate standard errors and test statistics based on these functions. Through illustrative examples we quantify the fact that inferences obtained from different non-ignorable non-response models can vary considerably. This lends support to the calls in the literature for caution in using such models.

Non-Ignorable Non-Response Models for Time-Ordered Categorical Variables

SUMMARY This paper presents an analysis of time-ordered categorical variables with incomplete data. The models allow for either an ignorable or a non-ignorable non-response model and take into account the time ordering of the categorical variables. All the necessary computations can be done with standard statistical software. In addition, we present a Bayesian analysis which uses information from previous years in the priors. The methods are presented in the context of an example, using data from the National Crime Survey. Although this paper deals with two-way tables, the methods are readily extended to higher dimensional tables.