Sensitivity analysis of incomplete longitudinal data departing from the missing at random assumption: Methodology and application in a clinical trial with drop-outs.

Statistical analyses of longitudinal data with drop-outs based on direct likelihood, and using all the available data, provide unbiased and fully efficient estimates under some assumptions about the drop-out mechanism. Unfortunately, these assumptions can never be tested from the data. Thus, sensitivity analyses should be routinely performed to assess the robustness of inferences to departures from these assumptions. However, each specific scientific context requires different considerations when setting up such an analysis, no standard method exists and this is still an active area of research. We propose a flexible procedure to perform sensitivity analyses when dealing with continuous outcomes, which are described by a linear mixed model in an initial likelihood analysis. The methodology relies on the pattern-mixture model factorisation of the full data likelihood and was validated in a simulation study. The approach was prompted by a randomised clinical trial for sleep-maintenance insomnia treatment. This case study illustrated the practical value of our approach and underlined the need for sensitivity analyses when analysing data with drop-outs: some of the conclusions from the initial analysis were shown to be reliable, while others were found to be fragile and strongly dependent on modelling assumptions. R code for implementation is provided.

Misspecification of the covariance structure in generalized linear mixed models.

When fitting marginal models to correlated outcomes, the so-called sandwich variance is commonly used. However, this is not the case when fitting mixed models. Using two data sets, we illustrate the problems that can be encountered. We show that the differences or the ratios between the naive and sandwich standard deviations of the fixed effects estimators provide convenient means of assessing the fit of the model, as both are consistent when the covariance structure is correctly specified, but only the latter is when that structure is misspecified. When the number of statistical units is not too small, the sandwich formula correctly estimates the variance of the fixed effects estimator even if the random effects are misspecified, and it can be used in a diagnostic tool for assessing the misspecification of the random effects. A simple comparison with the naive variance is sufficient and we propose considering a ratio of the naive and sandwich standard deviation out of the [3/4; 4/3] interval as signaling a risk of erroneous inference due to a model misspecification. We strongly advocate broader use of the sandwich variance for statistical inference about the fixed effects in mixed models.

[Case-cohort surveys]. / Les enquêtes cas-cohorte.

In this paper, we present the main tools for conception, implementation and analysis of case-cohort surveys. In particular, we describe the classical weighted estimators, the weighted approach recently suggested by Breslow et al. and the multiple imputation approach, an alternative to weighted analysis of case-cohort data. Variance estimators are also described. We show how to obtain the subcohort size. Finally, we mention the functions and procedures available in R, SAS and Stata software and we illustrate their implementation using simulated subcohorts from the PRIME cohort.

[The case series method]. / La méthode de la série de cas.

The case series method was developed by Farrington (1995) to investigate the strength of association between a time-varying exposure and an acute rare potentially recurrent event, using cases only. It can be used when the exposure can only be causally related to the event during a limited period of time. It has been widely used in pharmaco-epidemiology, particularly in the study of vaccine safety. The method is derived from a Poisson model by conditioning on the individual total number of events and its exposure history. As a consequence of this conditioning, the effects of fixed covariates cancel out, so that the method has a particular advantage as compared with cohort and case-control studies.

Relative-risk ratio was a useful measure of differential association in cohort and case-series studies.

OBJECTIVE: The framework consists of cohort or case-series studies with intermittent exposure and two types of events. The aim is to define and estimate an association measure between the exposure and the occurrence of one type of event rather than the other. STUDY DESIGN AND SETTING: The model and the estimation method are obtained by extending Farrington's approach for one type of recurrent event. The proposed association measure "RR(c)" is the ratio of the relative risks pertaining to each type of event. The estimated RR(c) and its confidence interval are derived under the independence assumption between the counts of the two types of events. The data that are analyzed are part of the data of a study on antimicrobial resistance in children. RESULTS: An interpretation of the RR(c) is proposed in terms of an odds ratio, which parallels a similar association measure defined in cross-sectional studies ("OR(c)"). The estimated value of the RR(c) agrees with the OR(c) reported in previous studies. CONCLUSION: The RR(c) appears as a useful tool for evaluating the risk of colonization (or infection) with resistant rather than susceptible bacteria following a previous intake of a given antibiotic conditional on colonization (or infection) with any bacteria.

Disability evolution in multiple sclerosis: how to deal with missing transition times in the Markov model?

Markov modeling of disability progression in multiple sclerosis requires knowledge of all times of transitions from a given level of disability to the next level, but such data are often missing. We address methodological challenges due to partly missing transition times. To estimate the effects of prognostic factors on the risk of transitions between three consecutive disability levels, two methods were used to deal with missing data. Listwise deletion limited the analysis to subjects with complete data. Multiple imputation of missing data revealed that data were missing at random (MAR mechanism) and imputed the missing transition times from the Weibull model. The results were then compared with the full data set with the actual times established through chart review. Multiple imputation estimates were systematically closer to those from the full data set than the listwise deletion estimates.

[Analysis of repeated binary data: sensitivity to missing data]. / Analyse de données binaires répétées: sensibilité aux données manquantes.

BACKGROUND: In longitudinal studies, it is extremely rare that all the planned measurements are actually performed. Missing data are often consecutive to drop-outs, but may also be intermittent. In both cases, the analysis of incomplete data necessarily requires assumptions that are generally unverifiable, and the need for sensitivity analyses has been advocated over the past few years. In this article, the attention will be given to longitudinal binary data. METHODS: A method is proposed, which is based on a log-linear model. A sensitivity parameter is introduced that represents the relationship between the response mechanism and the missing data mechanism. It is recommended not to estimate this parameter, but to consider a range of plausible values, and to estimate the parameters of interest conditionally on these plausible values. This allows to assess the sensitivity of the conclusion of a study to various assumptions regarding the missing data mechanism. RESULTS: This method was applied to a randomized clinical trial comparing the efficacy of two treatment regimens in patients with persistent asthma. The sensitivity analysis showed that the conclusion of this study was robust to missing data.

[Modeling incomplete observations]. / Modélisation d'observations incomplètes.

Incomplete observations, common in epidemiology as in many other fields, lead to problems of bias, precision and power. Using a simple example with 3 binary variables, we discuss situations where the observed odds ratio is biased. We present and compare the main strategies of analysis: complete observations modeling, missing data indicator, weighted analysis, simple imputation, multiple imputation, selection models, shared variable models.

Consultation of mental health professionals by French adolescents with probable psychiatric problems.

A national sample of school-attending adolescents aged 12-20 years (n = 8435) filled out an anonymous self-administered questionnaire. We considered a group of 868 adolescents with probable psychiatric problems (PPP) in order to identify personal, psychological and environmental factors as well as help-seeking behaviour associated with consultation of mental health professionals (MHP). Among the adolescents with PPP, 13.7% consulted MHP at least once during the year. Logistic regression analysis showed that consultation with MHP was more frequent for adolescents with multiple problems (OR = 2.3), functional physical disorders (OR = 1.9), family problems (OR = 2), separated parents (OR = 4.4) or multiple contacts with other doctors (GP, school doctor and other specialists, OR = 2.2, 3.6 and 2.6, respectively), and among those who confided in teachers or youth group advisors (OR = 2.2). Those who confided in peers consulted MHP less often (OR = 0.5). However, consultation with MHP was not associated with other sociodemographic or educational characteristics, with type of problem (internalized or externalized), or with confiding in one's parents.

[Modeling correlated data in epidemiology: mixed or marginal model?]. / Modélisation de données corrélées en épidémiologie: modèle mixte ou marginal?

Correlated observations (within centers, families, subjects,.) are common in epidemiology. Even when one is only interested in the modeling of means according to risk factors, it is also necessary to model the variance-covariance matrix of the observations in order to make correct inferences on the parameters of interest. All the more so when the aim of the survey is the measurement of these correlations or of the variance of the random effects from which they are assumed to originate. We discuss, within the framework of the linear and of the logistic models, the implications of two choices for the modeling of covariances. The mixed model shows the unobserved elements responsible for the similarity between certain observations. In a longitudinal survey, for instance, one can use a random effect, specific to each subject, expressing how much a subject's trajectory is translated as compared to what is expected according to its characteristics (age, sex,.). The marginal approach leads to modeling separately the means and the covariance matrix of the observations. The distinction between these two approaches is important for non linear models, in particular the logistic one. We insist on the interconnection between a mixed model formulation and a marginal one, as well as on the implication of the choice in terms of the parameters' interpretation.

[Use of GEE for modeling censored correlated data: application to the study of risk factors for withdrawal of totally implantable vascular access devices in cystic fibrosis]. / Utilisation des GEE pour la modélisation de données censurées corrélées: application à l'étude des facteurs de risque de retrait des chambres implantables chez le mucoviscidosique.

BACKGROUND: The proportional hazards model proposed by Cox for modeling censored data is not suited for correlated delays, for instance when several events can be observed on each subject. METHODS: To analyze correlated delays, we propose to use a log-linear marginal model equivalent to Cox model. Correlations are taken into account through the use of Liang and Zeger's Generalized Estimating Equations (GEE) and of their robust variance estimator. An advantage of this method is that it can be implemented through the SAS GENMOD procedure. When ties are observed, we propose to use multiple imputations, creating M data sets without ties from the original one. RESULTS: This method is applied to a retrospective survey on the risk of withdrawing totally implantable vascular access devices (TIVAD) because of complication in cystic fibrosis patients: 265 TIVAD implanted in 200 patients were observed. Risk factors were characteristics of the device or of the patient. Results obtained with the robust variance estimator and ten imputations show that the use of the device for taking blood (vs exclusive perfusion of antibiotics), polyurethane catheter (vs. silicon), use of counterpressure for upkeeping and pulmonary colonization by Pseudomonas Aeruginosa are significantly associated to withdrawal. Under the Cox model which does not account for the correlations, some conclusions differ because the robust variance of the estimators is smaller than the variance obtained under the working assumption of independent delays. CONCLUSION: This approach allows the modeling of correlated survival data with SAS software. Our results illustrate the necessity of accounting for existing correlations.

Predictive value of viral load and other markers for progression to clinical AIDS after CD4+ cell count falls below 200/microL. SEROCO & HEMOCO Study Group.

BACKGROUND: To assess the predictive value of biological and clinical events for progression to AIDS (1993 European classification) when the CD4+ cell count falls below 200/microL (CD4 threshold) in different exposure groups. To investigate whether such markers remain predictive independently of the serum HIV-1 RNA level at the CD4 threshold. METHODS: The predictive value of biological and clinical events occurring during the 24 months prior to the occurrence of CD4 threshold (n = 333) was quantified in a Cox model. Another Cox model was carried out in a subset of 77 patients in whom viral load from stored sera was available. Furthermore, changes in viral load during the 24 months preceding the CD4 threshold were assessed in a mixed model according to subsequent development of AIDS. RESULTS: Among the 333 patients, the slope of the CD4+ cell counts, the emergence of p24 antigen, persistent thrush, and age at the CD4 threshold were independent predictors of progression to clinical AIDS (44.7%). Among the subset of 77 patients, the HIV-1 RNA level at the CD4 threshold, persistent thrush and age remained independent predictors of progression to AIDS (45.5%). The increase of the HIV-1 RNA level was moderate, both in non-progressors (24.0% per year) and in those who subsequently developed AIDS (27.1% per year), (P = 0.93). Viral load was consistently higher in the latter group (P = 0.002). CONCLUSION: At a late stage of infection, age and persistent thrush remain predictive of progression to AIDS, independently of viral load.

A mixed model for repeated dilution assays.

We propose a generalized linear mixed model to estimate and test marginal effects on titers repeatedly measured by serial dilution assays. The link is log-log and the titer is assumed to follow a gamma distribution. The parameters are estimated by generalized estimating equations. The marginal effects are tested by means of Wald and score tests, using the robust estimator of the variance. This approach avoids the problems arising from assays leading to nonestimable individual titers. Simulations were used to compare the Wald and score tests to Wilcoxon and Student t-tests. This method is applied to the comparison of the antiviral efficiency of three treatments against HIV.

Psychosocial factors associated with help-seeking behavior among depressive adolescents.

A group of 3,287 French school pupils between the age of 12 to 20 years, of whom 14.4% had sought consultation for depression, were investigated in order to analyse the factors related to the type of medical help obtained. Multivariate analyses showed that severe emotional distress alone did not explain the help-seeking behavior. Thus, among those adolescents with the same anxio-depressive level, girls, older adolescents, adolescents with parents living apart, with health worries, and adolescents often absent from school more often sought help for depression. Socio economic status, however, did not correlate with a higher level of consultation for depression. Adolescents attending medical services for depression had a higher consultation rate, for any reason, with general practitioners and with school nurses, in whom they confided more often than their counterparts. Nonmedical professionals also seemed to contribute to accessing to medical help for depression. Overall, those who consulted for depression, in very limited numbers turned to mental health services (8.4%).

[The center effect in multicenter studies: fixed or random?]. / L'effet centre dans les études multicentriques: fixe ou aléatoire?

The multiplication of multicentric clinical studies leads to a generalization of the use of a center effect in statistical analyses. In the simple situation where the dependent variable is quantitative and there is only a simple factor of interest, the formalisation of such a protocol corresponds to a two way ANOVA (with the factor of interest and the center factor) with generally a term of interaction. Even such a simple situation leads to interrogations, one of them being to determine whether the centre effect is fixed or random. In the first situation, the conclusion of the study will concern only the centres evaluated in the study, while in the second, the conclusion will concern the whole population from which the centers have been randomly selected. Such a generalizability has a cost: a loss of power in determining the effect of the factor of interest (the loss being as important as the interaction factor of interest x centre factor is large) and the necessity of sophisticated statistical software. This paper will focus on successively: the notion of ANOVA, of fixed or random effect (with the conceptual, formal and numerical differences that distinguish these two approaches). A numerical example is proposed.

HIV/HTLV-I coinfection and clinical grade at diagnosis.

A total of 963 HIV-infected patients have been identified or followed up in Martinique since 1985. Medical files were used to retrieve information about age, sex, circumstances of diagnosis, HTLV-I status, and HIV clinical grade at first examination according to CDC criteria from 1987. Complete information was available for 774 patients. At the first clinical examination, the clinical grade of 65 coinfected patients was more severe than that of the monoinfected patients (GIV versus GII, OR = 2.60, p < 0.01), but after adjustment for age and sex, this odds ratio was reduced 1.57. Although this study cannot invalidate the hypothesis of a faster progression toward AIDS of coinfected than of monoinfected patients, it shows that one or several other mechanisms contribute to the different grades of severity at the first clinical examination observed between these two categories of patients. We believe that HTLV-I infection acquired during adulthood is a marker of high-risk behavior and that it might be associated with early or multiple HIV infections.

Smoking, CD45R0+ (memory), and CD45RA+ (naive) CD4+ T cells.

The relationship between smoking and percentages and counts of T-cell subsets, in particular CD45R0+ (memory) and CD45RA+ (naive) CD4+ lymphocytes, have been assessed in a sample of 311 middle-aged men working in the Paris area. Percentages of lymphocytes with the phenotypes CD4+, CD8+, CD4+CD45R0+, and CD4+CD45RA+ were determined using double immunofluorescence labeling. All cell counts, including CD4+CD45R0+, and CD4+CD45RA+ lymphocytes increased significantly with tobacco consumption, as did the percentages of lymphocytes with the CD4+CD45R0+ and CD4+ phenotypes. The increase in the percentage of lymphocytes with the CD4+CD45RA+ phenotype, although not significant, was similar to that observed in the percentage of CD4+CD45R0+ lymphocytes. When the proportions of CD4+ cells with the CD45R0+ or the CD45RA+ phenotype were considered, no specific modification of any of these two sub-populations was observed: the effect of smoking on memory and naive Cd4+ lymphocytes seemed to be equivalent.