Large Sample Theory for Semiparametric Regression Models with Two-Phase, Outcome Dependent Sampling

Outcome-dependent, two-phase sampling designs can dramatically reduce the costs of observational studies by judicious selection of the most informative subjects for purposes of detailed covariate measurement. Here we derive asymptotic information bounds and the form of the efficient score and influence functions for the semiparametric regression models studied by Lawless, Kalbfleisch, and Wild (1999) under two-phase sampling designs. We show that the maximum likelihood estimators for both the parametric and nonparametric parts of the model are asymptotically normal and efficient. The efficient influence function for the parametric part aggress with the more general information bound calculations of Robins, Hsieh, and Newey (1995). By verifying the conditions of Murphy and Van der Vaart (2000) for a least favorable parametric submodel, we provide asymptotic justification for statistical inference based on profile likelihood.

Estimation with Interval Censored Data in Longitudinal Studies

In biostatistical applications interest often focuses on the estimation of the distribution of a time-until-event variable T. If one observes whether or not T exceeds an observed monitoring time at a random number of monitoring times, then the data structure is called interval censored data. We extend this data structure by allowing the presence of a possibly time-dependent covariate process that is observed until end of follow up. If one only assumes that the censoring mechanism satisfies coarsening at random, then, by the curve of dimensionality, typically no regular estimators will exist. To fight the curse of dimensionality we follow the approach of Robins and Rotnitzky (1992) by modeling parameters of the censoring mechanism. We model the right-censoring mechanism by modeling the hazard of the follow up time, conditional on T and the covariate process. For the monitoring mechanism we avoid modeling the joint distribution of the monitoring times by only modeling a univariate hazard of the pooled monitoring times, conditional on the follow up time, T, and the covariates process, which can be estimated by treating the pooled sample of monitoring times as i.i.d. In particular, it is assumed that the monitoring times and the right-censoring times only depend on T through the observed covariate process. We introduce inverse probability of censoring weighted (IPCW) estimator of the distribution of T and of smooth functionals thereof which are guaranteed to be consistent and asymptotically normal if we have available correctly specified semiparametric models for the two hazards of the censoring process. Furthermore, given such correctly specified models for these hazards of the censoring process, we propose a one-step estimator which will improve on the IPCW estimator if we correctly specify a lower-dimensional working model for the conditional distribution of T, given the covariate process, that remains consistent and asymptotically normal if this latter working model is misspecified. It is shown that the one-step estimator is efficient if each subject is at most monitored once and the working model contains the truth. In general, it is shown that the one-step estimator optimally uses the surrogate information if the working model contains the truth. It is not optimal in using the interval information provided by the current status indicators at the monitoring times, but simulations in Peterson, van der Laan (1997) show that the efficiency loss is small.

Locally Efficient Estimation with Current Status Data and High Dimensional Covariates

In biostatistical applications, interest often focuses on the estimation of the distribution of time T between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed monitoring time, then the data is described by the well known singly-censored current status model, also known as interval censored data, case I. We extend this current status model by allowing the presence of a time-dependent process, which is partly observed and allowing C to depend on T through the observed part of this time-dependent process. Because of the high dimension of the covariate process, no globally efficient estimators exist with a good practical performance at moderate sample sizes. We follow the approach of Robins and Rotnitzky (1992) by modeling the censoring variable, given the time-variable and the covariate-process, i.e., the missingness process, under the restriction that it satisfied coarsening at random. We propose a generalization of the simple current status estimator of the distribution of T and of smooth functionals of the distribution of T, which is based on an estimate of the missingness. In this estimator the covariates enter only through the estimate of the missingness process. Due to the coarsening at random assumption, the estimator has the interesting property that if we estimate the missingness process more nonparametrically, then we improve its efficiency. We show that by local estimation of an optimal model or optimal function of the covariates for the missingness process, the generalized current status estimator for smooth functionals become locally efficient; meaning it is efficient if the right model or covariate is consistently estimated and it is consistent and asymptotically normal in general. Estimation of the optimal model requires estimation of the conditional distribution of T, given the covariates. Any (prior) knowledge of this conditional distribution can be used at this stage without any risk of losing root-n consistency. We also propose locally efficient one step estimators. Finally, we show some simulation results.

Locally Efficient Estimation with Bivariate Right Censored Data

Estimation for bivariate right censored data is a problem that has had much study over the past 15 years. In this paper we propose a new class of estimators for the bivariate survival function based on locally efficient estimation. We introduce the locally efficient estimator for bivariate right censored data, present an asymptotic theorem, present the results of simulation studies and perform a brief data analysis illustrating the use of the locally efficient estimator.

Locally Efficient Estimation in Censored Data Models: Theory and Examples

In many applications the observed data can be viewed as a censored high dimensional full data random variable X. By the curve of dimensionality it is typically not possible to construct estimators that are asymptotically efficient at every probability distribution in a semiparametric censored data model of such a high dimensional censored data structure. We provide a general method for construction of one-step estimators that are efficient at a chosen submodel of the full-data model, are still well behaved off this submodel and can be chosen to always improve on a given initial estimator. These one-step estimators rely on good estimators of the censoring mechanism and thus will require a parametric or semiparametric model for the censoring mechanism. We present a general theorem that provides a template for proving the desired asymptotic results. We illustrate the general one-step estimation methods by constructing locally efficient one-step estimators of marginal distributions and regression parameters with right-censored data, current status data and bivariate right-censored data, in all models allowing the presence of time-dependent covariates. The conditions of the asymptotics theorem are rigorously verified in one of the examples and the key condition of the general theorem is verified for all examples.

Low self-esteem prospectively predicts depression in adolescence and young adulthood

Low self-esteem and depression are strongly correlated in cross-sectional studies, yet little is known about their prospective effects on each other. The vulnerability model hypothesizes that low self-esteem serves as a risk factor for depression, whereas the scar model hypothesizes that low self-esteem is an outcome, not a cause, of depression. To test these models, the authors used 2 large longitudinal data sets, each with 4 repeated assessments between the ages of 15 and 21 years and 18 and 21 years, respectively. Cross-lagged regression analyses indicated that low self-esteem predicted subsequent levels of depression, but depression did not predict subsequent levels of self-esteem. These findings held for both men and women and after controlling for content overlap between the self-esteem and depression scales. Thus, the results supported the vulnerability model, but not the scar model, of self-esteem and depression.

Disentangling the effects of low self-esteem and stressful events on depression: Findings from three longitudinal studies

Diathesis-stress models of depression suggest that low self-esteem and stressful events jointly influence the development of depressive affect. More specifically, the self-esteem buffering hypothesis states that, in the face of challenging life circumstances, individuals with low self-esteem are prone to depression because they lack sufficient coping resources, whereas those with high self-esteem are able to cope effectively and consequently avoid spiraling downward into depression. The authors used data from 3 longitudinal studies of adolescents and young adults, who were assessed 4 times over a 3-year period (Study 1; N = 359), 3 times over a 6-week period (Study 2; N = 249), and 4 times over a 6-year period (Study 3; N = 2,403). In all 3 studies, low self-esteem and stressful events independently predicted subsequent depression but did not interact in the prediction. Thus, the results did not support the self-esteem buffering hypothesis but suggest that low self-esteem and stressful events operate as independent risk factors for depression. In addition, the authors found evidence in all 3 studies that depression, but not low self-esteem, is reciprocally related to stressful events, suggesting that individuals high in depression are more inclined to subsequently experience stressful events.

Is low self-esteem a risk factor for depression? Findings from a longitudinal study of Mexican-origin youth

We examined the relation between low self-esteem and depression using longitudinal data from a sample of 674 Mexican-origin early adolescents who were assessed at age 10 and 12 years. Results supported the vulnerability model, which states that low self-esteem is a prospective risk factor for depression. Moreover, results suggested that the vulnerability effect of low self-esteem is driven, for the most part, by general evaluations of worth (i.e., global self-esteem), rather than by domain-specific evaluations of academic competence, physical appearance, and competence in peer relationships. The only domain-specific self-evaluation that showed a prospective effect on depression was honesty-trustworthiness. The vulnerability effect of low self-esteem held for male and female adolescents, for adolescents born in the United States versus Mexico, and across different levels of pubertal status. Finally, the vulnerability effect held when we controlled for several theoretically relevant 3rd variables (i.e., social support, maternal depression, stressful events, and relational victimization) and for interactive effects between self-esteem and the 3rd variables. The present study contributes to an emerging understanding of the link between self-esteem and depression and provides much needed data on the antecedents of depression in ethnic minority populations

Understanding the link between low self-esteem and depression

Although it is well documented that low self-esteem and depression are related, the precise nature of the relation has been a topic of ongoing debate. We describe several theoretical models concerning the link between self-esteem and depression, and review recent research evaluating the validity of these competing models. Overall, the available evidence provides strong support for the vulnerability model (low self-esteem contributes to depression), weaker support for the scar model (depression erodes self-esteem), and little support for alternative accounts such as the diathesis-stress model. Moreover, the vulnerability model is robust and holds across gender, age, affective-cognitive versus somatic symptoms of depression, European background versus Mexican-origin participants, and clinical versus nonclinical samples. Research on further specifications of the vulnerability model suggests that the effect is (a) partially mediated by rumination, (b) not influenced by other characteristics of self-esteem (i.e., stability and contingency), and (c) driven predominantly by global rather than domain-specific self-esteem. The research has important theoretical implications because it counters the commonly repeated claim that self-esteem has no long-term impact. Moreover, the research has important practical implications, suggesting that depression can be prevented, or reduced, by interventions that improve self-esteem.

The development of self-esteem

In this article, we review new insights gained from recent longitudinal studies examining the development of self-esteem and its influence on important life outcomes. The evidence supports the following three conclusions. First, self-esteem increases from adolescence to middle adulthood, peaks at about age 50 to 60 years, and then decreases at an accelerating pace into old age; moreover, there are no cohort differences in the self-esteem trajectory from adolescence to old age. Second, self-esteem is a relatively stable, but by no means immutable, trait; individuals with relatively high (or low) self-esteem at one stage of life are likely to have relatively high (or low) self-esteem decades later. Third, high self-esteem prospectively predicts success and well-being in life domains such as relationships, work, and health. Given the increasing evidence that self-esteem has important real-world consequences, the topic of self-esteem development is of considerable societal significance.

Low self-esteem is a risk factor for depressive symptoms from young adulthood to old age

Data from two large longitudinal studies were used to analyze reciprocal relations between self-esteem and depressive symptoms across the adult life span. Study 1 included 1,685 participants aged 18 to 96 years assessed 4 times over a 9-year period. Study 2 included 2,479 participants aged 18 to 88 years assessed 3 times over a 4-year period. In both studies, cross-lagged regression analyses indicated that low self-esteem predicted subsequent depressive symptoms, but depressive symptoms did not predict subsequent levels of self-esteem. This pattern of results replicated across all age groups, for both affective–cognitive and somatic symptoms of depression, and after controlling for content overlap between the self-esteem and depression scales. The results suggest that low self-esteem operates as a risk factor for depressive symptoms at all phases of the adult life span.

Life-span development of self-esteem and its effects on important life outcomes

We examined the life-span development of self-esteem and tested whether self-esteem influences the development of important life outcomes, including relationship satisfaction, job satisfaction, occupational status, salary, positive and negative affect, depression, and physical health. Data came from the Longitudinal Study of Generations. Analyses were based on 5 assessments across a 12-year period of a sample of 1,824 individuals ages 16 to 97 years. First, growth curve analyses indicated that self-esteem increases from adolescence to middle adulthood, reaches a peak at about age 50 years, and then decreases in old age. Second, cross-lagged regression analyses indicated that self-esteem is best modeled as a cause rather than a consequence of life outcomes. Third, growth curve analyses, with self-esteem as a time-varying covariate, suggested that self-esteem has medium-sized effects on life-span trajectories of affect and depression, small to medium-sized effects on trajectories of relationship and job satisfaction, a very small effect on the trajectory of health, and no effect on the trajectory of occupational status. These findings replicated across 4 generations of participants— children, parents, grandparents, and their great-grandparents. Together, the results suggest that self-esteem has a significant prospective impact on real-world life experiences and that high and low self-esteem are not mere epiphenomena of success and failure in important life domains.

Tracking the trajectory of shame, guilt, and pride across the life span

The authors examined age differences in shame, guilt, and 2 forms of pride (authentic and hubristic) from age 13 years to age 89 years, using cross-sectional data from 2,611 individuals. Shame decreased from adolescence into middle adulthood, reaching a nadir around age 50 years, and then increased in old age. Guilt increased from adolescence into old age, reaching a plateau at about age 70 years. Authentic pride increased from adolescence into old age, whereas hubristic pride decreased from adolescence into middle adulthood, reaching a minimum around age 65 years, and then increased in old age. On average, women reported experiencing more shame and guilt; Blacks reported experiencing less shame and Asians more hubristic pride than other ethnicities. Across the life span, shame and hubristic pride tended to be negatively related to psychological well-being, and shame-free guilt and authentic pride showed positive relations with well-being. Overall, the findings support the maturity principle of personality development and suggest that as people age they become more prone to experiencing psychologically adaptive self-conscious emotions, such as guilt and authentic pride, and less prone to experiencing psychologically maladaptive ones, such as shame and hubristic pride.