Bootstrapping latent variable models for binary response

Abstract

Estimated asymptotic variances for the estimates of the parameters in a logit-probit model for binary response data are unreliable for moderate sized samples. We show how bootstrapping gives a bctter idea of the sampling distribution of the estimators, and can also allow an assessment of the reliability of the scoring of individuals on the latent scale. lntroductionWhen a one-factor logit-probit model is fitted by maximum likelihood to binary (0, I ) response data, one very often wishes to score the observed response patterns on the scale of the latent variable. We prefer to score with the estimated conditional mean of the latent variable for the given response pattern, though Bartholomew (1984) has suggested scoring the response patterns by their component scores: thesc are defined as the sum of the estimated discrimination parameters for items responding 1. A difficulty with both methods of scoring is that little is known about the sampling properties of the estimators of the discrimination parameters. Usually one depends on the first-order asymptotic distribution theory for maximum likelihood estimators which suggests that the discrimination parameters have a sampling distribution which is asymptotically normal. The covariance matrix of the estimators is estimated from the observed information matrix. For logit-probit models for binary response with their usually large number of parameters it is not clear that sample sizes are great enough in practice to justify the use of this procedure. One would like to know whether the distributions of the estimators are approximately normal. One also needs a guide to the reliability of the scoring of the response patterns, and this is difficult to obtain from standard asymptotic theory.A common way to investigate questions of this type when exact mathematical results are not feasible is to generate bootstrap samples, and use these as a substitute for the t Requests for reprints

Refining the assessment of disrupted maternal communication: Using item response models to identify central indicators of disrupted behavior

“…It is well known that maximum likelihood estimation of latent (trait) models with binary or ordered categorical data present modeling challenges (Albanese & Knott, 1994; Sterba et al, 2010). This problem is further magnified in rare-event behavioral data, where many indicators demonstrate a preponderance of nonevents (i.e., the disrupted maternal behavior does not occur).…”

“…Sparse data response patterns may lead to extreme parameter and/or standard error estimates, which are unstable (de Menezes, 1999) or frequently drift into inadmissible regions (Swaminathan, Hambleton, Sireci, Xing, & Rizavi, 2003). Albanese and Knott (1994) showed that estimated asymptotic variances of the parameter estimates in a one-factor model for binary data are unreliable. They arrived at a better idea of the sampling distribution of the parameter estimates by bootstrapping.…”

On fitting latent class models for binary data: The estimation of standard errors

“…However, when we consider the literature on latent class models, only a small number of references have addressed the topic (for example, Bartholomew (1987), while focusing on latent variable models, and Tanner (1993), when referring to estimates from E-M algorithm s). Albanese & Knott (1994), focusing upon latent trait models, and speci cally the one-factor logit model for binary data, have shown that the estimated asymptotic variances of the parameter estimates are unreliable. They arrived at a better idea of the sampling distribution of the parameter estimates by bootstrappin g. In their study, they generated 100 pseudoreplicates from the original data, by sampling individuals with replacement, and from the model, by using the maximum likelihood estimates of the parameters.…”

“…On the contrary, when one refers to the literature, the lack of reliability of asymptotic estimates in the presence of extreme parameter values has been reported. For example, in the context of the one-factor logit-probit model, the asymptotic estimates were found to be unreliable (Albanese & Knott, 1994); in an investigation of possible adjustments to the log-likelihoo d ratio statistic, problem s in coping with very small parameters were experienced (Holt & MacReady, 1989).…”

Contribuição da estatística para a compreensão de alguns aspectos do desenvolvimento infantil: análise de dados comportamentais provenientes de estudos do desenvolvimento de bebês

“…Foram obtidas também as idades do aparecimento/desaparecimento de comportamentos considerados relevantes. As dificuldades para efetuar as análises quantitativas, devido ao grande volume de dados gerados nos estudos de Panico (1984Panico ( ,1987 (Agresti 1990, Bishop et al 1980, Conover 1980, Cox 1970, Mead 1989, Siegel 1975 de Diament (1976) (Albanese 1990, Anderson 1976, Bartholomew 1984, Bock 1975, Everitt 1978, Johnson e Wichem 1984, Knott & Albanese 1992, Morrison 1976…”