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This paper presents two new model-based goodness-of-fit tests for the ordered stereotype model applied to an ordinal response variable. The proposed tests are based on the Lipsitz test, which partitions the subjects into G groups following the popular Hosmer-Lemeshow test for binary data. The tests construct an alternative model where group effects are added into the null model. If the model fits the data well then the null model is correct, and there should be no group effects. One of the main advantages of the ordered stereotype model is that it allows us to determine a new uneven spacing of the ordinal response categories, dictated by the data. The two proposed tests use this new adjusted spacing. One test uses the form of the original ordered stereotype model, and the other uses an ordinary linear model. We demonstrate the performance of both tests under a variety of scenarios. Finally, the results of the application in three examples are presented.

Effect of low-input habitats on biodiversity in Swiss agricultural landscapes.

In the present paper a mixed generalized estimating/pseudo­score equations (GEPSE) approach together with a distribution free multiple imputation technique is proposed for the estimation of regression and correlation structure parameters of multivariate probit models with missing values for an ordered categorical time invariant variable. Furthermore, a generalization of the squared trace correlation (R_T^2) for multivariate probit models, denoted as pseudo R_T^2, is proposed. A simulation study was conducted, simulating a probit model with an equicorrelation structure in the errors of an underlying regression model and using two different missing mechanisms. For a low `true' correlation the difference between the GEPSE, a generalized estimating equations (GEE) and a maximum likelihood (ML) estimator were negligible. For a high `true' correlation the GEPSE estimator turned out to be more efficient than the GEE and very efficient relative to the ML estimator. Furthermore, the pseudo R_T^2 was close to R_T^2 of the underlying linear model. The mixed approach is illustrated using a psychiatric data set of depressive inpatients. The results of this analysis suggest, that the depression score at discharge from a psychiatric hospital and the occurence of stressful life events seem to increase the probability of having an episode of major depression within a one­year interval after discharge. Furthermore, the correlation structure points to short­time effects on having or not having a depressive episode, not accounted for in the systematic part of the regression model.

In this paper it is shown that using the maximum likelihood (ML) principle for the estimation of multivariate probit models leads to consistent and normally distributed pseudo maximum likelihood regression parameter estimators (PML estimators) even if the `true' correlation structure of the responses is misspecified. As a consequence, e.g. the PML estimator of the random effects probit model may be used to estimate the regression parameters of a model with any `true' correlation structure. This result is independent of the kind of covariates included in the model. The results of a Monte Carlo experiment show that the PML estimator of the independent binary probit model is inefficient relative to the PML estimator of the random effects binary panel probit model and two alternative estimators using the `generalized estimating equations' approach proposed by Liang and Zeger (1986), if the `true' correlations are high. If the `true' correlations are low, the differences between the estimators are small in finite samples and for the models used. Generally, the PML estimator of the random effects probit panel model is very efficient relative to the GEE estimators. Therefore, if correlations between binary responses cannot be ruled out and the `true' structure of association is unknown or infeasible to estimate, the PML estimator of the random effects probit model is recommended.

In this paper it is shown that the use of non-singular block invariant matrices of covariates leads to `generalized estimating equations' estimators (GEE estimators; Liang, K.-Y.&Zeger, S. (1986). Biometrika, 73(1), 13-22) which are identical regardless of the `working' correlation matrix used. Moreover, they are efficient (McCullagh, P. (1983). The Annals of Statistics, 11(1), 59-67). If on the other hand only time invariant covariates are used the efficiency gain in choosing the `correct' vs. an `incorrect' correlation structure is shown to be negligible. The results of a simple simulation study suggest that although different GEE estimators are no more identical and are no more as efficient as an ML estimator, the differences are still negligible if both time and block invariant covariates are present.

In the present paper a mixed approach is proposed for the simultaneously estimation of regression and correlation structure parameters in multivariate probit models using generalized estimating equations for the former and pseudo-score equations for the latter. The finite sample properties of the corresponding estimators are compared to estimators proposed by Qu, Williams, Beck and Medendorp (1992) and Qu, Piedmonte and Williams (1994) using generalized estimating equations for both sets of parameters via a Monte Carlo experiment. As a `reference' estimator for an equicorrelation model, the maximum likelihood (ML) estimator of the random effects probit model is calculated. The results show the mixed approach to be the most robust approach in the sense that the number of datasets for which the corresponding estimates converged was largest relative to the other two approaches. Furthermore, the mixed approach led to the most efficient non-ML estimators and to very efficient estimators for regression and correlation structure parameters relative to the ML estimator if individual covariance matrices were used.

The present paper deals with the comparison of the performance of different estimation methods for regression models with correlated binary responses. Throughout, we consider probit models where an underlying latent continous random variable crosses a threshold. The error variables in the unobservable latent model are assumed to be normally distributed. The estimation procedures considered are (1) marginal maximum likelihood estimation using Gauss-Hermite quadrature, (2) generalized estimation equations (GEE) techniques with an extension to estimate tetrachoric correlations in a second step, and, (3) the MECOSA approach proposed by Schepers, Arminger and Küsters (1991) using hierarchical mean and covariance structure models. We present the results of a simulation study designed to evaluate the small sample properties of the different estimators and to make some comparisons with respect to technical aspects of the estimation procedures and to bias and mean squared error of the estimators. The results show that the calculation of the ML estimator requires the most computing time, followed by the MECOSA estimator. For small and moderate sample sizes the calculation of the MECOSA estimator is problematic because of problems of convergence as well as a tendency of underestimating the variances. In large samples with moderate or high correlations of the errors in the latent model, the MECOSA estimators are not as efficient as ML or GEE estimators. The higher the `true' value of an equicorrelation structure in the latent model and the larger the sample sizes are, the more is the efficiency gain of the ML estimator compared to the GEE and MECOSA estimators. Using the GEE approach, the ML estimates of tetrachoric correlations calculated in a second step are biased to a smaller extent than using the MECOSA approach.