Analysis of Covariance Structures and Probabilistic Binary Choice Data

In this chapter, we bring together the notions of measurement error discussed in Chaps. 3 and 4 with the structural modeling of simultaneous relationships presented in Chap. 6. We demonstrate that a bias is introduced when estimating the relationship between two variables measured with error if that measurement error is ignored. We then present a methodology for estimating the parameters of structural relationships between variables that are not observed directly: analysis of covariance structures. We focus on the role of the measurement model as discussed in Chap. 4 with the confirmatory factor analytic model.

Studies of families, twins, and adoptees have helped to quantify the genetic contributions to and overlaps between depression, anxiety, phobias and alcoholism, and to refine the boundaries of the schizophrenia spectrum. Analyses of covariance structures in twin data have confirmed genetic susceptibility and recent life stresses as the major determinants of depression. Genetic modelling of family data on schizophrenia and bipolar disorder indicates three or more common genes each having a small multiplicative effect on risk, although rare major genes may be present in some families. Linkage studies have localised genes for familial Alzheimer's disease on chromosomes 14 and 21; disease mutations on these chromosomes have since been isolated. Association studies have identified susceptibility (or protective) genes for Alzheimer's disease and alcoholism. Several tentative linkage and association findings in schizophrenia and bipolar disorder require further study.

Research studying robustness of maximum likelihood (ML) statistics in covariance structure analysis has concluded that test statistics and standard errors are biased under severe non-normality. An estimation procedure known as asymptotic distribution free (ADF), making no distributional assumption, has been suggested to avoid these biases. Corrections to the normal theory statistics to yield more adequate performance have also been proposed. This study compares the performance of a scaled test statistic and robust standard errors for two models under several non-normal conditions and also compares these with the results from ML and ADF methods. Both ML and ADF test statistics performed rather well in one model and considerably worse in the other. In general, the scaled test statistic seemed to behave better than the ML test statistic and the ADF statistic performed the worst. The robust and ADF standard errors yielded more appropriate estimates of sampling variability than the ML standard errors, which were usually downward biased, in both models under most of the non-normal conditions. ML test statistics and standard errors were found to be quite robust to the violation of the normality assumption when data had either symmetric and platykurtic distributions, or non-symmetric and zero kurtotic distributions.

To estimate the total (CVt), within-subject (CVw) and between-subject (CVb) variation in free-living energy expenditure as measured by the doubly-labelled water (DLW) technique. To examine the limitation of the DLW measurement of energy expenditure for evaluating reported energy intake. To estimate the probable minimum and maximum 'habitual' energy expenditures for a sustainable lifestyle. Review and analysis of individual data from 25 studies with repeat DLW measurements of energy expenditure (EE). Pooled mean CVw derived from 21 studies was 11.8% for EE and 12.3% for physical activity level (PAL). Multiple regression analysis of CVw in 25 studies found a positive association with time span between measurements. At zero time CVw for EE was 8.2% rising to 9.6% at 13 weeks and 15.4% at 52 weeks. At the same time points CVw for PAL was 9.1%, 10.0% and 13.4% respectively. Pooled mean CVt was 13.0% for EE and 10.7% for PAL. CVb calculated from pooled mean CVt and CVw was 20.6% for EE and 7.2% for PAL. 95% confidence limits of PAL in 11 age-sex groups averaged 1.2 to 2.2. The analysis supported previous estimates of 8% for within-subject variation in DLW measurements including analytic plus inherent biologic variation. Variation that included changes in weight, season and activity increased with increased time between measurements to about 15% at a time span of 12 months. Confidence limits of agreement between EE and reported energy intake were estimated to range from +/-15% to +/-32%. Estimates of the range of usual EE for normally active persons ranged from 1.3 to 2.2.

In this study, we have considered the modeling and analyses of binary data. We modeled binary data with categorical predictors, using logistic regression to develop a statistical method. We found that ANOVA-type analyses often performed unsatisfactory, even when using arcsine-square-root trans-formations. We concluded that such methods are not appropriate, especially in cases where the fractions were close to 0 or 1. The logistic transformation of fraction data could be a promising alternative, but it is not desirable in the statistical sense. The major purpose of this paper is to demonstrate that logistic regression with an ANOVA-model like parameterization aids our understanding and provides a somewhat different, but sound, statistical background. We examined a simple real world example to show that we can efficiently test the significance of regression parameters, look for interactions, estimate related confidence intervals, and calculate the difference between the mean values of the referent and experimental subgroups. This paper demonstrates that precise confidence interval estimates can be obtained using the proposed ANOVA-model like approach. The method discussed here can be extended to any type of experimental fraction data analysis, particularly for experimental design.

Covariance structures analysis is often used in nursing research to appraise statistical models reflecting complex human health processes. The model selection approach in covariance structures analysis is designed to select the "best" model from a specified set of theoretically defensible, competing alternatives, all of which are viewed as approximations. Model selection criteria explicitly incorporate both model misfit in the population and sampling error to evaluate the set of models. The result is that interpretability of model parameters and goodness-of-fit are enhanced simultaneously. Relative merits of the model selection approach are identified in light of technical concerns, parsimony, and use of scientific theory in nursing.

This study was conducted to examine factors associated with postpartum depression and abusive behavior in mothers with infants. Data were collected from baby check-ups in Japan and 413 participants were included in an analysis using: (i) an Original Questionnaire; (ii) the Zung Self-rating Depression Scale (ZSDS); (iii) the Parental Bonding Instrument; and (iv) the Childcare Anxiety Scale (CAS). Covariance structural analyses were performed to examine interconnections among the Parental Bonding Instrument subscales, CAS subscales ('worry about parenting', 'burden of nursing time', 'difficulty of bonding'), ZSDS, 'fear of being abusive', and 'abusive behavior'. Of the 413 mothers, 14.5% showed higher than moderate levels of depression (ZSDS > or = 50). In covariance structural analyses, 'depression' was strongly influenced by 'worry about parenting' in all variances, but was not associated with 'abusive behavior'. 'Worry about parenting' also had a strong influence on 'fear of being abusive', but did not affect 'abusive behavior'. Low 'maternal care' had most influence on 'difficulty of bonding', and 'difficulty of bonding' only affected 'abusive behavior'. The outcome of this study suggests that excessive worrying related to postpartum depression, 'fear of being abusive', and bonding difficulty are primary predictors of child abuse. Postpartum depression was not a predictor for abusive behavior after exclusion of the impact of bonding difficulties on abusive behavior. Therefore, the correlation between postpartum depression and abusive behavior identified in previous reports may have been influenced by bonding difficulties.

The purpose of this study was to clarify the role of community factors in parents' quality of child-nurturing life (QCNL). We developed a questionnaire to evaluate the degree of QCNL and determine the structural factors related to QCNL as community factors related to parents' QCNL derived from focus group interviews and the Delphi technique. The questionnaire also included the battery of the self-rating depression scale and Tsumori-Inage Infant's Developmental Test. Using the questionnaire, we then conducted a quantitative survey of parents whose children attended nursery schools in Kumamoto Prefecture. Factor analysis, calculation of the mean score and/or ratio to each item, Pearson's correlation coefficient, t test, multiple regression analysis, and covariance structure analysis were performed. The questionnaire we developed consisted of seven items with 75 elements, involving ten elements as community factors. Subjects included 699 parents (mean age 33.6 ± 5.4 years) and 965 children (age range 0-6 years). Factor analysis revealed that community factors consisted of five factors, such as "lifestyle rooted in the ground," "balance of housekeeping and work," "community network," "amenity," and "regeneration of life". These factors may be dominant in a rural area. Finally, we developed a structural model with "community factors," QCNL, QOL, and "child growth" by covariance structural analysis. The analysis revealed that community factors had a positive relation to parents' QCNL (r = 0.81, p < 0.001) and that parental SDS score had a negative relation to parents' QCNL (r = -0.59, p < 0.001). The analysis did show that community factors were positively related to the sound growth of children. The covariance structure analysis revealed that community factors were associated with parents' QCNL, SDS, and "child growth."

Several studies have shown increased plasma concentrations of neutrophil elastase in complex with alpha 1-protease inhibitor and/or lactoferrin in inflammatory conditions, and serial measurements have been advocated in order to follow disease activity. However, data on the magnitude of the within-subject variation are necessary for evaluation of the significance of changes in results obtained on analysis of serial samples. Within-subject variation of elastase/alpha 1-protease inhibitor complexes and lactoferrin over a short time was studied in six young men who had blood samples drawn every 4 h over 2 days. Within-subject variation over a longer time was investigated in 12 healthy adults, 6 females and 6 males, who had blood samples drawn in the morning once a week for 10 weeks. From five of the females and five of the males, blood samples were also drawn every morning on 5 consecutive days during 1 week. Within-subject variations over hours, days, and weeks were estimated as 0.050, 0.124, and 0.148 for elastase/alpha 1-protease inhibitor complexes and as 0.101, 0.119, and 0.143 for lactoferrin. A tendency towards variation of LAC with menstrual cycle in fertile females was noticed. From within-subject variation, between-subject variation and analytical variation, indices of individuality were calculated as 1.1 and 1.8 for elastase/alpha 1-protease inhibitor complexes and lactoferrin, respectively. This means that within-subject variation for lactoferrin is quite small compared to between-subject variation, and the usefulness of reference limits is very limited, when interpreting results from individual patients. For elastase/alpha 1-protease inhibitor complexes, the use of reference limits might be more appropriate, although still not optimal.

Seven subjects were tested, each on eight separate sessions, for brainstem auditory evoked responses to monaural right, monaural left, and binaural stimulus presentations. Comparisons of between-subject vs. within-subject variability of the absolute latencies of vertex-positive peaks expressed in terms of the coefficient of variation indicate that: within-subject stability is greater than between-subject stability for the five brainstem peaks; between-subject variability shows significant differences due to peak but not to ear of presentation; within-subject variability shows significant differences due to both peak and ear; comparisons of within-subject variability over time show significant differences due to peak but not to time; patterns of individual variation within the brainstem series are characterized by increases in stability of peak latencies over time, and by replicability of stability profiles over time. Other measures of latency and amplitude based on this series of responses are planned for a subsequent report.

The analysis of covariance structures (Jöreskog, 1973) is adapted to the simultaneous maximum likelihood estimation of genetical and environmental factor loadings and specific variances. The goodness of fit is tested by chi square and standard errors of parameter estimates can be obtained. Any linear model used in univariate genetical analyses can be extended to the multivariate case. Most biological hypotheses about the relationships between variables can be specified by a variety of factor models. Individual parameters can be given fixed values or set to zero and hypotheses concerning the congruence of genetical and environmental correlations can be tested. The method is illustrated with published twin data on cognitive abilities.

This study is the first to empirically validate the social self-concept component of the R. J. Shavelson, J. J. Hubner, and G. C. Stanton (1976) model. The primary purpose was to test for each of 3 age groups--preadolescents (Grade 3), early adolescents (Grade 7), and late adolescents (Grade 11)-3 hypotheses bearing on the structure of social self-concept within the context of this model: (a) that it is multidimensional, (b) that it is hierarchically ordered, and (c) that it becomes increasingly differentiated with age. Given evidence of a hierarchical social self-concept structure, a secondary focus of the study was to determine the extent to which this pattern held across age. On the basis of the analysis of covariance structures within the framework of confirmatory factor analysis, results revealed a multidimensional social self-concept structure that becomes increasingly differentiated and a hierarchical ordering that becomes better defined with age. Overall, findings were consistent with both the R. J. Shavelson et al. (1976) conceptualization of self-concept structure and developmental processes that underlie self-concept formation.

A general method is presented for estimating variance components when the experimental design has one random way of classification and a possibly unbalanced fixed classification. The procedure operates on a sample covariance matrix in which the fixed classes play the role of variables and the random classes correspond to observations. Cases are considered which assume (i) homogeneous and (ii) nonhomogeneous error variance, and (iii) arbitrary scale factors in the measurements and homogeneous error variance. The results include maximum-likelihood estimations of the variance components and scale factors, likelihood-ratio tests of the goodness-of-fit of the model assumed for the design, and large-sample variances and covariances of the estimates. Applications to mental test data are presented. In these applications the subjects constitute the random dimension of the design, and a classification of the mental tests according to objective features of format or content constitute the fixed dimensions.

Analysis of covariance structures is the common term for a number of techniques for analyzing multivariate data in order to detect and assess latent (unobserved) sources of variation and covariation in the observed measurements. The techniques of covariance structure analysis are general and flexible in that they can handle many types of covariance structures useful especially in the behavioral and social sciences. Although these techniques can be used for exploratory analysis, they have been most successfully applied to confirmatory analysis where the type of covariance structure is specified in advance. A covariance structure of a specified kind may arise because of a specified substantive theory or hypothesis, a given classificatory design for the measures, known experimental conditions, or because of results from previous studies based on extensive data. Sometimes the observed variables are ordered through time, as in longitudinal studies, or according to linear or circular patterns, as in Guttman’s (1954) simplex and circumplex models, or according to a given causal scheme, as in path analysis.

This paper states and proves the asymptotic properties of constrained generalized least squares estimators in the analysis of covariance structures in multiple populations with arbitrary distributions of variables. Asymptotic chi-square tests are also presented to permit evaluation of the goodness-of-fit of models. The currently known results for multiple population models based on variables that are multivariate normally distributed are obtained as a special case.