It is well-known that structural equation modeling (SEM) can represent a variety of traditional multivariate statistical models. This fact does not necessarily mean that SEM should be used for the traditional models. It is often said that a general model is more difficult to handle than a specific model developed for a given situation. In this paper, we shall clarify relative advantages between SEM and several traditional statistical models. Rather than comparison in mathematical properties, we shall discuss how and when SEM outperforms corresponding traditional models in practical situations. Special attention is paid to statistical analysis of a scale score, the sum of indicator variables determined by factor analysis. In particular, we shall study relative advantages between (i) confirmatory factor analysis and exploratory factor analysis, (ii) multiple indicator analysis and correlational and regression analysis of scale scores, (iii) analysis of factor means and analysis of variance of scale scores, and (iv) path analysis and multiple regression analysis.
Kano (this volume) states that it is possible to obtain an accurate estimate of disattenuated correlation even when the number of items per construct is small. This statement seems to support the common practice in the use of structural equation models (SEM) in which the number of indicator variables per construct are kept very small in order to insure a high degree of model fit. The present paper points out that latent variables with a small number of indicator variables tend to suffer from low generalizability and low validity, and thereby impair the value of the research finding. This issue exemplifies the fact that the relevance of goodness of model fit should be critically examined in reference to the purpose of research. An issue of the correction for attenuation is also examined in some detail in order to understand the nature of the correction implemented in the SEM framework.
Although the regression analysis among estimated factor scores may be criticized due to the attenuation of the estimated regression coefficients, this classical procedure can well separate the inferences of structural models from some propagation of information from the measurement model by approximately applying the conditionality principle. The author justifies and modifies the traditional approach by deriving the structural model with additional paths from measurement errors of the indicator variables to the response variables whose formal inference is nearly equivalent to the classical approaches by factor scores.
The purpose of this paper is to show a method of describing a corporate image fluctuation using factor scores derived from simultaneous analysis in multiple populations with structured means. A confirmatory factor analysis (CFA) model was applied to the independent, random samples collected in survey research conducted once a year since 1988. As a result, a multiple population CFA model, which constrained the model form, the values of factor loadings and the variances-covariance matrix of exogenous variables to be the same in all groups (1988-1997), fitted to the data well. In traditional exploratory factor analysis (EFA) models, it has been difficult to compare factor scores in different years owing to the lack of a fixed factor pattern across groups. However, structural equation modeling (SEM) settled this problem.
In this rejoinder, special attentions are paid to error covariances and specific factors in the comparison between SEM and traditional methods. When a factor analysis model receives a poor fit, it does not make sense to simply remove important variables although inconsistent with the factor analysis model, as pointed out by the discussants. It is to be emphasized that a better way than removing the variables is to allow for error covariances, in order to overcome the inconsistency problem. The model with error covariances guarantees the invariance of estimation results over item selection. The discussants pointed out that an important difference between a scale score (sum of items) and a measurement model by effect indicators in SEM is that a scale score includes specific factors whereas a measurement model excludes them. Practitioners could use scale scores when they are interested in effects of specific factors as well as a common factor. It is argued, however, that appropriate use of error terms and a common factor in SEM can make better inference than the use of unidimensional scale scores, because the error terms of effect indicators contain information on specific factors and they can individually evaluate the effects of the common factor and the specific factors in SEM. Other related topics are also discussed.
Power-normal distribution is a parametric family of distributions including log-normal and normal distributions as special cases, based on the power-transformation proposed by Box & Cox (1964). In this paper, basic properties of the bivariate power-normal distribution, which is an extension of the power-normal distribution to a two-dimensional case, are considered through some numerical examples and a mid-sized simulation from a viewpoint of the precision of parameter estimates and normality of the bivariate power-transformed distribution. Thus, in order to consider the effect of magnitude of truncation of the bivariate power-transformed distribution on parameter estimates, the two algorithms are used to estimate the parameters; the first allows the truncation, the second, does not. The result shows that the shapes of the bivariate power-normal distribution have the effect on the parameter estimates, and that for practical usage, the difference between the two algorithms with/without the truncation could be ignored.
Various types of ranking are in evidence throughout the real world. In this paper, we consider in particular, the field of sports, where ranking is most common. The 2002 FIFA World Cup Korea/Japan, which took place in June 2002, and the winner was Brazil. Since Brazil's FIFA world ranking was 2 just before the tournament, this result seemed quite plausible. On the other hand, however, France (ranking 1), Argentine (ranking 2) and Portugal (ranking 5) were not able to go to Stage 2 (best 16). These facts raise a question as to whether the FIFA ranking is credible in predicting the winners. In order to assess the credibility of such ranking, we introduce a simple model in this paper. Our model, which is an extension of the so-called Bradley-Terry model, contains an unknown parameter which indicates the credibility of the ranking. For the 2002 FIFA World Cup data, we estimated the credibility parameter and concluded that the FIFA world ranking for the 2002 tournament data was more credible in Stage 2 than in Stage 1.
It is known that the execution of a price and non-price promotion has a strong influence on the sales of a brand sold in a supermarket. Usually, we can easily obtain information on the degree of price promotion from POS data. On the other hand, unless the investigator collects information on the execution of non-price promotion in every retail store, we can not obtain this information. In this article, we consider the problem of identifying whether or not non-price promotion is conducted. We treat a non-price promotion execute/non-execute as a ‘state’. In that case, we assume that there is an unknown stationary probability matrix which describes the probability of a transition between states. Each state is characterized by a different stationary time series with unknown parameters. The objective of the analysis is to identify the regression model and to assign a state probability to each time instant. Finally, we give a high precision estimator of a past non-price promotion based on the proposed model.
In various research fields, a pretest-posttest experiment is an important and frequently employed method to evaluate treatment effects. In a pretest-posttest experiment, the measurements are made both at baseline and at follow-up in order to eliminate individual variability. There exist several methods that have been proposed to estimate the treatment effect and/or to test that there is no effects at all. In pretest-posttest designs, screening is often made upon the baseline measurements: that is only the individuals who meet some pre-specified requirement are allowed to enter the experiment. In such cases, we have to take into account the effect of the regression to the mean in the analysis to obtain proper conclusions. In this review article, various topics are discussed which are related to pretest-posttest designs and to the regression to the mean as well. We give mathematical results concerning moments of truncated normal distributions, which are useful to understand the underlying theory of the pretest-posttest data. Extensive references are also given.