The Analysis of Count Data: A Gentle Introduction to Poisson Regression and Its Alternatives

Count data reflect the number of occurrences of a behavior in a fixed period of time (e.g., number of aggressive acts by children during a playground period). In cases in which the outcome variable is a count with a low arithmetic mean (typically < 10), standard ordinary least squares regression may produce biased results. We provide an introduction to regression models that provide appropriate analyses for count data. We introduce standard Poisson regression with an example and discuss its interpretation. Two variants of Poisson regression, overdispersed Poisson regression and negative binomial regression, are introduced that may provide more optimal results when a key assumption of standard Poisson regression is violated. We also discuss the problems of excess zeros in which a subgroup of respondents who would never display the behavior are included in the sample and truncated zeros in which respondents who have a zero count are excluded by the sampling plan. We provide computer syntax for our illustrations in SAS and SPSS. The Poisson family of regression models provides improved and now easy to implement analyses of count data.

[Supplementary materials are available for this article. Go to the publisher's online edition of Journal of Personality Assessment for the following free supplemental resources: the data set used to illustrate Poisson regression in this article, which is available in three formats—a text file, an SPSS database, or a SAS database.]     

SPSS macros to compare any two fitted values from a regression model

In regression models with first-order terms only, the coefficient for a given variable is typically interpreted as the change in the fitted value of Y for a one-unit increase in that variable, with all other variables held constant. Therefore, each regression coefficient represents the difference between two fitted values of Y. But the coefficients represent only a fraction of the possible fitted value comparisons that might be of interest to researchers. For many fitted value comparisons that are not captured by any of the regression coefficients, common statistical software packages do not provide the standard errors needed to compute confidence intervals or carry out statistical tests—particularly in more complex models that include interactions, polynomial terms, or regression splines. We describe two SPSS macros that implement a matrix algebra method for comparing any two fitted values from a regression model. The !OLScomp and !MLEcomp macros are for use with models fitted via ordinary least squares and maximum likelihood estimation, respectively. The output from the macros includes the standard error of the difference between the two fitted values, a 95% confidence interval for the difference, and a corresponding statistical test with its p-value.     

Improved Regression Calibration

The likelihood for generalized linear models with covariate measurement error cannot in general be expressed in closed form, which makes maximum likelihood estimation taxing. A popular alternative is regression calibration which is computationally efficient at the cost of inconsistent estimation. We propose an improved regression calibration approach, a general pseudo maximum likelihood estimation method based on a conveniently decomposed form of the likelihood. It is both consistent and computationally efficient, and produces point estimates and estimated standard errors which are practically identical to those obtained by maximum likelihood. Simulations suggest that improved regression calibration, which is easy to implement in standard software, works well in a range of situations.     

On the correlation of a naturally and an artificially dichotomized variable

A method is suggested for estimating the correlation of a naturally (X) and an artificially (Y) dichotomized variable. It is assumed that a normal random variable (L) underlies the artificially dichotomized variable. The proposed correlation coefficient recovers the product moment correlation coefficient between X and L from a fourfold table of X and Y. The suggested correlation coefficient ν is contrasted with the phi correlation and the biserial η. The biserial η was proposed by Karl Pearson and is conceptually related to the new correlation coefficient. However, in addition, Pearson's biserial η invokes the assumption that the marginal distribution of L is normal, which contradicts its basic assumptions and thus does not recover the true correlation of L and X. Finally, an approximation is provided to simplify the calculation of ν and its standard error.     

Biases and Standard Errors of Standardized Regression Coefficients

The paper obtains consistent standard errors (SE) and biases of order O(1/n) for the sample standardized regression coefficients with both random and given predictors. Analytical results indicate that the formulas for SEs given in popular text books are consistent only when the population value of the regression coefficient is zero. The sample standardized regression coefficients are also biased in general, although it should not be a concern in practice when the sample size is not too small. Monte Carlo results imply that, for both standardized and unstandardized sample regression coefficients, SE estimates based on asymptotics tend to under-predict the empirical ones at smaller sample sizes.     

The Impacts of Ignoring a Crossed Factor in Analyzing Cross-Classified Data

Cross-classified random-effects models (CCREMs) are used for modeling nonhierarchical multilevel data. Misspecifying CCREMs as hierarchical linear models (i.e., treating the cross-classified data as strictly hierarchical by ignoring one of the crossed factors) causes biases in the variance component estimates, which in turn, results in biased estimation in the standard errors of the regression coefficients. Analytical studies were conducted to provide closed-form expressions for the biases. With balanced design data structure, ignoring a crossed factor causes overestimation of the variance components of adjacent levels and underestimation of the variance component of the remaining crossed factor. Moreover, ignoring a crossed factor at the kth level causes underestimation of the standard error of the regression coefficient of the predictor associated with the ignored factor and overestimation of the standard error of the regression coefficient of the predictor at the (k?1)th level. Simulation studies were also conducted to examine the effect of different structures of cross-classification on the biases. In general, the direction and magnitude of the biases depend on the level of the ignored crossed factor, the level with which the predictor is associated at, the magnitude of the variance component of the ignored crossed factor, the variance components of the predictors, the sample sizes, and the structure of cross-classification. The results were further illustrated using the Early Childhood Longitudinal Study-Kindergarten Cohort data.     

Determination of the Regression Coefficients and Their Associated Standard Errors in Hierarchical Regression Analysis

Hierarchical regression analysis is potentially a very useful statistical technique for establishing the significance of sets of predictor variables. However, when a hierarchical analysis which is based on theory is performed, some estimation procedures for the regression coefficients and their associated standard errors are potentially inappropriate. Specifically, the hierarchical regression equations, the incremental or hierarchical tests, and the parameter estimation of this procedure may not correspond. This problem is investigated by the development of four approaches (simultaneous, stagewise, orthogonal, and hierarchical) of estimation to the analysis. For each method, regression, coefficient estimators and their standard errors are determined. By comparison of these approaches, the use of orthogonalized sets of predictor variates or a modification to a series of simultaneous analyses are recommended as the most sensible technique for a theory driven hierarchical analysis.     

A default Bayesian hypothesis test for correlations and partial correlations

We propose a default Bayesian hypothesis test for the presence of a correlation or a partial correlation. The test is a direct application of Bayesian techniques for variable selection in regression models. The test is easy to apply and yields practical advantages that the standard frequentist tests lack; in particular, the Bayesian test can quantify evidence in favor of the null hypothesis and allows researchers to monitor the test results as the data come in. We illustrate the use of the Bayesian correlation test with three examples from the psychological literature. Computer code and example data are provided in the journal archives.     

Computing and comparing correlation and regression coefficients using a pocket calculator

Programs suitable for pocket calculators using reverse Polish notation are described. Program 1 computes regression coefficients, correlation coefficient, and standard error of estimate for paired data. Program 2 performs at test to compare the slopes of two regression lines. Program 3 computes F ratios to test the departure of a regression slope from zero and to test linearity of the regression. Programs 4 and 5 test the significance of (differences between independent and correlated correlation coefficients, respectively.     

Flexible Approaches to Computing Mediated Effects in Generalized Linear Models: Generalized Estimating Equations and Bootstrapping

In behavioral research, interest is often in examining the degree to which the effect of an independent variable X on an outcome Y is mediated by an intermediary or mediator variable M. This article illustrates how generalized estimating equations (GEE) modeling can be used to estimate the indirect or mediated effect, defined as the amount by which the regression coefficient of X on Y changes after adjusting for M. Advantages of this method are: (a) it applies to the class of generalized linear models, including linear, logistic, and Poisson regression as special cases; (b) it allows multiple independent variables and mediators in the same model; and (c) asymptotically valid standard errors and confidence intervals are obtained using standard software. This methodology is compared with the bootstrap, another general methodology that can be applied to the same broad class of models, and is evaluated using simulation in both linear and logistic regression scenarios. The methods are utilized to examine the degree to which the effect of low birthweight status on internalizing symptoms at age 20 is mediated through IQ at age 8.     

Exact Analysis of Squared Cross-Validity Coefficient in Predictive Regression Models

In regression analysis, the notion of population validity is of theoretical interest for describing the usefulness of the underlying regression model, whereas the presumably more important concept of population cross-validity represents the predictive effectiveness for the regression equation in future research. It appears that the inference procedures of the squared multiple correlation coefficient have been extensively developed. In contrast, a full range of statistical methods for the analysis of the squared cross-validity coefficient is considerably far from complete. This article considers a distinct expression for the definition of the squared cross-validity coefficient as the direct connection and monotone transformation to the squared multiple correlation coefficient. Therefore, all the currently available exact methods for interval estimation, power calculation, and sample size determination of the squared multiple correlation coefficient are naturally modified and extended to the analysis of the squared cross-validity coefficient. The adequacies of the existing approximate procedures and the suggested exact method are evaluated through a Monte Carlo study. Furthermore, practical applications in areas of psychology and management are presented to illustrate the essential features of the proposed methodologies. The first empirical example uses 6 control variables related to driver characteristics and traffic congestion and their relation to stress in bus drivers, and the second example relates skills, cognitive performance, and personality to team performance measures. The results in this article can facilitate the recommended practice of cross-validation in psychological and other areas of social science research.     

Familiality estimates from restricted samples

In a recent paper here, Benbow, Zonderman, and Stanley (1983) report that the coefficient of regression of offspring IQ on parental IQ is much lower among the gifted than in the population at large. Thus, Benbow, Stanley, Kirk, and Zonderman conclude in a second paper, the gifted resemble their parents less than do people in general. In this paper, I show that this result is an artifact of the particular estimator of the regression coefficient employed by Bendoq, Zonderman, and Stanley. The leastsquares estimator, which they employ, is severely biased downward, if the sample on the dependent variable is restricted to the upper tail of the distribution, and this is precisely the nature of Benbow et al.'s sample. That is to say, in a bivariate normal distribution with constant regression coefficient, samples restricted to values of the dependent variable (here, child's IQ) above a certain value will always produce a lower regression coefficient than unrestricted samples drawn from the entire but same distribution. I introduce an unbiased estimator that can be calculated from the sample statistics reported in the Benbow, Zonderman, and Stanley article and find that the coefficient of regression of gifted child's IQ on parental IQ is, in fact, higher than the regression coefficients reported in the literature for unrestricted samples. That is, Benbow et al.'s data suggest that the gifted in fact resemble their parents more than do persons in general.     

Alpha's standard error (ASE): an accurate and precise confidence interval estimate

This research presents the inferential statistics for Cronbach's coefficient alpha on the basis of the standard statistical assumption of multivariate normality. The estimation of alpha's standard error (ASE) and confidence intervals are described, and the authors analytically and empirically investigate the effects of the components of these equations. The authors then demonstrate the superiority of this estimate compared with previous derivations of ASE in a separate Monte Carlo simulation. The authors also present a sampling error and test statistic for a test of independent sample alphas. They conclude with a recommendation that all alpha coefficients be reported in conjunction with standard error or confidence interval estimates and offer SAS and SPSS programming codes for easy implementation.     

Predicting the point at which to change interventions: Borrowing a metric from econometrics

Conclusions The field of applied behavior analysis is a conservative one, a field that has developed many of its own methods. Despite our success, we have much to learn from other fields. One such field may be economics in general and econometrics in particular. Behavior analysis, education, and psychology are, of course, far less sophisticated in statistics than is economics — a fact of which we might be proud (see Baer, 1977). While the procedure presented here is not sophisticated in the mathematical sense, it might be valuable to us in determining when to change interventions, a decision for which we currently have no metric. The procedure seems easy to translate to ABA, has been well-tested and is a standard procedure in econometrics, is easy to use, and may present a reasonable alternative to our present way of deciding when to change interventions.     

测验同质性系数及其区间估计

在决定将多维测验分数合并成测验总分时, 应当考虑测验同质性。如果同质性太低, 合成总分没有什么意义。同质性高低可以用同质性系数来衡量。用来计算同质性系数的模型是近年来受到关注的双因子模型(既有全局因子又有局部因子), 测验的同质性系数定义为测验分数方差中全局因子分数方差所占的比例。本文用Delta法推导出计算同质性系数的标准误公式, 进而计算其置信区间。提供了简单的计算同质性系数及其置信区间的程序。用一个例子说明如何估计同质性系数及其置信区间, 通过模拟比较了用Delta法和用Bootstrap法计算的置信区间, 发现两者差异很小。     

A shortened version of Raven’s standard progressive matrices for children and adolescents

Numerous developmental studies assess general cognitive ability, not as the primary variable of interest, but rather as a background variable. Raven’s Progressive Matrices is an easy to administer non-verbal test that is widely used to measure general cognitive ability. However, the relatively long administration time (up to 45 min) is still a drawback for developmental studies as it often leaves little time to assess the primary variable of interest. Therefore, we used a machine learning approach – regularized regression in combination with cross-validation – to develop a short 15-item version. We did so for two age groups, namely 9 to 12 years and 13 to 16 years. The short versions predicted the scores on the standard full 60-item versions to a very high degree r = 0.89 (9–12 years) and r = 0.93 (13–16 years). We, therefore, recommend using the short version to measure general cognitive ability as a background variable in developmental studies.     

Regression effect in judgments to determine equal response contours

The data of Sheldon’s experiment (1973) on equal onset contours for vibrotactile stimuli display a marked regression effect. This effect is one of bias and is not to be confused with statistical regression toward the mean. It resembles the effect found by Stevens (1955, 1961) and others in data from scale-matching experiments, but the present effect involves bias toward the level of the standard on the dimension which the subject is adjusting to achieve his match, rather than being related to the entire stimulus range. Both regression effects appear to increase with the difficulty of the matching task.