Stephen M. Willis

A PC program for performing multigroup longitudinal comparisons using the Potthoff-Roy analysis and orthogonal polynomials

A PC-program performing the Potthoff-Roy (PR) multigroup (G-sample) analysis of longtidinal data is described and illustrated. This program and the underlying statistical model are useful in the comparison of several longitudinal samples. Applications include the study of growth, development, adaptation, aging, and treatment effects (in short, any phenomenon in which the passage of time is important) for which serial data are available. Specifically, this method fits polynomials to the average growth curves in the samples, and tests hypotheses concerning the curves themselves and the individual coefficients of the polynomials. The program features the utilization of orthogonal polynomial regression coefficients (OPRCs) and is written in GAUSS, a relatively inexpensive yet comprehensive matrix programming language. It is documented that using OPRCs to comprise the within-individual or time design matrix has several advantages over the more usual choice of the successive-powers-of-t form of this matrix and an example of one important such advantage is provided. GAUSS was employed to make the program readily-accessible (i.e., executable code) to biomedical investigators. The GAUSS compiler is not required to run this program. Information regarding the availability of the program is provided in the Appendix.

Computation of Foulkes and Davis' nonparametric tracking index using GAUSS

Foulkes and Davis (1981) define tracking as the maintenance of relative rank over a given time span. This paper outlines the development of their statistic, based on a set of individual growth profiles, which estimates the degree of tracking observed in a one‐sample longitudinal data set and shows how confidence intervals for the corresponding population parameter may be constructed. An example using a measure of skeletal growth is given and a GAUSS program to do the computations is provided. (Information on obtaining the GAUSS program is provided in the Appendix.) Properties of this statistical approach to tracking are contrasted with another non‐parametric method based on Cohens kappa statistic.

PC program for comparing tracking indices in several independent groups

A method for computing a measure of tracking based on Cohens kappa statistic for one‐sample longitudinal data sets was previously described and implemented. This paper shows how one may test the equality of several kappas, each computed from an independent longitudinal sample. Thus, it is possible to formally compare groups of individuals with regard to stability in growth (or adaptive) patterns. Relative assessments of predictability in growth outcomes in different populations can be made with this approach. Also, when a common value of kappa is not contradicted by the data, a method to estimate this value and obtain a confidence interval for it is shown. A menu‐driven GAUSS program for carrying out the procedure is described and made available. The method and program are illustrated with three samples of Guatemalan children.

Regression imputation of missing values in longitudinal data sets

A stand-alone, menu-driven PC program, written in GAUSS, which can be used to estimate missing observations in longitudinal data sets is described and male available to interested readers. The program is limited to the situation in which we have complete data on N cases at each of the planned times of measurement t1, t2,..., tT; and we wish to use this information, together with the non-missing values for n additional cases, to estimate the missing values for those cases. The augmented data matrix may be saved in an ASCII file and subsequently imported into programs requiring complete data. The use of the program is illustrated. Ten percent of the observations in a data set consisting of mandibular ramus height measurements for N = 12 young male rhesus monkeys measured at T = 5 time points are randomly discarded. The augmented data matrix is used to determine the lowest degree polynomial adequate to fit the average growth curve (AGC); the regression coefficients are estimated and confidence intervals for them are determined; and confidence bands for the AGC are constructed. The results are compared with those obtained when the original complete data set is used.

Rao's polynomial growth curve model for unequal-time intervals: A menu-driven gauss program

For lack of alternatives, longitudinal data are often analyzed with cross-sectional statistical methods, for instance, t-tests, ANOVA and ordinary least-squares regression. Appropriate statistical software has been generally unavailable to investigators using serial records to study growth and development or treatment effects. In an earlier paper (Schneiderman and Kowalski, Am. J. Phys. Anthropol., 67 (1985) 323-333.) we described a suitable method, Raos polynomial growth curve model (Rao, Biometrika, 46 (1959) 49-58), and provided an SAS computer program for the analysis of a single sample of complete longitudinal data. This method included the computation of an average polynomial growth curve, its 95% confidence band, its coefficients and corresponding confidence intervals. The present paper extends this method to accommodate a sample with observations made at unequal time-intervals. Significant improvements in the accessibility, operation and user-friendliness of the program have been made, facilitated by recent advances in microcomputer technology. This stand-alone GAUSS program (no compiler necessary) runs on PC-compatibles and is available at a nominal cost. In this report we provide an overview of the statistical model, the general structure of the program, and give an example in which a developmental variable (human upper incisor angulation) is analyzed. Ease of installation and use, speed of execution and color graphic displays of growth curves and confidence bands, and most importantly, suitability to longitudinal data, make this method/program a potentially valuable tool for those interested in growth, development, and treatment effects in humans and other species. Some areas in which this method will have immediate applications are orthodontics, maxillofacial surgery and pediatrics.

PC program for estimating polynomial growth, velocity and acceleration curves when subjects may have missing data

A stand-alone, menu-driven PC program, written in GAUSS386i, for estimating polynomial growth, velocity, and acceleration curves from longitudinal data is described, illustrated and made available to interested readers. Missing data are accommodated: we assume that the study is planned so that individuals will have common times of measurement, but allow some of the sequences to be incomplete. The degrees, Di, adequate to fit the growth profiles of the N individuals are determined and the corresponding polynomial regression coefficients are calculated and can be saved in ASCII files which may then be imported into a statistical computing package for further analysis. Examples of the use of the program are provided.

PC program for assessing the effect of a treatment when subjects are growing: the randomized parallel groups design.

A method for separating the effects of a treatment from those of normal development in the case of a randomized parallel groups design with pre- and post-treatment measures is described and implemented. The program allows the user to enter either summary statistics (published data are often in this form), or the pre- and post-treatment measurements for each individual. The program is illustrated using data reflecting the extent to which a treatment can be expected to impede normal growth, but the method and program are more general than this. All that is required is that the measurement be one that normally increases over time.

A GAUSS program for computing the Foulkes-Davis tracking index for polynomial growth curves

We have previously published a GAUSS program for computing the Foulkes-Davis tracking index, gamma, from a one-sample longitudinal data set when no assumptions were made concerning the structure of the individual growth curves (Schneiderman et al., Am J Hum Biol, 4 (1992) 417-420). In this paper we consider the computation of the Foulkes-Davis index assuming that each individual growth curve may be adequately represented by a polynomial function in time and a GAUSS program performing these computations is made available. As with the two other tracking indices we have described, gamma and kappa (Schneiderman et al., Am J Hum Biol, 2 (1990) 475-490), this one can be used to evaluate regularity in patterns of growth or adaptation. An example is presented where statural growth in the same three groups considered in the earlier papers are analyzed. The small disparities between these and the earlier results are discussed in view of the different assumptions of the models and the differences in how they operationalize the concept of tracking.

Implementation of exact and approximate randomization tests for polynomial growth curves

Two stand-alone, menu-driven PC programs, written in GAUSS386i, which compare groups of growth curves in a completely randomized design using either (a) exact or (b) approximate randomization tests, are described, illustrated, and made available to interested readers. The programs accommodate missing data in the context of studies planned to have common times of measurement, but where some of the measurement sequences are incomplete. The measurement whose growth is being monitored need not have a Gaussian distribution. We consider the hypothesis that the mean growth curves in G groups are the same; and either compute the exact P value (exact test), or estimate, and provide a confidence interval for, the P value (approximate test).

PC program implementing an alternative to the paired t-test which adjusts for regression to the mean

In many biomedical research contexts, treatment effects are estimated from studies based on subjects who have been recruited because of high (low) measurements of a response variable, e.g., high blood pressure or low scores on a stress test. In this situation, simple change scores will overestimate the treatment effect; and the use of the paired t-test may find significant change due not to the treatment per se but, rather, due to regression towards the mean. A PC program implementing a procedure for adjusting the observed change for the regression effect in simple pre-test-post-test experiments is described, illustrated, and made available to interested readers. The method is due to Mee and Chua (Am Stat, 45 (1991) 39-42) and may be considered as an alternative to the paired t-test which separates the effect of the treatment from the so-called regression effect.