Assessing the effect of a treatment when subjects are growing at different rates

The analysis of covariance is often used in the context of premeasure/postmeasure designs to compare treatment and control groups in both randomized [1] and nonrandomized [2] studies. The intent is to adjust the difference between the changes in the 2 groups for any difference which might exist at baseline, i.e., for any difference between the premeasures in the 2 groups. An important assumption underlying the use of the analysis of covariance is that the slopes of the lines for the regression of the postmeasure on the premeasure in the 2 groups are equal. In this paper we describe a program which can be used to test the hypothesis of equal slopes; and performs an alternative analysis which does not depend on this assumption. This is done in the context of comparing treatment and control groups with respect to a measurement subject to natural maturation as in [3]. Equal slopes in this context means equal growth rates; unequal slopes implies that the 2 groups are growing at different rates. The method, known as the Johnson-Neyman procedure [4] is, however, more general than this, and can be used in any two-sample comparison where an alternative to the usual analysis of covariance is deemed appropriate. The procedure identifies a `region of significance' which is especially useful in practice. This region consists of a set of values of the premeasure for which the treatment and the control groups are significantly different with respect to the postmeasure.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31282/1/0000188.pd

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 made 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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30933/1/0000603.pd

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 growth and 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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31275/1/0000181.pd

Clustering on the basis of longitudinal data

A menu-drive PC program, ZDIST, for computing the distances between the estimated polynomial growth curves of subjects who have been followed longitudinally is described, illustrated, and made available to interested readers. These distances can be computed on the basis of the individual growth curves themselves and/or from estimates of individuals' growth velocity and acceleration curves. The resulting distance matrices can be saved in ASCII format and subsequently imported into any clustering program which accepts this type of input, e.g. SYSTAT.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30607/1/0000244.pd

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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31205/1/0000107.pd

ANCOVA for nonparallel slopes: the Johnson-Neyman technique

The Johnson-Neyman (JN) procedure, as originally formulated (Stat Res Mem, 1 (1936) 57-93), applies to a situation in which measurements on 1 dependent (response) variable, X, and 2 independent (predictor) variables, Z1 and Z2, are available for the members of 2 groups. The expected value of X is assumed to be a linear function of Z1 and Z2, but not necessarily the same function for both groups. The JN technique is used to obtain a set of values for the Z variables for which one would reject, at a specified level of significance [alpha] (e.g., [alpha] = 0.05), the hypothesis that the 2 groups have the same expected X values. This set of values, or `region of significance,' may then be plotted to obtain a convenient description of those values of Z1 and Z2 for which the 2 groups differ. The technique can thus be described as a generalization of the analysis of covariance (ANCOVA) which does not make the assumption that the regression coefficients for the regression of X on the covariates, Z1 and Z2, are equal in the groups being compared. In this paper we describe, illustrate and make available a menu-driven PC program (TXJN2) implementing the JN procedure.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31210/1/0000112.pd

A PC program for classification into one of several groups on the basis of longitudinal data

A stand-alone, menu-driven PC program, ZCLASS, written in GAUSS386i, for classifying subjects into one of several distinct, existing groups on the basis of longitudinal data is described, illustrated, and made available to interested readers. The program accepts data from studies where common times of measurement are planned, but missing data are accommodated in that one or more measurement sequences may be incomplete.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31477/1/0000399.pd

GTRACK: A PC program for computing Goldstein's growth constancy index and an alternative measure of tracking

This paper reviews Goldstein's `growth constancy index,' [xi], a measure of tracking which can be used to determine whether or not individuals maintain their relative positions in the distribution of a given measurement as that distribution changes over time. We suggest that [xi] is an appropriate measure of tracking when the (standardized) measurements arise in the context of a Model I ANOVA, but that the intraclass correlation coefficient, rl, may be preferred when a Model II ANOVA is applicable. We also describe -- and make available -- a PC program which allows the user to choose between Model I and Model II, and computes the appropriate tracking index and confidence intervals for the corresponding parameter.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31409/1/0000326.pd

A PC program for growth prediction in the context of Rao's polynomial growth curve model

We consider the problem of growth prediction in the context of Rao's [1] one-sample polynomial growth curve model and provide a PC program, written in GAUSS, to perform the associated computations. Specifically, the problem considered is that of estimating the value of the measurement under consideration for a "new" individual at the Tth time point given measurements on that individual at T-1 previous points in time and the values of the measurement on N "similar" individuals at all T time points. The times of measurement t1, t2, ..., tT need not be equally spaced, but we assume that each of the N individuals comprising the normative sample were measured at these times. The method and the program are illustrated using the leave-one-out method on a sample of N = 12 male rhesus monkeys whose mandibular ramus height was measured five times at yearly intervals.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30069/1/0000439.pd