Preliminary tests of equality of variances used before a test of location are no longer widely recommended by statisticians, although they persist in some textbooks and software packages. The present study extends the findings of previous studies and provides further reasons for discontinuing the use of preliminary tests. The study found Type I error rates of a two-stage procedure, consisting of a preliminary Levene test on samples of different sizes with unequal variances, followed by either a Student pooled-variances t test or a Welch separate-variances t test. Simulations disclosed that the twostage procedure fails to protect the significance level and usually makes the situation worse. Earlier studies have shown that preliminary tests often adversely affect the size of the test, and also that the Welch test is superior to the t test when variances are unequal. The present simulations reveal that changes in Type I error rates are greater when sample sizes are smaller, when the difference in variances is slight rather than extreme, and when the significance level is more stringent. Furthermore, the validity of the Welch test deteriorates if it is used only on those occasions where a preliminary test indicates it is needed. Optimum protection is assured by using a separate-variances test unconditionally whenever sample sizes are unequal.

The notion that noriparnmelric methods are required as a replacement of parametric statistical methods when the scale of measurement in a research study does not achieve a certain level was discussed in light of recent developments in representational measurement theoiy. A new approach to examining the problem via computer simulation was introduced. Some of the beliefs that have been widely held by psychologists lor several decades were examined by means of a computer simulation study that mimicked measurement of an underlying empirical structure and performed two-sample Student /-tests on the resulting sample data. It was concluded that there is no need to replace parametric statistical tests by nonparamelric methods when the scale of measurement is ordinal and not iuierval-

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It is widely believed that measures of gain, growth, or change, expressed as simple differences between pretest and posttest scores, are inherently unreliable. It is also believed that gain scores lack predictive validity with respect to other criteria. However, these conclusions are based on misleading assumptions about the values of parameters in familiar equations in classical test theory. The present paper examines modified equations for the validity and reliability of difference scores that describe applied testing situations more realistically and reveal that simple gain scores can be more useful in research than commonly believed.Over a quarter century ago, Cronbach & Furby (1970) and many other authors (e.g., Gulliksen, 1950;Lord & Novick, 1968) concluded that simple differences between pretest and posttest scores have questionable value in behavioral and social science research. Yet this conclusion seems incompatible with the intuition of researchers in many disciplines who assume that measures of gains, changes, differences, growth, and the like are meaningful in experimentation, program evaluation, educational accountability studies, and the investigation of developmental growth and change.During the past two decades, many researchers using difference (or gain) scores have had difficulty justifying the use of such measures, even when they appear to yield interesting and reproducible findings. However, recent research on this topic has provided results favorable to simple gain scores (e.