In research investigations, it is not unusual to encounter situations in which one or more of the assumptions underlying the F test are seriously violated. One way to deal with these situations is to use some suitable non-parametric procedure such as the Kruskal-Wallis one-way ANOVA for independent groups using at least ordinal data. If a repeated measures or a within design is being used, one could use the Friedman Rank Test for Correlated Samples statistic (a one-way, within ANOVA for ordinal data).
Another approach would be to change the scale of measurement from either nominal or ordinal to an approximation of an interval scale by using one of the following four transformations:
(1) Square Root Transformation.
(2) Logarithmic Transformation.
(3) Reciprocal Transformation.
(4) Arcsine Transformation.
The SQUARE ROOT TRANSFORMATION would be used when our original data are measured by means of a NOMINAL SCALE such as frequencies or the use of discrete values such as yes/no, right/wrong, etc. Thus, we can apply a standard anova to our obtained frequencies only after they are transformed. We would transform instead of analyzing our data by means of Chi-square.
The LOGARITHMIC TRANSFORMATIONS would be used when the original data are measured on an ORDINAL SCALE (rankings or rating scale). Thus, once again, we can use a standard ANOVA on these ranks only after transforming them. Of course, we could also use either the Kruskal-Wallis or the Friedman rank test if we wished. However, always remember that non-parametric procedures are always less powerful than their parametric counterparts.
The RECIPROCAL TRANSFORMATION is used when the data are located on a RATIO SCALE and are time measures (especially reaction time studies or threshold measures). Once transformed, these time measures can be analyzed with a standard ANOVA procedure.
The ARCSINE TRANSFORMATION is applied to data that are in the form of either proportions or percentages. After transformation, the proportions may be fully analyzed by any of the standard ANOVAs.
If we have data that need transformation, we must first make a decision regarding the specific type (of the 4 possible) which will be appropriate for the data. The ANOVA is then carried out on these transformed scores and, after the statistical analysis is completed, the outcome of the analysis may be retransformed for the final interpretation. However, frequently the retransformation is not really important to the overall understanding of the ANOVA results.
Ordinarily, a statement regarding the kind of transformation used with your data must be clearly stated in the results section of the experiment report.
SALZWEDEL