If the data for one or more of the samples to be analyzed by a one-way analysis of variance (ANOVA) come from a population whose distribution violates the assumption of normality, or outliers are present, then the ANOVA on the original data may provide misleading results, or may not be the most powerful test available. In such cases, transforming the data or using a nonparametric tests may provide a better analysis.
- Transformations (a single function applied to each data value) are applied to correct problems of nonnormality or unequal variances. For example, taking logarithms of sample values can reduce skewness to the right. Transforming all the samples to remedy nonnormality often results in correcting heteroscedasticity (unequal variances). The same transformation should be applied to all samples. Unless scientific theory suggests a specific transformation a priori, transformations are usually chosen from the "power family" of transformations, where each value is replaced by x**p, where p is an integer or half-integer, usually one of:
- -2 (reciprocal square)
- -1 (reciprocal)
- -0.5 (reciprocal square root)
- 0 (log transformation)
- 0.5 (square root)
- 1 (leaving the data untransformed)
- 2 (square)
For p = -0.5 (reciprocal square root), 0, or 0.5 (square root), the data values must all be positive. To use these transformations when there are negative and positive values, a constant can be added to all the data values such that the smallest is greater than 0 (say, such that the smallest value is 1). (If all the data values are negative, the data can instead be multiplied by -1, but note that in this situation, data suggesting skewness to the right would now become data suggesting skewness to the left.) To preserve the order of the original data in the transformed data, if the value of p is negative, the transformed data are multiplied by -1.0; e.g., for p = -1, the data are transformed as x --> -1.0/x. Taking logs or square roots tends to "pull in" values greater than 1 relative to values less than 1, which is useful in correcting skewness to the right. Transformation involves changing the metric in which the data are analyzed, which may make interpretation of the results difficult if the transformation is complicated. If you are unfamiliar with transformations, you may wish to consult a statistician before proceeding.
- Nonparametric tests:
- Nonparametric tests are tests that do not make the usual distributional assumptions of the normal-theory-based tests. For the one-way ANOVA, the most common nonparametric alternative tests are the Kruskal-Wallis test and the median test. Although the Kruskal-Wallis test does not assume normality of the distributions for the sample populations, it does assume that the populations have the same distribution, except for a possible difference in the population medians. Thus the Krukal-Wallis test will not address the problem of inequality of variances. Also, as with the one-way ANOVA, it is assumed that the samples are independent of each other, and that there is independence within each sample. If the sampled values do indeed come from populations with normal distributions, then the one-way ANOVA is the most powerful test of the equality of the means, meaning that no other test is more likely to detect an actual difference among the means. (If a distribution is symmetric, its mean and median are both equal to the center of symmetry. Since the normal distribution is symmetric, the one-way ANOVA can also be viewed as testing for differences among the sample medians, if the normality assumption holds.) If the population distributions are not normal, however, the Kruskal-Wallis test may be more powerful at detecting differences between the sample medians. Because the Kruskal-Wallis test is nearly as powerful as the one-way ANOVA in the case of data from a normal distribution, and may be substantially more powerful in the case of nonormality, the Kruskal-Wallis test is well suited to analyzing data when outliers are suspected, even if the underlying distributions are close to normal.