If the data for one or both of the samples to be analyzed by a t test come from a population whose distribution violates the assumption of normality, or outliers are present, then the t test 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 test 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 the two samples to remedy nonnormality often results in correcting heteroscedasticity (unequal variances). The same transformation should be applied to both 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 unpaired two-sample t test, the most common nonparametric alternative tests are the Wilcoxon rank-sum test (equivalent to the Mann-Whitney U test), and the median test. Although the rank-sum test does not assume normality of the distributions for the two sample populations, it does assume that the two populations have the same distribution, except for a possible difference in the two population medians. Thus the rank-sum test will not address the problem of inequality of variances. Also, as with the t test, it is assumed that the two 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 t test is the most powerful test of the equality of the two means, meaning that no other test is more likely to detect an actual difference between the two 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 t test can also be viewed as testing whether the difference between the two sample medians is 0, if the normality assumption holds.) If the population distributions are not normal, however, the rank-sum test may be more powerful at detecting differences between the sample medians. Because the rank-sum test is nearly as powerful as the unpaired t test in the case of data from a normal distribution, and may be substantially more powerful in the case of nonormality, the rank-sum test is well suited to analyzing data when outliers are suspected, even if the underlying distributions are close to normal.