PROPHET StatGuide: Do your data violate t test assumptions?

If the populations from which data to be analyzed by a t test were sampled violate one or more of the t test assumptions, the results of the analysis may be incorrect or misleading. For example, if the assumption of independence is violated, then the two-sample unpaired t test is simply not appropriate, although another test (perhaps the paired t test) may be appropriate. If the assumption of normality is violated, or outliers are present, then the t test may not be the most powerful test available, and this could mean the difference between detecting a true difference or not. A nonparametric test or employing a transformation may result in a more powerful test. If the population variances are unequal, the Welch-Satterthwaite t test provides means of performing a t test adjusted for the inequality of the variances. Often, the effect of an assumption violation on the t test result depends on the extent of the violation (such as the how unequal the population variances are, or how skewed one or the other population distribution is). Some small violations may have little practical effect on the analysis, while other violations may render the t test result uselessly incorrect or uninterpretable. In particular, small or unbalanced sample sizes can increase vulnerability to assumption violations.

Potential assumption violations include:

Implicit factors:
A lack of independence within a sample is often caused by the existence of an implicit factor in the data. For example, values collected over time may be serially correlated (here time is the implicit factor). If the data are in a particular order, consider the possibility of dependence. (If the row order of the data reflect the order in which the data were collected, an index plot of the data [data value plotted against row number] can reveal patterns in the plot that could suggest possible time effects.)

Lack of independence:
Whether the two samples are independent of each other is generally determined by the structure of the experiment from which they arise. Obviously correlated samples, such as a set of pre- and post-test observations on the same subjects, are not independent, and such data would be more appropriately tested by a two-sample paired test. If you are unsure whether your samples are independent, you may wish to consult a statistician or someone who is knowledgeable about the data collection scheme you are using.

Values may not be identically distributed because of the presence of outliers. Outliers are anomalous values in the data. Outliers tend to increase the estimate of sample variance, thus decreasing the calculated t statistic and lowering the chance of rejecting the null hypothesis. They may be due to recording errors, which may be correctable, or they may be due to the sample not being entirely from the same population. Apparent outliers may also be due to the values being from the same, but nonnormal, population. The boxplot and normal probability plot (normal Q-Q plot) may suggest the presence of outliers in the data.

The t statistic is based on the sample mean and the sample variance, both of which are sensitive to outliers. (In other words, neither the sample mean nor the sample variance is resistant to outliers, and thus, neither is the t statistic.) In particular, a large outlier can inflate the sample variance, decreasing the t statistic and thus perhaps eliminating a significant difference. A nonparametric test may be a more powerful test in such a situation. If you find outliers in your data that are not due to correctable errors, you may wish to consult a statistician as to how to proceed.

The values in a sample may indeed be from the same population, but not from a normal one. Signs of nonnormality are skewness (lack of symmetry) or light-tailedness or heavy-tailedness. The boxplot, histogram, and normal probability plot (normal Q-Q plot), along with the normality test, can provide information on the normality of the population distribution. However, if there are only a small number of data points, nonnormality can be hard to detect. If there are a great many data points, the normality test may detect statistically significant but trivial departures from normality that will have no real effect on the t statistic (since the t statistic will converge in probability to the standard normal distribution by the law of large numbers).

For data sampled from a normal distribution, normal probability plots should approximate straight lines, and boxplots should be symmetric (median and mean together, in the middle of the box) with no outliers. If the sample sizes are approximately equal, and not too small, then the t statistic will not be much affected even if the population distributions are skewed, as long they have approximately the same skewness. If the sample sizes are not approximately equal, then the t statistic will be skewed in the same direction as shown by the smaller sample. Unless the sample sizes are small (less than 10), light-tailedness or heavy-tailedness will have little effect on the t statistic.

Robust statistical tests operate well across a wide variety of distributions. A test can be robust for validity, meaning that it provides P values close to the true ones in the presence of (slight) departures from its assumptions. It may also be robust for efficiency, meaning that it maintains its statistical power (the probability that a true violation of the null hypothesis will be detected by the test) in the presence of those departures. The t test is fairly robust for validity against nonnormality, but it may not be the most powerful test available for a given nonnormal distribution, although it is the most powerful test available when its test assumptions are met. In the case of nonnormality, a nonparametric test or employing a transformation may result in a more powerful test.

Unequal population variances:
The inequality of the population variances can be assessed by examination of the relative size of the sample variances, either informally (including graphically), or by a variance test such as the F test. The effect of inequality of variances is mitigated when the two sample sizes are equal, so that the t test is fairly robust against inequality of variances if the sample sizes are equal. The effect of inequality of the variances is most severe when the sample sizes are unequal and the smaller sample is associated with the larger variance. Prophet performs the Welch-Satterthwaite t test for unequal variances when the two sample variances fail the F test for equality. The Welch-Satterthwaite t test has about the same robustness properties as the standard t test does when the variances are equal. If both nonnormality and unequal variances are present, employing a transformation may be preferable. A nonparametric test like the Wilcoxon rank-rum test still assumes that the population variances are comparable.

Patterns in plot of data:
If the assumptions for the samples' population distributions are correct, the plot of each sample's values against its mean (or its sample ID) should suggest a horizontal band across the graph. Because there are only two unique sample means or sample ID values, this type of graph will consist of two vertical "stacks" of data points; the stacks should be about the same length. Outliers may appear as anomalous points in the graph. A fan pattern like the profile of a megaphone, with a noticeable flare either to the right or to the left as shown in the picture (one of the "stacks" of data points is much longer than the other), suggests that the variance in the values increases in the direction the fan pattern widens (usually as the sample mean increases), and this in turn suggests that a transformation may be needed.

Side-by-side boxplots of the two samples can also reveal lack of homogeneity of variances if one boxplot is much longer than the other, and reveal suspected outliers.

Special problems with small sample sizes:
If one or both of the sample sizes is small, it may be difficult to detect assumption violations. With small samples, violation assumptions such as nonnormality or inequality of variances are difficult to detect even when they are present. Also, with small sample size(s) the t test offers less protection against violation of assumptions.

Even if none of the test assumptions are violated, a t test with small sample sizes may not have sufficient power to detect a significant difference between the two samples, even if the means are in fact different. The power curve presented in the results of the t test indicates how likely the test would be to detect an actual difference between the means. The shallower the power curve, the bigger the actual difference would have to be before the t test would detect it. The power depends on variance, the selected significance (alpha-) level of the test, and the sample size. Power decreases as the variance increases, decreases as the significance level is decreased (i.e., as the test is made more stringent), and increases as the sample size increases. With very small samples, even samples from populations with very different means may not produce a significant t test statistic unless the sample variance is small. If a statistical significance test with small sample sizes produces a surprisingly non-significant P value, then a lack of power may be the reason. The best time to avoid such problems is in the design stage of an experiment, when appropriate minimum sample sizes can be determined, perhaps in consultation with a statistician, before data collection begins.

Special problems with unbalanced sample sizes:
The t test is fairly robust against inequality of variances if the sample sizes are equal. If the sample sizes are not approximately equal, and especially if the larger sample variance is associated with the smaller sample size, then the calculated t statistic may be dominated by the sample variance for the smaller sample. Also, if the sample sizes are not approximately equal, the t statistic will be skewed in the same direction of skewness, as shown by the smaller sample; if the two samples show very different skewnesses, the t statistic will be skewed even when the sample sizes are equal. However, unless the sample sizes are small (less than 10), light-tailedness or heavy-tailedness will have little effect on the t statistic.

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Last modified: March 17, 1997

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