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### What Did Student Really Do? (What Student Did, Part II)

When Student's t test for independent samples is run, every statistics package reports two results. They may be labeled

 equal variances assumed equal variances not assumed common variance separate variances pooled variance separate variances

Which results should be used?

The variances mentioned in the table and the output are population variances. One thing Student did was to say that if the population variances were known to be equal or could be assumed to be equal, exact tests and confidence intervals could be obtained by using his t distribution. This is the test labeled equal variances assumed, common variance or pooled variance. The term pooled variance refers to way the estimate of the common variance is obtained by pooling the data from both samples.

The test labeled equal variances not assumed or separate variances is appropriate for normally distributed individual values when the population variances are known to be unequal or cannot be assumed to be equal. This is not an exact test. It is approximate. The approximation involves t distributions with non-integer degrees of freedom. Before the ready availability of computers, the number of degrees of freedom was awkward to calculate and the critical values were not easy to obtain, so statisticians worried about how much the data could depart from the ideal of equal variances without affecting the validity of Student's test. It turned out the t test was extremely robust to departures from normality and equal variances.

Some analysts recommended performing preliminary statistical tests to decide whether the data were normally distributed and whether population variances were equal. If the hypothesis of equal population variances was rejected, the equal variances not assumed form of the test would be used, otherwise equal variances assumed version would be used. However, it was discovered that Students t test for independent samples was so robust that the preliminary tests would have analysts avoiding the equal variances assumed form when it was in no danger of it giving misleading results. These preliminary tests often detect differences too small to affect Student's t test. The analogy most often given is that using preliminary tests of normality and equality of variances to decided whether it was safe to use the equal variances assumed version of the t test was like sending out a rowboat to see whether it was safe for the ocean liner. Today, common practice is to avoid preliminary tests. Important violations of the requirements will be detectable to the naked eye without a formal significance test.

Rupert Miller, Jr., in his 1986 book Beyond ANOVA, Basics of Applied Statistics, {New York: John Wiley & Sons] summarizes the extent to which the assumptions of normality and equal population variances can be violated without affecting the validity of Student's test.

• If sample sizes are equal, (a) nonnormality is not a problem and (b) the t test can tolerate population standard deviation ratios of 2 without showing any major ill effect. The worst situation occurs when one sample has a much larger variance and a much smaller sample size than the other. For example, if the variance ratio is 5 and the sample size ratio is 1/5, a nominal P value of 0.05 is actually 0.22.
• Serious distortion of the P value can occur when the skewness of the two populations is different.
• Outliers can distort the mean difference and the t statistic. They tend to inflate the variance and depress the value and corresponding statistical significance of the t statistic.

Still, which test should be used?

Frederick Mosteller and John Tukey, on pages 5-7 of Data Analysis and Regression [Reading, MA: Addison-Wesley Publishing Company, Inc., 1997] provide insight into what Student really did and how it should affect our choice of test.

The value of Student's work lay not in great numerical change, but in:
• recognition that one could, if appropriate assumptions held, make allowances fo the "uncertainties" of small samples, not only in Student's original problem, but in others as well;
• provision of a numerical assessment of how small the necessary numerical adjustment of confidence points were in Student's problem...
• presentation of tables that could be used--in setting confidence limits, in making significance tests--to assess the uncertainty associated with even very small samples.
Besides its values, Student's contribution had its drawbacks, notably:
• it made it too easy to neglect the proviso "if appropriate assumptions held";
• it overemphasized the "exactness of Student's solution for his idealized problem";
• it helped to divert the attention of theoretical statisticians to the development of "exact" ways of treating other problems; and
• it failed to attack the "problem of multiplicity": the difficulties and temptation associated with the application of large numbers of tests to the same data.
The great importance given to exactness of treatment is even more surprising when we consider how much the small differences between the critical values of the normal approximation and Student's t disappears, especially at and near the much-used two-sided 5% point, when, as suggested by Burrau (1943), we multiply t by the constant required to bring its variance to 1, namely, [ {(f - 2) /f}].

The separate variances version rarely differs from common variance. When it does, there's usually a problem with the common variances version.

When sample sizes are large, the Central Limit Theorem takes over. The behavior of the separate variances t statistic is described by the normal distribution regardless of the distribution of the individual observations. The two populations of individual observations need not have the same variances. They need not even be normal.

If the separate variances t test is always valid for large samples and if the common variances test is probably invalid when the two tests disagree in small samples, why not use the separate variances version exclusively? Some statisticians seem to advocate this approach. The primary advantage of the common variances test is that it generalizes to more than two groups (analysis of variance).

When within group variances are unequal, it often happens that the standard deviation is proportional to the mean. For example, instead of the within group standard deviation being a fixed value such as 5 mg/dl, it is often a fixed percentage. If the standard deviation were 20% of the mean, one group might have values of 10 give or take 2, while the other might have values of 100 give or take 20. In such cases, a logarithmic transformation will produce groups with the same standard deviation. If natural logarithms are use, the common within group standard deviation of the transformed data will be equal to the ratio of the within group standard deviation to the mean (also known as the coefficient of variation).

The purpose of transforming data is not to achieve a particular result. Transformations are not performed to make differences achieve statistical significance. Transformations allow us to use standard statistical techniques confident they are appropriate for the data to which they're applied. That is, transformations are applied not to achieve a particular result, but to insure the results we obtain will be reliable. The following small dataset illustrates some of the issues. Serum progesterone levels were measured in subjects randomized to receive estrogen or not. The group with the higher serum progesterone levels also has the greater spread. The equal variances assumed t test has an observed significance level of 0.022; the unequal variances assumed t test has an observed significance level of 0.069. When a (natural) logarithmic transformation (ln) is applied to the data, the within group standard deviations are both around 0.50, which is approximately the ratio of the SD to the mean, and both P values are 0.012. The conclusion is that the geometric mean of progesterone levels of those given this dose of progesterone is between 1.4 and 7.9 times the levels of those on placebo (95% CI). Insofar as the geometric mean is a good approximation to the median, the previous sentence might be restated in terms of medians.

The analysis is presented in the common log scale (log), also. All test statistics and signifcance levels from the two log transformed analyses are identical. The numerical values of the summary statistics such as mean, SD, and confidence intervals differ, but are equivalent. The equivalence can be seen by taking the appropriate anti-logarithm of each number. For example, the mean value of log(SPROG) in the Estrogen group is 2.3917 while the mean value of ln(SPROG) is 5.5072, but both 102.3917 and e5.5072 are equal to 246.4. The within-group SD is equal to the coefficient of variation for the natural log transformation only.

This is a very small dataset, so small that tests for the inequality of population variances do not achieve statistical significance, even though one SD is three times larger than the other. Still, it possesses the essential features of one group having a much larger standard deviation and the standard deviations being proportional to mean response, so it is worthy of consideration.

+
 Estrogen N Mean SD SEM SPROG No 5 81.8000 40.4747 18.1008 Yes 4 271.5000 139.5337 69.7669 ln(SPROG) No 5 4.2875 .5617 .2512 Yes 4 5.5072 .5076 .2538 log(SPROG) No 5 1.8620 .2439 .1091 Yes 4 2.3917 .2204 .1102

 Levene's Test for Equality of Variances t-test for Equality of Means 95% Confidence Interval of the Difference F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference Lower Upper SPROG Equal variances assumed 2.843 .136 -2.935 7 .022 -189.7000 64.6229 -342.5088 -36.8912 Equal variances not assumed -2.632 3.406 .069 -189.7000 72.0767 -404.3650 24.9650 ln(SPROG) Equal variances assumed .639 .450 -3.372 7 .012 -1.2197 .3617 -2.0750 -.3644 Equal variances not assumed -3.416 6.836 .012 -1.2197 .3571 -2.0682 -.3712 log(SPROG) Equal variances assumed .639 .450 -3.372 7 .012 -.5297 .1571 -.9012 -.1583 Equal variances not assumed -3.416 6.836 .012 -.5297 .1551 -.8982 -.1612

Gerard E. Dallal