Student's T Test and Wilcoxon Test in Clinical Research

Student’s ‘t’ test

Student’s ‘t’ test is probably still the most popular of all statistical tests. The test compares two mean (average) values to judge if they are different or not. The Student’s ‘t’ test is the most sensitive test for interval data, but it also requires the most stringent assumptions. The variables/data are assumed to be normally distributed. If there is any reason to doubt this assumption, use another, distribution-free, test (e.g., Wilcoxon Test).

The following ‘t’ tests are commonly used :

1. One sample t test
• The mean (test) of a single group is compared with a hypothetical value (control).
2. Paired t
• When the ‘paired design’ is used, paired t is applied (eg. BP measured before(control) and after(test) a drug administration in a single group of 10 subjects)
3. Unpaired t
• For comparing two individual groups (eg. Height of two groups of 10 subjects each)

The conditions for applying t test are as follows :

1. The sample must be chosen randomly
2. The data must be quantitative (measurable e.g. height, BP)
3. The data should follow normal distribution
4. The size is ideally <30 in each group
   
5. Populations should have equal SD. (SD of one group should not be more than twice higher or half lesser than the other)
6. The t test used must be appropriate for the design. Paired t for the paired design and Unpaired t for comparing two group means

What if the above conditions are not met?

1. If the SDs are very different Welch correction can be used.
2. If the distribution of data are non-normal or skewed, use transformation techniques before applying a t test. Else go for a non-parametric test.
3. If you are not sure of the normality of the data, use a non-parametric test.
4. If the size in each group >30, go for normal test.

t’ test will give unreliable results if the above conditions are not met.

Wilcoxon two-sample test (Mann-Whitney U-test)

It is in fact a Student t-test performed on ranks. It is a most useful test to see whether the values in two samples differ in size. It resembles the Median-Test in scope, but it is much more sensitive. In fact, for large numbers it is almost as sensitive as the Two Sample Student t-test. For small numbers with unknown distributions this test is even more sensitive than the Student t-test. Ranks are nothing but ‘order numbers’ after measurements have been sorted by magnitude. The smallest measurement is given rank 1, and so on. Unlike t test, the assumptions are none really (the values are assumed to be normally distributed for applying t test).

This test used when

1. two groups are to be compared
2. the scale is ordinal (for data types of ranks and scores)
3. data are not normally distributed
4. the distribution is not known or doubtful

The Wilcoxon Matched-Pairs Signed-Ranks Test

This is a most useful test to see whether the members of a pair differ in size. It resembles the Sign-Test in scope, but it is much more sensitive. In fact, for large numbers it is almost as sensitive as the Student t-test. For small numbers with unknown distributions this test is even more sensitive than the Student t-test.

Assumptions:

The distribution of the difference (d) between the values within each pair (x, y) must be symmetrical, the median difference must be identical to the mean difference.
As members of a pair are assumed to have identical distributions, their differences should always have a symmetrical distribution, so this assumption is not very restrictive.

This test used when

1. Paired design is used (one group – data collected before & after drug administration)
2. the scale is ordinal (for data types of ranks and scores)
3. data are not normally distributed
4. the distribution is not known or doubtful

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