Statistics Tests

1. Paired-sample t test is based on the assumption of normality of distribution – Although no particular distribution needed for the paired sample but certain level of symmetry is assumed for the population distribution of the paired differences.
2. Wilcoxon signed rank sum test- When we work on small samples with probability of non-normality of the paired observation differences, it is best to use Wilcoxon matched pairs signed ranks test, which yields the test statistics that uses ranking with no concern about normality of distribution. As with the paired sample t-test, we are interested in measuring the differences before and after the treatment. The procedure for the Wilcoxon signed rank sum test is as follows:

- The differences are calculated and then ranked in ascending order, ignoring the sign of the difference.
- Zeroes are ignored (and the sample size is adjusted accordingly).
- Those with the same difference are given an average ranking. (For example, if the eighth and ninth values are the same, they are ranked as 8.5.)
- The ranks are then given the sign of the difference.
- Ranks are summed for all those with negative differences and those with positive differences.
- "The smallest value is then looked up in the appropriate table for the number of pairs included in the sign ranking.

Two Way Analysis of Variance-Two Way Analysis of Variance is a way of studying the effects of two factors separately (their main effects) and (sometimes) together (their interaction effect)

3. Wilcoxon rank sums test: This test compares two different sample distributions to see if they come from the same non-normal population probability distribution. This is the non-parametric equivalent of the t-test.

4. Mann-Whitney U test: this is a non-parametric test identical to the above (No. 2) but with more complex calculation.the non-parametric analogue of the independent two-sample t-test is the Mann-Whitney test. For both the general test and its two-sample version, the null hypothesis is that the medians (and not the means) are equal, against the general alternative that at least one differs from the others. We compare the medians, rather than the means, because the data will probably not be symmetric if we are using a non-parametric test, so the value of the mean will be artificially inflated or deflated. The Mann-Whitney test test this null hypothesis by transforming the data into pooled ranks (that is, they start by assigning rank 1 to the smallest observation in the pooled sample, and so on) and then calculating a test statistic from these ranks. Both tests appear in the non-parametric sub-menu of SPSS.

5. Sign test available in SPSS also calculate similar significant points when we apply non-parametric test on two independent sample. This tests hypotheses regarding the median of a distribution and is independent of distribution type.

6. Kolmogorov-Smirnov test: This test is used to compare two different samples to see if they could be from the the same population or to see if a sample distribution is of a certain hypothesized type.

7. Tukey's method depends on equal sample sizes and so is less widely applicable.

8. Scheffe's method uses a generalised procedure for all of the possible linear combinations of treatment means (called contrasts) but these results in wider confidence levels than the other two methods. Scheffé’s (1953) method of simultaneous inference construct bands called Scheffé confidence bands to be distinct from the usual s.e. bands which is representing a shift from the mean of the distribution in proportion to its variance.

9. while individual coefficients may be imprecisely estimated (low t-statistics), the joint effect could still be quite precisely estimated (high F-statistics).

10. Welsh test correction is applied to t test, when the assumption of normality is not supported. The Welch function on t-tests corrects for unequal variances

11. Kruskal-Wallis H-test: The non-parametric equivalent of ANOVA F-test is the Kruskal-Wallis test, which is a generalisation of the Mann-Whitney test for more than two groups. If the assumptions regarding normality and the variances being equal break-down then the appropriate non-parametric test is the Kruskal-Wallis H-test.

12. The non-parametric equivalent of Anova test where the assumptions of normality and similar variance is absent - called Friedman test. Friedman’s Test compare observations repeated on the same subjects. Unlike the parametric repeated measures ANOVA or paired t-test, this non-parametric makes no assumptions about the distribution of the data (e.g., normality).

13. Bonferroni, uses t tests to perform pair-wise comparisons between group means, but controls overall error rate by setting the error rate for each test to the experiment-wise error rate divided by the total number of tests. Hence, the observed significance level is adjusted for the fact that multiple comparisons are being made.

14. Non-parametric two related sample test: Stratification in case control studies- the results may then be pooled across the strata to arrive at conclusive evidence for any association. This can be achieved by using the Mantel-Haenszel χ2 test.

PARAMETRIC TESTS:
Two Way Analysis of Variance: Two Way Analysis of Variance is a way of studying the effects of two factors separately (their main effects) and (sometimes) together (their interaction effect).