Statistical Tests and Assumptions

Research questions and statistics

The most popular research questions include:

1. whether two variables (n = 2) are correlated (i.e., associated)
2. whether multiple variables (n > 2) are correlated
3. whether two groups (n = 2) of samples differ from each other
4. whether multiple groups (n >= 2) of samples differ from each other
5. whether the variability of two or more samples differ

Each of these questions can be answered using the following statistical tests:

1. Correlation test between two variables
2. Correlation matrix between multiple variables
3. Comparing the means of two groups:
   • Student’s t-test (parametric)
   • Wilcoxon rank test (non-parametric)
4. Comparing the means of more than two groups
   • ANOVA test (analysis of variance, parametric): extension of t-test to compare more than two groups.
   • Kruskal-Wallis rank sum test (non-parametric): extension of Wilcoxon rank test to compare more than two groups
5. Comparing the variances:
   • Comparing the variances of two groups: F-test (parametric)
   • Comparison of the variances of more than two groups: Bartlett’s test (parametric), Levene’s test (parametric) and Fligner-Killeen test (non-parametric)

Assumptions of statistical tests

Many of the statistical methods including correlation, regression, t-test, and analysis of variance assume some characteristics about the data. Generally they assume that:

• the data are normally distributed
• and the variances of the groups to be compared are homogeneous (equal).

These assumptions should be taken seriously to draw reliable interpretation and conclusions of the research.

These tests - correlation, t-test and ANOVA - are called parametric tests, because their validity depends on the distribution of the data.

Before using parametric test, some preliminary tests should be performed to make sure that the test assumptions are met. In the situations where the assumptions are violated, non-paramatric tests are recommended.

Assessing normality

1. With large enough sample sizes (n > 30) the violation of the normality assumption should not cause major problems (central limit theorem). This implies that we can ignore the distribution of the data and use parametric tests.
2. However, to be consistent, we can use Shapiro-Wilk’s significance test comparing the sample distribution to a normal one in order to ascertain whether data show or not a serious deviation from normality (Ghasemi and Zahediasl 2012).

Assessing equality of variances

The standard Student’s t-test (comparing two independent samples) and the ANOVA test (comparing multiple samples) assume also that the samples to be compared have equal variances.

If the samples, being compared, follow normal distribution, then it’s possible to use:

• F-test to compare the variances of two samples
• Bartlett’s Test or Levene’s Test to compare the variances of multiple samples.

Summary

This chapter introduces the most commonly used statistical tests and their assumptions.