# Introduction

One of the primary goals research is to examine the relationships between variables, usually with the goal of determining causal relationships. Often, a first step is to determine whether two groups have statistically different means across some variable of interest. Note: just because the sample means for two groups are literally different does not mean that they are statistically different in the population as a whole. The numerical difference may be an artifact of sampling, which is why a difference of means test is required.

The possible outcomes for difference of means tests is that the means of two groups are statistically different or are not statistically different.

The Research (or Alternative) Hypothesis posits that there is a statistically significant difference between the means of the two groups. The Null Hypothesis posits that there is not a statistically significant difference between the means of the two groups.

If there is a statistical difference, then we say that we reject the null hypothesis and accept the alternative hypothesis. Conversely, if there is not a statistical difference, then we say that we accept the null hypothesis and reject the alternative hypothesis.

## Type I and Type II Errors

XXX Consider fearless deletion in favor of the Testing Hypotheses discussion (which should be a separate page).

Two types of error are possible when conducting statistical tests such as difference of means tests.

A Type I Error is rejecting the null hypothesis when it is, in fact, true and accepting the alternative hypothesis when it is, in fact, false.  This is also known as a false positive; that is, incorrectly finding a difference of means when there really is not one.
A Type II Error is accepting the null hypothesis when it is, in fact, false and rejecting the alternative hypothesis when it is, in fact, true.  This is also known as a false negative; that is, failing to find a statistical difference in means when there really is one.

# Independent samples (unpaired) t test

## Extension to more than two groups

For comparisons across more than two groups (or categories of the independent variable), you can use either ANOVA or linear regression.

# Paired samples t test

## Beyond two waves

If testing for differences beyond a simple "pre-test, post-test" design (for example, for multiple stages of interventions), an appropriate technique would be Repeated measures ANOVA.

# The difference of proportions test

## Alternatives to the difference of proportions test

Nominal and ordinal differences of proportions can also be tested using the chi-square test.

# References

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