Lucinda used ANOVA to test if there were statistically significant differences between the five conditions of her independent variable. There were 8 participants in each condition. Her obtained F-test (df between = 4, df within = 30) was F = 10.897/3.72 = 2.914, p < .05. When she conducts her tests to know which groups are significantly different, how far apart must the group means be in order to be significantly different?
To determine how far apart the group means must be in order to be significantly different, Lucinda needs to perform post-hoc tests. Post-hoc tests are used to compare all possible pairs of group means to identify which pairs are significantly different from each other.
One commonly used post-hoc test is the Tukey's Honestly Significant Difference (HSD) test. This test determines the minimum difference between group means that is needed to be considered statistically significant.
To perform the Tukey's HSD test, Lucinda needs to follow these steps:
Step 1: Calculate the standard error of the mean (SE):
- SE = √(MS_within / n)
where MS_within is the mean square within groups obtained from the ANOVA and n is the number of participants in each group.
Step 2: Calculate the critical value (q) using the formula:
- q = q(α, df_between, df_within)
where α is the significance level (e.g., 0.05), df_between is the degrees of freedom for the numerator (df_between = k - 1, where k is the number of groups), and df_within is the degrees of freedom for the denominator (df_within = N - k, where N is the total sample size and k is the number of groups).
Step 3: Calculate the Honestly Significant Difference (HSD) value using the formula:
- HSD = q * SE
Step 4: Compare the differences between each pair of group means. If the difference is greater than the HSD value, it is considered statistically significant.
Therefore, by following these steps, Lucinda can determine the specific difference needed between group means to be considered statistically significant.