# Chapter 7. Research Questions About Means in Three or More Groups

- Special statistical tests are needed when more than two groups are studied or when a group is measured on several variables.
- Analysis of variance, or ANOVA, is a statistical method that divides the variance in an observation into the variance among groups and the rest of the variance, called the within-group or error variance.
- The F test used to compare two variances in Chapter 6 is used to compare the variance among groups to the error.
- An example of the way ANOVA is calculated from the definitional formulas is helpful in understanding the logic behind the test.
- The terms used in ANOVA are important, but the details of the computations are given for illustration only, and computer programs are used for all ANOVA procedures.
- One-way ANOVA is the appropriate method when more than two groups are studied on one variable.
- As with the
*t*test, certain assumptions must be made to use ANOVA, and equal variances is one of the most important. - Making many comparisons among groups increases the chances of a type I error, that a difference is concluded when there is none.
- Investigators can decide ahead of time what specific comparisons they want to make.
- The Bonferroni procedure is a common way to compensate for making many comparisons among groups; it works by reducing the size of α for each comparison, essentially increasing the difference needed to be significant.
- Some multiple comparison methods, called post hoc, are done only if the ANOVA results are statistically significant.
- Tukey’s test is one of the most highly recommended post hoc tests for comparing mean differences.
- The Scheffé post hoc test is the most conservative (requiring a larger difference to be significant), but it is also the most versatile.
- The Newman–Kuels post hoc test is used frequently in basic science research.
- Dunnett’s procedure is the test of choice if the only comparisons being made are between the mean in a control group and the means in other groups.
- Two-way ANOVA analyzes two factors instead of just one, as in one-way ANOVA. It also permits the analysis of the interaction between two factors.
- ANOVA designs involving more that two factors are possible, generally called factorial designs.
- Confounding variables can be accommodated by the ANOVA randomized block design.
- Repeated-measures ANOVA is a common procedure in medical research;
it is analogous to the paired
*t*test with more than two groups and is also called the split-plot design. - Nonparametric ANOVA methods include Kruskal–Wallis for one-way designs and Friedman two-way ANOVA for repeated measures. These methods are analogous to the Wilcoxon procedures and are used when the assumptions for ANOVA are not met.
- The chi-square test can be used to compare more than two proportions and to determine if there is an association between two factors, each of which can have two or more levels. It is a simple extension of the chi-square test we discussed in Chapter 6.
- As with research questions involving one or two groups, power analysis is ...

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