Brief Summary of ANOVA Review Points

Here are some key points from today's class, reviewing ANOVA. (Web documents alluded to below are available in the right-hand links column.)

• A one-way ANOVA compares means of three or more groups on one factor (e.g., comparing GPA at graduation among physical-science, social-science, and arts/humanities majors).
• As seen in the linked overview of conducting one-way ANOVA by hand, "between-group" compares conditions (e.g., did participants given one word-memorization strategy memorize more words on average than participants given other types of instructions?). 
• The "within-group" or "error" section of the results refers not to error in the sense of mistakes, but in the sense of imperfect potency of a given instruction. In other words, the fact that not all participants given rhyming (for example) as a strategy for memorizing words ended up memorizing the exact same number of words reflects this imperfection or "error." 
• The larger the ratio of the between-group mean-square to the error mean-square, the larger the F ratio and the greater likelihood of statistical significance. (The F-statistic, by the way, is named after Sir Ronald Fisher, the inventor of ANOVA.)
• In the old days, one would have to look at an F-table to see if a given result attained significance (such as the one on this site for p < .05). Nowadays, however, the computer output will tell you the significance of your results.
• A two-way ANOVA yields three types of mean-comparison effects. In the example we worked out in SPSS, gender (men, women) and marital status (married, widowed, divorced, separated, or single/never-married) were the factors or independent variables, and attitude toward Broadway musicals was the dependent variable. We thus had a 2 X 5 ANOVA. The results display thus included:
• Main-effect of gender: Did men's mean attitude (collapsing over all marital statuses) differ from women's mean attitude (collapsing over all marital statuses)?
• Main-effect of marital status: Did married persons' mean attitude toward Broadway musicals (collapsing over men and women) differ from widowed persons' mean attitude, divorced persons' mean attitude, etc.?
• Interaction of gender X marital status: Did any one or more cells representing combinations of gender and marital status (e.g., divorced men or single women) stand out from the other cells in their mean attitudes toward Broadway musicals?
• The linked document from David Lane shows how to use graphs to discern whether you have main-effects and/or interactions.
• There are two further topics we didn't cover today, but will review briefly next Tuesday:
• Just as a t-test has two versions for comparing two independent, non-overlapping groups (such as Democrats vs. Republicans) and for comparing paired conditions/groups (such as men and women who are heterosexually married to each other; or participants who receive medicine for four weeks and placebo for four weeks), ANOVA has analogous alternatives (here and here).
• Significant effects in ANOVA tell us only that at least two means differ significantly within a condition. If the main-effect of marital status described above were significant, we would not immediately know which two or more of the five marital-status groups differed from each other on attitude toward Broadway musicals. For example, were married people more favorable than divorced people? Separated people more favorable than widowed people? When a factor has more than two conditions, we must follow up significant ANOVA results with contrasts and comparisons between cells (see links).
• We can't forget to sing the song "ANOVA Man" by Mark Glickman at the next class (link to CAUSE Fun Resources)! Note the song's reference to "mu," the population mean on a given variable in a given condition. Even though we conduct ANOVA's with sample means in our own studies, the inference is back to the larger population!