Summary comments on the nature of ANALYSIS OF VARIANCE:

• The technique tests for significant differences between the means of k groups based on two different estimates of population VARIANCE.
  • One estimate is based on the variance within each of k samples.
  • The other is based on the variance between (among) the sample means.
• The F test is simply a ratio between these two estimates:

F =	estimate of variance based on BETWEEN mean differences
	estimate of variance based on WITHIN group differences

• If this ratio is small, then the two estimates tend to agree, and we conclude that the observed differences in means reflect differences allowable by drawing random samples from the same population.
•  If the ratio is "large," however, we conclude that the differences among our groups do not simply reflect random error in sampling.

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Interpretation of the F ratio:

• An F ratio less than 1 is never signficant for rejecting the null hypothesis.
• That would show more variation within groups than between groups, and thus the groups would explain no variation at all in X.
  • Ratios larger than 1 may be significant: one must find out by checking the table of the F distribution corresponding to the chosen alpha level.
    • Degrees of freedom (df) for the "between" estimate (the larger) run along the top of the F table
    • df for the "within" estimate run down the side.
  • An observed value larger than the one in the table means the difference is significant at that alpha value.
•  Relationship of F to t:
  • Conceptually: F is a generalization of T-test for two groups
  • Computationally:
    • When there are two groups (and thus df BSS = 1)
    • t = square root of F (or t² = F) 

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Analysis of variance with SPSS:

• SPSS Programs for analysis of variance
  • Under the Analyze Menu, choose Compare Means
    • Then select One-Way ANOVA
      • Press the "Options" button and check "Descriptive" and "Means plot"
        • Transfer your dependent variable into the "Dependent List"
          • Transfer your independent (discrete) variable into the "Factor" list
• One-way analysis of variance programs follow a common form, reading from right to left:
  • Source of variation
    • df
      • sums of squares
        • mean square
          • F ratio
            • significance 

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Analysis of variance in research: a simple example:

• Lacy, "Political Knowledge of College Activist Groups: SDS, YAF, and YD"
•  Intellectual Problem: previous research has shown that liberal students were better informed than non-liberals
  • But this research compared activist and non-activist students
  • The conservative activist students need to be studied too
  • Existing studies suggest no difference in intelligence between these two groups
• Lacy's Research Design and Data Analysis
  • Studied 15 YAF, 39 SDS, and 33 YD at University of Houston
  • Compared them on test of political knowledge
• Analysis of variance for knowledge of American government showed no significant differences among the group means