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ANOVA

Analysis of Variance (ANOVA) is a procedure comparing sample means of several groups to see if the population means are different. One-way analysis of variance is most easily explained by contrasting it with t-tests. Whereas t-tests compare only two distributions, analysis of variance is able to compare many. If, for instance, a sample of students takes a 10-point quiz and we wish to see whether women or men scored higher on this quiz, a t-test would be appropriate. There is a distribution of women's scores and a distribution of men's scores, and a t-test will tell if the means of these two distributions differ significantly from each other. If, however, we wished to see if any of five ethnic groups' (White, Black, Native, Asian and Hispanic) scores differed significantly from each other on the same quiz, it would require one-way analysis of variance to accomplish this. If we were to run such a test, one-way ANOVA could tell us if there are significant differences within any of the comparisons of the five groups in our sample. Further tests (such as the Scheffe test or Tukey test) are necessary to determine between which groups significant differences occur.

The previous paragraph briefly described analysis of variance. What does "one-way" part mean? If using one-way ANOVA, you are allowed to have exactly one dependent variable (always continuous) and exactly one independent variable (always categorical). The independent variable illustrated above (ethnicity) is one variable but it has several levels. In our example it has five: White, Black, Native, Asian and Hispanic. In general, ANOVA models may have a maximum of one dependent variable but they may have two or more independent variables (sometimes called factors). Two independent variables is the case of two-way ANOVA, three variables is the case of three-way ANOVA, and so on. In MANOVA, multivariate analysis of variance, there may be multiple dependent variables and multiple independent variables.

The explanation that follows gives a conceptual feel for what analysis of variance is attempting to accomplish. The mean (average) quiz scores for each of the ethnic groups are compared with each other: Natives with Asians, Natives with Blacks, Natives with Whites, Natives with Hispanics, Asians with Blacks, Asians with Whites, Asians with Hispanics, Blacks with Whites, Blacks with Hispanics and Whites with Hispanics. ANOVA will generate a significance value indicating whether there are significant differences within the comparisons being made. The significance value does not indicate where the difference is or what the differences are, but a Tukey test, a Scheffe test, a least-significant difference test (LSD) or a Bonferroni test can identify which groups differ significantly from each other.

ANOVA REFERENCES

D'Agostino, R., Sullivan, L., & Beiser, A. (2005). Introductory Applied Biostatistics. Cengage Learning.

Weerahandi, S. (2004). Generalized Inference in Repeated Measures : Exact Methods in MANOVA and Mixed Models. Wiley-Interscience, Hoboken, New Jersey.

Teller, G. R. (1999). Mathematical Statistics: A Unified Introduction. New York: Springer.

Huberty, C. J., & Olejnik, S. (2006). Applied MANOVA and Discriminant Analysis (2nd ed). Wiley-Interscience, Hoboken, New Jersey.

Dean, A. M., & Voss, D. (2000). Design and Analysis of Experiments. Springer-Verlag New York.

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