# Analysis of Variance (ANOVA)

"’Analysis of variance’ is in some ways a misleading name for a collection of statistical methods and models that deal with differences in the means of a variable across groups of observations. While ‘analysis of means’ may be a better name, the methods all employ ratios of variances in order to establish whether the means differe, and the name analysis of variance is here to stay. The name is often abbreviated to ANOVA, whch a student of ours for a long time thought was the name of an Italian statistician" (Iversen & Norpoth, 1987, p. 7).

## Two-way ANOVA

"Two-way ANOVA also introduces a concept not known in a one-way analysis: the concept of interaction. This refers to the way in which a category of one explanatory variable combines with a category of the other explanatory variable to produce an effect on the dependent variable that goes beyond the sum of the separate effects" (Iverson & Norpoth, 1987, p. 91).

## Dirty Words, Alcohol and Memory. Factorial Analysis of Variance

This hypothetical study investigated the effect of alcohol (no alcohol versus strong alcohol) on the ability of 20 introductory psychology students to correctly recall either a list of non-dirty words or a list of highly dirty words. All participants were randomly assigned to conditions. This design was 2 X 2 and completely randomized. The dependent variable was the number of repetitions required for perfect recall. Here are the results of the study:

 A1B1 Strong Alcohol/ Non-Dirty Words 21 22 19 18 21 A2B1 No Alcohol/ Non-Dirty Words 19 20 18 17 21 A1B2 Strong Alcohol/ Highly Dirty Words 2 5 6 3 4 A2B2 No Alcohol/ Highly Dirty Words 7 7 5 6 8

a. Graph these results, using a line graph

c. What means could you calculate to explore your interpretation?

Calculating the cell and column means would help you get a clearer sense of whether there is a main effect or effects, and wteher there is an interaction:

 Alcohol (column) Words (row) Strong No Non-dirty words 20.2 19 19.6 Highly dirty words 4 6.6 5.3 12.1 12.8 12.45

A complete statistical analysis requires additional calculations, some of which are shown below:

 Alcohol Words Strong No Non-dirty words 21 22 19 18 21 =20.2 19 20 18 17 21 =19 SX = 196 SX2 = 3992 n = 10 = 19.6 Highly dirty words 2 5 6 3 4 =4 7 7 5 6 8 =6.6 SX = 53 SX2 = 313 n = 10 = 5.3 SX = 121 SX2 = 2212 n = 10 = 12.1 SX = 128 SX2 = 2093 n = 10 = 12.8 SX = 249 SX2 = 4305 N = 20

The following is a typical summary table, showing the results of a complete analysis of variance on this data. Note that there is a significant main effect for the Words factor. There is also a significant interaction effect.

## Summary Table:

 Source SS df MS F p Alcohol 2.45 1 2.45 1.0089 .312 Words 1022.45 1 1022.45 454.42 0.000 Words x Alcohol 18.05 1 18.05 8.022 0.012 Error 36.00 16 2.250 Total 1078.95 19