Analysis of Variance (ANOVA)The F distributionGood for two or more groups

The F distributionF is a ratio of two independent estimates of the variance of the populationConsequently, it depends on the analysis (separating into parts) of the variance in a set of scores.We have already analyzed the variance in a set of scores when we did t tests

The analysis of varianceThe numerator of the t test is the difference between two means. Since the variance is the average of the differences from a mean, the difference between two means drawn from the same set of scores is one estimate of variance.The denominator of the t test, although expressed as a standard deviation, is also an estimate of variance.

The F ratioA ratio of two estimates of variance drawn from the same set of scores is called an F ratio.F = larger estimate of variance smaller estimate of variance

F = variance estimate 1 of s2 variance estimate 2 of s2

Why use ANOVA?We sometimes have independent variables with more than two groups:Grape Kool-Aid, Lemon Kool-Aid, and waterOne marijuana cigarette, two, three, fourTQM in place for one month, two, threeFour styles of governmentWith more than two groups, multiple t tests would be necessary.Multiple t tests inflate the Type I error rate.

Logic of ANOVA1. Find the total variance in a set of data.2. Analyze the variance intoa. The part due to the treatment (plus people)b. The part due to people without treatment (individual differences and error)3. Form an F ratio of the two parts

The ANOVA summary tableTo keep track of the ANOVA process, start with a summary table:

Source SS df MS ( s2) F p Decision Between WithinTotal

Notation summaryN is the number of scores altogether.n is the number of scores in a single group.k is the number of groups.g is a particular group.The variance, s2 , is also symbolized MS, which stands for Mean Square.

SS totalTo get SStotal, simply group all of the scores for all of the groups together and find the SS as if there were only one group:

SStotal = SX2 - (SX)2 / N

Or SStotal = SX2 (G2 / N)

SS within groupsSS within groups is due to individual differences and error.To compute SSwithin,calculate SS normally for each group separately, and add them up.SSwithin = S(SX2 - (SX)2/ n) =SS1+SS2++SSk = SX2 - (SX1)2 (SX2)2 (SXk)2

SS between groupsTo get SSbetween, treat the sum of each group as a score, and apply the be-bop version of the sum of squares song:SSB =

= (SX1)2 (SX2)2 (SXk)2 - (SX)2 N

Completing the one-way ANOVASource SS df MS ( s2) F p Between k-1 SSB/dfB MSB/MSW Within N-k SSW/dfWTotal N-1

To obtain p, either use the table to find the critical value of F, or SPSS to find the probability of the obtained value of F.

Making your decisionTo decide whether the obtained F ratio is significant, compare it to the table value (the critical value). If the obtained F is equal to or larger than the critical F, reject H0..05 levelTable value: Length of pierComputed value: Length of walkFishy zone:Reject H0

Decision rules for ANOVAFind the critical values for F from the table. Use dfB for the (between) column, and dfW for the (within) row in the table.The first table has the critical value at the .05 level, and the second table for the .01 level.In the p column of the ANOVA summary table, enter

More ANOVA itemsFor two group studies, t2 = FTwo-way factorial ANOVAMain effects: Combines two (or more) studiesFully crossed factors: A x BInteraction effect: The effect of one independent variable depends on the level of the other independent variable

Try this oneHere are the number of errors on an analysis of variance problem for each of nine students after drinking grape Kool-Aid, Frankenberry Punch, or water.GKA FBP H2O 5 8 10 3 7 11 4 7 12

Effect size measuresSignificant t and F ratios show that there is a real effect of the treatment, a real difference between the groups that cannot be explained by chance.Effect size measures show how big the effect of the treatment is.C.O.D. is one effect size measure: C.O.D. = r2 = SSB /SSTotal = h2

Effect sizes in samples vs. populationsh2 is a sample estimate of the proportion of the variance in the dependent variable that is accounted for by the independent variable.For population estimates of effect size, use a different statistic, w2

Computing w2For the t test for independent samples,w2 = t2 1 t2 + N 1

For the analysis of variance, w2 = SSB (k 1)MSW SSTotal + MSW

Concluding detailsANOVA assumptions are the same as the assumptions for the t-test for independent samples.Try to have each group/sample be the same size, that is, have equal ns. However, the ANOVA computations will work with unequal sample sizes.

Reporting ANOVA in APA formatReport the mean and standard deviation for each sample/group in a table, and refer to it in a sentence reporting the F ratio:Table 1 contains the means and standard deviations for the IQs, which were significantly different, F(3, 36) = 9.47, p < .05.

Table 1. Tested IQ.