Measures of Effect Size (Strength of Association)

Measures of effect size in ANOVA are measures of the degree of association between and effect (e.g., a main effect, an interaction, a linear contrast) and the dependent variable. They can be thought of as the correlation between an effect and the dependent variable.  If the value of the measure of association is squared it can be interpreted as the proportion of variance in the dependent variable that is attributable to each effect. Four of the commonly used measures of effect size in AVOVA are:  Eta squared (h²), partial Eta squared (h_p²), omega squared (w²), and the Intraclass correlation (r_I).   Eta squared and partial Eta squared are estimates of the degree of association for the sample.  Omega squared and the intraclass correlation are estimates of the degree of association in the population.  SPSS for Windows 9.0 (and 8.0) displays the partial Eta squared when you check the display effect size option. This set of notes describes the similarities and differences between these measures of association.

The measures of association will be calculated for the study of the effects of drive and reward on performance in an oddity task that was used as the example in the notes for a 2-way ANOVA (GLM: 2-way).  The analysis of variance table with the corresponding Eta squared scores for each effect is shown in Table 1.

Eta squared (h²)

Eta squared is the proportion of the total variance that is attributed to an effect.   It is calculated as the ratio of the effect variance (SS_effect) to the total variance (SS_total) --

h² = SS_effect / SS_total

The values used in the calculations for each h² along with the h_p² from the ANOVA output are shown in Table 2.

The reward by drive interaction was significant in this analysis, F(2,18) = 3.927, p = .038. Using h² as the measure of effect size, the interaction between drive and reward accounted for 24% of the total variability in the performance score.  Using h_p² as the measure of association, the interaction between drive and reward accounted for 30% of the total variability in the performance score.

If you decide to calculate h² rather than use the values of  h_p² displayed by SPSS then will you need to be careful about selecting the SS_total to be used in the calculation of  h².  The values reported in the "Total" row of the ANOVA table include the following SS: 

(a) the SS for each of the effects, 
(b) the SS for the error term, and 
(c) the SS for the intercept.  

 The value for SS_total in the h² formula includes the SS for each of the effects and the error term, but it does not include the SS for the intercept.  The values reported in the "Corrected Total" row of the ANOVA table include the SS for each of the effects and for the error term but they do not include the SS for the intercept.  Thus, the SS for the "Corrected Total" row should be used in the calculation of  h².

A pie chart can be used to graphically display proportion of total variance that is attributable to each effect, see Figure 1.  The entire circle represents the (corrected) total sums of squares. Each slice of the pie is an effect or the SS for error.  The percent of the pie represented by each slice is the effect size, h2.   The sums of squares for error represents more than half (54%) of the total variability in the dependent variable scores. The only significant effect, the drive by reward interaction, accounts for 24% of the variability in the dependent variable scores. In a balanced design with equal ns the sum of the h2 for the effects is the total amount of variance in the dependent variable that is predictable from the independent variables.  In this example the independent variables account for 46% (3.9% + 18.4% + 23.6%) of the variability  in the dependent variable scores.

Figure 1.  Relative effect sizes (Eta squared) for the drive, reward, and the drive by reward interaction.

Statistical Issues:

One of the problems with h² is that the values for an effect are dependent upon the number of other other effects and the magnitude of those other effects.  For example, if a third independent variable had been included in the design, then the effect size for the drive by reward interaction probably would have been smaller, even though the SS for the interaction might be the same.  Similarly, if the SS for reward had been larger and there was no change in the SS for the interaction effect, then the interaction Eta squared would have been smaller.  For that reason many people prefer an alternative computational procedure called the partial Eta squared.  SPSS reports the partial Eta squared rather than Eta squared. Partial Eta squared is described in the next section. Some authors (e.g., Tabachnick & Fidell, 1989) call partial Eta squared an "alternative" computation of Eta squared.

Graphics Note: 

Here are the steps used to create the pie chart in Figure 1.

1.  Create an SPSS data file with two variables, effect, and SSeffect.  The values entered into the data file are shown in Table 3. The values for effect include the three effects and the interaction.  The values for SSeffect are the sums of squares for each effect.

The SSeffect variable was used as a "weight cases" variable.

2. Select Graphs -> Interactive -> Pie -> Simple

Move the variable effect to the Slice By: window.  Move the variable Count[$count] to the Slice Summary: window. Press OK to create the pie chart.

Some editing was then done to delete some extra information for each slide.

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Partial Eta squared, h_p²

The partial Eta squared is the proportion of the the effect + error variance that is attributable to the effect.  The formula differs from the Eta squared formula in that the denominator includes the SS_effect plus the SS_error rather than the SS_total  --

h_p² = SS_effect / (SS_effect + SS_error)

The values from the ANOVA output that were used in the calculations for each  h_p² are shown in Table 4.

Graphic representation of partial Eta square measures requires set of pie charts, one for each effect.

Figure 2.  Partial Eta squared values for the drive, reward, and drive by reward effects.

Statistical Issues:

The sums of the partial Eta squared values are not additive. They do not sum to the amount of dependent  variable variance accounted for by the independent variables.   It is possible for the sums of the partial Eta squared values to be greater than 1.00.  In general, Eta squared values describe the amount of variance accounted for in the sample.  An estimate of the amount of variance accounted for in the population is omega squared.

Graphics Note: 

Here are the steps used to create the pie chart in Figure 2.

1.  Create an SPSS data file with six variables, drive, SSdrive, reward, SSreward, int (interaction), and SSint.  The values entered into the data file are shown in Table 5. There are two values for drive, reward, and int: 1 = the name of the effect (drive, reward, or reward*drive) and 2 = error.  The sums of squares for the effect and its error term are entered in the variables SSdrive, SSreward, and SSint.

2. Create one pie chart for each effect. First, weight cases by the SS for the effect (e.g., SSdrive). Select Graphs -> Interactive -> Pie -> Simple. Using the effect name (e.g., drive) as the Slice by: variable, and Count[$count] as the Slice Summary: variable.  And create the pie chart.  Repeat these steps for the reward pie chart and the drive by reward pie chart.

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Omega squared, w²

Omega squared is an estimate of the dependent variance accounted for by the independent variable in the population for a fixed effects model.  The between-subjects, fixed effects,  form of the w² formula is --

w² = (SS_effect - (df_effect)(MS_error)) / MS_error + SS_total

(Note: Do not use this formula for repeated measures designs)

The values from the ANOVA table that are used in the calculation of  w² are shown in Table 5.

Graphic representation of w² would be done with separate pie charts, similar to the way h_p² was presented above.

Because h² and h_p² are sample estimates and w² is a population estimate, w² is always going to be smaller than either  h² or h_p². The three measures of association are shown in Table 6.

Formulas for a random effects model are also available, see Kirk, 1982.

Intraclass correlation (r_I )

The intraclass correlation an estimate of the degree of association between the independent variable and the dependent variable in the population for a random effects model. Because it is for a random effects model it is not commonly used in psychology experiments.  The formula for  r_Iis --

r_I= (MS_effect - MS_error) / (MS_effect + (df_effect)(MS_error)) .

The model in the example used in this set of notes is a fixed effect model so the intraclass correlation will not be computed.

The square of the intraclass correlation is an estimate of the amount of dependent variable variance accounted for by the independent variable.

Kirk, R. E. (1982). Experimental design: Procedures for the behavioral sciences (2nd ed.). Belmont, CA: Brooks/Cole.

Tabachnick, B. G., & Fidell, L. S. (1989). Using multivariate statistics (2nd ed.). New York: Harper & Row.