Measures of Effect Size (Strength of Association)

1.  Overview
2.  Measures for Analysis of Variance

3.  References

 


1. Overview

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 (hp2), omega squared (w2), and the Intraclass correlation (rI).   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.

Table 1. ANOVA with partial Eta squared measures
 

Source

Type III Sum of Squares

df

Mean Square

F

Sig.

Eta Squared

Corrected Model

280.000

5

56.000

3.055

.036

.459

Intercept

2400.000

1

2400.000

130.909

.000

.879

DRIVE

24.000

1

24.000

1.309

.268

.068

REWARD

112.000

2

56.000

3.055

.072

.253

DRIVE * REWARD

144.000

2

72.000

3.927

.038

.304

Error

330.000

18

18.333

     

Total

3010.000

24

       

Corrected Total

610.000

23

       

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2.  Measures for Analysis of Variance

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 (SSeffect) to the total variance (SStotal) --

h2 = SSeffect / SStotal

The values used in the calculations for each h2 along with the hp2 from the ANOVA output are shown in Table 2.

Table 2. Values used in computing Eta squared.
Effect SSeffect SStotal
(Corrected Total)
h2 hp2
Drive 24.000 610.000 .039 .068
Reward 112.000 610.000 .184 .253
Reward * Drive 144.000 610.000 .236 .304

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 hp2 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  hp2 displayed by SPSS then will you need to be careful about selecting the SStotal 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 SStotal 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.efsize_fig1.jpg (62341 bytes)

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.

Table 3. Case Summaries
  Effect SSeffect
1 drive 24.00
2 reward 112.00
3 drive*reward 144.00
4 error 330.00

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, hp2

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 SSeffect plus the SSerror rather than the SStotal  --

hp2 = SSeffect / (SSeffect + SSerror)

The values from the ANOVA output that were used in the calculations for each  hp2 are shown in Table 4.

Table 4. Values used in computing partial Eta squared.
Effect SSeffect SSerror SSeffect + SSerror hp2
Drive 24.000 330.000 354.000 .068
Reward 112.000 330.000 442.000 .253
Reward * Drive 144.000 330.000 474.000 .304

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.

efsize_fig20.jpg (15662 bytes) efsize_fig3100.jpg (13808 bytes) efsize_fig40.jpg (11725 bytes)

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.

Table 5. Case Summaries
 

DRIVE

SSdrive

REWARD

SSreward

INT

SSint

1

SSdrive

24.00

SSreward

112.00

SSdrive*reward

144.00

2

SSerror

330.00

SSerror

330.00

SSerror

330.00

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 = (SSeffect - (dfeffect)(MSerror)) / MSerror + SStotal

(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.

Table 5. Values used in computing Omega squared.
Effect SSeffect dfeffect MSerror SStotal
(Corrected)
SSeffect-(dfeffect)(MSerror)
---------------------------------
MSerror + SStotal
w2
Drive 24.000 1 18.333 610.000 5.667 / 628.333 .001
Reward 112.000 2 18.333 610.000 75.334 / 628.333 .120
Reward * Drive 144.000 2 18.333 610.000 107.334 / 628.333 .171

Graphic representation of w2 would be done with separate pie charts, similar to the way hp2 was presented above.

Because h2 and hp2 are sample estimates and w2 is a population estimate, w2 is always going to be smaller than either  h2 or hp2. The three measures of association are shown in Table 6.

Table 6. A comparison of effect size values
Effect h2 hp2 w2
Drive .039 .068 .001
Reward .184 .253 .120
Reward * Drive .236 .304 .171

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


Intraclass correlation (rI )

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  rIis --

rI= (MSeffect - MSerror) / (MSeffect + (dfeffect)(MSerror)) .

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.

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3. References

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.

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Lee A. Becker, 1998-1999  -