Comparison of means
Cohen's d
Hedges's g
Glass's Δ
Point/biserial correlation
Estimated from data or published summary statistics
Variance explained by regression and ANOVA
Eta-squared and partial eta-squared (η2)
Epsilon-squared and partial epsilon-squared (ε2)
Partial statistics estimated from data
Overall statistics from data or published summary statistics
With confidence intervals
esize, esizei, and estat esize calculate measures of effect size for (1) the difference between two means and (2) the proportion of variance explained.
Say we have data on mothers and their infants' birthweights. We want to calculate the effect size on birthweight of smoking during pregnancy:
. webuse lbw
(Hosmer & Lemeshow data)
. esize twosample bwt, by(smoke) all
Effect size based on mean comparison
Obs per group:
Nonsmoker = 115
Smoker = 74
| Effect size | Estimate [95% conf. interval] | |
| Cohen's d | .3938497 .0985333 .6881322 | |
| Hedges's g | .3922677 .0981375 .685368 | |
| Glass's Delta 1 | .3756723 .0787487 .6709925 | |
| Glass's Delta 2 | .4283965 .1267939 .7272194 | |
| Point-Biserial r | .1897497 .0482935 .3199182 | |
We find that the difference in average birthweight is about 0.4 standard deviations.
We can reasonably assume birthweight is normally distributed; thus the reported confidence intervals are appropriate in this case.
In many cases, normality cannot reasonably be assumed. In such cases, we can obtain bootstrapped confidence intervals:
. bootstrap r(d) r(g), reps(200) nowarn seed(111): esize twosample bwt, by(smoke)
(running esize on estimation sample)
Bootstrap replications (200): .........10.........20.........30.........40.........50.........60..
> .......70.........80.........90.........100.........110.........120.........130.........140.....
> ....150.........160.........170.........180.........190.........200 done
Bootstrap results Number of obs = 189
Replications = 200
Command: esize twosample bwt, by(smoke)
_bs_1: r(d)
_bs_2: r(g)
| Observed Bootstrap Normal-based | ||
| coefficient std. err. z P>|z| [95% conf. interval] | ||
| _bs_1 | .3938497 .1391761 2.83 0.005 .1210697 .6666298 | |
| _bs_2 | .3922677 .138617 2.83 0.005 .1205833 .663952 | |
When you have summary statistics but not the underlying data, as you might when reading a journal article, you can use Stata's immediate command. Let's pretend our birthweight example was published. The hypothetical article recorded that for the 115 mothers who did not smoke, the average birthweight was 3,054.957 grams (sd=752.409) and that for the 74 smokers, the average was 2772.297 grams (sd=659.8075). We type
. esizei 115 3054.957 752.409 74 2772.297 659.807541
Effect size based on mean comparison
Obs per group:
Group 1 = 115
Group 2 = 74
| Effect size | Estimate [95% conf. interval] | |
| Cohen's d | .3938508 .0985343 .6881333 | |
| Hedges's g | .3922687 .0981385 .685369 | |
We can use the estat esize postestimation command to calculate effect sizes after fitting ANOVA models.
We fit a full factorial model of newborn birthweight on mother's smoking status and whether the mother saw a doctor during her first trimester:
. anova bwt smoke##ftv
Number of obs = 189 R-squared = 0.0860
Root MSE = 716.273 Adj R-squared = 0.0347
| Source | Partial SS df MS F Prob>F | ||
| Model | 8592836.5 10 859283.65 1.67 0.0897 | ||
| smoke | 192977.93 1 192977.93 0.38 0.5405 | ||
| ftv | 2200463.6 5 440092.72 0.86 0.5107 | ||
| smoke#ftv | 2322255.1 4 580563.77 1.13 0.3432 | ||
| Residual | 91322462 178 513047.54 | ||
| Total | 99915299 188 531464.35 |
We can obtain the proportion of variability explained (effect sizes) measured by η2, ε2, or ω2. Here is the default η2 measure:
. estat esize Effect sizes for linear models
| Source | Eta-squared df [95% conf. interval] | |
| Model | .0860012 10 . .1214932 | |
| smoke | .0021087 1 . .0351462 | |
| ftv | .0235286 5 . .054553 | |
| smoke#ftv | .0247986 4 . .0642531 | |
Reported are full and partial η2 values along with their confidence intervals. We could have added the epsilon or omega option to instead request the ε2 or ω2 measure.
We can also use the estat esize postestimation command to calculate effect sizes after fitting linear models.
We replace the insignificant drvisit variable with the continuous variable age and fit the model using linear regression.
. regress bwt smoke##c.age
| Source | SS df MS | Number of obs = 189 | |
| F(3, 185) = 4.55 | |||
| Model | 6859112.22 3 2286370.74 | Prob > F = 0.0042 | |
| Residual | 93056186.4 185 503006.413 | R-squared = 0.0686 | |
| Adj R-squared = 0.0535 | |||
| Total | 99915298.6 188 531464.354 | Root MSE = 709.23 |
| bwt | Coefficient Std. err. t P>|t| [95% conf. interval] | |
| smoke | ||
| Smoker | 797.9369 484.3249 1.65 0.101 -157.5731 1753.447 | |
| age | 27.60058 12.14868 2.27 0.024 3.632806 51.56835 | |
| smoke#c.age | ||
| Smoker | -46.51558 20.44641 -2.28 0.024 -86.85368 -6.177479 | |
| _cons | 2408.383 292.1796 8.24 0.000 1831.951 2984.815 | |
This time, we request the ω2 estimates of effect size:
. estat esize, omega Effect sizes for linear models
| Source | Omega-squared df | |
| Model | .0532781 3 | |
| smoke | .0090843 1 | |
| age | -.0044019 1 | |
| smoke#c.age | .0218418 1 | |
Reported are full and partial ω2 values.
If we did not have the data to estimate this model but instead found the regression fit published in a journal, we could still estimate the overall η2, ε2, and ω2 from the model's degrees of freedom and the summary statistic that F(3, 185) = 4.55. We could type
. esizei 3 185 4.55 Effect sizes for linear models
| Effect Size | Estimate [95% conf. interval] | |
| Eta-squared | .0687138 .0079234 .1364187 | |
| Epsilon-squared | .0536119 | |
| Omega-squared | .0533434 | |
The ω2 agrees to three decimal places. Had we typed 4.5454107 rather than 4.55, we would have had full agreement to the shown eight decimal places.
See the manual entry.
