East Carolina University
Department of Psychology


Effect of n1/n2 on Estimated d and rpb

    When comparing two means, the most commonly employed effect size estimators are g (estimated d) and the point-biserial r.  Each has it advocates and its critics.  One of the factors that one should consider when choosing which to employ is the effect of disparate sample sizes on the two estimators, which I illustrate below.

Equal Sample Sizes

        First we look at an analysis of two samples where the sample sizes are equal.

T-TEST GROUPS=A(1 2)
  /MISSING=ANALYSIS
  /VARIABLES=Y1
  /CRITERIA=CI(.9500).

Group Statistics
  A N Mean Std. Deviation Std. Error Mean
Y1 1 20 5.5000 2.30560 .51555
2 20 7.8000 2.30560 .51555

Independent Samples Test
  t-test for Equality of Means
t df Sig. (2-tailed) Mean Difference
Y1 Equal variances assumed -3.155 38 .003 -2.30000
 

    Notice that the two means differ by one standard deviation (2.3).  That is, estimated d = 1.00, a large effect (Cohen's benchmark for a large effect was d = .8).

    Now we compute the point biserial.


CORRELATIONS
  /VARIABLES=Y1 WITH A
  /PRINT=TWOTAIL NOSIG
  /MISSING=PAIRWISE.

Correlations
  A
Y1 Pearson Correlation .456**
Sig. (2-tailed) .003
N 40
**. Correlation is significant at the 0.01 level (2-tailed).
 

    No matter how we look at it, our effect here is large.

(Very) Unequal Sample Sizes

    Now let us look at the analysis on a data set where the sample sizes differ considerably.  The standard deviations and the mean differ very little from those in the first data set.


T-TEST GROUPS=B(1 2)
  /MISSING=ANALYSIS
  /VARIABLES=Y2
  /CRITERIA=CI(.9500).

Group Statistics
  B N Mean Std. Deviation Std. Error Mean
Y2 1 100 5.5000 2.25854 .22585
2 4 7.7750 2.24109 1.12055

Independent Samples Test
  t-test for Equality of Means
t df Sig. (2-tailed) Mean Difference
Y2 Equal variances assumed -1.976 102 .051 -2.27500

   

    The means still differ by one standard deviation -- estimated d = 1.01, a large effect (Cohen's benchmark for a large effect was d = .8).

    Now we compute the point biserial.

 

CORRELATIONS
  /VARIABLES=Y2 WITH B
  /PRINT=TWOTAIL NOSIG
  /MISSING=PAIRWISE.

 

Correlations
  B
Y2 Pearson Correlation .192
Sig. (2-tailed) .051
N 104
 

    Although the estimated d indicates we have a large effect, the point biserial indicates that we have a small to medium effect.

 How is the ratio of sample sizes having such an effect on the value of the point biserial?

   If you would like to read more about the differences between estimated d and the rpb as estimators of effect size, I recommend the following article:  McGrath, R. E., & Meyer, G. J.  (2006).  When effect sizes disagree:  The case of r and d. Psychological Methods, 11, 386-401.

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This page most recently revised on 4. April 2010