It would be helpful at this point to translate Cohen's d values into R2 values to learn how much variance is explained by differing d values. To convert the two, we apply the following transformation:
Table 2.7 contains conversions for r increments of 0.10, 0.20, 0.30, etc.
To get a better feel for the relationship between Cohen's d and r2, we obtain a plot of their values (Figure 2.15).
As can be gleamed from Figure 2.15, the relationship between the two effect size measures is not exactly linear and increases rather sharply for rather large values (the curve is somewhat exponential).
Suppose a researcher would like to estimate required sample size for a two‐sample t‐test, for a relatively small effect size, d = 0.41 (equal to r of 0.20), at a significance level of 0.05, with a desired power level of 0.90. We compute:
> pwr.t.test (n =, d =0.41, sig.level =.05, power =.90, type = c(“two.sample”)) Two-sample t test power calculation n = 125.9821 d = 0.41 sig.level = 0.05 power = 0.9 alternative = two.sided NOTE: n is number in *each* group
Thus, the researcher would require a sample size of approximately 126. As R emphasizes, this sample size is per group, so the total sample size required is 126(2) = 252.
Table 2.7 Conversions for r → r2→ d.11
| r | r 2 | d |
|---|---|---|
| 0.10 | 0.01 | 0.20 |
| 0.20 | 0.04 | 0.41 |
| 0.30 | 0.09 | 0.63 |
| 0.40 | 0.16 | 0.87 |
| 0.50 | 0.25 | 1.15 |
| 0.60 | 0.36 | 1.50 |
| 0.70 | 0.49 | 1.96 |
| 0.80 | 0.64 | 2.67 |
| 0.90 | 0.81 | 4.13 |
| 0.99 | 0.98 | 14.04 |
Figure 2.15 Relationship between Cohen's d and R‐squared.
2.23 PAIRED‐SAMPLES t‐TEST: STATISTICAL TEST FOR MATCHED‐PAIRS (ELEMENTARY BLOCKING) DESIGNS
Oftentimes in research, we are able to sample observations that are matched on one or more variables or characteristics. For instance, consider the hypothetical data in Table 2.8.
Table 2.8 Matched-Pairs Design
| Treatment 1 | Treatment 2 | |
|---|---|---|
| Block 1 | 10 | 8 |
| Block 2 | 15 | 12 |
| Block 3 | 20 | 14 |
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