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is quite large. For example, reading off the plot in Figure 2.14, to detect ρ = 0.10, at even a relatively low power level of 0.60, one requires upward of almost 500 participants. This might explain why many studies that yield relatively small effect sizes never get published. They often have insufficient power to reject their null hypotheses. As effect size increases, required sample size drops substantially. For example, to attain a modest level of power such as 0.68 for a correlation coefficient of 0.5, one requires only 21.5 participants, as can be more clearly observed from Table 2.6 which corresponds to the power curves in Figure 2.14 for power ranging from 0.60 to 0.69.

      Hence, one general observation from this simple power analysis for detecting ρ is that size of effect (in this case, ρ) plays a very important role in determining estimated sample size. As a general rule, across virtually all statistical tests, if the effect you are studying is large, a much smaller sample size is required than if the effect is weak. Drawing on our analogy of the billboard sign that reads “H₀ is false,” all else equal, if the sign is in large print (i.e., strong effect), you require less “power” in your prescription glasses to detect such a large sign. If the sign is in small print (i.e., weak effect), you require much more “power” in your lenses to detect it.

      2.22.1 Estimating Sample Size and Power for Independent Samples t‐Test

      For an independent‐samples t‐test, required sample size can be estimated through R using pwr.t.test :

      > pwr.t.test (n =, d =, sig.level =, power =, type = c(“two.sample”, “one.sample”, “paired”))

      where, n = sample size per group, d = estimate of standardized statistical distance between means (Cohen's d), sig.level = desired significance level of the test, power = desired power level, and type = designation of the kind of t‐test you are performing (for our example, we are performing a two‐sample test).

Table 2.6 Power Estimates as a Function of Sample Size and Estimated Magnitude Under Alternative Hypothesis

Exact – Correlation: Bivariate Normal Model Tail(s) = Two, Correlation ρ H0 = 0, α err prob = 0.05
		Correlation ρ H1 = 0.1	Correlation ρ H1 = 0.2	Correlation p HI = 0.3	Correlation ρ HI = 0.4	Correlation ρ HI = 0.5
#	Power(1‐β err prob)	Total Sample Size	Total Sample Size	Total Sample Size	Total Sample Size	Total Sample Size
1	0.600000	488.500	121.500	53.5000	29.5000	18.5000
2	0.610000	500.500	124.500	54.5000	30.5000	18.5000
3	0.620000	511.500	126.500	55.5000	30.5000	19.5000
4	0.630000	523.500	129.500	56.5000	31.5000	19.5000
5	0.640000	535.500	132.500	58.5000	32.5000	19.5000
6	0.650000	548.500	135.500	59.5000	32.5000	20.5000
7	0.660000	561.500	138.500	60.5000	33.5000	20.5000
8	0.670000	574.500	142.500	62.5000	34.5000	21.5000
9	0.680000	587.500	145.500	63.5000	34.5000	21.5000
10	0.690000	601.500	148500	64.5000	35.5000	22.5000

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,

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