Practical Issues in Sample Size Determination for Correlation Coefficient Inference

Abstract

Introduction An important issue in planning any study that will require inference for a correlation coefficient is the determination of the appropriate sample size to use. The usual procedure of choosing n based on the power of the test of the hypothesis that the population correlation is zero often results in correlations of little practical importance being declared "significant," and confidence intervals that are too wide to be of any practical use. In this article, alternative methods for determining sample size are presented and compared to the "usual" procedure. Example Suppose that one is planning a study involving bivariate normal data in which statistical inference is to be performed for a Pearson Correlation Coefficient (PCC) and that the consensus of previous research in the area is that the population correlation is no smaller than 0.40. Reference to sample size tables for the "usual" t-test of the correlation coefficient [1] indicates that a sample of n= 46 will yield 80% power for detecting departures from zero as small as ρ = 0.40 when α = 0.05 (Table 1). For the sake of argument, suppose that the value of the sample PCC (denoted here after by r) from a subsequent sample of 46 is exactly equal to 0.40. This yields a 2-tailed p-value of 0.006 and a 95% confidence interval of (0.12, 0.62). Although these results indicate statistical significance, their practical significance is unclear because the confidence interval is too wide to draw any reasonable conclusion about the true magnitude of ρ. For example, Hebel and McCarter [2] classify 0.0 ≤ |ρ | ≤ 0.2 as "negligible," 0.2 < | ρ | < 0.5 as "weak," 0.5 ≤ | ρ | ≤ 0.8 as "moderate," and 0.8 < | ρ | ≤ 1.0 as "strong." Thus, using their classification scheme, all we can conclude from a confidence interval of (0.12, 0.62) is that ρ is somewhere between "negligible" and "moderate" (inclusive). If one prefers to interpret correlation coefficients in terms of effect size, Cohen [1] suggests that one classify | ρ | = 0.1 as a "small" effect size, | ρ | = 0.3 as "medium," and | ρ | = 0.5 as "large." Using this scheme, all that a confidence interval of (0.12, 0.62) tells us is that the effect size of | ρ | is somewhere between "small" and "large" (inclusive). One of the alternative approaches proposed in this article is to select n on the basis of the desired width of the resulting Confidence Interval (C.I.) for ρ rather than the power of the test of H 0 : ρ= 0. For the aforementioned example, Table 2 indicates that a sample size of n = 273 is required to yield a 95% C.I. of width 0.20 using a "planning value" of r = 0.40. Assuming that a value of exactly r = 0.40 is obtained from a subsequent sample of 273, the resulting p-value is <0.001 and the 95% C.I. is (0.30, 0.50). While this result also indicates statistical significance, the C.I. is sufficiently narrow to indicate that the population correlation between the two variables is "weak" according to the classification scheme of Hebel and McCarter.