Sample Size Tables for Correlatio...
MULTIVARIATE BEHAVIORAL RESEARCH 309 Multivariate Behavioral Research, 38 (3), 309-323 Copyright �� 2003, Lawrence Erlbaum Associates, Inc. Sample Size Tables for Correlation Analysis with Applications in Partial Correlation and Multiple Regression Analysis James Algina University of Florida Stephen Olejnik University of Georgia Tables for selecting sample size in correlation studies are presented. Some of the tables allow selection of sample size so that r (or r2, depending on the statistic the researcher plans to interpret) will be within a target interval around the population parameter with probability .95. The intervals are .05, .10, .15, and .20 around the population parameter. Other tables allow selection of sample size to meet a target for power when conducting a .05 test of the null hypothesis that a correlation coefficient is zero. Applications of the tables in partial correlation and multiple regression analyses are discussed. SAS and SPSS computer programs are made available to permit researchers to select sample size for levels of accuracy, probabilities, and parameter values and for Type I error rates other than those used in constructing the tables. For about 40 years, data analysts have been recommending to researchers in the behavioral sciences that an effect size measure should be reported in addition to a test for statistical significance (Cohen, 1965 Hays, 1963). In recent years there has been increased support for reporting effect sizes. Three examples of the increased support are as follows. 1. According to The Publication Manual of the American Psychological Association (2001) ���it is almost always necessary to include some index of effect size or strength of relationship in your Results section.��� (p. 25). 2. The Editor of Journal of Applied Psychology requires an explanation for failing to report an effect size (Murphy, 1997). 3. The journal Educational and Psychological Measurement requires effect size estimates to be reported (Thompson, 1994). The practice of reporting effect sizes has also received support from the APA Task Force on Statistical Inference (Wilkinson and the Task Force on Statistical Inference, 1999). The emphasis on reporting effect sizes and strength of relationship implies that researchers should plan studies not only to have sufficiently powerful hypothesis tests but also to have sufficiently accurate estimates of effect sizes.
J. Algina and S. Olejnik 310 MULTIVARIATE BEHAVIORAL RESEARCH One of the most widely used statistics in the social and behavioral sciences is the Pearson product moment correlation coefficient r. An interesting characteristic of this statistic is that it serves as an index of strength of linear association for two variables as well as the basis for testing the null hypothesis that the population correlation coefficient is equal to some specified value, most often zero. Since the correlation coefficient is itself an effect size, it follows that a study, employing the correlation coefficient, should be planned so the correlation coefficient will be estimated with adequate accuracy and hypotheses about the correlation coefficient will be tested with sufficient power. For example, suppose a validity study is conducted in which the population correlation coefficient is thought to be .30. A sample size of 70 is used. In this study, power for a one-tailed test that the correlation coefficient is zero would be .82. And, there would be approximately a 95% chance that the sample correlation coefficient would be in the interval (.1345, .4655). Although the study has is good power, it would have insufficient accuracy if one regards a sample correlation coefficient as small as .14 or as large as .47 as misleading. In this case, the sample size of 70 is not large enough to ensure adequate estimation accuracy. Estimating the correlation coefficient with adequate accuracy is important not only when estimation of the strength of association between pairs of variables in the study is the ultimate goal of the study, but also when the data will be used in additional analyses such as regression or structural equation modeling. Even in the latter types of studies, the strength of association between pairs of variables is typically of some interest. The purpose of the present study is to present a table that will facilitate selection of sample sizes to ensure that correlation coefficients are estimated with adequate accuracy. Because some researchers use r2 as an effect size to measure strength of association, we also present a table that will facilitate selection of sample sizes to ensure estimation accuracy for squared correlation coefficients. In addition we investigate the accuracy of two sample size selection methods that are approximate, but are very easy to apply. And for the sake of completeness, we present power tables, so that researchers can select a sample size that meets the twin goals of adequate power and accuracy. Method When the goal is to estimate with sufficient accuracy, we need to find the smallest sample size n necessary for the sample correlation coefficient to fall into a prescribed interval with a prescribed probability:
J. Algina and S. Olejnik MULTIVARIATE BEHAVIORAL RESEARCH 311 (1) prob[max(���1, ��� c) r min( + c, 1)] p. In Equation 1, max(���1, ��� c) and min( + c, 1) are the ends of an interval into which the researcher wants r to fall and p is the probability of that event. The quantity c operationalizes the researcher���s view of adequate estimation accuracy. For example, if a researcher believes the population correlation coefficient is likely to be .45 and wants the sample correlation coefficient to be within .10 with probability of at least .95 then Equation 1 becomes prob[(.45 ��� .10) r (.45 + .10)] .95 and the problem is to find n so that this interval is met. This type interval used in Equation 1 is widely used in survey sampling in order to select sample size (see, for example, Jaeger, 1984 Kish, 1965 or Sudman, 1978). The following steps were used to find the required sample size for a given and c. Beginning with a provisional sample size of n = 3, find the probability that r is in the interval in Equation 1. If the probability exceeds the target, the sample size is sufficient. If the provisional sample size is not sufficient, increase the provisional sample size by one and carry out the computations again. The probability that r is in the required interval was calculated by finding the area under the density function of r, under assumed bivariate normality, for the limits in Equation 1. The area can be found by using numerical integration methods. We wrote a Mathematica program to compute the areas. The program used the GaussKronrod method, which is the default in Mathematica. The GaussKronrod method is an adaptive Gauss quadrature method with error estimation based on evaluation at Kronrod points (Wolfram, 1999). The required sample size was found for all combinations of from .00 to .95 and c = .05 to .20 both in steps of .05. Because we used a simple search procedure, the numeric integration was time consuming for some of the combinations of and c. While this problem could have been solved by using a more efficient search strategy, the numeric integration also had to be monitored carefully for problems in the integration. In some cases the value for the probability in Equation 1 increased as n increased and then declined again in other cases the value of the probability increased by a large amount with small increases in n. In still other cases Mathematica produced various warnings indicating problems in the integration. All of these problems were solved by increasing the precision with which r was represented in the calculations and the precision of the computations in NIntegrate, the procedure used in Mathematica to carry out the integration. These changes increased the time required for the computations.