1998). Cohen (1988, p. 75) argues that standardized regression coefficients provide a useful way to discuss power for the linear regression setting. The standardized regression coefficient corresponds to the raw slope (in our case, a1) multiplied by the standard deviation of exposure and divided by the standard deviation of the error term. In the simple linear regression setting, the standardized regression coefficient corresponds precisely to the Pearson correlation between X and Y. Because the standardized regression coefficient is a unitless quantity, power calculations are simplified considerably and involve only sample size. According to Table 3.4 of Cohen (1988), the New Zealand study would have had high power (85% or greater) to detect correlations of approximately ± 0.2 or larger, and the Seychelles and the Faroe Islands studies would have had power to detect smaller correlations of approximately ±0.1 or more. Figure 7-2 graphs the power that each study would have had to detect various values of the standardized regression coefficients. The Faroe Islands study, being the largest study, has the highest power, and the New Zealand study has the lowest.
To further aid in interpreting the power calculations summarized in Figure 7-2, we have computed the standardized regression observed in the studies. Table 7-1 shows the standardized regression coefficients for the significant outcomes in the Faroe Islands and New Zealand studies. Five of the eight effects reported in the Faroe Islands study were very small, ranging from −0.05 to 0.08. The power to detect such small effects in the Seychelles study — even with a sample of 700 children — was only about 50% (see Figure 7-2). Thus, some of the inconsistency between the findings of the Faroe Islands and the Seychelles studies could be due to limited power to detect very small effects, even in these very large samples. On the other hand, these analyses cannot explain the failure of the main Seychelles study to detect the neuropsychological effects of the magnitude reported in the New Zealand study, because the Seychelles study should have had adequate power to detect those effects.
There is at least one important caveat to the power considerations discussed above. Standard power calculations for the linear model setting are based on the assumption of a true linear relationship between exposure and outcome. In a real world dose-response setting, such as encountered for MeHg, there is likely to be some nonlinearity. That means that the observed level of statistical significance in a study might depend less on the total sample size than on the spread of the exposure