What p -hacking really looks like: A comment on Masicampo and LaLande (2012)

Masicampo and Lalande (2012; M&L) assessed the distribution of 3627 exactly calculated p-values between 0.01 and 0.10 from 12 issues of three journals. The authors concluded that "The number of p-values in the psychology literature that barely meet the criterion for statistical significance (i.e., that fall just below .05) is unusually large". "Specifically, the number of p-values between .045 and .050 was higher than that predicted based on the overall distribution of p." There are four factors that determine the distribution of p-values, namely the number of studies examining true effect and false effects, the power of the studies that examine true effects, the frequency of Type 1 error rates (and how they were inflated), and publication bias. Due to publication bias, we should expect a substantial drop in the frequency with which p-values above .05 appear in the literature. True effects yield a right-skewed p-curve (the higher the power, the steeper the curve, e.g., Sellke, Bayarri, & Berger, 2001). When the null-hypothesis is true the p-curve is uniformly distributed, but when the Type 1 error rate is inflated due to flexibility in the data-analysis, the p-curve could become left-skewed below pvalues of .05. M&L (and others, e.g., Leggett, Thomas, Loetscher, & Nicholls, 2013) model pvalues based on a single exponential curve estimation procedure that provides the best fit of p-values between .01 and .10 (see Figure 3, right pane). This is not a valid approach because p-values above and below p=.05 do not lie on a continuous curve due to publication bias. It is therefore not surprising, nor indicative of a prevalence of p-values just below .05, that their single curve doesn't fit the data very well, nor that Chi-squared tests show the residuals (especially those just below .05) are not randomly distributed.