*In this blog, I'll explain how p-hacking will not lead to a peculiar prevalence of p-values just below .05 (e.g., in the 0.045-0.05 range) in the literature at large, but will instead lead to a difficult to identify increase in the Type 1 error rate across the 0.00-0.05 range. I re-analyze data by Masicampo & LaLande (2012), and try to provide a better model of the p-values they have observed through simulations in R (code below). I'd like to thank E.J. Masicampo and Daniel LaLande for sharing and allowing me to share their data, as well as their quick response to questions and a draft of this post. Thanks to Ryne Sherman for his p-hack code in R, to Nick Brown for asking me how M&L's article related to my previous blog post and for comments and suggestions on this post, and Tal Yarkoni for feedback on an earlier draft.*

Masicampoand LaLande (2012; M&L) assessed the distribution of all 3627

*p*-values between 0.01 and 0.10 in 12 issues of the three journals. Exact*p*-values were calculated from*t*-tests and*F*-tests. The authors conclude “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:
1) the number of
studies examining true effect and false effects

2) the power of
the studies that examine true effects

3) the frequency
of Type 1 error rates (and how they were inflated)

4) 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*-value distribution (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*-value distribution could become left-skewed below*p*-values of .05.
M&L
model

*p*-values based on a single exponential curve estimation procedure that provides the best fit of*p*-values between .01 and .10. 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.
P-hacking
does not create a peak in

*p*-values*just*below .05. Actually,*p*-hacking does not even have to lead to a left-skewed*p*-value distribution. If you perform multiple independent tests in a study, the*p*-value distribution is uniform, as if you had performed 5 independent studies. The right skew emerges through dependencies in the data in a repeated testing procedure. For example, collecting data, performing a test, collecting additional data, and analyzing the old and additional data together.
Figure
1 (left pane) shows

*p*-value distributions when H0 is true and researchers perform a test after 50 participants, and collect 10 additional participants for a maximum of 5 times, leading to a 15.4% Type 1 error rate, or (right pane) compare a single experimental condition to one of five possible control conditions, leading to a Type 1 error rate of 18.2%. Only 5000 out of 100000 studies should yield a*p*<.05, but we see an increase (above 500 for each of the 10 bins) across the range from .00 to .05.
Identifying
a prevalence of Type 1 errors in a large heterogeneous set of studies is,
regrettably, even more problematic due to the

*p*-value distribution of true effects. In Figure 2 (left) we see a*p*-distribution of 100000 experiments with 50% power. Adding the 200000 experiments simulated above give the*p*-value distribution on the right. Even when only 1/3^{rd}of the studies examines a true effect with a meager 51% power, it is already impossible to observe a strong left-skewed distribution.
Do
frequencies of

*p*-values just below*p*=.05 observed by M&L indicate extreme*p*-hacking in a field almost devoid of true effects? No. The striking illustration of the prevalence of*p*-values in Psychological Science just below .05 (Figure 3, right, from M&L) from the blind rater is not apparent in the data coded by the authors themselves (Figure 3, left, re-analysis based on the data kindly provided by M&L). It is unclear what has led to the difference in coding by the authors and the blind rater, but the frequencies between .03 and .05 look like random variation due to the relatively small number of observations.
There
is also no evidence of a pre-valence of

*p*-values just below .05 when analyzing all*p*-values collected by M&L. The authors find no peak when dividing the*p*-values in bins of .01, .005, or .0025, and there is only a slight increase in the .04875-.05 range. There should be a similar increase in the .0475-.04875 range, which is absent, and it is easier to explain this pattern by random variation than by*p*-hacking.As an example of a modeling approach of the

*p*-value distribution based on power, Type 1 error rates and

*p*-hacking, and publication bias, I’ve simulated 11000 studies with 41% power, taking into account publication bias by removing half of the studies with a

*p*>.05, and added 4500 studies examining no real effect with an inflated Type 1 error rate of 18.2% (using the example above where an experimental study is compared to one of 5 control conditions). The R-script is available below. I do not mean to imply these parameters (the relative number of studies examining true vs. non true effects, power, the effect of publication bias, and the inflated Type 1 error rate) are typical for psychology, or even the most likely values for these parameters. I only mean to illustrate that choosing values for these four parameters is able to quite accurately simulate the

*p*-value distribution actually observed by M&L. The true parameters require further empirical investigation. One interesting difference between my model and the observed

*p*-values is a slight

*increase*of

*p*-values just

*above*.05 – perhaps some leniency by reviewers to tend to accept studies with

*p*-values that are almost statistically significant.

Even
though 839 of the 3877 simulated

*p*-values are Type 1 errors (614 of which are only significant through*p*-hacking) there is no noticeable prevalence of*p*-values just below .05. It is clear that*p*-hacking can be a big problem even when it is difficult to observe. As such, the findings by M&L clearly indicate publication bias (a large drop in*p*-values >.05), but do not necessarily identify a surprising prevalence of*p*-values just below .05, when interpreted against a more realistic model of expected*p*-value distributions. Concluding that*p*-hacking is a problem by analyzing a large portion of the literature is practically impossible, unless there are a huge amount of studies that use extreme flexibility in the data analysis.
An
alternative to attempting to point out

You can download the data here (which E.J. Masicampo and Daniel LaLande gratiously allowed me to share). Store it at C:// and the code below will produce the left pane of Figure 3:

The R-code for the simulations in Figure 1 and 2 (Thanks to Ryne Sherman's

The R-code for the simulations in Figure 4:

References:

*p*-hacking in the entire psychological literature is to identify left-skewed*p*-curves in small sets of more heterogeneous studies. Better yet, we should aim to control the Type 1 error rate for the findings reported in an article. Pre-registration and/or replication (e.g., Nosek & Lakens, 2014) are two approaches that can improve the reliability of findings.You can download the data here (which E.J. Masicampo and Daniel LaLande gratiously allowed me to share). Store it at C:// and the code below will produce the left pane of Figure 3:

library(foreign) dataSPSS<-read.spss("C:/ML_QJEP2012_data.sav", to.data.frame=TRUE) hist(dataSPSS$PS, xaxt="n", breaks=20, main="P-values in Psychological Science from M&L", xlab=("Observed p-value")) axis(side = 1, at = c(0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10)) abline(v=0.05, lwd=2)

The R-code for the simulations in Figure 1 and 2 (Thanks to Ryne Sherman's

*p*-hack script):The R-code for the simulations in Figure 4:

References:

Masicampo, E. J., & Lalande, D. R. (2012). A peculiar
prevalence of

*p*-values just below. 05.*The Quarterly Journal of Experimental Psychology*,*65(11)*, 2271-2279.
Nosek, B. A.,
& Lakens, D. (2014). Registered reports: A method to increase the
credibility of published results.

*Social Psychology*,*43*, 137-141. DOI: 10.1027/1864-9335/a000192.
Sellke, T., Bayarri, M. J., & Berger, J. O. (2001). Calibration
of

*p*-values for testing precise null hypotheses.*The American Statistician*,*55*,62–71.