I wrote a commentary that made a very simple point. A published model assumed that the variance of z-scores is typically less than 1. I pointed out that this is not a reasonable assumption because the standard deviation of z-scores is at least one and often greater than 1, when studies vary in effect sizes, sample sizes, or both. This commentary was rejected. One reviewer even provided R-Code to make his or her case. Here is my rebuttal.

Here is the r-code provided by the reviewer. We see SDs of 0.59, 0.49 and 0.46. Based on these results, the reviewer thinks that setting a prior to a range of values between 0 and 1 is reasonable.

Let’s focus on the example that the reviewer claims is realistic for a p-value distribution for 80% power. The reviewer simulates this scenario with a beta distribution with shape parameters 1 and 31. The Figure shows the implied distribution of p-values. What is most notable is that p-values greater than .38 are entirely missing; the maximum p-value is .38.

In this figure 80% of p-values are below .05 and 20% are above .05. This is why the reviewer suggests that the pattern of observed p-values corresponds to a set of studies with 80% power.

However, the reviewer does not consider whether this distribution of p-values could arise from a set of studies where p-values are the result of the non-central parameter and sampling error that follows a sampling distribution.

To simulate studies with 80% power, we can simply use a standard normal distribution centered over 2.80. Sampling error will produce z-scores greater and smaller than the non-centrality parameter of 2.80. Moreover, we already know that the standard deviation of these tests statistics is 1 because z-scores have the standard normal distribution as a sampling distribution (a point made and ignored by the reviewers and editor).

We can know compute the two-tailed p-values for each z-test and plot the distribution of p-values. Figure 2 shows the actual distribution in black and the reviewer’s beta distribution in red.

It is visible that the actual distribution has a lot more p-values that are very close to zero, which corresponds to high z-scores. We can know transform the p-values into z-scores using the reviewers’ formula (for one-tailed tests).

y<-qnorm(mean=0,sd=1,p)

mean(y) #-2.54

sd(y) #1.11

We see that the standard deviation of these z-scores is greater than 1.

Using the correct formula for two-tailed p-values, we of course get the result that we already know to be true.

y = -qnorm(p/2)

mean(y) #2.80

sd(y) #1.00

It should be obvious that the reviewer made a mistake by assuming we can simulate p-value distributions with any beta-distribution. P-values cannot assume any distribution because the actual distribution of p-values is a function of the properties of the distribution of test-statistics that are used to compute p-values. With z-scores as test statistics it is well-known from intro statistics that sampling error follows a standard normal distribution, which is a normal distribution with a standard deviation of 1. Any transformation of z-scores into p-values and back into z-scores does not alter the standard deviation. Thus, the standard deviation has to be at least 1.

**Heterogeneity in Power**

The previous example assumed that all studies have the same amount of power. Allowing for heterogeneity in power, will further increase the standard deviation of z-scores. This is illustrated with the next example, where mean power is again 80%, but this time the non-centrality parameters vary with a normal distribution centered over 3.15 and a standard deviation of 1. Figure 3 shows the distribution of p-values which is even more extreme and deviates even more from the simulated beta-distribution by the reviewer.

Using the reviewer’s formula, we now get a standard deviation of 1.54, but if we use the correct formula for two-tailed p-values, we end up with 1.41.

y<-qnorm(mean=0,sd=1,p)

mean(y) #-2.90

sd(y) #1.54

y = -qnorm(p/2)

mean(y) #3.16

sd(y) #1.39

This value makes sense because we simulated variation in z-scores with two standard normal distributions. One for the variation in the non-centrality parameters and one for the variation in sampling error. Adding two variances, gives a joint variance of 1 + 2 = 2, and a standard deviation of sqrt(2) = 1.41.

**Conclusion**

Unless I am totally crazy, I have demonstrated that we can use simple intro stats knowledge to realize that the standard deviation of p-values converted into z-scores has to be at least 1 because sampling error alone produces a standard deviation of 1. If the set of studies is heterogeneous and power varies across studies, the standard deviation will be even greater than 1. A variance less than 1 is only expected in unrealistic simulations or when researchers use questionable research practices, which reduces variability in p-values (e.g., all p-values greater than .05 are missing) and therewith also the variability in z-scores.

A broader conclusion is that the traditional publishing model in psychology is broken. Closed peer-review is too slow and unreliable to ensure quality control. Neither the editor of a prestigious journal, nor four reviewers were able to follow this simple line of argument. Open review is the only way forward. I guess I will be submitting this work to a journal with open reviews, where reviewers’ reputation is on the line and they have to think twice before they criticize a manuscript.

Quote from above: “A broader conclusion is that the traditional publishing model in psychology is broken. Closed peer-review is too slow and unreliable to ensure quality control.”

From the information i picked up here and there in the last years, i have come to the conclusion that peer-review as it is performed in the “traditional” journal-editor-peer reviewer model makes no sense. I even think it’s unscientific, and unethical.

I would even go so far to state that everyone still participating in that sh#t should have science points deducted from their “i am a scientist” card.

In all seriousness: why are you still participating in that bullsh#t? I checked the website of your university, and it stated there that you are a professor. If that means you are “tenured”, i reason you can’t be fired for not publishing in “official” journals, nor should you have to play the possible “publish or perish” game anymore.

I think if, and how, scientists use and/or cite papers is all the peer-review that makes sense (and is needed).

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Yes, tenured and that gives me the freedom to blog, but pay raises are based on publications in peer-reviewed journals. So, I am taking a hit if I only blog and do not publish in peer-reviewed journals. Also, peer-review is not bad. I could make mistakes and nobody really comments on blogs. So, submitting work to a fair and open review by experts is not a bad thing. Glad we have progressive journals like Meta Psychology that value scientific accuracy over fake novelty and do not care about mistakes in published work.

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“(…) but pay raises are based on publications in peer-reviewed journals.”

Ah, that is new information to me!

Why isn’t that stuff mentioned in all the recent discussions about the mess in academia?

Why aren’t all these “open science/let’s improve things” people talking, and doing something, about that?

p.s. thank you for your comment that review by others could be useful. I agree with that, but i think that should be done differently. For instance, you could ask colleagues to do that, and possibly reward them with co-authorship in case of a truly useful review/contribution, etc.

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