P-Values – Statistical Bullshit

More Issues with P-Values

What did Cohen (1992) have to say in The Earth is Round (p < .05)?

Few statistical topics have spurred as much controversy as p-values. For this reason, I felt that another post on StatisticalBullshit.com about p-values could be helpful to all readers – any myself! I always learn new things about Statistical Bullshit when I review p-values, and I hope you learn a little bit from reading this post. If you have any questions or comments about this post (or anything else), please email me at MHoward@SouthAlabama.edu. I’ll do my best to reply ASAP.

An overview of Statistical Bullshit associated with p-values can start with Cohen’s (1994) classic article, “The Earth is Round (p < .05).” The title is meant to be a pun on p-values. Given that we cannot obtain multiple measurements about the roundness of the earth and apply statistical tests (under normal approaches), it is impossible to determine the statistical significance of the earth’s roundness – but we know it is round. So, does that mean that there are certain important conclusions that p-values cannot derive? Probably so!

The first line of Cohen’s abstract describes null hypothesis significance testing (NHST), which makes research inferences based p-values, as a “ritual” which makes a “mechanical dichotomous decision around a sacred .05 criterion.” Cohen makes his disdain for p-values and NHST obvious from the beginning. To immediately support his claims, in the introduction, Cohen also cites previous authors who state that the “everybody knows” the concerns of p-values, and they are “hardly original.” My favorite quotation is from Meehl who claims that NHST is “a potent but sterile intellectual rake who leaves in his merry path a long train of ravished maidens but no viable scientific offspring.” (1967, p. 265). Quite the statement! And the earliest of these citations is in 1938! For a long time, researchers have been foaming-at-the-mouth over p-values.

After the initial damnation of p-values, Cohen gets into the actual concerns. First, he notes that people often misunderstand p-values. He notes the logic of NHST, when p-values are significant, is:

If the null hypothesis is correct, then these data are highly unlikely.
These data have occurred.
Therefore, the null hypothesis is highly unlikely.

Cohen dislikes this logic, and he notes that this logic can derive problematic conclusions. In the article, Cohen provides a slightly unusual situation which denotes the problem in the above reasoning. I am going to provide a similar situation which results in the same conclusions, but it (hopefully) will make more sense to readers than Cohen’s example.

Most employees will not be on the Board of Directors in a company. But, let’s say that we sampled a random person, and they were on the Board of Directors within a company. Therefore, we can state the following, using the logic of NHST.

If a person is employed, then (s)he is probably not on the Board of Directors.
This person is on the Board of Directors.
Therefore, (s)he is probably not employed.

We know that, if an individual is on the Board of Directors, then they are employed. So, the logic of NHST led us to an inappropriate conclusion.

Second, Cohen notes that the meaning of p-values, “the probability that the data could have arisen if the null hypothesis is true,” is not the same as, “the probably that the null hypothesis is true given the data.” Unfortunately, researchers often mistake p-values for the latter, and lead to some problematic inferences.

To demonstrate the problem with this thinking, Cohen provides an example with schizophrenia. Assume that schizophrenia arises in two percent of the population, and assume we have a tool that has a 95% accuracy in diagnosing schizophrenia and even a 97% accuracy in declaring normality. Not bad! When we use the tool, our null hypothesis is that an individual is normal, and the research hypothesis is that the individual is schizophrenic. So, the (problematic) logic is: when the test is significant, the null hypothesis is not true.

However, when we calculate the math for a sample of 1000, some problems arise. Particularly, in a sample of 1000, the number of schizophrenics is most likely 20. The tool would correctly identify 19 of them and label one as normal. This result is not overly concerning. Alternatively, in the sample of 1000, the number of normal individuals is most likely 980. The tool would, most likely, correctly identify 951 normal individuals as normal (980 * .97); however, 29 individuals would be labeled as schizophrenic! So, of the 50 people identified as schizophrenia, 60 percent of them would actually be normal! The problem in this example that we assume that the null hypothesis is false given the data (a significant result), when we should think about the probably that the data could have arisen (five percent chance) given the null hypothesis is true.

Third, Cohen notes that the null hypothesis is almost always that no effect exists. But is this stringent enough to test hypotheses? Cohen argues that it is not. He notes that, given enough participants, anything can be significantly different from nothing. Also, given enough participants, the relationship of anything with anything else is greater than nothing. So, if authors find (p > .05), they can just add more participants until (p < .05).

Fourth, Cohen notes that, when only observing p-values, that results cannot be “very statistically significant” or “strongly statistically significant,” although authors like to make similar claims when the p-value is less than 0.01, 0.001, or 0.001. Instead, results can only be statistically significant when analyzing p-values. This is because p-values are derived from magnitude, variance, and sample size. If a p-value is significant at a 0.05 level and another at a 0.001 level, the results do not necessarily mean that the magnitude of effect was larger. Instead, the magnitude of the effect can be identical, and the sample sizes of the two groups are just larger. As the sample size does not reflect the effect itself, it is inappropriate to say that the former is only “statistically significant” and the other is “very statistically significant.” This is a mistake that I still see in many articles.

These four aspects represent the primary concerns of Cohen. In his article, he also suggests some modest solutions to p-values and NHST. Probably the most adopted is the use of confidence intervals. Confidence intervals indicate a range of values that we can expect the effect to fall within, based upon the magnitude of the effect and standard error. If a confidence interval contains zero, then it cannot be significantly different from random chance. In my experiences, most researchers view confidence intervals the same as p-values, and they see zeros as the on-off switch instead of (p < .05).

Once Cohen’s article was published, discussions of p-values grew even further. Several authors published responses. I won’t summarize all of them, but Cohen’s rejoinder (1995) downplayed most of the criticisms in a single-page article; however, one of the few responses that Cohen seemed to ponder was that of Baril and Cannon (1995). The authors criticized Cohen’s NHST examples and considered them “inappropriate” and “irrelevant.” Cohen replied that his examples were not intended to model NHST as used in the real world, but only to demonstrate how wrong conclusions can be when the logic of NHST is violated. Further, in response to a different article, Cohen claims that he does not question the validity of NHST but only its widespread misinterpretation.

My Two Cents

In general, I think p-values are alright. Before you gather your torches and pitchforks, let me explain. I see p-values as a quick sketch of studies’ results. A p-value can provide quick, easy information, but any researcher should know to look at effect sizes and the actual study results to fully understand the data. No one should see (p > .05) or (p < .05) and move on. There is so much more to know.

Also, while confidence intervals provide more information, their application usually provides the same results as p-values. Researchers often just look for confidence intervals outside of zero and move on.

Lastly, p-values are a great first-step into understanding statistics. They aren’t too scary, and understanding p-values can lead to understanding more complex topics.

Summary

A p-value indicates the probably that the observed data occurred due to random chance. Usually, if the p-value is above 0.05, we do not accept the result as being different from random chance. If it is below 0.05, we accept the result as being more than random chance. While p-values are good first-looks at data, everyone should know how to interpret other statistical results, as they are much better descriptions of the data.

Keep looking out for Statistical Bullshit! And let me know if you have any questions or topics that you’d like to see on StatisticalBullshit.com by emailing me at MHoward@SouthAlabama.edu!

Note: This was originally posted to MattCHoward.com, but I felt it was particularly relevant to StatisticalBullshit.com.

Is Big Worse than Bad?

Today’s post is about the concept of how being big could be worse than being bad in regards to Equal Employment Opportunity Enforcement Policies (EEOP).

So far on StatisticalBullshit.com, I’ve written about general Statistical Bullshit, Statistical Bullshit that I’ve come across in consulting, or Statistical Bullshit that readers have sent into the website. I don’t *think* that I’ve written about Statistical Bullshit that was pointed out by an academic article. For this reason, today’s post is about the concept of how being big could be worse than being bad in regards to Equal Employment Opportunity Enforcement Policies (EEOP). Most of the material for this post comes from Jacobs, Murphy, and Silva’s (2012) article, entitled “Unintended Consequences of EEO Enforcement Policies: Being Big is Worse than Being Bad,” which was published in the Journal of Business and Psychology. Rick Jacobs was my advisor at Penn State, so I am happy to have his article discussed on StatisticalBullshit.com. For more information about this concept, please email me at MHoward@SouthAlabama.edu or check out Jacobs et al. (2012).

As stated by Jacobs et al. (2012), “The Equal Employment Opportunity Commission (EEOC) is the chief Federal agency charged with enforcing the Civil Rights Acts of 1964 and 1991 and other federal laws that forbid discrimination against a job applicant or an employee because of the person’s race, color, religion, sex (including pregnancy), national origin, age (40 or older), disability, or genetic information” (p. 467). In other words, the EEOC ensures that businesses do not discriminate against protected classes, and this includes in business employment practices.

When a disproportionately low number of peoples from a protected class are hired, most often relative to the majority class, this is called disparate impact. But how do we know whether a “disproportionately low number” has occurred? In EEO cases, there are many methods, but the 80% rule and statistical significance testing are among the most popular.

The 80% rule specifies that disparate impact occurs when members of a protected class are hired at a rate that is less than 80% the rate of the majority class. Let’s take a look at the following example to figure out what this means:

	Hired	Applied	Ratio	80% Rule
Caucasian	20	40	1:2 (.50)	.50 * .80 = .40
African American	5	20	1:4 (.25)	.40 > .25

In this example, 40 Caucasian people and 20 African Americans applied for the same job. The organization selected 20 Caucasians and 5 African Americans for the job. This results in 50% of the Caucasians being hired, but only 25% of the African Americans being hired. To determine whether disparate impact occurred, we take .50 (ratio for Caucasians) and multiply it by .80 (80% rule). This results in .40. We then compare this number to the ratio of African Americans hired, .25. Since .40 is greater than .25, we can determine that disparate impact occurred on the basis of the 80% rule.

Although it may seem relatively simple, the 80% rule works quite well across most situations. But what about the other method – statistical significance testing?

Many different tests could be used to identify disparate impact, but the chi-square test may be the simplest. The chi-square test can be used to determine whether the association between two categorical variables is significant, such as whether the association between race and hiring decisions is significant. So, we can use this to test whether disparate impact may have occurred.

To do so, you can use the following calculator: https://www.graphpad.com/quickcalcs/contingency2/ . Let’s enter the data above, which would look like the following in a chi-square calculator:

	Hired	Not Hired
Caucasian	20	20
African American	5	15

The resultant p-value is .06, which is greater than .05. Not statistically significant! Although this is the exact same data as the 80% rule example, the chi-square test determined that it was not a case of disparate impact. Interesting!

But what happens when we double the size of each group? The 80% rule table would look like this:

	Hired	Applied	Ratio	80% Rule
Caucasian	40	80	1:2 (.50)	.50 * .80 = .40
African American	10	40	1:4 (.25)	.40 > .25

Again, the resultant ratio for Caucasians is .50, which is .40 when multiplied by .80 (80% rule). The resultant ratio for African Americans is .25, which is smaller than .40. This suggests that disparate impact occurred on the basis of the 80% rule.

On the other hand, let’s enter this data into the chi-square calculator, which would look like this:

	Hired	Not Hired
Caucasian	40	40
African American	10	30

The resultant p-value is .009, which is much less than .05. Statistically significant! While the ratios are identical for the two examples, the latter chi-square test determines that disparate impact occurred.

This is the idea behind “Being Big is Worse than Being Bad.” Although both examples had the same ratio, and thereby were just as bad, the chi-square test indicated that disparate impact only occurred in the latter example, which was bigger. Thus, significance testing has concerns when identifying disparate impact, because the sample size strongly influences whether a result is significant or not.

So, do we just apply the 80% rule? Not necessarily. Jacobs et al. (2012) call for “a more dynamic definition of adverse impact, one that considers sample size in light of other important factors in the specific selection situation” (p. 470), and they also call for a less-simplified view of disparate and adverse impact. While the 80% rule can certainly address certain problems that significance testing encounters, it cannot satisfy all the needs within this call.

Issues like these is why we need more statistics-savvy people in the world. Disparate and adverse impact are huge issues that impact millions of people. And many of these decisions are not made by statisticians. Instead, they are made by courts and companies. Even if you aren’t interested in becoming a statistician, the world still needs people that understand statistics – and know how to watch out for Statistical Bullshit in significance testing!

That’s all for today. If you have any questions, please email me at MHoward@SouthAlabama.edu. Until next time, watch out for Statistical Bullshit!

P-Hacking

Does it hurt to take a peek? Or just leave “unimportant” findings unreported?

For the first post on StatisticalBullshit.com, it seems appropriate to discuss one of the most common instances of Statistical Bullshit: p-hacking!

What is p-hacking? Well, let’s first talk about p-values.

A p-value is the probability that the observed data occurred due to random chance alone. For instance, when performing a t-test that compares two groups of data, such as performance for two work units, the p-value indicates the likelihood that the observed differences between the two groups occurred due to random chance alone. If the p-value is 0.05, for example, it indicates that there is a five percent likelihood that the observed results occurred due to random chance alone. So, if the p-value is reasonably small (most often < .05), then we can assume that some effect other than random chance alone caused the observed relationships – and we most often assume that our effect of interest was indeed the cause. In these instances, we say that the result is statistically significant.

For more information on p-values, visit my p-value page at MattCHoward.com.

So, what is p-hacking? P-hacking is when a researcher or practitioner looks at many relationships to find a statistically significant result (p < .05), and then only reports significant findings. For instance, a researcher or practitioner may collect data on seven different variables, and then calculate correlations between each of them. This would result in a total of 21 different correlations. They could find one or two significant relationships (p < .05), rejoice, and write-up a report about the significant finding(s). But is this a good practice? Definitely not.

Given 21 different correlations, we would expect one or two of them to be significant, even if all the variables were completely randomly generated (and hence should not be significantly related). This is because the p-value is the likelihood that a result is significant due to random chance alone. If we expect this random chance to occur five percent of the time, then 21 correlations would produce at least one significant result on average (21 * .05 = 1.05). So, although a result may be statistically significant, it does not always mean that a meaningful effect caused the finding.

Some readers may still be skeptical that random variables could produce significant findings. Let me give you another example. I recently completed two studies in which I had participants predict a completely random future event. As you probably assumed, no one was able to predict the future even better than random chance alone, which supports that any correct guesses were only achieved by random chance alone. So, any predictor variables should not be significantly related to the number of correct guesses. However, I found a statistically significant relationship between gender and the number of correct guesses (r = .21, p < .05), and women are able to predict the future better than men! Right?

Well, let’s look at the results of the two studies:

Study 1		Study 2
Predictor	Correlation with Number of Correct Guesses	Predictor	Correlation with Number of Correct Guesses
1.) Perceived Ability to Predict the Future	-.08	1.) Perceived Ability to Predict the Future	-.02
2.) Self-Esteem	-.07	2.) Self-Esteem	.09
3.) Openness	.02	3.) Openness	.04
4.) Conscientious	-.01	4.) Conscientious	.01
5.) Extraversion	-.07	5.) Extraversion	-.13
6.) Agreeableness	.03	6.) Agreeableness	.04
7.) Neuroticism	.04	7.) Neuroticism	-.01
8.) Age	.10	8.) Age	.07
9.) Gender	-.04	9.) Gender	.22**

** = p < .01

As you guessed, we cannot claim that women predict the future better than men based on these results. Given that 18 correlations were calculated, we would naturally assume that one would be significant due to random chance alone, which was likely the relationship between gender and the number of correct guesses.

So, what do we do about p-hacking?

Authors have presented a wide-range of possible solutions, but three appear to be the most popular:

Do-away with p-values altogether. A small number of academic journals have been receptive to this issue, and they often request that submitted papers include confidence intervals and discuss effect sizes instead.
Report all Many journals now require submissions to include a supplemental table that notes all measured variables not reported in the manuscript. A growing number of journals have even started to request that submissions include the full dataset(s) with all measured variables. A growing concern has also been expressed regarding authors that do not report entire studies because they did not support their results (this will likely be a future StatisticalBullshit.com topic).
Only test relationships specified prior to collecting data. Recent databases have been created in which researchers can publicly identify relationships to test in their data before it has been collected, and then only test these relationships once the data has been collected.
Adjust p-value cutoffs. Many corrections can be made to account for statistical significance due to random chance alone. Perhaps the most popular is the Bonferroni correction, in which the p-value cutoff is divided by the number of comparisons made. For instance, if you performed 10 correlation analyses, you would divide the p-value of .05 by 10, resulting in a new p-value cutoff of .005. Many researchers and practitioners view this as too restrictive, however.
Replicating your results can help ensure that a result was not due to random chance alone. Lighting rarely strikes twice, and the same completely random relationship rarely occurs twice.

While these solutions were developed in academia, they can also be applied to practice. For instance, if a work report includes statistical results, you should always ask (1) whether the document or presentation includes statistics other than p-values, such as correlations or t-values, (2) whether other analyses were conducted but not reported, (3) whether the reported relationships were intended to be tested, (4) whether a p-value adjustment is needed, and (5) whether the findings have been replicated using a new scenario or sample. Only after obtaining answers for these questions should you feel confident in the results!

Of course, there is still a lot more that could be said about p-hacking, but I believe that is a good introduction. If I missed your favorite method to address p-hacking, or anything else, please let me know by emailing MHoward@SouthAlabama.edu. Likewise, feel free to email about your own Statistical Bullshit stories or questions. P-values are one of my favorite (and most popular) Statistical Bullshit topics, so be ready for more posts about p-values in the future.

Until next time, watch out for Statistical Bullshit!