Statistics – Page 13 – Statistical Bullshit

Small Samples, Big Problems

Have you ever discussed statistical power or representative samples at work? Should you?

Often in business, we are restricted to relatively small samples. In fact, a recent publication in the Journal of Organizational Behavior suggest that the most common type of business is a microbusiness – often defined as a business with less than 10 employees (Brawley & Pury, 2017). As many readers already know, most all statistics require many more participants. For instance, the most common recommendation for a correlation analysis is a minimum of 30 participants, and more advanced statistics most often require even more participants – often in the 100s.

But what is really the harm in having a small sample size? Can the results really be that misleading? The answer is yes.

This post discusses two concerns of small samples: power and representativeness.

Power is the likelihood of a statistical analysis to discover a significant result if a significant result actually exists in the population…But what does that mean? Well, I’ll discuss this much more in-depth in a later post, but sample size is an important component to calculating statistical significance. Even if an effect is extremely strong in the population, a statistical test using a small sample size will not identify that effect as statistically significant. Weird, right?

Let’s use this example: Imagine that we are studying pretty strong effect that has a population correlation of .40, such as the relationship between self-efficacy and job performance. To study this relationship, let’s say that we use a microbusiness – one with eight employees – and we measure self-efficacy and job performance with each employee. What is the likelihood that the resultant correlation between the two variables will be statistically significant, if we know the population correlation of the variables is .40? Well, the likelihood that the result will be statistically significant is only 15%! We would fail to reject the null more than every four out of every five times!

Crazy! This example demonstrates one important reason to have a large sample size – you cannot identify significant results even if they should be significant. To learn more about this phenomenon, I suggest reading more about statistical power (Cohen, 1992a, 1992b; Murphy et al., 2014) and playing with a sample size/power calculator (http://www.sample-size.net/correlation-sample-size/).

Next, let’s discuss having a representative sample. Even if we have more employees, let’s say 150, there is a chance that our sample is not representative of the population. If a sample is representative, it accurately reflects the members of the population. Often, we assume that a randomly selected sample is representative, but this is not always the case. Certain people may not volunteer to take your survey, and that may skew your results…But how bad can it be?

Well, let’s look at the self-efficacy and job performance example again with a correlation of .40. If we had a representative sample of 300 people, the result might look something like this:

Not too bad – the regression line shows a clear, increasing relationship. Now, let’s take 150 of these people and graph the results again:

Woah! Big difference! Now the correlation between the two is literally .00, and we only removed half of the participants. What happened?

As you guessed, I did not take a random subset of the 300 people. Instead, I selected only those that scored five or above on the self-efficacy measure, as you can see with the differing axis labels in the two charts. This resulted in the sample being non-representative (because everyone with a self-efficacy score under five was missing), and thereby the result was greatly different than the entire set of 300 people.

But could this ever happen in business? Yes! Imagine that you are feeling down about your work performance and unable to do the most basic tasks. Then, you see an email about a job survey to measure self-efficacy and performance. Would you take it? Maybe, but a lot of people would just delete the email in order to avoid facing their lackluster self-perceptions, abilities, and performance.

Also, who would typically take those surveys anyways? The grumpy employees that just want to do their work and go home? Or the goodie-goodies that do whatever their boss asks? I’d guess the latter, and the samples may not be representative of all these employees.

And think about those satisfaction surveys at restaurants. Yes, people that really hated the service or really loved the service will complete them…but what about all the people in the middle? Have you ever completed a satisfaction survey when the service was just okay? I’m guessing not, which resulted in the results being non-representative.

So, whenever you need to collect data, be sure to carefully consider your sample size – not only for statistical power, but also for representativeness. If you ignore these two aspects, then you could obtain results that are entirely misleading, and thereby implement policies that do nothing for your company – or worse!

Until next time, watch out for Statistical Bullshit! And email me at MHoward@SouthAlabama.edu if you have any questions, comments, or anything else!

References

Brawley, A. M., & Pury, C. L. (2017). Little things that count: A call for organizational research on microbusinesses. Journal of Organizational Behavior, 38, 917-920.

Cohen, J. (1992a). Statistical power analysis. Current directions in psychological science, 1(3), 98-101.

Cohen, J. (1992b). A power primer. Psychological bulletin, 112(1), 155.

Murphy, K. R., Myors, B., & Wolach, A. (2014). Statistical power analysis: A simple and general model for traditional and modern hypothesis tests. Routledge.

What is in a Mean?

When are mean comparisons appropriate? And when are they Statistical Bullshit?

This post is inspired by an interaction that I had while consulting. I was hired as a statistical analyst, and my duties included reviewing analyses that were already conducted internally. Most of the organization’s prior analyses were appropriate, but I noticed that certain assumptions were based on completely inappropriate mean comparisons. These assumptions led to needless practices that cost time and money – all because of Statistical Bullshit. Today, I want to teach you how to avoid these issues.

Let’s first discuss when mean comparisons are appropriate. Mean comparisons are appropriate if you (A) want to obtain a general understanding of a certain variable or (B) want to compare multiple groups on a certain outcome. In the case of A, you may be interested in determining the average amount of time that a certain product takes to make. From knowing this, you could then determine whether an employee is taking more or less time than the average to make the product. In the case of B, you may be interested in determining whether a certain group performed better than another group, such as those that went through a new training program vs. those that went through the old training program. The data from such a comparison may look something like this:

So, from this comparison, you may be able to suggest that the new training program is more effective than the old training program; however, you would need to run a t-test in be sure of this.

Beyond these two situations, there are several other scenarios in which mean comparisons are appropriate, but let’s instead discuss an example when mean comparisons are inappropriate.

Say that we wanted to determine the relationship between two variables. Let’s use satisfaction with pay (measured on a 1 to 7 Likert scale) and turnover intentions (also measured on a 1 to 7 Likert scale). As you probably already know, we could (and probably should) determine the relationship between these two variables by calculating a correlation. Imagine instead that you decided to calculate the mean of the two variables and the results looked like this:

Does this result indicate that there is a significant relationship between the two variables? In my prior consulting experience, the internal employee who ran a similar analysis believed this to be true. That is, the internal employee believed that two variables with similar means are significantly related; however, this couldn’t be further from the truth. Let’s look at the following examples to find out why.

Take the example that we just used – satisfaction with pay and turnover intentions. Which of the following scatterplots do you believe represents the data in the bar chart above?

Still don’t know? Here is a hint: The first chart represents a correlation of 1, the second represents a correlation of -1, the third represents a correlation of 0, and the fourth represents a correlation of 0. Any guesses?

Well, it was actually a trick question. Each figure could represent the data in the bar chart above, because the X and Y variables in each have a mean of 4.75…well, the last one is off by a few tenths, but you get my point.

So, if the means of two variables are equal, their relationship could still be anything – ranging from a large negative relationship, to a null relationship, to a large positive relationship. In other words, the means of two variables have nothing to do regarding their relationship.

But does it work the other way? That is, if the means of two variables are extremely different, could they still have a significant relationship?

Certainly! Let’s look at the following example using satisfaction with pay (still measured on a 1 to 7 Likert scale) and actual pay (measured in thousands of dollars).

As you can see, the difference in the means is so extreme that you can’t even see one bar! Now, let’s look at the following four scatterplots:

Seem a little familiar? As you guessed, the first represents a correlation of 1, the second represents a correlation of -1, the third represents a correlation of 0, and the fourth represents a correlation of 0. More importantly, each of these include a Y variable with a mean of 4.75 and an X variable with a mean of 47500. Although the means are extremely far apart, they have no influence on the relationship between two variables.

From these examples, it should be obvious that the mean of two variables has no influence on their relationship – no matter if the means are close together or far apart. Instead, it is the covariation between the pairings of the X and Y values that determine the significance of their relationship, which may be a future topic on StatisticalBullshit.com or even MattCHoward.com (especially if I get enough requests for it).

Now that you’ve read this post, what will you say if you are ever at work and someone tries to tell you that two variables are related because they have similar means? You should say STATISTICAL BULLSHIT! Then demand that they calculate a correlation instead…or a regression…or a structural equation model…or other things that we may cover one day.

That’s all for this post! Don’t forget to email any questions, comments, or stories. My email is MHoward@SouthAlabama.edu, and I try to reply ASAP. Until next time, watch out for Statistical Bullshit!

Regression Toward the Mean

Can you make a career on Statistical Bullshit?

Regression Toward the Mean is one of the most common types of Statistical Bullshit in industry. And, as the title quotation insinuates, some consultants have made an entire career swindling money from organizations through manipulating this statistical phenomenon. If you are currently in practice, or ever plan to be, read on to discover whether you are currently being swindled out of thousands – or possibly millions!

Businesses are always on a time-series. That is, most organizations are not worried about the profit that they turned today, but rather the profit that they will turn tomorrow. For this reason, many types of statistics and methodologies applied in business are meant to analyze longitudinal trends in order to predict future results.

Let’s take perhaps the simplest time-series design: a single variable measured on multiple occasions. In this example, let’s say that we are looking at overall company revenue in millions.

Month	Revenue
January	10.5
February	10
March	11
April	12.5
May	10.5
June	10
July	12
August	11
September	11
October	5

It seems that the average company revenue over the month was $11 million, but a severe drop occurred in October. What would you do if your company revenue looked something like this?

Most anyone would say panic and take extreme measures – and that is what most companies do. A company may replace the CEO, lay-off a large number of workers, or immediately implement a new corporate strategy. Let’s say that a company does all three for our example, and the result looks like this:

Success! The new CEO is a genius! The lay-offs worked! And the new corporate strategy is brilliant! Right? Well, maybe not.

The Regression Toward the Mean phenonmemon suggests that a time-series dataset will revert back to its average after an extreme value. In other words, when an extreme high- or low-value occurs, it is much more difficult to get any more extreme than it is to revert back to the average. So, in this instance, it is fully possible that the company’s actions successfully caused revenue to revert back to more normal values; however, it is perhaps just as likely that the revenue simply regressed back toward the mean naturally. So, the new changes (and money spent!) may have actually done very little or even nothing at all…but you can always be sure that the new CEO will take credit for it.

Let’s discuss another common example of Regression Toward the Mean in business. Imagine you are a floor manager at a factory, and your monthly number of dangerous incidents looks something like this.

Week	Incidents
January	5
February	4
March	8
April	6
May	6
June	4
July	2
August	8
September	5
October	20

Wow! Quite the spike in incidents! So, what do you do? Of course, you’d request for your CEO to bring in a safety expert to reduce the number of dangerous incidents, and I can guarantee that the results will look something like this:

Another success! The safety expert saved lives! You are brilliant! As you guessed, however, this may not be the case.

Once again, a Regression Toward the Mean effect may have occurred, and the number of safety incidents naturally reverted back to an average level. The money spent on the safety expert could have been used for other more fruitful purposes, but you can nevertheless take credit for saving your coworker’s lives.

Despite these two examples (and many many more that could be provided), not all instances of extreme values can be cured by waiting for the values to revert to more typical figures. Sometimes, an effect is actually occurring, and an intervention is truly needed to fix a problem. Without it, things could possibly get even worse.

So, what should you do when extreme values occur? Perform an intervention? Wait it out? In academia, the answer is simple. Most researchers have the luxury of collecting data from a control group that does not receive the intervention, and then comparing the data after a sufficient amount of time has passed. If the intervention group resulted in better outcomes than the control group, then the intervention was indeed a success. If the two groups have roughly equal outcomes, then the intervention had no effect.

Businesses do not have such luxuries. Decisions need to be made quickly and correctly – or else someone could lose their job (or their life!). For this reason, it is often common practice to go ahead and perform the intervention. If the values return to normal, then you seem like a genius. If they do not, then at least you tried. On the other hand, if you do nothing and the values return to normal, then you seem like a genius again. If they do not return to normal, however, then it seems like you ignored the severity of the issues. The table below summarizes this issue:

	Values Remain Extreme	Values Return to Normal
Do Nothing	You Ignored the Issue	You Succeeded!
Do Something	You Tried	You Succeeded!

Long story short, you should probably make an attempt to fix the issue, although it may simply be Statistical Bullshit in the end.

Before concluding, one last question should be answered about Regression Toward the mean: How exactly can people make a career on it?

Well, imagine that you are a safety consultant, and you receive several consulting offers at once. You look at the companies, and they all seem to have a relatively stable number of incidents; however, you notice one that is going through a period of elevated incidents. Now that you know about Regression Toward the Mean, you know that you should take this company’s offer. Not only will they (probably) be willing to spend lots of money, but you (probably) need to do very little to reduce the incident rate. Even if your safety suggestions are bogus, you can still appear to be a competent safety consultant. Although it may sound crazy, I think you would be surprised how often this occurs in the real-world.

That is all for Regression Toward the Mean. Do you have your own Regression Toward the Mean story? Maybe a question? Feel free to email me at MHoward@SouthAlabama.edu. Until next time, watch out for Statistical Bullshit!