The joys of inventing data

Have I gone over to the dark side? Cracked under pressure from the REF to resort to fabrication of results to secure that elusive Nature paper? Or had my brain addled by so many requests for information from ethics committees that I’ve just decided that its easier to be unethical? Well readers will be reassured to hear that none of these things is true. What I have to say concerns the benefits of made-up data for helping understand how to analyse real data.

In my field of experimental psychology, students get a thorough grounding in statistics and learn how to apply various methods for testing whether groups differ from one another, whether variables are associated and so on. But what they typically don’t get is any instruction in how to simulate datasets. This may be a historical hangover. When I first started out in the field, people didn’t have their own computers, and if you wanted to do an analysis you either laboriously assembled a set of instructions in Fortran which were punched onto cards and run on a mainframe computer (overnight if you were lucky), or you did the sums on a pocket calculator. Data simulation was just unfeasible for most people. Over the years, the landscape has changed beyond recognition and there are now windows-based applications that allow one to do complex multivariate statistics at the press of a button. There is a danger, however, which is that people do analyses without understanding them. And one of the biggest problems of all is a tendency to apply statistical analyses post hoc. You can tell people over and over that this is a Bad Thing (see Gould and Hardin, 2003) but they just don’t get it. A little simulation exercise can be worth a thousand words.

So here’s an illustration. Suppose we’ve got two groups each of 10 people, let’s say left-handers and right-handers. And we’ve given them a battery of 20 cognitive tests. When we scrutinise the results, we find that they don’t differ on most of the measures, but there’s a test of mathematical skill on which the left-handers outperform the right-handers. We do a t-test and are delighted to find that on this measure, the difference between groups is significant at the .05 level, so we write up a paper entitled "Left-handed advantage for mathematical skills" and submit it to a learned journal, not mentioning the other 19 tests. After all, they weren’t very interesting. Sounds OK? Well, it isn’t. We have fallen into the trap of using statistical methods that are valid for testing a hypothesis that is specified a priori in a situation where the hypothesis only emerged after scrutinising the data.

Let’s generate some data. Most people have access to Microsoft Excel, which is perfect for simple simulations. In row 1 we put our column labels, which are group, var1, var2, …. var 20.

In column A, we then have ten zeroes followed by ten ones, indicating group identity. We then use random numbers to complete the table. The simplest way to do this is to just type in each cell:

   =RAND()

This generates a random number between 0 and 1.

A more sophisticated option is to generate a random z-score. This creates random numbers that meet the assumption of many statistical tests that data are normally distributed. You do this by typing:

   =NORMSINV(RAND())

At the foot of each column you can compute the mean and standard deviation for each group, and Excel automatically computes a p-value based on the t-test for comparing the groups with a command such as:

=TTEST(B2:B11,B12:B22,2,2)

See this site if you need an explanation of this formula.

So the formulae in the first three columns look like this (rows 4-20 are hidden): 

Copy this formula across all columns. I added conditional formatting to row 27 so that ‘significant’ p-values are highlighted in yellow (and it just so happens with this example that the generated data gave a p-value less than .05 for column C).

Every time you type anything at all on the sheet, all the random numbers are updated: I’ve just added a row called ‘thisrun’ and typing any number in cell B29 will re-run the simulation.  This provides a simple way of generating a series of simulations and seeing when p-values fall below .05. On some runs, all the t-tests are nonsignificant, but you’ll quickly see that on many runs one or more p-values are below .05. In fact, on average, across numerous runs, the average number of significant values is going to be one because we have twenty columns, and 1/20 = .05. That’s what p < .05 means! If this doesn’t convince you of the importance of specifying your hypothesis in advance, rather than selecting data for analysis post hoc, nothing will.

This is a very simple example, but you can extend the approach to much more complicated analytic methods. It gets challenging in Excel if you want to generate correlated variables, though if you type a correlation coefficient in cell A1, and have a random number in column B, and copy this formula down from cell C2, then columns B and C will be correlated by the value in cell A1:

=B2*A$1+NORMSINV(RAND())*SQRT(1-A$1^2)

NB, you won’t get the exact correlation on each run: the precision will increase with the number of rows you simulate.

Other applications, such as Matlab or R, allow you to generate correlated data more easily. There are examples of simulating multivariate normal datasets in R in my blog on twin methods.

Simulation can be used not just for exploring a whole host of issues around statistical methods. For instance, you can simulate data to see how sample size affects results, or how results change if you fail to meet assumptions of a method. But overall, my message is that data simulation is a simple and informative approach to gaining understanding of statistical analysis. It should be used much more widely in training students.

Reference

Good, P. I., & Hardin, J. W. (2003). Common errors in statistics (and how to avoid them). Hoboken, NJ: Wiley.