## You Can’t Trust Your PhD Supervisor:)

Before I continue, let me be clear that during my PhD Graham taught me everything I consider worth knowing about the process of science, theory, psychopathology and academic life. So, I tend to listen to what he says (unless it’s about marriage or statistics) very seriously. The two points I want to focus on are:

2.  Do an experiment but make up or severely massage the data to fit your hypothesis. This is an obvious one, but is something that has surfaced in psychological research a good deal recently (http://bit.ly/QqF3cZ;http://nyti.ms/P4w43q).

I’d really like to do a study looking at more experienced researcher’s basic understanding of assumptions (a follow up to Hoekstra’s study on a more experienced sample, and with more probing questions) just to see whether my suspicions are correct. Maybe I should email Hoekstra and see if they’re interested because, left to my own devices, I’ll probably never get around to it.

Anyway, my point is that I think it’s not just deliberate fraud that creates false knowledge, there is also a problem of well-intentioned and honest folk simply not understanding what to do, or when to do it.

3.  Convince yourself that a significant effect at p=.055 is real. How many times have psychologists tested a prediction only to find that the critical comparison just misses the crucial p=.05 value? How many times have psychologists then had another look at the data to see if it might just be possible that with a few outliers removed this predicted effect might be significant? Strangely enough, many published psychology papers are just creeping past the p=.05 value – and many more than would be expected by chance! Just how many false psychology facts has that created? (

This is a massive over-simplification because an effect of p = .055 is ‘real’ and might very well be ‘meaningfiul’. Conversely, an effect with p < .001 might very well be meaningless. To my mind it probably matters very little if a p is .055 or .049. I’m not suggesting I approve of massaging your data, but really this point illustrates how wedded psychologists are to the idea of effects somehow magically becoming ‘real’ or ‘meaningful’ once p drifts below .05. There’s a few points to make here:

First, all effects are ‘real’. There should never be a decision being made by anyone about whether an effect is real or not real. They’re all real. It’s just that some are large and some are small. There is a decision about whether an effect is meaningful, and that decision should be made within the context of the research question.

Second, I think an equally valid way to create ‘false knowledge’ is to publish studies based on huge samples reporting loads of small and meaningless effects that are highly significant. Imagine you look at the relationship between statistical knowledge and eating curry. You test 1 million people and find that there is a highly significant negative relationship, r = -.002, p < .001. You conclude that eating curry is a 'real' effect – it is meaningfully related to poorer statistical knowledge. There are two issues here: (1) in a sample of 1 million people the effect size estimate will be very precise, and the confidence interval very narrow. So we know the true effect in the population is going to be very close indeed to -.002. In other words, there is basically no effect in the population – eating curry and statistical knowledge have such a weak relationship that you may as well forget about it. (2) anyone trying to replicate this effect in a sample substantially smaller than 1 million is highly unlikely to get a significant result. You've basically published an effect that is 'real' if you use p < .05 to define your reality, but is utterly meaningless and won't replicate (in terms of p) in small samples.

Third, there is a wider problem than people massage their ps. You have to ask why people massage their ps. The answer to that is because psychology is so hung up on p-values. Over 10 years since the APA published their report on statistical reporting (Wilkinson, 1999) there has been no change in the practice of applying the all-or-nothing thinking of accepting results as ‘real’ if p < .05. It's true that Wilkinson's report has had a massive impact in the frequency with which effect sizes and confidence intervals are reported, but (in my experience which is perhaps not representative) these effect sizes are rarely interpreted with any substance and it is still the p-value that drives decisions made by reviewers and editors.

This whole problem would go away if ‘meaning’ and ‘substance’ of effects were treated not as a dichotomous decision, but as a point along a continuum. You quantify your effect, you construct a confidence interval around it, and you interpret it within the context of the precision that your sample size allows. This way, studies with large samples could no longer focus on meaningless but significant effects, instead the researcher could say (given  the high level of precision they have) that the effects in the population (the true effects if you like) are likely to be about the size that they observed and interpret accordingly. In small studies, rather than throwing out the baby with the bathwater, large effects could be given some creditability but with the caveat that the estimates in the study lack precision. This is where replication is useful. No need to massage data – researchers just give it to the reader as it is, interpret it and apply the appropriate caveats etc. One consequence of this might be that rather than publishing a single small study with massaged data to get p < .05, researchers might be encouraged to replicate their own study a few times and report them all in a more substantial paper. Doing so would mean that across a few studies you could show (regardless of p) the likely size of the effect in the population.

That turned into a bigger rant than I was intending ….

### References

Haller, H., & Kraus, S. (2002). Misinterpretations of Significance: A Problem Students Share with Their Teachers? MPR-Online, 7(1), 1-20.
Wilkinson, L. (1999). Statistical Methods in Psychology Journals: Guidelines and Explanations. American Psychologist, 54(8), 594-604.

## Assumptions Part 1: Normality

…. I didn’t grow a pair of breasts. If you didn’t read my last blog that comment won’t make sense, but it turns out that people like breasts so I thought I’d mention them again. I haven’t written a lot of blogs, but my frivolous blog about growing breasts as a side effect of some pills was (by quite a large margin) my most viewed blog. It’s also the one that took me the least time to write and that I put the least thought into. I think the causal factor might be the breasts.

This blog isn’t about breasts, it’s about normality. Admittedly the normal distribution looks a bit like a nipple-less breast, but it’s not one: I’m very happy that my wife does not sport two normal distributions upon her lovely chest. I like stats, but not that much …

### Assumptions

Anyway, I recently stumbled across this paper. The authors sent a sample of postgrads (with at least 2 years research experience) a bunch of data analysis scenarios and asked them how they would analyze the data. They were interested in whether or not, and how these people checked the assumptions of the tests they chose to use. The good news was that they chose the correct test (although given all of the scenarios basically required a general linear model of some variety that wasn’t hard). However, not many of them checked assumptions. The conclusion as that people don’t understand assumptions or how to test them

I get asked about assumptions a lot. I also have to admit to hating the chapter on assumptions in my SPSS and R books. Well, hate is a strong word, but I think it toes a very conservative and traditional line. In my recent update of the SPSS book (out early next year before you ask) I completely re-wrote this chapter. It takes a very different approach to thinking about assumptions.

Most of the models we fit to data sets are based on the general linear model, (GLM) which means that any assumption that applies to the GLM (i.e., regression) applies to virtually everything else. You don’t really need to memorize a list of different assumptions for different tests: if it’s a GLM (e.g., ANOVA, regression etc.) then you need to think about the assumptions of regression. The most important ones are:

• Linearity
• Normality (of residuals)
• Homoscedasticity (aka homogeneity of variance)
• Independence of errors.

### What Does Normality Affect?

For this post I’ll discuss normality. If you’re thinking about normality, then you need to think about 3 things that rely on normality:

1. Parameter estimates: That could be an estimate of the mean, or a b in regression (and a b in regression can represent differences between means). Models have error (i.e., residuals), and if these residuals are normally distributed in the population then using the method of least squares to estimate the parameters (the bs) will produce better estimates than other methods.
2. Confidence intervals: whenever you have a parameter, you usually want to compute a confidence interval (CI) because it’ll give you some idea of what the population value of the parameter is. We use values of the standard normal distribution to compute the confidence interval: using values of the standard normal distribution makes sense only if the parameter estimates actually come from one.
3. Significance tests: we often test parameters against a null value (usually we’re testing whether b is different from 0). For this process to work, we assume that the parameter estimates have a normal distribution. We assume this because the test statistics that we use (such as the t, F and chi-square), have distributions related to the normal. If parameter estimates don’t have a normal distribution then p-values won’t be accurate.

### What Does The Assumption Mean?

People often think that your data need to be normally distributed, and that’s what many people test. However, that’s not the case. What matters is that the residuals in the population are normal, and the sampling distribution of parameters is normal. However, we don’t have access to the sampling distribution of parameters or population residuals; therefore, we have to guess at what might be going on by testing the data instead.

### When Does The Assumption Matter?

However, the central limit theorem tells us that no matter what distribution things have, the sampling distribution will be normal if the sample is large enough. How large is large enough is another matter entirely and depends a bit on what test statistic you want to use. So bear that in mind. However, oversimplifying things a bit, we could say:

1. Confidence intervals: For confidence intervals around a parameter estimate to be accurate, that estimate must come from a normal distribution. The central limit theorem tells us that in large samples, the estimate will have come from a normal distribution regardless of what the sample or population data look like. Therefore, if we are interested in computing confidence intervals then we don’t need to worry about the assumption of normality if our sample is large enough. (There is still the question of how large is large enough though.) You can easily construct bootstrap confidence intervals these days, so if your interest is confidence intervals then why not stop worrying about normality and use bootstrapping instead?
2. Significance tests: For significance tests of models to be accurate the sampling distribution of what’s being tested must be normal. Again, the central limit theorem tells us that in large samples this will be true no matter what the shape of the population. Therefore, the shape of our data shouldn’t affect significance tests provided our sample is large enough. (How large is large enough depends on the test statistic and the type of non-normality. Kurtosis for example tends to screw things up quite a bit.) You can make a similar argument for using bootstrapping to get a robust if p is your thing.
3. Parameter Estimates: The method of least squares will always give you an estimate of the model parameters that minimizes error, so in that sense you don’t need to assume normality of anything to fit a linear model and estimate the parameters that define it (Gelman & Hill, 2007). However, there are other methods for estimating model parameters, and if you happen to have normally distributed errors then the estimates that you obtained using the method of least squares will have less error than the estimates you would have got using any of these other methods.

### Summary

If all you want to do is estimate the parameters of your model then normality doesn’t really matter. If you want to construct confidence intervals around those parameters, or compute significance tests relating to those parameters then the assumption of normality matters in small samples, but because of the central limit theorem we don’t really need to worry about this assumption in larger samples. The question of how large is large enough is a complex issue, but at least you know now what parts of your analysis will go screwy if the normality assumption is broken..

This blog is based on excerpts from the forthcoming 4th edition of ‘Discovering Statistics Using SPSS: and sex and drugs and rock ‘n’ roll’.