ProfAndyField

Assumptions Part 2: Homogeneity of Variance/Homoscedasticity

My last blog was about the assumption of normality, and this one continues the theme by looking at homogeneity of variance (or homoscedasticity to give it its even more tongue-twisting name). Just to remind you, I’m writing about assumptions because this paper showed (sort of) that recent postgraduate researchers don’t seem to check them. Also, as I mentioned before, I get asked about assumptions a lot. Before I get hauled up before a court for self-plaigerism I will be up front and say that this is an edited extract from the new edition of my Discovering Statistics book. If making edited extracts of my book available for free makes me a bad and nefarious person then so be it.

Assumptions: A reminder

Now, I’m even going to self-plagiarize my last blog to remind you that most of the models we fit to data sets are based on the general linear model, (GLM). This fact 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 Homoscedasticity Affect?

Like normality, if you’re thinking about homoscedasticity, then you need to think about 3 things:

  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). if we assume equality of variance then the estimates we get using the method of least squares will be optimal. 
  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. 
  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. 

When Does The Assumption Matter?

With reference to the three things above, let’s look at the effect of heterogeneity of variance/heteroscedasticity:

  1. Parameter estimates: If variances for the outcome variable differ along the predictor variable then the estimates of the parameters within the model will not be optimal. The method of least squares (known as ordinary least squares, OLS), which we normally use, will produce ‘unbiased’ estimates of parameters even when homogeneity of variance can’t be assumed, but better estimates can be achieved using different methods, for example, by using weighted least squares (WLS) in which each case is weighted by a function of its variance. Therefore, if all you care about is estimating the parameters of the model in your sample then you don’t need to worry about homogeneity of variance in most cases: the method of least squares will produce unbiased estimates (Hayes & Cai, 2007). However, if you even better estimates, then use weighted least squares regression to estimate the parameters. 
  2. Confidence intervals: unequal variances/heteroscedasticity creates a bias and inconsistency in the estimate of the standard error associated with the parameter estimates in your model (Hayes & Cai, 2007). As such, your confidence intervals and significance tests for the parameter estimates will be biased, because they are computed using the standard error. Confidence intervals can be ‘extremely inaccurate’ when homogeneity of variance/homoscedasticity cannot be assumed (Wilcox, 2010). 
  3. Significance tests: same as above. 

Summary

If all you want to do is estimate the parameters of your model then homoscedasticity doesn’t really matter: if you have heteroscedasticity then using weighted least squares to estimate the parameters will give you better estimates, but the estimates from ordinary least squares will be ‘unbiased’ (although not as good as WLS). 
If you’re interested in confidence intervals around the parameter estimates (bs), or significance tests of the parameter estimates then homoscedasticity does matter. However, many tests have variants to cope with these situations; for example, the t-test, the Brown-Forsythe and Welch adjustments in ANOVA, and numerous robust variants described by Wilcox (2010) and explained, for R, in my book (Field, Miles, & Field, 2012) 

Declaration


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

References

  • Field, A. P., Miles, J. N. V., & Field, Z. C. (2012). Discovering statistics using R: And sex and drugs and rock ‘n’ roll. London: Sage. 
  • Hayes, A. F., & Cai, L. (2007). Using heteroskedasticity-consistent standard error estimators in OLS regression: An introduction and software implementation. Behavior Research Methods, 39(4), 709-722. 
  • Wilcox, R. R. (2010). Fundamentals of modern statistical methods: substantially improving power and accuracy. New York: Springer.

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ProfAndyField

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 p 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’.