In this Unit we introduced hypothesis tests using an example where our estimate of how the statistic was distributed, due to random variation under the null hypothesis, arose directly from the experimental design. In other words, our estimate of variation derived directly from the way in which those observations were obtained. This is known as an **exact test**.

In many ways exact tests are the easiest type of test to understand, partly because, with the advent of computer simulation models they can often be evaluated without the use of complicated or obscure mathematical formulae - and in doing so, their assumptions can be exposed. However, because computation was difficult and expensive until quite recently, considerable effort has been invested in developing tests requiring minimal calculations - by the end-user at least. Of these, the most popular are known as 'parametric tests'.

### Parametric

In principle, **parametric** methods assume data are normally distributed - or, less commonly, that your data represent one of the mathematically tractable frequency distributions which are closely related to the normal distribution. Under this assumption the distribution of a number of statistics can be estimated. These statistics include means, differences between means, and ratios between variances.

In order to estimate the distribution of the statistic of interest, you require an estimate of the parameters of the distribution your data represent - in the case of the normal distribution - its mean and standard deviation.

A wide variety of parametric methods are currently available. Provided their assumptions are fully met, parametric tests are just as powerful as exact methods.

In practice of course, no data are ever normal, and perfect parametric normal populations only exist in mathematical and simulation models. Consequently, the term **approximately normal** is often applied - even though there is no quantitative criterion as to what 'approximately' might mean. Testing data for normality does not help in this respect - even though it is commonly done. Failing to show data are non-normal does not mean they are normal - it only shows you did not have enough power to show they were non-normal.

Fortunately, when calculated from large samples, a number of statistics (such as the mean) converge towards a normal distribution - irrespective of how their data are distributed. As a result, the crucial question is not how the data are distributed - but how the statistics are. More generally therefore, statisticians worry about how 'errors' are distributed. Be that as it may, even if your data are perfectly normal, many statistics have distributions very different from the smooth continuous normal family. Nevertheless, these distributions are commonly approximated by a parametric distribution.

Parametric models have one further advantage that is particularly important for hypothesis tests, the shape of their tails depend upon estimated means and standard deviations - which, provided your data represent a normal population, places comparatively little weight upon the more extreme observations in your sample. Unfortunately, as we note in Unit 8, if your data represent a non-normal population this model can also be horribly misleading.

Parametric tests vary in their sensitivity to non-normality. If the distribution is significantly non-normal, there are two options:

If you are testing the difference between means, you can 'normalise' the error distribution by taking means of larger samples. More data also increase the power of tests, and allow smaller differences to become 'significant'. However, gathering more data is not always possible.

Transform your data (as discussed in Unit 6) to 'normalise' the error distribution. However, take care that the transformations agree with both the assumptions of your test, and the biology you are investigating. Not all distributions can be transformed to normal, especially where there are a lot of zeros.

*What do you do if the distributions cannot be normalised, or if you have categorical data?*

### Non-parametric ('distribution-free')

**Non parametric** tests do not assume the data have any particular distribution, and can analyse data where no other test is applicable. Confusingly, the term **non parametric** is also applied to tests that assess statistics according to rank rather than location.

Non-parametric tests generally involve much less computation than parametric tests. Some biologists prefer non-parametric tests because they do not have to consider whether data are normally distributed, and their conclusions are therefore more 'robust'.

However, because transforming continuous data to ranks wastes information, non-parametric tests are less powerful than parametric ones. Therefore, provided their assumptions are met, parametric tests can demonstrate smaller differences than non-parametric methods. Notice however that, if you should not assume your statistic has a predefined parametric distribution, this power may be an illusion.

Non-parametric comparisons fall into roughly three types:

Tests based upon the rank, rather than the relative magnitude, of your observations. These may also be used for ranked categorical data. They generally use frequency distributions generated by mathematical statisticians. Despite this, the test statistics are usually quite easy to calculate.

Tests based upon the statistics inherent in proportions. These can be used upon any measurement that can exist in one of two possible states. For example whether an observation is positive or negative, black or white, absent or present. These often rely upon the binomial distribution for small samples, or the normal distribution for larger ones.

Likelihood ratio tests are based upon comparing proportions as probabilities. These comparisons are used for data that cannot be ranked, but may exist in several states. For example comparing proportions of tick species infesting young goats. The distribution of the test statistic often approximates to a conventional parametric distribution.

Confusingly, many nonparametric statistics are assumed to approximate to a parametric distribution - and are therefore assessed using critical values, rather than by relative rank. In other words, although the statistic (and its test) is described as being nonparametric, the test's model is parametric.

We cover a number of non-parametric tests in Units 9, and 10.