Hypothesis testing

Null hypothesis testing (often described just as hypothesis testing is very commonly used in epidemiological investigations, and may be used in both analytic studies (for example, assessing whether disease experience differs between different exposure groups), and in descriptive studies (for example, if assessing whether disease experience differs from some suspected value). As in most studies, only a sample of individuals is taken, it is not possible to definitively state whether or not there is a difference between the two exposure groups. Hypothesis tests provide a method of assessing the strength of evidence in favour or against a true difference in the underlying population. However, despite their widespread use, the results of hypothesis tests are often misinterpreted.

Concept behind null hypothesis testing

Hypothesis tests provide a systematic, objective method of data analysis, but do not actually answer the main question of interest (which is commonly along the lines of 'is there a difference in disease experience between individuals with exposure x and individuals without exposure x?'). Rather, hypothesis tests answer the question 'if there is no difference in disease experience between individuals with or without exposure x, what is the probability of obtaining the current data (or data more 'extreme' than this)?' As such, hypothesis tests do not inform us whether or not there is a difference, but instead they offer us varying degrees of evidence in support of or against a situation where there is no difference in the population under investigation. This situation of 'no difference' is known as the 'null hypothesis', and should be stated whenever a null hypothesis test is performed.

Null hypothesis testing relies on the fact that the likely values obtained in a sample taken from a population in which the null hypothesis is true (i.e. there is no difference in disease experience between groups) can be predicted. That is, although it is most likely that a sample from this population will also show no difference, it is not impossible that, through chance, a sample will contain more diseased animals in the 'exposed' group than in the 'unexposed' group, for example. Through knowledge of the number of animals sampled and the prevalence of disease in the population, the probability of getting any particular pattern of data from this sample can be ascertained. This probability is known as the p-value, and is the main outcome of interest from a hypothesis test. A p-value of 1.00 would suggest that if there was no difference in disease experience according to exposure in the population and repeated samples were taken from this population and performed hypothesis tests on these, all of these samples would be expected to show this pattern (or 'more extreme' - i.e. show more of a difference). However, a p-value of 0.001 would suggest that there is only a 0.1% chance of seeing these data (or more extreme) if there was no true difference in the population. Note that although 0.001 is a very low p-value, it cannot be used to prove that the null hypothesis is false, or that the 'alternative hypothesis' (of their being a difference) is true - it can only be stated that it gives 'strong evidence against the null hypothesis'.

Significance levels

Although the approach described above (of varying degrees of evidence against the null hypothesis) is the most statistically correct interpretation of p-values, it is often not practical to apply this in epidemiological analysis. For example, an investigator may want to identify a number of exposures which appear to be associated with the outcome in order to investigate these further - the approach described above would not allow this (as it will never lead to an association being proven. As such, many studies will use significance levels as a method of interpretation of p-values as 'significant' or 'not significant'. Commonly, a p-value of 0.05 or less is used to denote a 'significant' association. This means that if the there is a 5% chance or less of observing the data in question (or more extreme) if the null hypothesis is true, then the association will be denoted as 'significant'. Of course, there remains a 5% chance that there is no true difference in the population, and this can be a problem when testing large numbers of exposures (as shown in the cartoon here). Because of this, great care should be taken whenever using significance levels to interpret hypothesis tests, and the actual p-value should always be presented.

Limitations of null hypothesis tests

One considerable limitation of hypothesis testing is described above: namely, hypothesis tests do not relate to the main question of interest (whether or not there is a true difference in the population), and only provide degrees of evidence in favour or against there being no true difference. Another limitation is that there will always be a difference of some magnitude between the two groups, even if this is of no relevance. Consider a cohort study where 1 million nondiseased individuals are followed up to see whether or not exposure to substance x is associated with disease. It may be that in this whole population of 1 million animals, 10.0% of exposed individuals develop the disease and that 9.9% of unexposed individuals develop the disease. Of course, this difference is not of any biological relevance, and yet there is a difference there (as this is a whole population rather than a sample, we would not conduct a hypothesis test). As the size of any sample increases, the ability to detect a true difference increases. As there will be a 'true difference' (however small) in most populations, This means that with large sample sizes, the chance of a hypothesis test detecting a difference increases (indeed, some statisticians view hypothesis testing as a method of determining whether or not the sample size is sufficient to detect a difference). This problem can be reduced by ensuring that the appropriate measure of effect is always presented along with the hypothesis test p-value. In this example, the

Hypothesis testing and study power

Approach to hypothesis testing

The approach to hypothesis testing first requires making the assumption that there is no difference between the two groups (which is known as the null hypothesis). Statistical methods are then employed in order to evaluate the probability that the observed data would be seen if the null hypothesis was correct (known as the p-value). Based on the resultant p-value, a decision can be made as to whether the support for the null hypothesis is sufficiently low so as to give evidence against it being correct. It is important to note, however, that the null hypothesis can never be completely disproved based on a sample - only evidence can be gained in support or against it. However, based on this evidence, investigators will often come to a conclusion that the null hypothesis is either 'accepted' or 'rejected'.

In any hypothesis test, there is a risk that the incorrect conclusion is made - which will either take the form of a type I or a type II error, as described below. Note that no single hypothesis test can be affected by both type I and type II errors, as they are each based on different assumptions regarding the source population. However, as the true state of the source population will not be known, both types of errors should be considered when interpreting a hypothesis test (and when calculating the required sample size).

Type I error

This type of error refers to the situation where it is concluded that a difference between the two groups exists, when in fact it does not. The probability of a type I error is often denoted with the symbol α. As this type of error is based on a situation in which the 'null hypothesis' is correct, it is associated with the p-value given in a hypothesis test, which is often set at 0.05 to indicate 'significance'. This means that there is a 5% chance of a type I error (which in the case of hypothesis testing, is interpreted as 'if the null hypothesis was correct, we would expect to see this difference or greater only 5% of the time - meaning that there is [weak] evidence against the null hypothesis being correct).

Type II error

This type of error refers to the situation where it is concluded that no difference between two groups exists, when in fact it does. The probability of a type II error is often denoted with the symbol β. As the 'power' of a study is defined as the probability of detecting the difference when it does exist, it can be calculated as (1-β).