Biostatistics in Research Practice: Non-parametric tests Dr Victoria Allgar

What is a non-parametric test ? Methods of analysis that do not assume a particular family of distributions for the data.

When to use a non- parametric test Non-parametrics are distribution free Non-parametrics are distribution free Data may be rank ordered (Ordinal data) Data may be rank ordered (Ordinal data) Data may be from small samples Data may be from small samples There may be non-normal distribution of the variables (Skewed data) There may be non-normal distribution of the variables (Skewed data) Outliers may be present Outliers may be present

Non-parametric v Parametric tests Usually only perform one analysis of a data set choosing between parametric and non-parametric methods. Usually only perform one analysis of a data set choosing between parametric and non-parametric methods. It is usual to use a parametric method, unless there is a clear indication that it is not valid. It is usual to use a parametric method, unless there is a clear indication that it is not valid. It is important to realise that if we apply different tests to the same data then we do not expect them to give the same answer, but in general two valid methods will give similar answers. It is important to realise that if we apply different tests to the same data then we do not expect them to give the same answer, but in general two valid methods will give similar answers. Non-parametric tests are less powerful than the equivalent parametric test (especially in small samples) and will tend to give a less significant (larger) p-value Non-parametric tests are less powerful than the equivalent parametric test (especially in small samples) and will tend to give a less significant (larger) p-value

Dealing with ordinal data Non-parametric tests are usually based on Order Statistics and Ranks: Non-parametric tests are usually based on Order Statistics and Ranks: –ORDER STATISTICS : the observations arranged in increasing order of size. –RANKS : their places in this order

Ordering data Data Ordered Ranked

Data Ordered Ranked

Data Ordered Ranked

Wilcoxon Signed Rank Test Non-parametric equivalent for testing or estimating location for a one sample problem (equivalent to one sample t-test) OR for paired samples (equivalent to the paired t-test) Non-parametric equivalent for testing or estimating location for a one sample problem (equivalent to one sample t-test) OR for paired samples (equivalent to the paired t-test)Assumptions A random sample of n independent observations OR independent random pairs are taken. A random sample of n independent observations OR independent random pairs are taken. The variable of interest is the difference (d) : The variable of interest is the difference (d) : –For the one sample problem d = Observed value - Hypothesised value –For paired data d = X - Y - where X is value at time 1 and Y is value at time 2. The level of measurement is at least ordinal. The level of measurement is at least ordinal.

Examples Anxiety levels pre and post operation Anxiety levels pre and post operation Pain levels pre and post operation Pain levels pre and post operation Yorkshires’ fruit and veg consumption vs recommended 5 a day Yorkshires’ fruit and veg consumption vs recommended 5 a day BP pre and post exercise BP pre and post exercise

Friedman Test The assumption that the residuals have a Normal distribution cannot be assessed before fitting the model. The assumption that the residuals have a Normal distribution cannot be assessed before fitting the model. Sometimes, however, it can be seen from the raw data that the model will not fit well. In particular, wide variation in the standard deviations for each row and column will suggest problems with the parametric ‘two-way ANOVA’. Sometimes, however, it can be seen from the raw data that the model will not fit well. In particular, wide variation in the standard deviations for each row and column will suggest problems with the parametric ‘two-way ANOVA’. Therefore, we have a non-parametric equivalent of the two way ANOVA that can be used for data sets which do not fulfill the assumptions of the parametric method. Therefore, we have a non-parametric equivalent of the two way ANOVA that can be used for data sets which do not fulfill the assumptions of the parametric method. The method, which is sometimes known as Friedman’s two way analysis of variance, is purely a hypothesis test. The method, which is sometimes known as Friedman’s two way analysis of variance, is purely a hypothesis test.

Examples Time periods: Time periods: –Pre op, post op and 12 months –Baseline, week 2, week 12

Mann Whitney U test Non-parametric equivalent of two sample t-test. Non-parametric equivalent of two sample t-test. The Mann-Whitney test is used to compare two sets of data from independent groups. The Mann-Whitney test is used to compare two sets of data from independent groups. It is the most commonly used alternative to the independent samples t-test. It is the most commonly used alternative to the independent samples t-test. The values from both samples are combined and then the data is ranked from smallest to largest. The rank of 1 is assigned to the smallest value, 2 to the next smallest and so on. If the ranks are tied, then the average rank is used. The values from both samples are combined and then the data is ranked from smallest to largest. The rank of 1 is assigned to the smallest value, 2 to the next smallest and so on. If the ranks are tied, then the average rank is used. Assumptions Assumptions –There are two independent random variables (X and Y), of size n and m. –The variable of interest is a continuous random variable. –The two populations differ only with respect to location.

Examples Comparing two groups e.g. Comparing two groups e.g. –Anxiety between men and women –Control group and an intervention group

Kruskal Wallis Test Just as the one way analysis of variance is a more general form of the t-test, there is a one for the non-parametric Mann-Whitney test. Just as the one way analysis of variance is a more general form of the t-test, there is a one for the non-parametric Mann-Whitney test. The Kruskal-Wallis test is an obvious mathematical extension of the Mann-Whitney test. The Kruskal-Wallis test is an obvious mathematical extension of the Mann-Whitney test. Assumptions Assumptions –There are three or more independent random variables (X1, X2, X3……….Xn, ), of size n1,n2, n3……….nn –The variable of interest is ordinal or a continuous random variable which is non-normal. –The populations differ only with respect to location.

Examples Comparing more than 2 groups, e.g. Comparing more than 2 groups, e.g. –Contol group and two intervention groups: Satisfaction with procedure –Age groups: 18-30; 30-50, 50+ –Social class groups

Choosing an appropriate method of analysis Number of groups of observations Number of groups of observations Independent or dependent groups of observations Independent or dependent groups of observations The type of data The type of data The distribution of data The distribution of data The objective of the analysis The objective of the analysis

Choice of Test