MD-Logic Artificial Pancreas System

Abstract

analyse categorical data in small samples IN THE PREVIOUS TUTORIAL we have outlined some simple methods for analysing binary data, including the comparison of two proportions using the Normal approximation to the binomial and the Chi-squared test. 1 However, these methods are only approximations, although they are good when the sample size is large. When the sample size is small we can evaluate all possible combinations of the data and compute what are known as exact P-values. FISHER'S EXACT TEST When one of the expected values (note: not the observed values) in a 2 × 2 table is less than 5, and especially when it is less than 1, then Yates' correction can be improved upon. In this case Fisher's Exact test, proposed in the mid-1930s almost simultaneously by Fisher, Irwin and Yates, 2 can be applied. The null hypothesis for the test is that there is no association between the rows and columns of the 2 × 2 table, such that the probability of a subject being in a particular row is not influenced by being in a particular column. If the columns represent the study group and the rows represent the outcome, then the null hypothesis could be interpreted as the probability of having a particular outcome not being influenced by the study group, and the test evaluates whether the two study groups differ in the proportions with each outcome. An important assumption for all of the methods outlined, including Fisher's Exact test, is that the binary data are independent. If the proportions are correlated then more advanced techniques should be applied. For instance in the leg ulcer example of the previous tutorial, 1 if there were more than one leg ulcer per patient, we could not treat the outcomes as independent. The test is based upon calculating directly the probability of obtaining the results that we have shown (or results more extreme) if the null hypothesis is actually true, using all possible 2 × 2 tables that could have been observed, for the same row and column totals as the observed data. These row and column totals are also known as marginal totals. What we are trying to establish is how extreme our particular table (combination of cell frequencies) is in relation to all the possible ones that could have occurred given the marginal totals. This is best explained by a simple worked example. The data in table 1 come from an RCT comparing intra-muscular magnesium injections with placebo for the treatment of chronic fatigue syndrome. 3 Of the 15 patients who had the intra-muscular magnesium injections 12 felt better (80 per cent) whereas, of the 17 on placebo, only three felt better (18 per cent). There are 16 different ways of rearranging the cell frequencies for the table whilst keeping the marginal totals the same, as illustrated in figure 1 (right). The result that corresponds to our observed cell frequencies is (xiii). The general form of table 1 is given in table 2, and under the null hypothesis of no association Fisher showed that the probability of obtaining the frequencies a, b, c and d in table 2 is