# gofsumm - Statistics Categorical Data Introduction By...

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Statistics: Categorical Data Introduction By categorical data, we mean that a population is assigned to one of k categories on the basis of one or more measurements. There is a particular name used for a set of counts tabulated in some systematic way: con- tingency table . Initially we will be concerned with 1 × k contingency tables: a population classified according to k categories. Dealing with categorical data statistically is typically harder than dealing with ordinary data, and we must devise a number of procedures to deal with them. We focus our attention on the counts in each category. There are three basic questions we might ask about how a population is allocated into categories. 1. Do the counts agree broadly with some hypothetical situation? 2. If not, what are the deviations from the pattern suggested by the hypothetical situ- ation? 3. If any, why might there be such deviations? Goodness of fit tests We have observed frequencies. By convention, we call these O i for group i . We calculate expected frequencies for some hypothetical situation (usually there will be some fluctuation about these expected values, due to the usual variability associated with random sampling). By convention, we call these E i for group i . Basic goodness-of-fit tests are carried out as follows. 1. Calculate the expected number in each category; this will depend on the hypothetical situation envisaged. 2. For each category, calculate the difference between the observed and expected fre- quency O i - E i , and then square this difference. Finally, divide this square by the expected frequency to give ( O i - E i ) 2 /E i . This is a measure of the relative discrep- ancy between the observed and hypothesized frequency for each category. 3. Add these measures for each category. This gives the test statistic, k i =1 ( O i - E i ) 2 E i . 4. Write down the degrees of freedom , k - 1, the number of categories minus 1. 1

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5. Perform a chi-square test using tables and the appropriate degrees of freedom. If the observed value of the test statistic is larger than some critical value (say at 5%), then the result of the experiment is “significant at the 5% level”, and we conclude that the expected frequencies do not match the observed frequencies. In terms of formal hypothesis testing we might write the null and alternative hypotheses as H 0 : The differences between the observed and expected frequencies are
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