![]() As the name indicates, the idea is to assess whether the pattern or distribution of responses in the sample "fits" a specified population (external or historical) distribution. The test of hypothesis with a discrete outcome measured in a single sample, where the goal is to assess whether the distribution of responses follows a known distribution, is called the χ 2 goodness-of-fit test. To ensure that the sample size is appropriate for the use of the test statistic above, we need to ensure that the following: min(np 10, n p 20. This is done by multiplying the observed sample size (n) by the proportions specified in the null hypothesis (p 10, p 20. These expected frequencies are determined by allocating the sample to the response categories according to the distribution specified in H 0. When we conduct a χ 2 test, we compare the observed frequencies in each response category to the frequencies we would expect if the null hypothesis were true. The test above statistic formula above is appropriate for large samples, defined as expected frequencies of at least 5 in each of the response categories. χ 2 (chi-square) is another probability distribution and ranges from 0 to ∞. The observed frequencies are those observed in the sample and the expected frequencies are computed as described below. In the test statistic, O = observed frequency and E=expected frequency in each of the response categories. We find the critical value in a table of probabilities for the chi-square distribution with degrees of freedom (df) = k-1. Test Statistic for Testing H 0: p 1 = p 10, p 2 = p 20. The formula for the test statistic is given below. ![]() We then determine the appropriate test statistic for the hypothesis test. ) where k represents the number of response categories. Specifically, we compute the sample size (n) and the proportions of participants in each responseĬategory (. We select a sample and compute descriptive statistics on the sample data. In one sample tests for a discrete outcome, we set up our hypotheses against an appropriate comparator. The comparator is sometimes called an external or a historical control. The known distribution is derived from another study or report and it is again important in setting up the hypotheses that the comparator distribution specified in the null hypothesis is a fair comparison. The procedure we describe here can be used for dichotomous (exactly 2 response options), ordinal or categorical discrete outcomes and the objective is to compare the distribution of responses, or the proportions of participants in each response category, to a known distribution. Discrete variables are variables that take on more than two distinct responses or categories and the responses can be ordered or unordered (i.e., the outcome can be ordinal or categorical). Here we consider hypothesis testing with a discrete outcome variable in a single population.
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