![]() ![]() 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. Identify the appropriate hypothesis testing procedure based on type of outcome variable and number of samples.Appropriately interpret results of chi-square tests.Learning ObjectivesĪfter completing this module, the student will be able to: We will consider chi-square tests here with one, two and more than two independent comparison groups. Specifically, the test statistic follows a chi-square probability distribution. The technique to analyze a discrete outcome uses what is called a chi-square test. We could use the same classification in an observational study such as the Framingham Heart Study to compare men and women in terms of their blood pressure status - again using the classification of hypertensive, pre-hypertensive or normotensive status. For example, in some clinical trials the outcome is a classification such as hypertensive, pre-hypertensive or normotensive. The specific tests considered here are called chi-square tests and are appropriate when the outcome is discrete (dichotomous, ordinal or categorical). The hypothesis is based on available information and the investigator's belief about the population parameters. ![]() This module will continue the discussion of hypothesis testing, where a specific statement or hypothesis is generated about a population parameter, and sample statistics are used to assess the likelihood that the hypothesis is true. For example, if testing for a fair, six-sided die, there would be five degrees of freedom because there are six categories or parameters (each number) the number of times the die is rolled does not influence the number of degrees of freedom.Ĭhi-squared distribution, showing X 2 on the x-axis and P-value on the y-axis.Boston University School of Public Health The degrees of freedom are not based on the number of observations as with a Student's t or F-distribution. Pearson's chi-squared test ( χ 2 is the number of categories. ![]() For broader coverage of this topic, see Chi-squared test. ![]()
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