The next step of McNemar's test depends on the approach you want to use:
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Hence, by contradiction, big values of χ² provide evidence in favor of the alternative hypothesis H₁. Indeed, if p b = p c, then in our data b should be roughly equal to c, which translates into small values of χ². Note that, under the null hypothesis, we expect to observe small test statistic values. Under the null hypothesis and if b and c are large enough, then χ² follows approximately the chi-squared distribution with 1 degree of freedom. The formula employs only the off-diagonal fields from the contingency table: When we've got the contingency table ready, we can quickly and easily compute McNemar's test statistic. The alternative hypothesis is that these proportions differ significantly, and so the treatment (course/TV spot) does have some effect.Ī quick tip: don't focus that much on formulas, but try to remember the examples - they are beneficial when it comes to recalling the correct interpretation of McNemar's test for paired proportions.
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H₁: p b ≠ p c McNemar's test interpretation Simplifying these equations, we get a slightly simpler form of our hypotheses, which you can often see in textbooks: The alternative hypothesis is that the marginal distributions are not equal, i.e., that the above equation does not hold: P c + p d = p b + p d, but this follows immedately from the previous equation). Namely, it states that the row marginal and column marginal are equal - people often call this the hypothesis of marginal homogeneity: Now, the null hypothesis of McNemar's test is a claim about the marginal distributions. Let's denote by p a, p b, p c, p d the theoretical probability of their respective group:Īs probabilities always sum to one, we obviously have: And once we have the contingency tables, we'll be able to state the hypotheses of McNemar's test more formally. When we want to perform McNemar's test, the first step is to prepare the contingency table. Like in the t-test, the crucial assumption of McNemar's test is that we have a simple random sample (i.e., a set of independent, identically distributed random variables). Have you heard about the (calc:2458)? McNemar's test has a similar aim, but it uses a dichotomous dependent variable, while the t-test needs a continuous variable. no) to vote for candidates before and after watching their TV spot. fail) of a group of students prior to and following a special preparatory curse.
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Other examples of when we can use McNemar's test: So, for our patients, we can have a result of some medical test that is either positive or negative. Additionally, we have the values of a dichotomous variable (i.e., a variable with only two categories). More precisely, we call on McNemar's test when we have two paired groups, for instance, patients before and after treatment. As you might've already heard, this statistical test deals with paired binomial data. Let's first discuss what the assumptions of McNemar's test are.