# 卫生经济学 Health Economics ECON32202T

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The general form of Bayes’ theorem (sometimes called Bayes’ rule) is:
$$p(A \mid X)=\frac{p(X) \times p(A)}{p(X \mid A) \times p(A)+p(X \mid \sim A) \times p(\sim A)}$$
where
$p(A)$ is the prior (our prior knowledge, for example, of the prevalence of cancer in the population as a whole);
$p(A \mid X)$ is the posterior probability (a revised estimate of the probability of $A$, given $X$, in our example, of there being cancer, given that the test result was positive);
$p(X \mid A)$ is the conditional probability of $X$, given $A$ (in our example, of a positive test when a patient has cancer); $p(X \mid \sim A)$ is the conditional probability of $X$, given not- $A$ (in our example, of a positive test when a patient does not have cancer).

$\chi^{2}$ (chi-squared or ‘chi-square’ – statisticians are not agreed) is a statistical test based on a comparison between a test statistic and a critical value from a chi-squared distribution. A chi-squared variable can be regarded as the sum of a number of squared independent normal variables, each with zero mean and unit variance. The number of such squared terms is the number of degrees of freedom of the $\chi^{2}$ distribution. A chi-squared test can be used to test the null hypothesis that two or more population distributions do not differ. When comparing observed values with those expected under the null hypothesis, it is the sum of the ratio of the squared differences between observed $(O)$ and expected $(E)$ values to the expected value:
$$\chi^{2}=\sum \frac{\left[O_{i}-E_{i}\right]^{2}}{E_{i}}$$
There are two well-known versions, the Pearson $\chi^{2}$ test and the MantelHaenszel test. See Statistical Significance.