Chi Squared Tests (AS)

Chi-Squared Tests fall under the branch of hypotenesis testing in statistics, primarily aimed at determining if there is a significant association between two categorical variables in a sample.
These tests are based on the chi-squared distribution, which is a theoretical probability distribution that is particularly important because of its relationship to the standard normal distribution.
The null hypothesis in Chi-Squared Tests assumes that there is no relationship or dependence between the categories or variables being tested.
The alternative hypothesis assumes that there is a relationship or dependence between the variables being tested.
The Chi-Squared test statistic is computed based on observed and expected frequencies. Observed frequency refers to actual data collected and expected frequency is what we would expect if the null hypothesis were true.
In order to conduct a Chi-Squared Test, you must calculate the expected frequencies. This is usually done by multiplying the row total for a category by the column total for a category and then dividing by the overall total.
After getting the observed and expected frequencies, the Chi-Squared test statistic is calculated. The formula used is: Σ [ (Observed frequency - Expected frequency)^2 / Expected frequency ]. The summation is carried out for all classes or categories.
The calculated test statistic is compared to the critical value from the Chi-Squared distribution table. The degrees of freedom for the test are equal to the number of categories minus one.
If the calculated test statistic is greater than the critical value, we reject the null hypothesis.
Assumptions with using a Chi-Squared Test include: Variables are categorical (ordinal/nominal), random sampling from the population and sample size is sufficiently large (rule of thumb: all expected frequencies should be equal to or greater than 5).
The Chi-Squared Test is a powerful statistical test, but like any test, it is not foolproof and has limitations. A significant test result does not imply a large effect size or practical importance. Also, the test is sensitive to sample size - large samples may lead to significant results even if the observed difference is not practically important.
It is important to provide context to the Chi-Squared Test results by calculating the Effect Size. Common measures of association include Cramer’s V and Phi Coefficient.
In Chi-Squared Tests, the strength and direction of the relation between variables is measured using the Contingency Coefficient C and Cramer’s V.
Lastly, it is crucial to remember that Chi-Squared Tests can only be used for data that is frequency or count data. For measurement data (interval/ratio), other statistical techniques are needed.