Introduction to Test of Independence

Independence means that the probability of any event involving both variables can be obtained by multiplying their individual probabilities. In a test of independence, you determine whether two variables in a contingency table are independent or dependent. For example, alcohol consumption and road accident fatality are independent if knowing one doesn't affect the probability of the other.

A contingency table arranges data in rows and columns representing two categorical variables. For instance, rows might represent sobriety or intoxication, while columns represent fatality or nonfatality of road accidents. The table contains observed frequencies from randomly selected samples, organized in a two-way format to display the relationship between the two variables being tested.

The expected frequency for each cell in a contingency table must be at least 5. This requirement ensures the validity of the chi-square test statistic. Expected frequencies are calculated using a formula involving the row totals, column totals, and overall sample size to determine what frequencies would occur if the variables were truly independent.

The chi-square test statistic is calculated by comparing observed and expected frequencies using a formula that sums the squared differences between them, divided by expected values. The test statistic measures how far apart observed and expected values are. If they differ greatly, the test statistic becomes large and falls in the right tail of the chi-square distribution.

Degrees of freedom determine which chi-square distribution to use when finding critical values and P-values. For a test of independence, degrees of freedom are calculated as (r - 1) × (c - 1), where r is the number of rows and c is the number of columns in the contingency table. This value is essential for interpreting the test results accurately.

A test of independence is always right-tailed because the chi-square test statistic is always positive and increases when observed and expected values diverge. Large test statistics fall in the right tail of the chi-square curve, indicating strong evidence against independence. The test evaluates whether the statistic exceeds a critical value in this right tail region.

The final step is performing a hypothesis test to determine whether the two variables are independent. This involves comparing the calculated chi-square test statistic to the critical value or examining the P-value. If the test statistic exceeds the critical value or the P-value is below the significance level, you reject independence and conclude the variables are dependent.