Lecture 9: Contingency Analysis

What is a chi-squared test contingency test used for?

A chi-squared contingency test is used to determine whether an association exists between two or more categorical variables.

What are the steps involved in conducting a chi-squared contingency test?

1) Set up your hypotheses:

• Null hypothesis (H₀): The events of one categorical variable are independent of the other.

• Alternative hypothesis (H_A): The events of one categorical variable are dependent on the other.

2) Calculate expected counts: (Row total × Column total) / Table total

3) Check the assumptions:

• No expected frequency < 1.

• ≤ 20% of categories have expected frequencies < 5.

If assumptions are met, proceed with the test. If not, combine categories or consider an alternative test.

4) Calculate the test statistic.

5) Look up the critical value:

For a contingency table with r rows and c columns: df = (r − 1) (c − 1)

Use df and significance level (α) to find the critical value in the chi-squared distribution table.

6) Apply the decision rule:

• If χ²_observed ≥ χ²_critical → Reject H₀

• If χ²_observed < χ²_critical → Fail to reject H₀

What is Fisher’s exact test used for?

Fisher’s exact test is used to test for an association between two categorical variables when the assumptions of the chi-squared test of independence are not met and the data are arranged in a 2 × 2 contingency table.

What is an odds ratio (OR), and how is it calculated?

An odds ratio (OR) compares the odds of a focal outcome between two groups (treatment versus control).

Odds are the ratio of successes to failures, where success is the focal outcome.

Formula:

Interpretation:

• OR = 1: No difference in odds between groups.

• OR > 1: Higher odds of the focal outcome in the treatment group.

• OR < 1: Lower odds of the focal outcome in the treatment group.

What is relative risk (RR), and how is it calculated?

Relative risk (RR) compares the probability (risk) of a focal outcome between two groups (treatment versus control).

Interpretation:

• RR = 1: No difference in risk between groups.

• RR > 1: Higher risk of the focal outcome in the treatment group.

• RR < 1: Lower risk of the focal outcome in the treatment group.

When should you use relative risk versus odds ratio?

• Relative risk (RR): Use when you can calculate the probability of an outcome in each exposure group (e.g., treatment versus control).

• Odds ratio (OR): Use when you cannot directly calculate the probability of an outcome, particularly when participants are selected based on their outcome (e.g., diseased versus healthy).