Chi-Square Test

The Chi-square test is a non-parametric statistical procedure used to analyze unordered categorical or nominal data.
It operates by comparing the actual observed frequencies of an event against the theoretical expected frequencies, assuming the null hypothesis is true.
The test evaluates the deviation between these observed and expected counts to determine if the difference is statistically significant or simply due to random chance.
It is exclusively a test of significance and provides no information regarding the strength of association or the magnitude of the effect size.

Test of Independence (Association): Used to assess whether two categorical variables (e.g., a specific exposure and a clinical outcome) are related or completely independent of each other.
Goodness-of-Fit Test: Assesses the degree of agreement between an empirically observed frequency distribution and a theoretically predicted frequency distribution.
Test for Trend: Applied to a contingency table where one variable has two categories and the other has multiple ordered (ordinal) categories to assess differences in the trend of proportions.

Data Format: Data must be presented as raw counts or frequencies, not as percentages or proportions.
Independence of Observations: Each subject must contribute data to only one cell in the contingency table, meaning there are no repeated measurements on the same individual.
Minimum Expected Frequencies (Rule of 5): For the test to be mathematically valid, at least 80% of the expected cell frequencies must be greater than 5, and no expected frequency should be less than 1.
If the sample size is small (e.g., less than 40) or the expected frequency rule is violated, Fisher's exact test should be used instead of the Chi-square test.
Yates' continuity correction is often applied in 2x2 tables to improve the approximation of the discrete probability of observed frequencies to the continuous Chi-square distribution, thereby reducing the risk of a Type I false-positive error.

Expected Frequency: For any given cell in a contingency table, this is calculated by multiplying the respective row total by the column total, and dividing the product by the overall total sample size.
Test Statistic Formula: The statistic is calculated as the sum of all cells using the formula $\sum \frac{(O - E)^{2}}{E}$ , where O is the observed frequency and E is the expected frequency.
Degrees of Freedom (df): Calculated as $(R o w s - 1) \times (C o l u m n s - 1)$ . For a standard 2x2 contingency table, the degrees of freedom is 1.

Clinical Setting: A researcher wishes to investigate if there is an association between passive smoking (the exposure) and coronary death (the outcome).
Hypotheses:
- The null hypothesis ( $H_{0}$ ) states that the row variables and column variables are independent, meaning passive smoking status has no bearing on the risk of coronary death.
- The alternative hypothesis ( $H_{1}$ ) states that there is a significant association between passive smoking and coronary death.
Contingency Table Setup: A cohort of 250 patients is surveyed and divided based on their exposure and outcome into a 2x2 contingency table.

Patient Status	Coronary Death (Positive Outcome)	No Coronary Death (Negative Outcome)	Total
Passive Smoking (Exposure Positive)	50	100	150
No Passive Smoking (Exposure Negative)	20	80	100
Total	70	180	250

Interpretation and Analysis: The expected frequencies are calculated for each cell; for example, the expected deaths in passive smokers would be $(150 \times 70) / 250 = 42$ .
The Chi-square statistic is computed by summing the squared differences between the observed and expected frequencies, divided by the expected frequencies across all four cells.
This calculated test statistic is then compared against the critical value from the standard Chi-square distribution table for 1 degree of freedom.
If the computed statistic is larger than the critical value (yielding a p-value < 0.05), the null hypothesis is rejected, leading to the clinical conclusion that there is a statistically significant association between passive smoking and coronary death.