Chi-square Tests: Applications and Interpretation in Epidemiology

Chi-square Tests

Introduction to Chi-square Tests

Chi-square tests are a family of statistical tests used to determine whether observed frequencies differ significantly from expected frequencies under a specific hypothesis. They are widely used in categorical data analysis, especially in biological and epidemiological studies.

• Key Purpose: To assess whether differences between observed and expected counts are due to chance or indicate a meaningful association.

• Common Applications: Goodness-of-fit, test of independence, and test of homogeneity.

West Nile Virus Case Study

Background and Research Questions

West Nile Virus (WNV) is a mosquito-borne virus. Most infected individuals are asymptomatic or have mild symptoms, but some develop severe neurological disease. Epidemiological questions include:

• Are all age groups equally likely to be diagnosed with WNV?

• Are all age groups equally likely to die from WNV?

Chi-square Goodness-of-Fit Test

Comparing Observed and Expected Distributions

The goodness-of-fit test compares the observed distribution of a categorical variable to an expected distribution based on a known population or theoretical model.

• Null Hypothesis (H0): The observed frequencies match the expected frequencies (e.g., age distribution among WNV cases matches the general population).

• Expected Frequency: For each category, calculated as:

$\text{Expected frequency} = \text{Row probability} \times \text{Grand total}$

Example Table: Age Distribution of WNV Cases vs. Census Data

Age Group	Cases	Relative Frequency	Census Data 2010*	Expected Frequency
0 - 44	240	0.31	0.727	0.727 × 773 = 562.0
45 - 64	335	0.43	0.187	0.187 × 773 = 144.5
65 - 100	198	0.26	0.086	0.086 × 773 = 66.5
Unknown	6	-	-	-
Total (valid)	773	1	1	773

Calculating the Chi-square Statistic

• For each category, calculate:

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

• Oi: Observed frequency in category i

• Ei: Expected frequency in category i

Example Calculation Table

Age Group	Observed	Expected	Obs - Exp	Std. Residuals (z-scores)	Component to $\chi^2$
0 - 44	240	562.0	-322	-13.6	185
45 - 64	335	144.5	190.5	15.8	249.6
65 - 100	198	66.5	131.5	16.1	259.2
Total	773	773	0	-	693.8

• Degrees of Freedom: $df = \text{number of categories} - 1$

• In this example: $df = 3 - 1 = 2$

Interpreting the Results

• If the calculated $\chi^2$ statistic is much larger than expected under the null hypothesis (with a small p-value), we reject the null hypothesis.

• Example: $\chi^2 = 693.8$, $df = 2$, $p$-value < 0.001 indicates a significant difference between observed and expected age distributions among WNV cases.

• Standardized Residuals: Values above 2 or below -2 indicate categories that are over- or under-represented, respectively.

Summary of Findings

• Seniors (65+) and middle-aged (45-64) are significantly over-represented among WNV cases.

• Younger individuals (0-44) are significantly under-represented.

Types of Chi-square Tests

Overview and Comparison

• Goodness-of-Fit Test: Compares observed frequencies to expected frequencies from a known distribution (one categorical variable).

• Test of Independence: Assesses whether two categorical variables are independent in a contingency table.

• Test of Homogeneity: Compares distributions of a categorical variable across different populations.

Comparison Table

Test Type	Purpose	Data Structure	Degrees of Freedom
Goodness-of-Fit	Compare observed to expected frequencies (one variable)	One categorical variable	# categories - 1
Test of Independence	Test association between two variables	Contingency table (two variables)	(# rows - 1) × (# columns - 1)
Test of Homogeneity	Compare distributions across groups	Contingency table (two variables)	(# rows - 1) × (# columns - 1)

Chi-square Test Calculations

Step-by-Step Procedure

1. State the null and alternative hypotheses.

2. Calculate expected frequencies for each cell:

$\text{Expected frequency} = \text{Row total} \times \text{Column total} / \text{Grand total}$

3. Compute the chi-square statistic:

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

4. Determine degrees of freedom:

$df = (\text{number of rows} - 1) \times (\text{number of columns} - 1)$

5. Compare the calculated statistic to the critical value or use the p-value to draw a conclusion.

Properties of the Chi-square Distribution

• Right-skewed distribution; shape depends on degrees of freedom.

• Always an upper-tail test (chi-square statistic cannot be negative).

• Larger statistics indicate greater deviation from the null hypothesis.

Interpretation and Application

Key Points for Interpretation

• Standardized residuals (z-scores) help identify which categories contribute most to the chi-square statistic.

• Significant results indicate that observed frequencies differ from expected frequencies more than would be expected by chance.

• Always check assumptions: expected frequencies should generally be at least 5 in each cell.

Example Application: WNV and Age

• Observed: Seniors and middle-aged groups are over-represented among WNV cases; young people are under-represented.

• Interpretation: There is a statistically significant association between age and WNV diagnosis.

Additional info:

• Chi-square tests are non-parametric and do not require normality.

• They are sensitive to sample size; large samples can detect small differences as significant.