Chi-Square Calculation Sequence

To accurately assess the association between two categorical variables using the Chi-square test, follow these sequential steps:

1. Contingency Table: Tabulate Data (D)

The initial step involves organizing your raw data into a contingency table. This table displays the observed frequencies for each combination of categories of the two variables. For example, if studying the association between smoking habits (Smoker, Non-smoker) and lung disease (Yes, No), the contingency table would show counts for each of these four combinations.

2. Expected Frequencies Calculation (B)

Next, you need to calculate the expected cell frequencies for each cell in the contingency table. These are the frequencies you would expect if there were no association (i.e., if the null hypothesis of independence were true). The formula used is:

$$E_{ij} = \frac{(\text{Row } i \text{ Total}) \times (\text{Column } j \text{ Total})}{\text{Grand Total}}$$

Where '$E_{ij}$' is the expected frequency for the cell in row '$i$' and column '$j$', and '$(\text{Row } i \text{ Total})$', '$(\text{Column } j \text{ Total})$', and '$(\text{Grand Total})$' are the respective totals.

3. Chi-Square Statistic Calculation (A)

With both observed ('$O_{ij}$') and expected ('$E_{ij}$') frequencies calculated, the next step is to compute the Chi-square ($\chi^2$) statistic. This statistic quantifies the discrepancy between the observed data and what is expected under the null hypothesis. The formula is:

$$\chi^2 = \sum_{i} \sum_{j} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$$

The summation is performed over all cells in the contingency table.

4. Degrees of Freedom Assessment (C)

Before interpreting the Chi-square statistic, you must determine the degrees of freedom ('df'). This value is crucial for finding the critical value from the Chi-square distribution table. For a contingency table, the degrees of freedom are calculated as:

$$\text{df} = (r - 1)(c - 1)$$

Where '$r$' is the number of rows and '$c$' is the number of columns in the contingency table.

5. Decision Making: Compare Values (E)

The final step is to compare the calculated $\chi^2$ statistic with a critical value obtained from the Chi-square distribution table, using the previously determined degrees of freedom and a chosen significance level (e.g., $\alpha = 0.05$).

• If the calculated $\chi^2$ value is greater than the critical value, you reject the null hypothesis, suggesting a significant association between the variables.
• If the calculated $\chi^2$ value is less than or equal to the critical value, you fail to reject the null hypothesis, indicating no significant association.

This comparison allows you to make a statistical decision about the relationship between the two categorical variables.