Chi Squared Test For Trend

TL;DR

• Tests whether observed counts in a contingency table differ from counts expected under no relationship between two categorical variables.
• Involves computing expected frequencies, summing (observed − expected)^2/expected across cells, and comparing the result to a Chi-squared critical value.
• Commonly applied in fields such as sociology and psychology.

Definition

The Chi-squared test for trend is a statistical test used to determine if there is a significant relationship between two variables by comparing observed counts in a contingency table to expected counts under the assumption of no relationship.

Explanation

Steps to perform the Chi-squared test for trend, as described in the source:

1. Set up a contingency table showing the number of observations in each category for both variables.
2. Calculate expected frequencies for each cell using the formula:

$$E = \frac{\text{Row Total} \times \text{Column Total}}{\text{Table Total}}$$

3. Compute the Chi-squared statistic by summing, over all cells, the squared differences between observed and expected frequencies divided by the expected frequency:

$$\chi^2 = \sum \frac{(O - E)^2}{E}$$

4. Determine the degrees of freedom:

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

5. Compare the calculated Chi-squared statistic to the critical value from a Chi-squared table for the chosen significance level and the calculated degrees of freedom.
   • If the statistic is greater than the critical value, conclude there is a significant relationship between the variables.
   • If the statistic is less than the critical value, do not conclude a significant relationship; observed differences may be due to chance.

Examples

Example contingency table (relationship between gender and political party affiliation):

Gender	Republican	Democrat	Total
Male	20	10	30
Female	15	25	40
Total	35	35	70

Example calculations from the table:

• Expected frequency for male Democrats:

$$E = \frac{30 \times 35}{70} = 15.5$$

• Chi-squared contribution for male Democrats:

$$\frac{(20 - 15.5)^2}{15.5} = 1.29$$

Repeat the same process for each cell and sum the results to obtain the final Chi-squared statistic. Degrees of freedom for this example:

$$(2 - 1) \times (2 - 1) = 1$$

Use cases

• Applied in sociology and psychology to examine relationships between two categorical variables.

Notes or pitfalls

• The test result is interpreted by comparing the calculated Chi-squared statistic to the critical value from a Chi-squared table at the chosen significance level and appropriate degrees of freedom.
• A statistic greater than the critical value indicates a significant relationship; a statistic less than the critical value indicates the observed differences may be due to chance.

Related terms

• Contingency table
• Degrees of freedom