Comprehensive Guide to Chi-Square Inference

Chi-Square ($\chi^2$) tests

Analyze variables with multiple categories and measure deviations of observed data from expected data.

Formula for Chi-Square statistic

$\chi^2 = \sum \frac{(Observed - Expected)^2}{Expected}$

Right-Skewed

The characteristic shape of the Chi-Square distribution, more skewed with fewer degrees of freedom.

Positive Values Only

Chi-Square values are always non-negative since deviations are squared.

Goodness of Fit Test

Determines if the distribution of a single categorical variable matches a claimed population distribution.

Conditions of Goodness of Fit

Random sample, 10% Condition, and all expected counts must be at least 5.

Hypotheses in Goodness of Fit Test

$H0$: distribution matches; $Ha$: distribution does not match.

Degrees of Freedom (df) in Chi-Square

For Goodness of Fit: $df = number\ of\ categories - 1$.

Expected Counts for Fair Die

If fair, each face of a die would have an expected count of 10 when rolled 60 times.

Test for Homogeneity

Compares the distribution of a categorical variable across two or more populations.

Degrees of Freedom in Homogeneity Test

$df = (number\ of\ rows - 1) \times (number\ of\ columns - 1)$.

Expected Count Formula for Two-Way Tables

$Expected\ Count = \frac{(Row\ Total) \times (Column\ Total)}{Table\ Total}$.

Test for Independence

Determines if there is an association between two categorical variables within a single population.

Distinctive Feature of Independence Test

Involves one single sample cross-classified by two variables.

Hypotheses for Independence Test

$H0$: no association; $Ha$: association exists.

Quick Check Routine for Chi-Square tests

Examine total 'N' and the phrasing of the hypotheses.

Common Mistake: Large Counts

Error in checking observed counts instead of expected counts.

Proportion vs. Counts

Chi-Square tests must be performed on counts, not percentages.

Correct Phrasing for Hypotheses

Hypotheses should be in words rather than symbols for Homogeneity/Independence.

Rejecting the Null Hypothesis

We say we do not have sufficient evidence to conclude distributions are different.

Contributions to Chi-Square Sum

Identify cells with the largest observed-expected deviations when rejecting $H_0$.

Test outcome for P-value

If P-value < $\alpha$, we reject $H_0$.

Chi-Square Curves and Degrees of Freedom

As degrees of freedom increase, the Chi-Square distribution becomes more symmetric.

Expected Count Condition

All expected counts in a test must be at least 5 to be valid.

Study Design: Homogeneity vs. Independence

The difference lies in how the data was collected—multiple samples for Homogeneity, one sample for Independence.

Chi-Square Distribution Characteristics

The distribution is positively skewed and only takes positive values.

Significance of Deviations

Any significant deviation from expected counts increases the Chi-Square statistic.