Unit 8: Inference for Categorical Data: Chi-Square

Categorical data

Data in which each observation falls into a label or group (e.g., party, brand, color, yes/no) rather than a numerical measurement.

Categorical inference

Using sample counts/proportions to judge whether an observed categorical pattern is plausibly due to chance variation under a proposed model.

Counts

The number of observations in each category/cell; the primary summary measure used in chi-square procedures.

Proportions

Counts divided by the total sample size; used to describe categorical distributions but chi-square calculations require counts.

Observed count (O)

The actual sample count in a category (one-way table) or cell (two-way table).

Expected count (E)

The count predicted in a category/cell if the null hypothesis model were true.

Null hypothesis (H0) in chi-square

A “blueprint” model specifying how counts should fall into categories in the long run (distribution, equal distributions across groups, or independence).

Alternative hypothesis (Ha) in chi-square

The claim that the observed categorical pattern does not match the null model (at least one proportion differs / distributions differ / variables are associated).

Goodness-of-fit test

A chi-square test that checks whether one categorical variable follows a claimed distribution of proportions in a population.

Test for homogeneity

A chi-square test that compares the distribution of one categorical response variable across two or more populations/treatments (based on separate samples or random assignment).

Test for independence

A chi-square test that evaluates whether two categorical variables are associated within one population (based on one sample measuring both variables).

One-way table

A table of counts for a single categorical variable across k categories (used in goodness-of-fit).

Two-way table

A table of counts for two categorical variables arranged in r rows and c columns (used in homogeneity or independence).

Margins (row/column totals)

The totals for each row and each column in a two-way table, used to compute expected counts.

Grand total (n)

The total number of observations in the table; used in expected-count formulas.

Chi-square distribution

A family of right-skewed distributions on nonnegative values used to model chi-square test statistics when $H_0$ is true (approximately).

Right-skewed

A distribution shape with a long tail to the right; chi-square distributions are always right-skewed (less skewed as df increases).

Degrees of freedom (df)

Using a chi-square distribution to approximate the sampling distribution of $\chi^2$ under $H_0$; works better when expected counts are sufficiently large.

df for goodness-of-fit

For $k$ categories in a one-way table, $df = k - 1.$

df for two-way tables

For an $r \times c$ table, $df = (r - 1)(c - 1).$

Chi-square test statistic (χ²)

A measure of overall mismatch between observed and expected counts: $\chi^2 = \sum (O - E)^2/E.$

Cell contribution to χ²

The amount a single category/cell adds to $\chi^2$: $(O - E)^2/E.$

Standardized residual

A directional diagnostic for a cell: $(O - E)/\sqrt{E}$; positive means observed $>$ expected, negative means observed $<$ expected.

Pearson residual

Another name often used for standardized residuals in chi-square output: (O − E)/√E.

Right-tailed test (chi-square)

Chi-square p-values come from the right tail because only large χ² values indicate strong evidence against H0.

P-value (chi-square)

The probability, assuming $H_0$ is true, of getting a $\chi^2$ statistic at least as large as the one computed.

Significance level (α)

The cutoff probability used to decide whether to reject $H_0$ (e.g., 0.05).

Reject H0

Decision made when $p\text{-value} < \alpha$; conclude the data provide evidence against the null categorical model.

Fail to reject H0

Decision made when p-value $> \alpha$; conclude there is not convincing evidence against the null model (not proof $H_0$ is true).

Statistical significance

A result is statistically significant when the p-value is small enough to reject H0, indicating the pattern is unlikely under the null model.

Practical significance

Whether the size/impact of the observed differences matters in context; can differ from statistical significance, especially with large samples.

Chi-square approximation

Using a chi-square distribution to approximate the sampling distribution of χ² under H0; works better when expected counts are sufficiently large.

Large expected counts condition

A requirement for reliable chi-square inference; common AP rule of thumb: all expected counts are at least 5.

Random condition

A chi-square inference condition requiring data from a random sample (or random assignment in an experiment) to support broader inference.

Independence condition

A chi-square inference condition requiring observations to be independent; often supported by sampling design and the 10% condition.

10% condition

When sampling without replacement, the sample size should be less than 10% of the population to help justify independence.

One person, one cell rule

Each individual should contribute to exactly one cell in the table; violations (e.g., repeated measures) break independence and can mislead p-values.

Pooling (combining categories)

Combining categories to increase expected counts when some are too small; must be logically defensible and requires recomputing E and df.

Sparse table

A two-way table with many small expected counts (often from too many categories), which can make chi-square approximations unreliable.

Follow-up analysis (after significant χ²)

Additional work (e.g., residuals or conditional proportions) to determine which cells/categories drive the significant result.

Conditional distribution

Proportions of one variable within each category of another variable (e.g., sleep distribution within each screen-time group) used to describe associations.

Association (in chi-square independence)

A relationship where the distribution of one categorical variable differs across levels of another; evidence occurs when independence is rejected.

Independence (probability statement)

A and B are independent if $P(A \cap B) = P(A)P(B);$ in tables, it implies similar conditional distributions.

Expected count in goodness-of-fit

For category $i$ with claimed proportion $p_i$ in sample size $n,$ $E_i = n \cdot p_i.$

Expected count in two-way tables

For cell (i,j), $E_{ij} = \frac{(\text{row total}_i \times \text{column total}_j)}{n}$ under $H_0$ for homogeneity or independence.

Critical-value approach

A method that compares the computed $\chi^2$ statistic to a cutoff from the chi-square distribution (with given $df$) instead of relying only on a p-value.

Scope of inference

What conclusions are justified based on design: random sampling supports generalization; random assignment supports causation.

Random assignment

Assigning individuals to treatments by chance; allows causal conclusions about how treatment affects a categorical response distribution.

Observational study

A study with no random assignment; can show association but does not justify causal conclusions.

AP-style conclusion

A conclusion that states reject/fail to reject, references p-value vs α, describes the claim in context (variables/population), and respects scope (no unwarranted causation).