6 Chi Square and proportion test

[Supplement] What question does a chi-square goodness-of-fit test answer?

Whether categorical counts match a specified set of probabilities.

[Supplement] What expected count is used for category j in a goodness-of-fit test?

E_j = n p_j.

[Supplement] What is the Pearson chi-square statistic?

chi² = sum_j (O_j - E_j)² / E_j.

[Supplement] What are the goodness-of-fit degrees of freedom for k categories with fully specified probabilities?

k - 1, if no parameters are estimated from the same data.

[Supplement] Why can fitting null-distribution parameters affect goodness-of-fit degrees of freedom?

Estimating parameters uses information from the sample, potentially reducing the appropriate reference degrees of freedom.

[Supplement] What question does a chi-square independence test answer?

Whether two categorical variables are statistically independent in the target population.

[Supplement] How is a contingency-table expected count calculated under independence?

E_ij = row_total_i times column_total_j / grand_total.

[Supplement] What are the independence-test degrees of freedom for an r by c table?

(r - 1)(c - 1).

[Supplement] Should a chi-square count test be applied directly to percentages without sample sizes?

No. The test needs counts and their sampling information.

[Supplement] Why do small expected counts matter in chi-square testing?

They can make the asymptotic chi-square approximation unreliable.

[Supplement] What can be considered when a contingency table is sparse?

An exact test such as Fisher's test when suitable, or an appropriate simulation-based method.

[Supplement] Does a significant chi-square independence test establish causality?

No.

[Supplement] What is the R call for a basic contingency-table chi-square test?

chisq.test(table_object).

[Supplement] Why may an R chi-square statistic differ from an uncorrected hand calculation in a 2 by 2 table?

R may apply a continuity correction; check the function's options and reported method.

[Supplement] For counts (20,30,50) under three equal probabilities, what is the chi-square statistic?

Each expected count is 100/3; chi² = 14, with 2 degrees of freedom.

[Supplement] What is a one-sample proportion test used for?

Testing whether a population binary-event probability equals p0.

[Supplement] What is the sample proportion?

p_hat = x/n, where x is the number of successes.

[Supplement] What is the uncorrected large-sample one-proportion z statistic?

z = (p_hat - p0) / sqrt(p0(1-p0)/n).

[Supplement] Which proportion enters the null standard error in that test?

The null proportion p0, rather than p_hat.

[Supplement] What is the pooled proportion for a two-sample test of equal proportions?

p_pool = (x1+x2)/(n1+n2).

[Supplement] What is the uncorrected large-sample two-proportion z statistic under equal proportions?

z = (p1_hat-p2_hat) / sqrt(p_pool(1-p_pool)(1/n1+1/n2)).

[Supplement] Does a confidence interval for a proportion difference always use the same pooled standard error as the equality test?

No. Interval and test constructions can differ; identify the method used.

[Supplement] What assumptions support the usual independent-proportion tests?

Appropriate independent binary observations and sufficient expected successes and failures for the approximation.

[Supplement] Is the usual independent two-proportion test suitable for paired before/after binary outcomes?

No. Pairing requires a method that accounts for dependence.

[Supplement] What does prop.test() do in R?

Tests one or more proportions using an approximate method; continuity correction can affect results.

[Supplement] What does binom.test() provide for one binomial sample?

An exact binomial test and an associated confidence interval.

[Supplement] Does rejecting p = 0.5 reveal why the probability differs?

No. The test concerns the probability, not its causal explanation.