Chapter 9: Categorical data analysis

Categorical data analysis

Collection of tools used for nominal scale data.

Chi-square goodness of fit model

Tests if observed frequency distribution matches expected distribution.

Goodness-of-fit test

Determines if observed frequencies match expected frequencies.

Null hypothesis

Vector of equal probabilities for all categories.

Alternative hypothesis

Demonstrates that probabilities are not all identical.

Goodness-of-fit test statistic

Calculated statistic using observed and expected frequencies.

Expected frequencies

Sum of probabilities under the null hypothesis.

Sampling distribution of GOF statistic

Distribution of test statistic if null hypothesis is true.

Degrees of freedom

Count of independent quantities minus constraints.

Testing the null hypothesis

Determining the reject region based on calculated statistic.

X2 test of independence

Test for association between categorical variables.

Continuity correction

Adjustment for one degree of freedom tests.

Systematic problems

Issues with GOF statistic when N is small or df=1.

Assumptions of the test

Normality, independence, and sufficient expected frequencies.

One-sample z-test

Test for population mean using known standard deviation.

Null and alternative hypothesis

Comparing sample mean to null hypothesis.

Z score

Number of standard errors separating sample mean from predicted population mean.

Assumptions of the z-test

Normality, independence, and known standard deviation.

One-sample t-test

Test for population mean without known standard deviation.

T-distribution

Similar to normal distribution but with heavier tails.

Independent samples t-test

Compares means of two groups.

Pooled estimate of standard deviation

Weighted average of variance estimates.

Assumptions of the independent samples t-test

Normality, independence, and homogeneity of variance.

Paired samples t-test

Compares means of two groups with repeated measures.

Effect size

Measured using Cohen's d.

Checking normality of a sample

Using QQ plot or Shapiro-Wilk test.

Wilcoxon test

Nonparametric test for non-normal data.

Two-sample Mann-Whitney U test

Nonparametric test for comparing two groups.

One-sample Wilcoxon test

Nonparametric test for paired samples.

One-way ANOVA

Investigates differences in means.

Null and alternative hypothesis

Comparing means using variances.

From sum of squares to F-test

Calculating F-ratio from sum of squares.

Effect size

Measured using eta squared.

Multiple comparisons and post hoc tests

Correcting for multiple testing.

Bonferroni correction

Adjusting p-values for multiple comparisons.

Holm corrections

Sequential adjustment of p-values.

Assumptions of one-way ANOVA

Homogeneity of variance, normality, and independence.

Checking homogeneity of variance assumption

Using Levene or Brown-Forsythe test.

Kruskal-Wallis rank sum test

Nonparametric test for three or more groups.