BRM Chapter 24 - Two-Way Tables and the Chi-Square Test

Two-way table - definition

A table that classifies individuals according to two categorical variables, with rows for one variable's categories and columns for the other's categories.​

Row variable - definition

The categorical variable whose categories form the rows of a two-way table (for example, graduation status: graduated vs did not graduate).​

Column variable - definition

The categorical variable whose categories form the columns of a two-way table (for example, race/ethnicity).​

Marginal totals - definition

The "Total" row and "Total" column that show the overall distributions of each variable separately, combining over the other variable's categories.​

Using percents in two-way tables

It is often clearer to convert cell counts to percentages when comparing groups so that patterns in the relationship between variables are easier to see.​

Describing relationships in two-way tables

To describe an association, compute relevant percents (such as the percent graduating within each race) and compare them across rows or columns.​

Graduation and race example - pattern

In the graduation-by-race table, over 60% of white students and more than 70% of Asian students graduated in 6 years, but less than 40% of Black and American Indian/Alaska Native students did.​

Question of inference for two-way tables

When a sample table shows an association, we ask whether this reflects a real association in the population or could be due to random sampling variation.​

Cocaine treatment example - setup

In a randomized study, 72 cocaine addicts were assigned equally to three treatments (desipramine, lithium, placebo), and success was defined as not using cocaine.​

Cocaine treatment example - observed pattern

The proportion of subjects who did not use cocaine was much higher in the desipramine group than in the lithium or placebo groups.​

Null hypothesis for a two-way table

The null hypothesis states there is no association between the row and column variables; any differences in sample counts are due to chance alone.​

Null hypothesis - cocaine study

H0: There is no association between the treatment an addict receives and whether or not there is success in not using cocaine in the population of all cocaine addicts.​

Alternative hypothesis for a two-way table

The alternative hypothesis states there is an association between the row and column variables; the distribution of one variable differs across levels of the other.​

Alternative hypothesis - cocaine study

Ha: There is an association between the treatment an addict receives and whether or not there is success in not using cocaine in the population of all cocaine addicts.​

Expected counts - definition

The counts we would expect in each cell of the two-way table if the null hypothesis of no association were true, allowing for random variation.​

Expected counts - equal-group cocaine example

If the overall success rate is 24/72 = 1/3 and each treatment group has 24 subjects, we expect 8 successes and 16 failures in each treatment group under H0.​

General idea of the chi-square test

To test H0, compare observed cell counts with expected counts; large overall discrepancies provide evidence against "no association."​

Chi-square statistic - concept

A single number that measures how far the observed counts in all cells are from their expected counts, combining squared differences over all cells.​

Chi-square distribution - definition

The sampling distribution of the chi-square statistic when H0 is true; it takes only nonnegative values and is skewed to the right.​

Degrees of freedom for chi-square

For a two-way table with r rows and c columns, the chi-square test uses a chi-square distribution with (r − 1)(c − 1) degrees of freedom.​

Degrees of freedom - cocaine example

The cocaine table has 3 treatments and 2 outcomes, so df = (3 − 1)(2 − 1) = 2.​

Using chi-square critical values

Tables give critical values showing how large the chi-square statistic must be (for a given df) to be significant at levels such as 0.05 or 0.01.​

Cocaine study - chi-square result

With df = 2 and χ² = 10.5, the statistic exceeds the 0.01 critical value (9.21), so the association between treatment and success is significant at P < 0.01.​

Interpreting significant chi-square

The test shows strong evidence of some association; to see the nature of the relationship, look back at the table (desipramine performs better than the other treatments).​

Conditions for using the chi-square test

You can safely use the chi-square test when no more than 20% of expected counts are less than 5 and all expected counts are at least 1.​

Chi-square test - what it tells us

The chi-square test tells whether an observed association is statistically significant, not whether it is large or practically important.​

Simpson's paradox - idea

An association that holds within each of several groups can disappear or reverse when the data from all groups are combined into a single table.​

Medical helicopter example - overall pattern

Overall, 32% of helicopter patients died versus 24% of road-transport patients, suggesting helicopters are worse when seriousness of accidents is ignored.​

Medical helicopter example - within groups

When data are broken down by seriousness of accident, the death rate is lower for helicopter patients in both serious and non-serious accidents.​

Lurking variable in helicopter example

Seriousness of the accident is a lurking variable; helicopters are used more often for serious accidents, so combining all patients without this variable is misleading.​

Simpson's paradox - definition

When an association or comparison that holds within each of several groups reverses or disappears when the groups are combined, this is called Simpson's paradox.​

Lurking variables and categorical data

As with quantitative data, lurking variables can change or reverse observed associations between categorical variables in two-way tables.​

Statistics in summary - two-way tables

Categorical variables group individuals into classes; to display the relationship between two categorical variables, use a two-way table and compare appropriate percentages.​

Statistics in summary - Simpson's paradox

Lurking variables can make an observed association misleading; Simpson's paradox is an extreme case where combining groups reverses the association.​

Statistics in summary - chi-square test

The chi-square test compares observed and expected counts in a two-way table and uses the chi-square distribution to decide whether an observed association is statistically significant.