Chi-Square

when

both variables are categorical (nominal or ordinal) and want to know whether the proportions differ between groups (i.e. is one variable contingent (dependent) on another)

null

the two variables are independent, i.e. proportions are the same in every group (observed = expected)

alternative

the two variables are associated, i.e. proportions differ per group (observed ≠ expected)

contingency table

table that counts how many people fall into each combination of (two) categorical variables

cells

count for each combination (factor one x factor two)

margins

row and column totals along the edges (show how many people are in each category ignoring the other variable)

used to account for initial differences in effect size (i.e. calculate expected frequency under equal proportions assumption)

grand total

number of people in the sample

expected frequency

number of observation expected in each cell of a contingency table under the null hypothesis (i.e. equal proportion)

calculation

1. calculate the difference (observed - expected)

2. square the difference (avoids canceling out)

3. divide by expected frequency (standardise)

4. add up

df

(rows - 1) x (columns - 1)

assumptions

unique:

• variables are quantitative (at least one nominal)

• no expected frequencies < 1

• no more than 20% expected frequencies < 5

general:

• independent observations

violation of expected frequency → collapse or Fisher’s test

reporting

χ²(df) = value, p = value

to know which direction the effect is in, compare the observed with the expected counts → the biggest contributions (i.e. biggest difference) drive the results.