What does categorical data mean?

• A categorical variable puts people or things into groups
  • Examples: yes/no, pass/fail, psychology/neuroscience, junior/senior, Australia/USA
• This week’s focus is different from earlier weeks
  • Earlier: one categorical variable (goodness-of-fit), or categorical IV with numeric DV (t-tests), or numeric-numeric (correlation)
  • This week: categorical IV and categorical DV together
• The main question this week answers
  • Is there an association between two categorical variables?

Categorical vs numeric measurement

• Numeric measurement gives a score on a scale
  • Example: “How much affection do you show?” 0–10
• Categorical measurement puts you into a category
  • Example: “Are you affectionate?” yes/no
• If you have a choice when designing a study, numeric is usually better
  • Numeric captures more information
  • Numeric can be turned into categories later if you want
  • Categorical cannot be turned into a true numeric score
  • Numeric outcomes usually give more statistical power

Categorical dependent variables (DV)

• Numeric DV questions predict a score
  • Example: higher performance, more symptoms, greater engagement
• Categorical DV questions predict group membership
  • Example: likelihood of passing vs failing, disease vs no disease, convicted vs not convicted
• Categorical DVs can have more than two categories
  • Example: first preference vs other preference vs not listed

Recognising categorical variable research questions

• In these questions, you can usually spot the categories in the wording
  • “Are teenagers less likely to pass than older drivers?”
  • “Are people with family history more likely to develop depression?”
• Typical structure
  • IV: categorical group (teen vs older, attended vs not, history vs no history)
  • DV: categorical outcome (pass vs fail, depression vs no depression, multiple categories)

The key statistical analysis used for Categorical Data

• Main test: chi-square test of independence
• Same test is used whether the study is experimental or non-experimental
• The difference is how you interpret the result
  • Experimental IV (random assignment/manipulation) supports causal language
  • Non-experimental IV (naturally occurring groups) does not support causal language

How chi-square test of independence works

• You organise two categorical variables into a contingency table (a cross-tab)
  • Rows = groups/levels of IV
  • Columns = categories of DV
• The test compares
  • Observed counts (what you actually got)
  • Expected counts (what you would expect if there were no association)
• Assumption you must check
  • All expected counts should be at least 5
  • If any expected value is below 5, you should not run this chi-square test in the usual way
• Degrees of freedom
  • df = (rows − 1) × (columns − 1)
• The conclusion answers
  • Is there evidence of an association between the two variables?
  • Then describe the pattern using percentages or comparing observed vs expected

Expected counts (the rule you use)

• Expected count for a cell =
  • (Row total / Grand total) × Column total
• Practical tip
  • If calculating by hand, keep many decimals to avoid rounding errors

Effect sizes for chi-square tests (how big is the association?)

Effect size (W / V)	Interpretation
< 0.1	Negligible
0.1–0.3	Small
0.3–0.5	Moderate
≥ 0.5	Large

Important note!

A result can be statistically significant but have a negligible/small effect size

What to report for chi-square test of independence (APA-style)

• State the association result and the test
  • χ²(df, N = total) = value, p = value
• Then describe the pattern clearly
  • Percentages or proportions in each group
• Add effect size when possible
  • W or Cramer’s V and its size label (small/moderate/etc.)

Relevant Stata Commands

• Chi-square goodness-of-fit (one categorical variable)
  • Column of data:
    • csgof varname, expperc(pct1, pct2, ...)
  • Summary counts typed directly:
    • chitesti Obs_1 Obs_2 ... Obs_k \ exp_1 exp_2 ... exp_k
• Chi-square test of independence (two categorical variables)
  • Raw data in two columns:
    • tabulate var1 var2, exp row chi2 V
    • exp shows expected counts
    • row shows row percentages (often easiest for interpretation)
    • chi2 runs the chi-square test
    • V reports Cramer’s V (especially useful for 2×2 tables)
  • Summary table typed directly (2×2 layout):
    • tabi Obs_1 Obs_2 \ Obs_3 Obs_4, exp row chi2 V

How to choose the right categorical test

• One categorical variable, comparing observed counts to expected proportions
  • Use chi-square goodness-of-fit
• Two categorical variables, testing whether they are associated
  • Use chi-square test of independence
• Experimental vs non-experimental does not change the test
  • It changes how strong your conclusion can be (causal vs non-causal language)

So Overall…

• Categorical outcomes are about chance/likelihood of being in a category, not a score
• The main tool for two categorical variables is the chi-square test of independence
• Always check expected counts are at least 5
• Use percentages to explain the direction of the association
• Always look at effect size as well as p-value
• Your conclusion wording depends on whether the IV was experimentally manipulated or naturally occurring