Purpose

Public preview: This page shows the analysis output, figures, diagnostics, and reporting guidance without exposing the R code. The paid version includes the complete editable R Markdown source, reusable functions, all analysis code, synthetic data, documentation, and exported results.

This workflow shows how to analyze three categorical outcomes common in education dissertations: a binary indicator, an ordered proficiency level, and an unordered postsecondary pathway. It emphasizes outcome coding, reference categories, model assumptions, probability-based interpretation, sparse cells, clustering, and calibrated reporting. The synthetic data are designed for practice and do not represent real students or schools.

Learning outcomes

You will be able to select a model from the outcome’s measurement structure, fit binary logistic, proportional-odds, and multinomial models, translate coefficients into adjusted probabilities, evaluate major assumptions and diagnostics, and produce dynamic APA-style text.

User settings

Simulate or import data

Outcome and data audit

Coding and counts

Outcome frequencies and percentages
outcome category n percent
On track 0 1040 49.5
On track 1 1059 50.5
Proficiency Advanced 829 39.5
Proficiency Basic 345 16.4
Proficiency Below basic 553 26.3
Proficiency Proficient 372 17.7
Pathway Four-year college 770 36.7
Pathway Other 360 17.2
Pathway Two-year college 443 21.1
Pathway Workforce 526 25.1

The event coded 1 is “on track.” Proficiency levels retain their natural ordering. The pathway outcome has no defensible ordering, so two-year college is an explicit comparison category. Changing a reference category changes coefficient labels, but not fitted probabilities.

Missingness and sparse cells

Variables with missing observations
variable missing_n percent
attendance 89 4.2
Pathway by multilingual status: sparse-cell audit
pathway multilingual n flag
Two-year college No 306 FALSE
Two-year college Yes 137 FALSE
Four-year college No 529 FALSE
Four-year college Yes 241 FALSE
Workforce No 352 FALSE
Workforce Yes 174 FALSE
Other No 236 FALSE
Other Yes 124 FALSE

Complete-case modeling below is used to keep the tutorial focused. A dissertation should justify its missing-data strategy and ordinarily consider multiple imputation when assumptions are plausible.

Binary logistic regression

Fit and odds ratios

Binary logistic regression: adjusted odds ratios
term odds_ratio conf_low conf_high p_value
(Intercept) 0.025 0.008 0.084 0.000
prior_score 2.836 2.529 3.192 0.000
attendance 60.642 15.748 236.940 0.000
ses 1.319 1.196 1.457 0.000
multilingualYes 0.999 0.806 1.238 0.993
interventionYes 1.253 1.029 1.527 0.025
school_support 1.095 0.993 1.209 0.069

Odds ratios are multiplicative changes in odds, not percentage-point changes in probability. Continuous predictor units should be meaningful before interpreting them.

Adjusted probabilities

Discrimination and calibration

Binary-model performance
metric value
AUC 0.766
Brier score 0.197
Sensitivity 0.705
Specificity 0.677

Ordinal logistic regression

Proportional-odds model

Proportional-odds model
term odds_ratio conf_low conf_high p.value
prior_score 2.407 2.199 2.640 0.000
attendance 12.030 3.917 37.109 0.000
ses 1.231 1.134 1.337 0.000
multilingualYes 0.999 0.835 1.196 0.994
interventionYes 1.215 1.029 1.435 0.021
school_support 1.171 1.078 1.273 0.000

A positive coefficient indicates greater odds of being in a higher rather than any lower proficiency category, assuming that cumulative contrast is constant across cut points.

Assumption review

Nominal-effects test of the proportional-odds assumption
term Df logLik AIC LRT Pr(>Chi)
NA -2424.641 4867.283 NA NA
prior_score NA NA NA NA NA
attendance NA NA NA NA NA
ses NA NA NA NA NA
multilingual NA NA NA NA NA
intervention NA NA NA NA NA
school_support NA NA NA NA NA

Treat this test as one source of evidence. Large samples can flag small departures, and small samples can miss consequential ones. Examine category-specific fitted probabilities and consider partial proportional-odds or multinomial models when departures matter substantively.

Category probabilities

Multinomial logistic regression

Fit and relative risk ratios

Multinomial model relative risk ratios
outcome term relative_risk_ratio p_value
Four-year college (Intercept) 2.356 0.264
Four-year college prior_score 2.064 0.000
Four-year college attendance 0.629 0.591
Four-year college ses 1.200 0.004
Four-year college multilingualYes 1.060 0.674
Four-year college interventionYes 1.022 0.862
Four-year college school_support 1.270 0.000
Workforce (Intercept) 1.931 0.415
Workforce prior_score 0.862 0.032
Workforce attendance 0.552 0.513
Workforce ses 0.773 0.000
Workforce multilingualYes 0.973 0.850
Workforce interventionYes 1.066 0.634
Workforce school_support 1.127 0.072
Other (Intercept) 1.999 0.433
Other prior_score 0.749 0.000
Other attendance 0.324 0.257
Other ses 0.915 0.222
Other multilingualYes 1.077 0.641
Other interventionYes 1.028 0.850
Other school_support 1.055 0.468

Each relative risk ratio compares the named pathway with two-year college. It is not an odds ratio for the named category in isolation because all outcome probabilities compete and sum to one.

Adjusted pathway probabilities

Clustering and sensitivity

Students are nested within schools. The models above include a school-level predictor but do not by themselves account for residual dependence. Depending on the question and number of clusters, use cluster-robust standard errors, generalized estimating equations, or generalized mixed models. Avoid treating a small number of clusters as if they supported asymptotic cluster-robust inference.

Binary model with school-clustered standard errors
term Estimate Std. Error z value Pr(>|z|)
(Intercept) -3.676 0.658 -5.585 0.000
prior_score 1.042 0.055 18.868 0.000
attendance 4.105 0.734 5.593 0.000
ses 0.277 0.052 5.331 0.000
multilingualYes -0.001 0.104 -0.009 0.993
interventionYes 0.226 0.109 2.075 0.038
school_support 0.091 0.076 1.202 0.229

Dynamic reporting examples

In the synthetic binary analysis, intervention participation was associated with 1.25 times the adjusted odds of being on track, 95% CI [1.03, 1.53], p = .025. The model’s apparent-sample AUC was 0.77; this is descriptive and should not be presented as external validation. In the ordinal analysis, a one-SD increase in prior achievement corresponded to 2.41 times the cumulative odds of a higher proficiency category, 95% CI [2.20, 2.64], p < .001.

Reporting checklist

  • State how each category was coded and identify the event, order, and reference group.
  • Report cell counts, missingness, sparse combinations, and the analytic sample for each model.
  • Explain predictor units and any centering or standardization.
  • Report uncertainty and adjusted probabilities alongside odds or relative risk ratios.
  • Review linearity in the logit, influential observations, separation, calibration, and model-specific assumptions.
  • Address clustering, weights, repeated observations, and complex sampling when present.
  • Distinguish association, classification, and causal effects.

When these models are not enough

Use multilevel categorical models for meaningful residual clustering; generalized estimating equations for population-average correlated outcomes; discrete-time survival models for event timing; latent class models when the categories are unobserved; and causal methods when the estimand is an intervention effect rather than an adjusted association. Rare events, quasi-separation, survey weights, and complex samples require specialized planning.

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