Purpose

This workflow shows how the measurement scale of an outcome determines the regression model. Related predictors are used to model a continuous score, a count, an ordered rating, and a binary milestone so that coefficients, assumptions, predictions, and reporting can be compared directly.

The included data are entirely synthetic. Run the complete example before adapting it. Select a model from the outcome definition and study design—not from whichever result is most favorable.

Choose the model from the outcome

Outcome Example Model in this workflow Main effect scale
Continuous Assessment score Linear regression Mean difference
Count Advising visits Poisson or negative-binomial regression Expected count ratio
Ordinal Ordered progress rating Proportional-odds regression Cumulative odds ratio
Binary Milestone reached: yes/no Logistic regression Odds ratio

This workflow covers counts observed over a common opportunity period. Rates with unequal exposure require an offset. Repeated measures, clustered observations, and time-to-event outcomes need models designed for those structures.

User settings

Simulate and import the example

Rows: 500
Columns: 10
$ student_id       <chr> "S001", "S002", "S003", "S004", "S005", "S006", "S007~
$ age              <dbl> 22, 35, 30, 40, 22, 41, 27, 33, 37, 32, 23, 37, 36, 3~
$ first_generation <fct> No, No, No, No, Yes, Yes, Yes, No, No, No, No, No, Ye~
$ study_hours      <dbl> 7.5, 9.5, 7.1, 15.4, NA, 15.7, 6.5, 10.7, 16.7, 11.5,~
$ preparation      <dbl> 69.4, 59.6, 47.1, 56.5, 64.1, 94.2, 73.3, 64.4, 66.8,~
$ support          <fct> Enhanced, Enhanced, Standard, Enhanced, Standard, Enh~
$ score            <dbl> 68.1, 63.0, 53.2, 67.3, 64.7, 76.8, 65.1, 64.4, 75.8,~
$ advising_visits  <dbl> 0, 0, 4, 2, 1, 1, 9, 3, 0, 4, 1, 2, 3, 2, 9, 4, 3, 2,~
$ rating           <ord> Advanced, Proficient, Developing, Proficient, Beginni~
$ milestone        <int> 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1,~

Data audit

Analysis flow

stage n
Imported records 500
Complete cases 482
Excluded for missing analysis values 18
variable missing_n missing_percent
student_id 0 0.0
age 0 0.0
first_generation 0 0.0
study_hours 18 3.6
preparation 0 0.0
support 0 0.0
score 0 0.0
advising_visits 0 0.0
rating 0 0.0
milestone 0 0.0

Outcome distributions

Continuous outcome: linear regression

term estimate std.error statistic p.value conf.low conf.high
(Intercept) 27.679 3.155 8.774 0.000 21.480 33.878
preparation 0.425 0.031 13.861 0.000 0.365 0.485
study_hours 0.784 0.094 8.361 0.000 0.600 0.968
supportEnhanced 4.049 0.752 5.382 0.000 2.571 5.527
first_generationYes -2.688 0.766 -3.508 0.000 -4.194 -1.182
age 0.085 0.055 1.546 0.123 -0.023 0.193

Each coefficient is an adjusted mean difference in the score. The intercept describes the predicted score when numeric predictors equal zero, so it may not be substantively meaningful without centering.

Linear-model diagnostics

Also review independence from the design, influential cases, linearity for continuous predictors, and uncertainty estimates appropriate to the sampling structure. A flag prompts investigation; it is not an automatic deletion rule.

Count outcome: Poisson and negative-binomial regression

poisson_dispersion selected_model
1.807 negative-binomial
term estimate conf.low conf.high p.value
(Intercept) 2.637 1.412 4.927 0.002
preparation 0.992 0.985 0.998 0.010
supportEnhanced 0.903 0.774 1.054 0.195
first_generationYes 1.140 0.976 1.333 0.099
age 1.014 1.003 1.026 0.014

Exponentiated coefficients are expected count ratios because every record has the same observation opportunity in this example. Call them rate ratios only when the model includes a defensible exposure offset. The Poisson model assumes equality of the conditional mean and variance. The Pearson dispersion estimate above is used as a screening diagnostic; the worked example switches to a negative-binomial model when overdispersion is substantial. Excess zeros, exposure time, clustering, and repeated counts need additional consideration.

Ordinal outcome: proportional odds

term estimate conf.low conf.high p.value
preparation 1.048 1.034 1.063 0.000
study_hours 1.056 1.013 1.101 0.010
supportEnhanced 1.886 1.353 2.638 0.000
first_generationYes 1.078 0.773 1.504 0.658
age 0.991 0.968 1.016 0.484

An odds ratio above 1 indicates higher cumulative odds of being in a higher rating category, conditional on the other predictors. The proportional-odds model assumes that each predictor has the same log-odds effect at every cumulative split.

Proportional-odds diagnostic

term Df logLik AIC LRT Pr(>Chi)
preparation NA NA NA NA NA
study_hours NA NA NA NA NA
support NA NA NA NA NA
first_generation NA NA NA NA NA
age NA NA NA NA NA

Small likelihood-ratio p-values indicate that allowing a predictor to vary across cumulative splits improves fit, which raises concern about proportional odds. Interpret these tests with category counts, graphical checks, theory, and multiplicity in mind; do not automatically replace the model after one small p-value.

Binary outcome: logistic regression

term estimate conf.low conf.high p.value
(Intercept) 0.018 0.003 0.109 0.000
preparation 1.043 1.025 1.062 0.000
study_hours 1.068 1.013 1.128 0.016
supportEnhanced 1.692 1.112 2.591 0.015
first_generationYes 0.575 0.367 0.890 0.014
age 0.982 0.952 1.013 0.256

Odds ratios are not risk ratios. Predicted probabilities often communicate practical magnitude more clearly, especially when the outcome is common.

Calibration and predicted probability

Dynamic results template

Linear, count, proportional-odds, and logistic regression models were fitted to 482 complete synthetic records. Holding study hours, support condition, first-generation status, and age constant, a one-point increase in preparation was associated with a 0.42-point difference in assessment score, 95% CI [0.36, 0.48], p < .001. The linear model explained 38.8% of score variance.

The Poisson dispersion statistic was 1.81, so a negative-binomial model was used for advising visits. Each one-point increase in preparation corresponded to an expected count ratio of 0.99, 95% CI [0.99, 1], p = .010. Every record had the same observation opportunity; a rate model would require an exposure offset.

For the ordered progress rating, each one-point increase in preparation corresponded to 1.05 times the cumulative odds of a higher category, 95% CI [1.03, 1.06]. In the binary milestone model, the adjusted odds ratio was 1.04, 95% CI [1.02, 1.06], p < .001. Interpret these estimates on their model-specific scales and alongside predicted values, diagnostics, study design, and substantive importance.

Reporting checklist

  • Define the outcome, its coding, and why the selected model matches its scale.
  • State the analysis sample and missing-data approach.
  • Identify reference groups, units, centering, and transformations.
  • Report estimates with confidence intervals and model-appropriate effect scales.
  • Present predicted means or probabilities when they aid interpretation.
  • Document diagnostics, assumption concerns, sensitivity analyses, and limits on causal claims.

Exports

file
synthetic_regression_outcomes.csv
linear_model_results.csv
count_model_results.csv
ordinal_model_results.csv
proportional_odds_review.csv
logistic_model_results.csv
logistic_calibration.csv
run_settings.csv
package_versions.csv

References

  • Agresti, A. (2010). Analysis of Ordinal Categorical Data (2nd ed.). Wiley.
  • Fox, J. (2016). Applied Regression Analysis and Generalized Linear Models (3rd ed.). Sage.
  • Harrell, F. E. (2015). Regression Modeling Strategies (2nd ed.). Springer.
Dissertation Stats Helper · Reproducible statistical guidance for doctoral research · Match the model to the outcome, design, and research question.