Assignment 3 · Questions 1–2 · Multicategory responses
Coal-face exposure and lung disease severity
371 coal miners were graded by X-ray as normal, mild or severe pneumoconiosis and grouped by years worked at the coal face. The assignment interpreted two fits of the same data: one that ignores the ordering of the categories and one that uses it.
Two models for three categories
Nominal: baseline-category logits
Each abnormal category is compared with normal (nnet::multinom).
Ordinal: proportional odds
Cumulative logits share one slope (MASS::polr parameterisation).
Here is years at the coal face and the response is multinomial with probabilities for each exposure group.
Slide the exposure
Drag the slider or either chart. Dots are the observed proportions in each exposure group (area ∝ miners).
- Normal
- Mild
- Severe
| Model | Normal | Mild | Severe | Parameters | Deviance | AIC |
|---|---|---|---|---|---|---|
| Nominal | 82.78% | 9.15% | 8.07% | 4 | 417.450 | 425.450 |
| Proportional odds | 82.61% | 9.60% | 7.79% | 3 | 416.919 | 422.919 |
What the 2023 analysis found
Nominal model (Q1). The severe-vs-normal slope 0.1093 means each extra year multiplies the odds of severe rather than normal status by 1.1155. Over ten years that is a factor of 2.98, 95% CI (2.160, 4.119); for mild vs normal the factor is 2.31, 95% CI (1.709, 3.112). At 25 years the fitted probabilities are 0.8278, 0.0915 and 0.0807.
Ordinal model (Q2). A single slope of 0.0959 acts on every cumulative split: ten more years multiply the odds of being in a worse category by 2.61, 95% CI (2.065, 3.297). At 25 years: 0.8261, 0.0960, 0.0779.
The two models agree closely, and the proportional-odds model does it with one fewer parameter and a lower AIC (422.92 vs 425.45), so respecting the ordering is the better summary.
Proportional-odds estimates
| Term | Estimate | Std. error | t value |
|---|---|---|---|
| year (γ) | 0.0959 | 0.0119 | 8.03 |
| normal|mild (θ₁) | 3.9558 | 0.4097 | 9.66 |
| mild|severe (θ₂) | 4.8690 | 0.4411 | 11.04 |
Data: Ashford (1959), British Journal of Industrial Medicine 16, 268-278.
Added in 2026
Model checks and an optional AI explanation
Explain this output with AI
Optional · your own keySends only the numeric summary below to the AI provider you choose, from your browser. The figures on this page are the reference; the AI only paraphrases them and can be wrong. How AI is used
What would be sent (7 numbers, no data rows)
- beta (per year of exposure)
- 0.0959 · SE 0.01194
- Odds ratio of a worse category for 10 more years
- 2.609 · 95% CI (2.065, 3.297)
- Fitted probabilities at 25 years (normal, mild, severe)
- 0.826, 0.096, 0.0779
- AIC, proportional odds (3 parameters)
- 422.9
- AIC, nominal multinomial logit (4 parameters)
- 425.4
- Brant test of parallel slopes
- 0.0514 · chi-squared on 1 df, p = 0.821
- LR test, proportional odds vs separate slopes
- 0.000224 · on 1 df, p = 0.988
Plus the model description, data source and: A positive beta means more years at the coal face shift miners towards more severe categories. Observational data: associations, not causal effects. Original analysis: 2023 coursework, refitted in 2026.
Exact request text (system prompt and message)
Sent verbatim with the model id, a response schema and your key (in a request header, never in the text).
System prompt
You explain the output of a statistical model to a reader who knows basic statistics. The output comes from a student's 2023 coursework, refitted for a portfolio site. Rules: - Use only the numbers and facts in the JSON summary you are given. Do not invent numbers, studies, data or context. - Quote numbers exactly as they appear in the summary (you may round them, but never compute new quantities). - Report uncertainty where the summary gives it (confidence intervals, standard errors, p-values) and do not treat p > 0.05 as proof of no effect. - Do not make causal claims unless the summary's context says the design supports them. - If something a reader would want is not in the summary, say so in "caveats" instead of guessing. - Plain English, Australian spelling, no marketing tone. Keep it short: a summary of two or three sentences and two to five points. - List every number you quote in "numbers_used", written exactly as in your text. Return only the JSON object described by the schema.
Message
Explain this model output.
Summary (JSON):
{
"title": "Years at the coal face and pneumoconiosis severity (normal, mild, severe)",
"model": "Proportional-odds (cumulative logit) model, MASS::polr parameterisation: logit P(Y <= j) = zeta_j - beta * years",
"data": "Ashford (1959): coal miners grouped by years worked at the coal face",
"sample_size": "371 miners in 8 exposure groups",
"quantities": [
{
"label": "beta (per year of exposure)",
"value": 0.0959,
"note": "SE 0.01194"
},
{
"label": "Odds ratio of a worse category for 10 more years",
"value": 2.609,
"note": "95% CI (2.065, 3.297)"
},
{
"label": "Fitted probabilities at 25 years (normal, mild, severe)",
"value": "0.826, 0.096, 0.0779"
},
{
"label": "AIC, proportional odds (3 parameters)",
"value": 422.9
},
{
"label": "AIC, nominal multinomial logit (4 parameters)",
"value": 425.4
},
{
"label": "Brant test of parallel slopes",
"value": 0.0514,
"note": "chi-squared on 1 df, p = 0.821"
},
{
"label": "LR test, proportional odds vs separate slopes",
"value": 0.000224,
"note": "on 1 df, p = 0.988"
}
],
"context": [
"A positive beta means more years at the coal face shift miners towards more severe categories.",
"Observational data: associations, not causal effects.",
"Original analysis: 2023 coursework, refitted in 2026."
]
}