Multinomial regression (predicting categories)

In brief

A column with three or more categories without an order (for example means of transport: bus, bike, train) is predicted from other columns. Each category is compared with a reference category. Statable calculates it with R; for a target made of text or labelled groups with three or more levels it is the calculation of the question "Can something be predicted?", and the reference category is under "More options".

What for

When the categories have no order, there is no other sensible way: a linear regression would assume numbers with equal distances for the categories. The method shows how the odds of each category compared with the reference category change when a predictor rises.

Requirements

• Target column: three or more categories (text or labelled groups); an ordered column can also be calculated this way, for example when the parallel-lines assumption is violated.
• Reference category: Statable takes the first category (like R); SPSS NOMREG takes the last. It can be chosen under "More options". Ideally justify it on substantive grounds, for example a usual or the largest group. It appears in every sentence and in the table. A different choice changes the odds ratios (in SPSS they are then reciprocals), not the fit.
• Group columns as predictors: the reference group is the first level (as in R); SPSS takes the last, and the ORs are then reciprocals.
• Sample size: the model estimates separate coefficients for each category. Rule of thumb (Peduzzi et al. 1996): at least 10 cases per coefficient in the smallest category, otherwise a note appears.
• One assumption that Statable does not check: the odds of two categories relative to each other should not depend on which other categories exist (independence of irrelevant alternatives).

Reading the result

• First, per predictor, the likelihood ratio test: does it contribute to the prediction overall? Then the odds ratio (OR) per category compared with the reference category, with a 95% confidence interval and p. In multinomial regression, Exp(B) is also called "relative risk ratio" (RRR); Statable calls it "OR", like SPSS.
• Model test and Nagelkerke R² (pseudo R²; not a share explained).
• If the estimation does not finish reliably (usually a predictor separates the categories), a note stands first; the numbers are then not reliable.

Example (own data, 208 people, bus as the reference): age contributes significantly to the prediction of the means of transport overall (χ²(2) = 14.84, p < .001). With each additional year, the odds of the train versus the bus rise by a factor of 1.03 (OR = 1.03, 95% CI [1.01, 1.06], p = .003); for the bike there is no significant relationship with age (OR = 0.99, 95% CI [0.97, 1.02], p = .484). The model fits significantly better than a model without predictors (χ²(6) = 24.12, p < .001, Nagelkerke R² = .12).

Reporting the result

χ²(6) = 24.12, p < .001, Nagelkerke R² = .12; alter: χ²(2) = 14.84, p < .001; w: χ²(4) = 8.75, p = .068

In the text, name the reference category and give the odds ratios of the important comparisons. Do not word it causally.

Limits

• Many parameters: (categories − 1) × (coefficients + 1). With small samples the estimate quickly becomes uncertain.
• If the target column has an order, ordinal regression uses more information and needs fewer cases; multinomial regression is then only the fallback.
• Statable does not check the assumption of independence of irrelevant alternatives.

Source

Agresti, A. (2013). Categorical Data Analysis (3rd ed.), sec. 8.1. Wiley. Hosmer, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied Logistic Regression (3rd ed.), ch. 8. Wiley. Peduzzi, P., et al. (1996). A simulation study of the number of events per variable in logistic regression analysis. Journal of Clinical Epidemiology, 49, 1373-1379. Venables, W. N., & Ripley, B. D. (2002). Modern Applied Statistics with S (4th ed.), sec. 7.3. Springer.