Ordinal regression (predicting ordered levels)

In brief

An ordered column (for example satisfaction from "low" through "medium" to "high") is predicted from one or more other columns. Ordinal regression uses the order of the levels without assuming equal distances between them. Statable calculates it with R. It is chosen under "More options" → "Model" in the question "Can something be predicted?"; for an ordered target with 3 to 10 levels, Statable suggests it.

What for

Likert scales and grades are ordered; ordinal regression does not assume equal distances between the levels (linear regression does). It asks: by which factor do the odds of a higher rather than an equal or lower level rise when a predictor rises by one?

Requirements

• Target column: ordered levels (level "ordinal" or value labels). Statable suggests ordinal regression for 3 to 10 levels; more levels are possible up to 12, with a note that linear regression is then often more appropriate. The order comes from the data (numbers by their code) and is never alphabetical; Statable calculates text only if the order has been set under "Describe columns". Levels without cases are dropped.
• Predictor columns: numbers or group columns. For group columns the reference group is the first level (as in R and in logistic regression); SPSS takes the last, and the ORs are then reciprocals.
• Parallel-lines assumption: each predictor acts equally at all cut points between the levels. Statable checks it with the Brant test, overall and per predictor.
• At least 5 cases per level, otherwise a note appears ("estimate uncertain").

Reading the result

• Odds ratio (OR) per predictor with a 95% confidence interval and p: the factor by which the odds of a higher level rise, the same at every cut point. B > 0 (OR > 1) means: higher values of the predictor go with higher levels. For group columns it is the comparison with the first group; whether the column contributes as a whole is shown by the likelihood ratio test. 1 within the interval means: no relationship shown.
• The thresholds are in a table of their own. They belong to the model equation and are not interpreted.
• Model test (likelihood ratio test against a model without predictors) and Nagelkerke R². This R² is a pseudo R² and not a share explained as in linear regression.
• Brant test: a significant result stands first in the sentence: the effect is then not the same across all levels, and a single OR summarizes it only roughly. Statable then offers the multinomial regression (without order). If the test cannot be calculated with the data (for example an empty cell), Statable says so without stopping. With large samples any test reacts to small departures.

Example (Venables & Ripley 2002, data MASS::housing, 1,681 tenants): satisfaction (low, medium, high) from perceived influence on the management, building type and contact with others. For high compared with low influence, the odds of a higher satisfaction level are higher by a factor of 3.63 (OR = 3.63, 95% CI [2.83, 4.66], p < .001). The model fits significantly better than a model without predictors (χ²(6) = 169.73, p < .001, Nagelkerke R² = .11); the Brant test does not speak against parallel lines (χ²(6) = 8.53, p = .201).

Reporting the result

χ²(6) = 169.73, p < .001, Nagelkerke R² = .11; Infl (High): OR = 3.63, 95% CI [2.83, 4.66], p < .001; Brant χ²(6) = 8.53, p = .201

In the text, name the order of the levels and mention the parallel-lines assumption. Do not word it causally.

Limits

• Sign: Statable writes, like R (MASS::polr) and SPSS PLUM, logit P(Y ≤ j) = θⱼ − βx; B > 0 means higher levels. Stata (ologit) and SAS without "descending" flip the signs.
• Instead of the Brant test, SPSS reports a likelihood ratio test ("test of parallel lines"); the numbers differ.
• If a predictor separates the levels completely, the estimation fails or is not reliable; Statable reports it.
• Two levels are the case for logistic regression; Statable calculates it there anyway when the target has exactly two values.

Source

McCullagh, P. (1980). Regression models for ordinal data. Journal of the Royal Statistical Society B, 42, 109-142. Brant, R. (1990). Assessing proportionality in the proportional odds model for ordinal logistic regression. Biometrics, 46, 1171-1178. Agresti, A. (2010). Analysis of Ordinal Categorical Data (2nd ed.), ch. 3. Wiley. Venables, W. N., & Ripley, B. D. (2002). Modern Applied Statistics with S (4th ed.), sec. 7.3. Springer.