Ordinal logistic regression
Shown in the app as ordinal_logistic. Part of the method reference, under Regression.
When to use
Model an ORDERED categorical outcome (3+ ordered levels, e.g. low/medium/high) from predictors via proportional-odds logistic regression.
What you provide
- Outcome: one ordered categorical column.
- Predictors: one or more columns.
Options
Missing data
- Complete case (listwise deletion)
Multiple comparisons
- Not adjustable for this method. Where it makes post-hoc comparisons, the procedure applies its own correction.
Confidence level
Every interval in the result is reported at the level you choose:
- 90%
- 95%
- 99%
Related methods
Other methods under Regression:
- Linear regression
- Logistic regression
- Firth penalised logistic regression
- Poisson regression
- Negative-binomial regression
- Multinomial logistic regression
- Factors associated with outcome (continuous outcome)
- Factors associated with outcome (binary outcome)
Guides that discuss this method
How to read the output
Ordinal logistic regression (the proportional-odds model) analyses an outcome with ordered categories, for example a five-point rating or satisfaction rated low, medium or high, from one or more predictors. The categories must be in their true order: a column already stored with an order keeps it, numbers are ordered numerically, and text labels can be put in order in the analysis settings. Only rows with a value for the outcome and every predictor are used.
Coefficients and odds ratios. Each coefficient is on the log-odds scale. Exponentiated, it is an odds ratio for being in a higher category rather than a lower one, the same at every cut point of the scale. For a categorical predictor it compares each category with the first (an ordered predictor is compared category by category too, never fitted as a polynomial trend). The headline result is the first predictor’s odds ratio with its profile-likelihood confidence interval; you can choose a 90%, 95% or 99% level, and every interval uses it.
Thresholds. The model also estimates one threshold (cut point) between each pair of adjacent outcome categories, on the logit scale. They place the categories on the scale and are shown with their standard errors in the tables; they are rarely interpreted on their own.
Testing a whole predictor. Each predictor is also tested as a whole with a likelihood-ratio test, which is the test to read for a categorical predictor with several levels, and the whole model is tested against a model with only the thresholds.
The proportional-odds check. The model assumes each predictor’s effect is the same across every cut point. Brant’s test checks this overall and for each coefficient. When it fails, Celeus warns and names the coefficients whose effect changes along the scale; read their odds ratios as averages across the cut points.
Sources: McCullagh P (1980), Journal of the Royal Statistical Society, Series B 42(2):109-127, doi:10.1111/j.2517-6161.1980.tb01109.x. Brant R (1990), Biometrics 46(4):1171-1178, doi:10.2307/2532457.
Worked examples
Celeus's result on a published or illustrative dataset, next to an independent computation and, where one exists, a published value. The data themselves are not reproduced here.
MASS::housing (expanded one row per Freq)
- Source: Published dataset distributed with an R package
- Licence: MASS (GPL)
- Reference: Madsen (1976); Venables & Ripley (2002), Modern Applied Statistics with S, 4th ed. - the canonical worked proportional-odds (ordinal logistic) example
| Quantity | Celeus | Independent computation | Published value | Agrees |
|---|---|---|---|---|
| Estimate | 0.566394 | 0.566394 | - | Yes |
| p-value | 6.22819e-08 | 6.22819e-08 | - | Yes |
| Residual deviance | 3479.15 | 3479.15 | - | Yes |
ordinal::wine
- Source: Published dataset distributed with an R package
- Licence: ordinal (GPL)
- Reference: Randall (1989); Christensen, 'Cumulative Link Models for Ordinal Regression with the R Package ordinal' (the article shipped with the ordinal package) prints the coefficients, thresholds, likelihood-ratio tests and profile intervals of this model
| Quantity | Celeus | Independent computation | Published value | Agrees |
|---|---|---|---|---|
| Estimate | 2.50307 | 2.5031 | 2.5031 | Yes |
| p-value | 2.19493e-06 | 2.19461e-06 | - | Yes |
| Confidence interval | 1.50976, 3.59521 | 1.50976, 3.59523 | - | Yes |
| Effect size | 12.22 | 12.2203 | - | Yes |
| Effect size confidence interval | 4.52565, 36.42340 | 4.52566, 36.42390 | - | Yes |
| Test statistic for each model term | 26.928, 11.043 | 26.928, 11.043 | - | Yes |
| Whole-model likelihood-ratio statistic | 34.4543 | 34.4543 | - | Yes |
| Threshold (cut-point) estimate between each pair of adjacent ordered categories | -1.34437, 1.25080, 3.46687, 5.00639 | -1.34438, 1.25081, 3.46689, 5.00640 | - | Yes |
| Number of observations | 72 | 72 | - | Yes |
ordinal::wine (rating as integers 1-5)
- Source: Published dataset distributed with an R package
- Licence: ordinal (GPL)
- Reference: Randall (1989); independent computation (must equal the ordered-factor fit)
| Quantity | Celeus | Independent computation | Published value | Agrees |
|---|---|---|---|---|
| Estimate | 2.50307 | 2.5031 | - | Yes |
| p-value | 2.19493e-06 | 2.19461e-06 | - | Yes |
| Confidence interval | 1.50976, 3.59521 | 1.50976, 3.59523 | - | Yes |
| Effect size | 12.22 | 12.2203 | - | Yes |
| Effect size confidence interval | 4.52565, 36.42340 | 4.52566, 36.42390 | - | Yes |
| Test statistic for each model term | 26.928, 11.043 | 26.928, 11.043 | - | Yes |
| Whole-model likelihood-ratio statistic | 34.4543 | 34.4543 | - | Yes |
| Threshold (cut-point) estimate between each pair of adjacent ordered categories | -1.34437, 1.25080, 3.46687, 5.00639 | -1.34438, 1.25081, 3.46689, 5.00640 | - | Yes |
| Number of observations | 72 | 72 | - | Yes |
ordinal::wine (rating as text labels)
- Source: Published dataset distributed with an R package
- Licence: ordinal (GPL)
- Reference: Randall (1989); independent computation (must equal the ordered-factor fit)
| Quantity | Celeus | Independent computation | Published value | Agrees |
|---|---|---|---|---|
| Estimate | 2.50307 | 2.5031 | - | Yes |
| p-value | 2.19493e-06 | 2.19461e-06 | - | Yes |
| Confidence interval | 1.50976, 3.59521 | 1.50976, 3.59523 | - | Yes |
| Effect size | 12.22 | 12.2203 | - | Yes |
| Effect size confidence interval | 4.52565, 36.42340 | 4.52566, 36.42390 | - | Yes |
| Test statistic for each model term | 26.928, 11.043 | 26.928, 11.043 | - | Yes |
| Whole-model likelihood-ratio statistic | 34.4543 | 34.4543 | - | Yes |
| Threshold (cut-point) estimate between each pair of adjacent ordered categories | -1.34437, 1.25080, 3.46687, 5.00639 | -1.34438, 1.25081, 3.46689, 5.00640 | - | Yes |
| Number of observations | 72 | 72 | - | Yes |
ordinal::wine (judges 1 and 2)
- Source: Published dataset distributed with an R package
- Licence: ordinal (GPL)
- Reference: Randall (1989); independent computation
| Quantity | Celeus | Independent computation | Published value | Agrees |
|---|---|---|---|---|
| Estimate | 2.80486 | 2.80485 | - | Yes |
| p-value | 0.0139945 | 0.0139946 | - | Yes |
| Confidence interval | 0.750324, 5.344450 | 0.750135, 5.344360 | - | Yes |
| Effect size | 16.5247 | 16.5246 | - | Yes |
| Effect size confidence interval | 2.11769, 209.44200 | 2.11729, 209.42300 | - | Yes |
| Test statistic for each model term | 7.49456 | 7.49456 | - | Yes |
| Whole-model likelihood-ratio statistic | 7.49456 | 7.49456 | - | Yes |
| Threshold (cut-point) estimate between each pair of adjacent ordered categories | -1.17623, 0.18171, 1.66304, 3.40810 | -1.176210, 0.181715, 1.663040, 3.408100 | - | Yes |
| Number of observations | 16 | 16 | - | Yes |
ordinal::wine (rating removed in rows 1, 10 and 20)
- Source: Published dataset distributed with an R package
- Licence: ordinal (GPL)
- Reference: Randall (1989); independent computation on the 69 complete rows
| Quantity | Celeus | Independent computation | Published value | Agrees |
|---|---|---|---|---|
| Estimate | 2.40768 | 2.40767 | - | Yes |
| p-value | 6.03737e-06 | 6.03773e-06 | - | Yes |
| Confidence interval | 1.40669, 3.50552 | 1.40669, 3.50554 | - | Yes |
| Effect size | 11.1082 | 11.1081 | - | Yes |
| Effect size confidence interval | 4.08244, 33.29880 | 4.08242, 33.29950 | - | Yes |
| Test statistic for each model term | 24.2764, 10.8245 | 24.2764, 10.8245 | - | Yes |
| Whole-model likelihood-ratio statistic | 31.4648 | 31.4648 | - | Yes |
| Threshold (cut-point) estimate between each pair of adjacent ordered categories | -1.23001, 1.13719, 3.38253, 4.92260 | -1.23002, 1.13719, 3.38253, 4.92259 | - | Yes |
| Number of observations | 69 | 69 | - | Yes |
| Number of rows dropped for missing values | 3 | 3 | - | Yes |
MASS::housing (expanded one row per Freq)
- Source: Published dataset distributed with an R package
- Licence: MASS (GPL)
- Reference: Madsen (1976); Venables & Ripley (2002), Modern Applied Statistics with S, 4th ed.; independent computation
| Quantity | Celeus | Independent computation | Published value | Agrees |
|---|---|---|---|---|
| Estimate | 0.566394 | 0.566394 | - | Yes |
| p-value | 6.22819e-08 | 6.22818e-08 | - | Yes |
| Confidence interval | 0.394516, 0.738841 | 0.394516, 0.738841 | - | Yes |
| Effect size | 1.7619 | 1.7619 | - | Yes |
| Effect size confidence interval | 1.48367, 2.09351 | 1.48367, 2.09351 | - | Yes |
| Test statistic for each model term | 108.2390, 55.9101, 14.3062 | 108.2390, 55.9101, 14.3062 | - | Yes |
| Whole-model likelihood-ratio statistic | 169.728 | 169.728 | - | Yes |
| Threshold (cut-point) estimate between each pair of adjacent ordered categories | -0.496135, 0.690708 | -0.496135, 0.690708 | - | Yes |
| Number of observations | 1681 | 1681 | - | Yes |
carData::WVS
- Source: Published dataset distributed with an R package
- Licence: carData (GPL)
- Reference: World Values Survey 1995-97 (Australia, Norway, Sweden, USA), as shipped in carData; independent computation
| Quantity | Celeus | Independent computation | Published value | Agrees |
|---|---|---|---|---|
| Estimate | 0.176369 | 0.17637 | - | Yes |
| p-value | 0.000870167 | 0.000870094 | - | Yes |
| Confidence interval | 0.0725805, 0.2802400 | 0.0725805, 0.2802400 | - | Yes |
| Effect size | 1.19288 | 1.19288 | - | Yes |
| Effect size confidence interval | 1.07528, 1.32345 | 1.07528, 1.32345 | - | Yes |
| Test statistic for each model term | 11.09630, 5.43355, 4.51792, 250.88100, 51.11980 | 11.09630, 5.43355, 4.51792, 250.88100, 51.11980 | - | Yes |
| Whole-model likelihood-ratio statistic | 337.784 | 337.784 | - | Yes |
| Threshold (cut-point) estimate between each pair of adjacent ordered categories | 0.729764, 2.532480 | 0.729769, 2.532480 | - | Yes |
| Number of observations | 5381 | 5381 | - | Yes |
References
- McCullagh, P. (1980). Regression models for ordinal data. Journal of the Royal Statistical Society, Series B 42: 109-127. doi:10.1111/j.2517-6161.1980.tb01109.x doi:10.1111/j.2517-6161.1980.tb01109.x
- Venables, W. N., Ripley, B. D. (2002). Modern Applied Statistics with S. Springer, New York, 4th edition.