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Tutorial : proportional Odds linear regression


Lorsque qu’un critère de jugement qualitatif et ordonné est à l’étude (de type, échelle de mesure par exemple petit/moyen/grand ou faible/moyen/élevé), le modèle POLR- Proportional Odds Linear Regression est le plus adapté.
L’utilisation de ce modèle apparaît dans certains cas plus intéressant et instructif que de devoir dichotomiser l’échelle en deux classes (ex : jeune/vieux, faible/élevé, etc …) afin d’utiliser une régression logistique. En effet, l’utilisation de ce modèle suppose que la relation existante entre X et Y est indépendante du point où la dichotomisation est faite.McCullagh appelle cela l’hypothèse d’égalité des Odds-ratio [1]. La vérification de cette hypothèse peut se faire en dichotomisant chaque passage d’une classe à une autre, et en estimant les Odds-ratio des régressions logistiques correspondantes avec leur intervalle de confiance. L'hypothèse sera respectée si une certaine stabilité des Odds ratio est observée. L’idée est ainsi d’évaluer subjectivement l’égalité des odds-ratio pour chaque changement de seuil.


When the studied outcome is qualitative and ordinal (like measurement scale for example small/medium/large or low/medium/high), The Proportional Odds Linear Regression (POLR) is the most suitable model.
The use of this model seems more interesting compared to dichotomize the outcome in two modalities and using a classical logistic regression. The POLR model suppose that the relationship between the studied exposure and the outcome is independant according to the threshold used if the outcome was dichotomised. McCullagh called this assumption the equality assumption of the Odds Ratio [1]. The check of it can be done by dichotomising the outcome by all the possible thresold and estimating the Odds-ratios and their confidence intervals from corresponding logistic regression. The asumption will be satisfied if a roughly stability of the Odds is observed. This evaluation is done graphically. 

Demonstration in Plug-Stat

Suppose that we studied the relationship between the PaO2/FiO2 ratio < 200 at the admission in intensive care unit and the functional status at 3 months of patients, evaluated by the Glasgow Outcome Scale in 5 modalities (Low disability / Moderate disability / Severe disability / Persistent vegetative state / Death [2].
1. The first step presents the number of patients per modalities of the outcome according to the exposition. If one is few representative, it could be necessary to pool it with an adjacent modality.  In our example, we pool classes "Death" and "persistant vegetative state".

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2. The second step consists to take into account confounding factors, if necessary (clic here to obtain help about this selection).
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3. The third step is the checked of the equality oassumption for the odds ratios. In Plug-Stat, we proposed  the following table and figure to evaluate the stability of the Odds ratio according to the different cut-offs.
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On the left, we think that the assumption is not violated while on the right, it seems more debatable.
4. Once the hypothesis is checked, a common odds ratio is estimated and corresponds to the odd of 1 point deterioration in the outcome scale. In our example, the common adjusted odd ratio is estimated at 0.43 with a 95% confidence interval from 0.21 to 0.84 for patients with a PaO2/FiO2 ratio > 200 compmared to the others patients. Indeed, a PaO2/fiO2 ratio > 200 at the entry of patients seems to be a protective factor for the functional evolution of patients, evaluated by the Glasgow Outcome Scale.
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Here are a few examples of works which used a POLR model:
  • Does education offset the effect of maternal disadvantage on childhood anaemia in Tanzania? Evidence from a nationally representative cross-sectional study. Ojoniyi et al. BMC Pediatr. 2019.
  • Hazardous alcohol use among female heads-of-household in rural Mozambique. Wainberg et al. Alcohol. 2018.
  • Erythropoietin in traumatic brain injury (EPO-TBI): a double-blind randomised controlled trial. Nichol et al. Lancet. 2015
  • Randomized assessment of rapid endovascular treatment of ischemic stroke. Goyal et al. NEJM. 2015

  1. McCullagh P. Regression models for ordinal data. 1980; 109–142.
  2. Jennett B, Bond M. Assessment of outcome after severe brain damage. 1975;:480–484
  3. Valenta Z, Pitha J, Poledne R. Proportional odds logistic regression—effective means of dealing with limited uncertainty in dichotomizing clinical outcomes. Statist Med. Wiley-Blackwell; 2006;25(24):4227–34.