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The code below is to provide a reproducible example - however, this is not an R question, but a question about the formula/methodology (though if someone could provide the R code as well, that would be awesome).

I have a POLR model with a 4-level ordinal outcome.

library(MASS)

set.seed(2)
var1 <- sample(LETTERS[1:4], 100, replace = TRUE)
outcome <- sample(c("bad", "ok", "good", "great"), 100, replace = TRUE)
outcome <- factor(outcome, ordered = TRUE, levels = c("bad", "ok", "good", "great"))

df <- data.frame(var1, outcome)

mod1 <- polr(outcome ~ var1, Hess = TRUE, data = df)

mod1coeffs <- summary(mod1)$coefficients
pval <- (1-pnorm(abs(mod1coeffs[,"t value"]), 0, 1))*2
(mod1coeffs <- cbind(mod1coeffs, pval))

odds_ratio <- exp(mod1coeffs[, "Value"])
mod1coeffs <- cbind(mod1coeffs[,c("Value", "pval")], odds_ratio)
(mod1coeffs <- data.frame(mod1coeffs))

Output:

R>(mod1coeffs <- data.frame(mod1coeffs))
                 Value        pval odds_ratio
var1B      -0.41383640 0.394896754  0.6611091
var1C       0.05689622 0.911737772  1.0585460
var1D       0.37002901 0.461145889  1.4477766
bad|ok     -0.68350927 0.052608937  0.5048423
ok|good     0.06629081 0.846102876  1.0685374
good|great  1.15528574 0.001413507  3.1749305

I could report the outcome by saying that compared to Var1A, Var1B decreases the odds for a better outcome by 66%. However, this is in a business context and the first question I am going to get is "what is that in probabilities?". I am, of course, aware of the p = o / 1 + o formula, but as I understand it, that is for probability from odds, not odds-ratios. I have tried to read up on the internet on this, but all I'm getting is "to convert odds-ratios to probabilities you need more information, which is included in your model". What other information and where is it included? Can someone please help me calculate the probabilities? Ideally, I would like to say something like compared to Var1A, Var1B decreases the probability for a better outcome by xx%. Thank you very much!

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The information is in the intercepts, which you can find by typing

mod1

You then can use the formula to compute the probabilities

$P(DV = level) = \frac{1}{1 + e^{-(intercept + coef*var1}}$

plugging in appropriate values for intercept and coef.

With only one categorical independent variable, it seems like you could just make a table, but that won't work well for more complex models, or those with continuous variables.

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  • $\begingroup$ Thank you very much, Peter, for the response, much appreciated. Just a quick clarification: I assume that coef*var1 means the coefficient of var 1 at the right level, i.e. in the above case the coefficient of Var1B (-0.41383640)? $\endgroup$ Commented yesterday
  • $\begingroup$ Yes. That would be the coefficient to get the probability for level B $\endgroup$ Commented yesterday
  • $\begingroup$ Thank you very much, Peter. This has been huge help and I truly appreciate you taking the time to answer my question. $\endgroup$ Commented yesterday

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