# Intepreting and applying ordinal logistic regression coefficients to calculate probabilities?

Can someone help hint me how I can interpret ordinal logistic regression coefficients and how I can use the .L, .Q and .C terms to calculate probabilities? I am analysing a dataset, where people complain of chest or not pain when they receive external supplements for a vitamin X or not. so I performed a logistic regression, using categorized and ordered (so it is an ordinal categorical variable) values of vitamin X from these observations. So I have grouped these vitamin X values within my data as very low, sub-optimal, optimal and high values and ordered that way.

The data are as such:

chest_pain_status: binary, 0 means not having chest pain, 1 means having chest pain

suppl = binary, 0 means not supplementing and 1 means supplementing

vitx_levels: very_low < suboptimal < optimal< high

My approach is as follows:

mod <- glm(chest_pain_status ~ suppl*vitx_levels, dataset = data,family = binomial())

summary(mod)
Estimate       std.Error     z value          Pr( <|z|)
(intercept)             -2.7816        0.2993        -9.2927          0.000
suppl1                   0.9822        0.3209         3.0803          0.0021
vitx_levels.L           -0.1751        0.0634        -2.7391          0.0062
vitx_levels.Q           -0.1537        0.0552        -2.7689          0.0053
vitx_levels.C            0.0164        0.0460         0.3579          0.7203
suppl1:vitx_levels.L     0.16821       0.0739         2.7472          0.0229
suppl1:vitx_levels.Q     0.2433        0.0641         3.7932          0.00015
suppl1:vitx_levels.C    -0.0096        0.0537        -0.1787          0.8582


Question1: Please how do I interpret coefficients of vitx_levels.L, vitx_levels.Q and vitx_levels.C? From my googling I can understand that R tries to test linear, quadratic and cubic relationships between vitx_levels and the log odds ratios of developing chest pain. It therefore means that there is a significant linear and negative relationship between vitx_levels and chest_pain. Same for the quadratic term. I hope I interpreted it right? Any additional ideas?

Question2: What values could I substitute in to vitx_levels.L, vitx_levels.Q and vitx_levels.C If I wanted to create a formula like the one below where I will later transform y to get probabilities:

y <- -2.7816 + 0.9822 * suppl1 - 0.1751 * vitx_levels.L - 0.1537 * vitx_levels.Q + 0.0164 * vitx_levels.C +  0.16821 * suppl1:vitx_levels.L + 0.2433 * suppl1:vitx_levels.Q - 0.0096 * suppl1:vitx_levels.C


So for the function above, what values could I fit into vitx_levels.L, vitx_levels.Q and vitx_levels.C to be able to compute the value of y?

• Which way around are the coefficients? Is the control first or the diseased patients first? Where's the coefficient for chest pain - it's in your equation but not in the output? I assume "intercept" is the residual is that right?
– M__
Nov 10, 2023 at 3:41
• @M__. Thanks for your attention. Chest pain is the outcome and would not have a coefficient. Intercept is not the residual. Nov 10, 2023 at 11:34
• Yes of course - thanks. Where is the residual (every GLM should have one)?
– M__
Nov 10, 2023 at 14:01

Its a while since I've done GLM because ML is the method of choice.

Question1 Yes multi-linear regression is an approximate description of GLM.

Generally the weights simply mean whether a feature is positively or negatively associated with the phenomena. It's important to be sure it's the right way around (negatives and positives can flip, if that makes sense).

suppl1 is significantly and relatively strongly associated with symptoms, whilst L and Q vitamin levels appear to have some protective effects. Suppl1 has a dose dependent effect with L and Q having a lower correlation with the observed symptoms.

The dose dependency result therefore confirm suppl1 as associated with the symptoms. In all cases levels C are not significant, I assume this is a low level supplement.

Summary vitx_levels appear to have some protective effect.

I do stress it is important to ensure the "negative versus positive" is the right way round. The residual is important and I suspect you've omitted this in the GLM. The lower the residual the better the fit of the model.

BTW LD is different from GLM although they are related (title versus body of the question).

Question 2 Yes that is correct 0.9822 * 1 I assume you mean 0.9822 * suppl1 ... However all C weights are not significant, so you could omit them.

residual versus intercept ... I'm pretty certain that these are the same thing (its the c variable) and the intercept here appears quite large.

• I am very grateful for your very elaborate response. I wish to ask what are the possible values vitx_levels.L or vitx_levels.Q could take in the equation below y <- -2.7816 + 0.9822 * suppl1 - 0.1751 * vitx_levels.L - 0.1537 * vitx_levels.Q + 0.0164 * vitx_levels.C + 0.16821 * suppl1:vitx_levels.L + 0.2433 * suppl1:vitx_levels.Q - 0.0096 * suppl1:vitx_levels.C. For example, let us say I have a x=2 could I say for the  vitx_levels.Q term in the equation above that vitx_levels.Q = x^2 since vitx_levels.Q is the quadratic term? Nov 10, 2023 at 21:18
• @Charles Yes its predictive model
– M__
Nov 10, 2023 at 21:31