# Interpretation of the results of ordinal regression

My ordinal DV is estimation of danger (10-point Likert scale, 1 - least dangerous, 10 - most dangerous). One binary IV is gender (male, female). The other binary IV is type of image (fast, slow).

Following instructions on this web page, I conducted the ordinal regression to get the main effects and gender*image interaction in SPSS (I cannot use R or other programming software):

Danger = Rating of how dangerous they see each type of image

Gender 1 = Men

Gender 2 = Women

Image 1 = Fast

Image 2 = Slow

My interpretation of the table: Men (Gender 1) were less likely (the value is negative) than women to give a high rating of danger. Those exposed to image 1 were more likely to give a high rating of danger than those exposed to image 2. There is a significant interaction between gender and image, such as men who were exposed to Image 1, were less likely (the value is negative) to give a high rating of danger than men who were exposed to Image 2, and than women who were exposed to Image 1 and Image 2.

This is as far as I got. Could anyone confirm that that is correct? It seems pretty incomplete and nonsensical (especially the interaction bit).

I'm also meant to do (according to the instructions that I found) some calculations (by hand!) and some odds ratios (within each level of the factors or something) but I do not understand any of it and I would have to input some codes into SYNTAX in SPSS which I have no clue about. I would never be able to do that.

• It sounds like this question might be arising from instructions for an academic course assignment. If so, please replace one of your tags with the self-study tag and read the policy for handling such questions on this site. This web page on ordinal regression with SPSS might be helpful.
– EdM
Jul 8, 2022 at 17:23
• @EdM no no, it's not an academic course assignment! The instructions I'm using is just some pdf on how to do ordinal regression. It's the only resource showing interaction but as I have said there is usage of the SYNTAX. Thank you for the link; I have seen this page already, it is very confusing. It uses some matrix language. And they don't show how to do interaction in ordinal regression at all.
– user361794
Jul 8, 2022 at 18:21
• A link to the instructions that you're trying to follow and what extra calculations you need would help. Please provide that information by editing the question, as comments are easy to overlook and can be deleted. Note that the interpretation of an interaction in an ordinal model isn't that much different from that in a binary logistic regression. It's the difference in log-odds from what you would predict based on the predictor variables individually, with the log-odds being those for being in a higher level.
– EdM
Jul 8, 2022 at 18:57
• @EdM Thank you for your response. Yes I do not know how to do log-odds. I could only interpret the Parameter Estimates table but I'm not even sure if my interpretation above is correct or not. I was hoping someone could look at the output I have included and confirmed? I'm using this website: restore.ac.uk/srme/www/fac/soc/wie/research-new/srme/modules/… I managed until the middle of the page but then they start calculating stuff and use excel and syntax which I could not follow.
– user361794
Jul 8, 2022 at 20:50

Start by thinking about how you would do this in a standard ordinary least squares model with a continuous outcome. Say that your software reported an Intercept of 5 and coefficients of -1.689 for Gender = 1, 2.506 for Image = 1, and -1.925 for the [Gender = 1] * [Image = 1] interaction.

With the "dummy coding" evidently used here, the Intercept is the expected outcome at the reference levels of categorical independent variables. SPSS chooses the highest level of each as the reference by default. So the Intercept value of 5 is the expected outcome for Gender = 2 and Image = 2.

The coefficient of -1.689 for Gender = 1 is the difference from that intercept value when Gender = 1 and you still have Image = 2. The predicted value for Gender = 1 and Image = 2 is thus 5 - 1.689 = 3.311.

The coefficient of 2.506 for Image = 1 is the difference from the intercept value when Image = 1 and you still are at the reference level of Gender = 2. The predicted value for Image = 1 and Gender = 2 is thus 5 + 2.506 = 7.506.

The [Gender = 1] * [Image = 1] interaction coefficient of -1.925 is the extra difference from what you would otherwise predict from those two individual coefficients. That is, to get the prediction for [Gender = 1] and [Image = 1] in ordinary least squares, you start with the Intercept and add both the individual coefficients and the interaction coefficient. For this hypothetical example, you get an estimate of 5 - 1.689 + 2.506 - 1.925 = 3.892.

In the ordinal regression model the primary interest is in the regression coefficients themselves.* Instead of each coefficient representing a change in linear outcome associated with a change in a predictor, it represents the change in log-odds of being in a higher outcome category. Note that my hypothetical example used the same coefficients as your ordinal regression model. So your understanding:

Men (Gender 1) were less likely (the value is negative) than women to give a high rating of danger. Those exposed to image 1 were more likely to give a high rating of danger than those exposed to image 2.

is partly correct and needs to be modified to:

Men (Gender 1) were less likely (the value is negative) than women to give Image 2 a high rating of danger. Women (Gender 2) exposed to image 1 were more likely to give a high rating of danger than women exposed to image 2.

That's because, with the interaction, the coefficient for Gender (not really a "main effect" when there's an interaction) is the difference between men and women when viewing the reference Image, Image 2. The coefficient for Image (again, not really a "main effect") is the difference between Image 1 and Image 2 when viewed by the reference Gender, women. That's not specific to ordinal regression. When there's an interaction between predictors and predictors are coded this way (called dummy of treatment coding), then all single-predictor coefficients are presented for differences when their interacting predictors are at reference levels.

Interaction coefficients can be harder to express in words. One way to describe the negative interaction coefficient could be:

Men exposed to Image 1 were less likely to give it a high rating of danger than you would expect based on their reaction to Image 2 and women's reaction to Image 1.

That complexity of explaining an interaction coefficient is a reason to show illustrative examples at combinations of predictor values instead. With only 4 combinations that's pretty easy. Relative to the reference of women seeing Image 2, men seeing image 2 had a difference of - 1.689 in the log-odds of a higher danger rating, women seeing Image 1 had 2.506 higher log-odds of a higher danger rating, and men seeing Image 1 had a difference of -1.689 + 2.506 - 1.925 = -1.106 in log odds.

Exponentiating those log-odds differences give you corresponding odds ratios of higher danger ratings.

Calculations up to that point are pretty simple; you just think in terms of changes of log-odds (or odds ratios) instead of changes in linear outcomes. But you should combine such estimates with estimates of the error, which require taking the coefficient covariance matrix into account. There presumably are ways to do that in SPSS, but I don't use it and software-specific questions are off-topic on this site.

This answer goes into more detail about interactions in general and for generalized linear models like ordinal regression in particular. The UCLA OARC web page on ordinal regression in SPSS provides more information specific to ordinal regression and its implementation in SPSS. The SPSS syntax for calculating probabilities of specific outcome ratings given combinations of predictors does seem awkward, but the approach of starting with the probability for the highest outcome level and working downward from there with the level-specific intercepts and the level-independent regression coeffcients can be implemented by hand for point estimates.

*Although there is a relationship between the reported "Threshold" values and a set of intercepts, "In general, these are not used in the interpretation of the results" according to the UCLA OARC web page on ordinal regression in SPSS. In the SPSS output, the "Threshold" values are the negatives of corresponding intercepts for each level.

• I will have a close look at your logodds explanation. I have now been told ordinal regression is far too complex to interpret and that I could "try categorising the DV into 2 or 3 groups and using logistic regression instead" = can I categorise my ordinal DV that way?
– user361794
Jul 11, 2022 at 18:35
• @lisaarthur "The table above shows the main effect of gender and the main effect of image and an interaction between the two. No?" With an interaction there is NO single "main effect" of gender "without the image" whether you are doing regular, logistic, or ordinal regression. The effect of gender depends on the image that is presented, and vice versa. I wrote very carefully how each of those "main effects" needs to be interpreted given the interaction, with this dummy coding of the IV. There is no more difficulty interpreting log-odds in ordinal than in binary logistic regression.
– EdM
Jul 11, 2022 at 19:01
• @lisaarthur I added a paragraph to explain. This is a common misunderstanding with interaction terms. With the usual treatment/dummy coding of categorical predictors, a "main" coefficient for a predictor involved in an interaction is for a situation when the predictors interacting with it are at reference levels. So the Image coefficient in your table is for the situation with the reference Gender, women in this case.
– EdM
Jul 11, 2022 at 19:24
• My biggest query is in regards to what you said before: "Men (Gender 1) were less likely (the value is negative) than women to give Image 2 a high rating of danger. Women (Gender 2) exposed to image 1 were more likely to give a high rating of danger than women exposed to image 2" = I get the first sentence about men but not the "Women (Gender 2) exposed to image 1 were more likely to give a high rating of danger than women exposed to image 2" = Where did you get this from? Which line? I cannot see information on which image women rated as more dangerous.
– user361794
Jul 11, 2022 at 19:24
• Looks like we had comments crossing in time. Look at the paragraph I added under the portion of the answer that you didn't understand.
– EdM
Jul 11, 2022 at 19:26