# Can I combine dichotomous and continuous outcomes into a single regression model?

I am doing analysis on an educational product that aims to predict what impacts whether or not a student gets a question correct or incorrect. The DV includes item scores from four different question types. For two question types, True/False and Multiple choice, the value per item is dichotomous, i.e 0 or 1. However, for the other question types, the value returned for each item ranges from 0 to 1, i.e. the value is continuous. My understanding is that psychometricians refer to these types of items as polytomous. My question is whether all of these question types can be included in a single predictive model, or whether this violates certain assumptions. Also what type of model would you recommend? Linear regression, logistic regression with weighting, decision tree, something else?

For more context, some of the independent variables are length of the question, number of choices, subject, grade level, etc. (Edit starts here) For question types that had a score of 0 or 1, then a logistic regression would make sense. For the other "continuous" type, a linear regression may make sense. I would prefer to run a single predictive model to facilitate comparisons in my independent variables. For example, I'd like to know if math content is more difficult than English content. I would also control for question type. A proposed model might look something like:

score ~ question_type + subject + grade_level + item_seen_before

Regarding the levels of the controls and IVs, they look like:

• question_type: 4 levels
• subject: 2 levels--Math and English
• grade_level: 3 levels--elementary, middle, high school
• item_seen_before: Dichotomous--yes/no

Thanks in advance for any assistance.

• The student either gets the question right or wrong, correct?
– Dave
Sep 21, 2022 at 12:54
• For dichotomous questions, the value is 0 or 1, i.e. incorrect or correct. But for a multiple select question, for example, the score for a single item could be 0, .4, .6, .2, 1.0, etc. In other words, the range in scores for these types of questions can be from 0 to 1. If I just had question types that had a score of 0 or 1, then a logistic regression would make sense. For the other type, a linear regression would make sense. But I'd like to combine the two types into a single model...if possible.
– Mezy
Sep 21, 2022 at 13:06
• Why do you want a multivariate model instead of separate univariate models? Also, for your "continuous" items, are only those scores available, or could they take any value between 0 and 1? If the former, ordinal regression may be a good univariate models for those variables; if the latter, ordinal beta regression would be.
– Noah
Sep 21, 2022 at 14:38
• @Noah Having a single model would facilitate comparison between the IVs of interest. The "continuous" variables don't have an infinite range, but rather depend on the number of choices. Each choice is treated as a T/F question within a multiple select item, so an item with 4 choices will have different scores than an item with 8 or 9 choices. But the range will always be 0 to 1. I am not familiar with ordinal beta regression. Does that still sound like a viable option?
– Mezy
Sep 21, 2022 at 15:04
• Can you explain (in an edit to your question) what you mean by "facilitate comparison between the IVs of interest"? So it sounds like your "continuous" items are actually discrete, and could perhaps be analyzed using ordinal regression or separate binary regressions for each T/F sub-item. Ordinal beta regression would not be a good fit.
– Noah
Sep 21, 2022 at 15:15