# Multilevel Modeling in Linear Mixed Models versus Generalized Linear Mixed Models

I am analyzing a data set that includes several discrete and continuous outcome variables (DV). For the continuous DVs I intend to use Linear Mixed Models (LMM) processed in SPSS. For the discrete variables, I have a couple of DVs rated on a 1-100 scale, and other DVs rated on a 1-10 scale. I've been advised to use Generalized Linear Mixed Models (GLMM) for these discrete variables. I understand one of the main reasons is because these values cannot go below 1, or above 10. For my analysis, these limitations are not practically important, as I am interested in the amount of increase or decrease due to a set of predictors (which predictors have larger statistically significant coefficients). Bumping up against limits (negative values and values greater than 10 for the 1-10 ratings) do not appear to be of importance, only the predictor coefficients. The more I learn about GLMM, the more I understand that interpretation of coefficients can be a challenge with different link functions (logit, probit, etc). For example, Level-1 variances get rescaled based on the distribution and the specific set of predictors. With this in mind, three (3) questions:

1. Can analysis of discrete DVs (1-100 or 1-10 rating scales) be accomplished with LMM if the variables are considered "approximately continuous", and there is no concern for lower or upper value limits, and no concern about retaining discrete values (fractions OK)?
2. If GLMM is still required, would using a "Linear Model" target distribution (with normal distribution and identity link) provide a more straightforward interpretation of significant predictor coefficients, more like an LMM, with consistently-scaled level-1 residuals?
3. Which target distribution link/link function could be most appropriate for 1-100 and 1-10 rating scale data?

When dealing with mixed models or multilevel models for different types of outcome variables, the choice between Linear Mixed Models (LMM) and Generalised Linear Mixed Models (GLMM) depends significantly on the nature of your data and the assumptions you are willing to make. Let's address your three questions in turn:

Can analysis of discrete DVs (1-100 or 1-10 rating scales) be accomplished with LMM if the variables are considered "approximately continuous", and there is no concern for lower or upper value limits, and no concern about retaining discrete values (fractions OK)?

Yes, it is possible to use Linear Mixed Models (LMM) for discrete outcome variables if you consider these variables to be "approximately continuous" and if you are not concerned about the discrete nature or boundaries of these variables. In practice, many researchers treat variables measured on a sufficiently large scale (like 1-100) as continuous, particularly when the primary interest is in the relationships between variables (e.g., predictor coefficients) rather than the precise prediction of the outcome values.

However, this approach can sometimes be less appropriate for smaller scales (like 1-10) where the discreteness and boundaries might have more impact on the model's assumptions and results. For a 1-10 scale, using LMM might overlook some distributional characteristics and violate some assumptions of linearity and homoscedasticity. Nonetheless, if your primary interest is in the relative size of predictor coefficients and you are less concerned with perfect adherence to these assumptions, LMM can still be a pragmatic choice.

If a GLMM is still required, would using a "Linear Model" target distribution (with normal distribution and identity link) provide a more straightforward interpretation of significant predictor coefficients, more like an LMM, with consistently-scaled level-1 residuals?

Using a "Linear Model" target distribution in a GLMM with a normal distribution and identity link function essentially makes the GLMM equivalent to an LMM. This choice means you are treating your data as if it follows a normal distribution and interpreting your coefficients in a similar manner to an LMM.

This approach can provide a straightforward interpretation of predictor coefficients and consistently-scaled residuals, making it simpler and more analogous to an LMM.

Which target distribution/link function could be most appropriate for 1-100 and 1-10 rating scale data?

For your 1-100 and 1-10 rating scale data, the choice of distribution and link function in a GLMM depends on the specific nature of your data and the focus of your analysis:

• Normal Distribution with Identity Link: If you want straightforward interpretation similar to LMM, and you are treating your scales as "approximately continuous", this combination is appropriate. This treats the outcome as continuous and allows for negative and out-of-bound predictions, which you mentioned are not practically important for your analysis.

• Beta Distribution with Logit Link: If your 1-100 scale can be rescaled to (0,1) by dividing by 100, and/or dividing the 1-10 by 10, a beta distribution might be suitable as it naturally handles continuous outcomes bounded between 0 and 1. The logit link ensures that predictions remain within these bounds. Note that the fitted values will not typically be integers when the fitted values are back-transformed to the 1-100 or 1-10 scale.

• Proportional Odds Model: For ordinal outcomes, especially for smaller scales like 1-10, using an ordinal logistic regression within the GLMM framework could be more appropriate. This model respects the ordinal nature and boundaries of the scale.

• Thank you so much. This is exactly what I needed to press ahead. I have a lengthy syntax pre-built and adjustable for SPSS. I plan to run each set of predictors vs DV through LMM first, and then backcheck with GLMM using NORMAL DISTRIBUTION-IDENTITY (treat DV as 'approximately-continuous'), and then MULTINOMIAL DISTRIBUTION-LOGIT (treat DV as ordinal), and then compare all the results for the Predictor coefficients and statistical significance. Again thank you for your detailed and very helpful answer! Commented Jul 18 at 14:50
• You are most welcome. Glad that it helped :) Commented Jul 18 at 15:31