In a mixed-effects model with a random slope for a categorical variable, the random slope represents the variation in the effect of the categorical variable across the levels of the grouping variable (in your case, province of residence). The estimated coefficients for the random slope represent the deviation of the effect of immigration status for each province from the overall average effect of immigration status.

To interpret the results of such a model, you can examine the estimated coefficients for the fixed effect of immigration status and the random slope for immigration status. The fixed effect coefficients represent the overall average effect of immigration status on positive mental health, while the random slope coefficients represent the deviation of this effect for each province.

One reference that discusses interpreting random slopes in mixed-effects models is "A brief introduction to mixed effects modelling and multi-model inference in ecology" by Harrison et al. ³. Another reference that discusses interpreting random slopes is "Random slope models" by the Centre for Multilevel Modelling at the University of Bristol ⁴.

(1) A brief introduction to mixed effects modelling and multi-model .... https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5970551/.

(2) Random slope models | Centre for Multilevel Modelling | University of .... https://www.bristol.ac.uk/cmm/learning/videos/random-slopes.html.

(3) Correct interpretation of random slope variance in linear mixed effects .... https://stats.stackexchange.com/questions/593010/correct-interpretation-of-random-slope-variance-in-linear-mixed-effects-models.

(4) Random effect slopes in linear mixed models - Cross Validated. https://stats.stackexchange.com/questions/3757/random-effect-slopes-in-linear-mixed-models.

Lyudmil Antonov Thank you so much for your insightful answer. Can you please help me clear further a bit more about this? Suppose the categorical variable, that is immigration status, has three categories such as native-born people, recent immigrants, and long-residing immigrants, In the fixed effect model if I consider the native-born category as constant. After running the model if I get OR for recent immigrants is 1.25 and for Long term immigrants is 0.67. And in the random slope, after controlling for the native-born group, for recent immigrants is 2.15, and for long-residing immigrants is 1.56. How can I interpret the result? I am sorry for bothering you. I am not finding references related to the use of a categorical variable as a random slope and its interpretation process. Thank you so much once again.

In a fixed effect model, the odds ratio (OR) for recent immigrants of 1.25 means that the odds of having a higher score on the positive mental health scale are 1.25 times higher for recent immigrants compared to native-born people. Similarly, the OR for long-term immigrants of 0.67 means that the odds of having a higher score on the positive mental health scale are 0.67 times lower for long-term immigrants compared to native-born people.

In a random slope model, the interpretation of the coefficients is similar, but it is important to note that these coefficients represent the average effect across all provinces. The OR for recent immigrants of 2.15 means that on average, across all provinces, the odds of having a higher score on the positive mental health scale are 2.15 times higher for recent immigrants compared to native-born people. Similarly, the OR for long-term immigrants of 1.56 means that on average, across all provinces, the odds of having a higher score on the positive mental health scale are 1.56 times higher for long-term immigrants compared to native-born people.

However, since you have included a random slope for immigration status by province, this means that the effect of immigration status on positive mental health scale may differ across provinces. You can examine these differences by looking at the random effects estimates for each province, that is, compare each province separately with each of the other provinces.

Lyudmil Antonov thank you so much once again. By the sentence "You can examine these differences by looking at the random effects estimates for each province, that is, compare each province separately with each of the other provinces." are you indicating adding a random intercept for the provinces?

No, I apologize for any confusion. I was referring to examining the random effects estimates for the random slope of immigration status for each province. These estimates will show you how the effect of immigration status on the positive mental health scale varies across provinces.

Adding a random intercept for provinces would allow you to model differences in the average positive mental health scale score across provinces. This is separate from modeling the variation in the effect of immigration status across provinces using a random slope.

In your case, you could interpret the results by examining the estimates of the fixed effects of the immigration status variable on the positive mental health scale, as well as the estimates of the random slope for the province of residence. You could also examine the variance components of the random effects to determine the amount of variation in the effect of immigration status on positive mental health across the different provinces.

As for references, I recommend checking out the following articles:

These references provide a comprehensive introduction to mixed-effects modeling and offer guidance on interpreting the results of models with categorical variables and random slopes.

Lyudmil Antonov It is making sense. I am really grateful for you. Would you please have a look into the objectives and model plans (for STATA) and help me make it clear that I am applying your suggestions correctly?

1. to examine the extend to which the effect of immigration status (the variable contains three categories: Native-born, Recent Immigrants, and Long Residing Immigrants) on the mental health (the variable, MH, the Positive Mental Health scale) vary in term of region of residence (the variable contains 5 categories), regional economic condition (regional GDP, it is a discrete variable) and regional composition of minority people (the variable contains total number of visible minorities by region, the variable, VisMin, is also a discrete variable). I am planning to develop the model like: mixed MH c.GDP##ib1.immigrant c.VisMin##ib1.immigrant [pw=WTS_M] | | Province: ib4.Province#ib2.immigrant, covariance (unstructured) mle 2. to examine the extend to which the effect of post-migration stress (the variable is a scale) on the mental health (the variable, MH, the Positive Mental Health scale) vary in term of region of residence (the variable contains 5 categories), regional economic condition (regional GDP, it is a discrete variable) and regional composition of minority people (the variable contains total number of visible minorities by region, the variable, VisMin, is also a discrete variable). I am planning to develop the model like: mixed MH c.GDP##c.post-stress c.VisMin##c.post-stress [pw=WTS_M] | | Province: ib4.Province#c.post-stress, covariance (unstructured) mle

I know, I am bothering you. But I cannot make you understand how much insight I am getting from your suggestions. I have no words to express my gratitude to you.

Thank you for your kind words. Let me put it more clearly.

You have some statistical model. Each such model deals with two broad categories of variables: response variable(s) and explanatory (predictory) variables. Other names of these are dependent and independent variables. The explanatory variables are either categorical (called factors) or numeric (called covariates). Explanatory variables exert effects on the response variable. These effects are two types: fixed effects or random effects. A model that involves both fixed and random effects is called a mixed-effects model.

Now concretely for your model. Your response variable is Mental Health (MH). It is measured by a scale. The scale has some items (questions) and each item is measured by categories or numbers that are, in fact, categories (for example: very bad, bad, moderate, good, very good; or, in numbers 0, 1, 2, 3, 4 for a 5-category scale). Such type of response variable is called ordered categorical variable (the order is from small to big) and the model that should be used is of the type ordinal logistic regression.

Your explanatory variables are:

factors: immigration status (3 levels), region of residence (10 levels)

covariates: regional GDP, visible minorities (VisMin)

Of these explanatory variables, the factor "region of residence" has a random effect, and the other 3 explanatory variables have fixed effects. So, the model will be mixed-effects ordinal logistic regression.

In the R environment, functions that can perform ordinal logistic regression are polr (polr = proportional odds logistic regression), clm and clm2 (clm = cumulative link model), mixor (this function can fit mixed-effects ordinal logistic regression models, which include random effects for grouping factors). Unfortunately, I haven't worked in STATA to tell if or which functions are available for mixed-effects ordinal logistic regression.

Thank you so much, Lyudmil Antonov. It makes sense to me. the code would be meologit MH ib1.immigrant GDP Vismin c.GDP#ib1.immigrant c.VisMin#ib1.immigrant ib4, province#ib2.immigrant [pw=WTS_M] | | Province: ib2.immigrant, covariance (unstructured).

Here,

MH is a mental health scale

immigrant is a three-category immigration status variable (native-born, recent immigrant, and long-residing immigrant)

Province is a ten-category province of residence variable.

GDP is the provincial GDP per capita variable.

Vismin is the % of visible minority people in each province.

Here the focus is whether the effect of immigration status on mental health varies in terms of the province and provincial social (composition of vis minority people) and economic (regional GDP per capita).

Now, my last query is,

1. Can I use Province as the random intercept in the random effect side at the same time in the fixed effect side with the interaction with immigration status?

2. Do you think the above-mentioned codes make sense? here, the fixed effect part is the section before | | command; province: is the command for random intercept for the province, and ib1.immigrant after province: is the random slopes for immigration status.

Thank you once again for your continuous support.

You're welcome. I'm glad you found my previous response helpful. To answer your last queries:

1. Yes, you can use Province as the random intercept in the random effect side at the same time in the fixed effect side with the interaction with immigration status. This is called a random-intercept and random-slope model, which allows you to model the variation in both the intercept and the slope across different levels of Province. According to the third search result³, you can specify this model using `||` notation, such as `|| Province: immigrant`.

2. Your code seems to make sense, but I cannot verify its validity without seeing your data and output. You may want to check the syntax and options of meologit command from the second search result², which provides a detailed explanation and examples of how to use this command. You may also want to compare your results with other methods, such as xtlogit or gllamm, to see if they are consistent.

(1) Multilevel Models | Stata. https://www.stata.com/features/overview/multilevel-mixed-effects-models/.

(2) Mixed Effects Logistic Regression | Stata Data Analysis Examples. https://stats.oarc.ucla.edu/stata/dae/mixed-effects-logistic-regression/.

(3) Title stata.com Example 38g — Random-intercept and random-slope models .... https://www.stata.com/manuals/semexample38g.pdf.