# Conditional monotonicity in cumulative probit model

**URL:** <https://discourse.mc-stan.org/t/conditional-monotonicity-in-cumulative-probit-model/33612>\
**Category:** brms\
**Tags:** ordinal-response\
**Created:** [December 18, 2023, 9:28am UTC](https://discourse.mc-stan.org/t/conditional-monotonicity-in-cumulative-probit-model/33612 "2023-12-18T09:28:20Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![BorgeR](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/borger/32/17524_2.png) [@BorgeR](https://discourse.mc-stan.org/u/BorgeR)\
**Post date:** [December 18, 2023, 9:28am UTC](https://discourse.mc-stan.org/t/conditional-monotonicity-in-cumulative-probit-model/33612/1 "2023-12-18T09:28:20Z")

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- Operating System: Windows 11
- brms Version: 2.20.4

This is my first post on the Stan Forum and I am just discovering all the possibilities in brms and with Bayesian analysis in general.

I am working with an ordinal outcome variable and want to use a cumulative probit model. I have two predictors of interest, one categorical with 4 categories and one ordinal with 4 levels. My question concerns modeling interaction effects between the two using the technique for monotonic effects of ordinal predictors described in [Bürkner and Charpentier (2020)](https://bpspsychub.onlinelibrary.wiley.com/doi/abs/10.1111/bmsp.12195). Specifically, I want a model that assumes different monotonic effects of my ordinal predictor in each category of my categorical predictor (i.e., x is conditionally monotonic on z), which is achieved through cell mean coding in brms. According to [this post](https://discourse.mc-stan.org/t/enforcing-conditional-monotonicity/8288), in a linear model this is simply done with: `formula = 0 + z + z:mo(x)`. To my understanding and based on the error messages I am receiveing, this syntax will not work when specifying a cumulative probit model as it is not possible to remove the intercept.

Does anyone know or have any ideas for how to model conditional monotonicity in ordinal models?

Thanks,  
Randi

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<div class="post-metadata">

**Author:** ![Solomon](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/solomon/32/2495_2.png) [@Solomon](https://discourse.mc-stan.org/u/Solomon)\
**Post date:** [December 20, 2023, 3:00pm UTC](https://discourse.mc-stan.org/t/conditional-monotonicity-in-cumulative-probit-model/33612/2 "2023-12-20T15:00:34Z")

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The problem is you cannot remove the \beta\_0 intercept from an ordinal `brm()` model (which has to do with identification purposes). As a consequence, you cannot use the `y ~ 0 + ...` syntax. In your case, try using `formula = 1 + z + z:mo(x)` instead.

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<div class="post-metadata">

**Author:** ![BorgeR](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/borger/32/17524_2.png) [@BorgeR](https://discourse.mc-stan.org/u/BorgeR)\
**Post date:** [December 21, 2023, 9:13am UTC](https://discourse.mc-stan.org/t/conditional-monotonicity-in-cumulative-probit-model/33612/3 "2023-12-21T09:13:13Z")

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Thank you, this seems to give me what I want. For some reason, I only considered either the index variable approach or the dummy variable approach. Your suggestion keeps the b0 intercept, but we still get monotonic effects of x in each category of z (using some data from `psych::bfi` just for illustrating the output):

```no-highlight
 Family: cumulative 
  Links: mu = probit; disc = identity 
Formula: N1 ~ 1 + gender + gender:mo(education) 
   Data: testdf (Number of observations: 2558) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Population-Level Effects: 
                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept[1] -0.64 0.08 -0.79 -0.48 1.00 3618 2650
Intercept[2] 0.00 0.08 -0.15 0.17 1.00 3775 2614
Intercept[3] 0.40 0.08 0.25 0.57 1.00 3760 2700
Intercept[4] 0.97 0.08 0.81 1.13 1.00 3711 2790
Intercept[5] 1.57 0.08 1.41 1.74 1.00 3913 2980
gender2 0.20 0.10 0.02 0.40 1.00 3326 2822
gender1:moeducation -0.00 0.03 -0.06 0.05 1.00 3676 2708
gender2:moeducation -0.05 0.02 -0.10 -0.01 1.00 3122 2826

Simplex Parameters: 
                        Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
gender1:moeducation1[1] 0.26 0.20 0.01 0.74 1.00 4986 2232
gender1:moeducation1[2] 0.24 0.19 0.01 0.70 1.00 6081 2477
gender1:moeducation1[3] 0.24 0.19 0.01 0.68 1.00 5886 3220
gender1:moeducation1[4] 0.25 0.19 0.01 0.69 1.00 4902 2931
gender2:moeducation1[1] 0.24 0.17 0.01 0.62 1.00 3327 1798
gender2:moeducation1[2] 0.21 0.16 0.01 0.58 1.00 4082 2110
gender2:moeducation1[3] 0.33 0.19 0.03 0.72 1.00 4885 2653
gender2:moeducation1[4] 0.21 0.16 0.01 0.60 1.00 6592 2872

Family Specific Parameters: 
     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
disc 1.00 0.00 1.00 1.00 NA NA NA

Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

```

Unrelated, your [blog posts](https://solomonkurz.netlify.app/blog/2021-12-29-notes-on-the-bayesian-cumulative-probit/#ref-kurzStatisticalRethinkingBrms2020) on the cumulative probit model have been so helpful for my understanding of these kinds of models. Such a valuable resource – thank you for making such excellent and accessible posts.