Notation for multivariate model with varying effects

Interfaces brms
,
1

I have fitted a model of some two-alternative forced-choice data where responded chose from a pair of stimuli and three domains (y1, y2, y3).

I began with a brms model:

fmla_y1 <- bf(Y1 ~ 0 + X1 +
                      X2 +
                      X3 +
                      X4 +
                      (1 + X1 + X2 + X3 | p | Subject),
                 center = FALSE)

fmla_y2 <- bf(Y2 ~ 0 + X1 +
                      X2 +
                      X3 +
                      X4 +
                      (1 + X1 + X2 + X3 | p | Subject),
              center = FALSE)

fmla_y3 <- bf(Y3 ~ 0 + X1 +
                      X2 +
                      X3 +
                      X4 +
                      (1 + X1 + X2 + X3 | p | Subject),
              center = FALSE)

MV_Mod <- brm(fmla_y1 + fmla_y2 + fmla_y3 + set_rescor(rescor = FALSE),
               family = bernoulli(),
               data)

My attempt at the notation of this model is:

\begin{aligned} \text{Y1}_i & \sim \operatorname{Bernoulli} (p_i) \\ \operatorname{logit} (p_i) & = \alpha^\text{1}_{Subject_i} + \beta_{1} \text{X1}_{i} + \beta_{2} \text{X2}_{i} + \beta_{3} \text{X3}_{i} + \beta_{4} \text{X4}_{i} \\ \alpha_{Subject}^\text{1} & \sim \operatorname{Normal}(\overline{\alpha}^\text{1}_{Subject},\sigma_{Subject}) \\ \overline{\alpha}^\text{1}_{Subject} & \sim \operatorname{Normal}(0, 1) \\ \beta^\text{1}_{1,2,3,4} & \sim \operatorname{Normal}(0, 1) \\ \sigma_{Subject} & \sim \operatorname{Normal}(0, 1) \\ \text{Y2}_i & \sim \operatorname{Bernoulli} (p_i) \\ \operatorname{logit} (p_i) & = \alpha^\text{2}_{Subject_i} + \beta_{1} \text{X1}_{i} + \beta_{2} \text{X2}_{i} + \beta_{3} \text{X3}_{i} + \beta_{4} \text{X4}_{i} \\ \alpha_{Subject}^\text{2} & \sim \operatorname{Normal}(\overline{\alpha}^\text{2}_{Subject},\sigma_{Subject}) \\ \overline{\alpha}^\text{2}_{Subject} & \sim \operatorname{Normal}(0, 1) \\ \beta^\text{2}_{1,2,3,4} & \sim \operatorname{Normal}(0, 1) \\ \sigma_{Subject} & \sim \operatorname{Normal}(0, 1) \\ \text{Y3}_i & \sim \operatorname{Bernoulli} (p_i) \\ \operatorname{logit} (p_i) & = \alpha^\text{3}_{Subject_i} + \beta_{1} \text{X1}_{i} + \beta_{2} \text{X2}_{i} + \beta_{3} \text{X3}_{i} + \beta_{4} \text{X4}_{i} \\ \alpha_{Subject}^\text{3} & \sim \operatorname{Normal}(\overline{\alpha}^\text{3}_{Subject},\sigma_{Subject}) \\ \overline{\alpha}^\text{3}_{Subject} & \sim \operatorname{Normal}(0, 1) \\ \beta^\text{3}_{1,2,3,4} & \sim \operatorname{Normal}(0, 1) \\ \sigma_{Subject} & \sim \operatorname{Normal}(0, 1) \\ \begin{bmatrix} \alpha_{\text{Subject}}^\text{1} \\ \alpha_{\text{Subject}}^\text{2} \\ \alpha_{\text{Subject}}^\text{3} \\ \beta_{2} \\ \beta_{3} \\ \beta_{4} \end{bmatrix} & \sim \operatorname{MVNormal} \begin{pmatrix} \begin{bmatrix} \alpha^\text{1} \\ \alpha^\text{2} \\ \alpha^\text{3} \\ \beta_{2} \\ \beta_{3} \\ \beta_{4} \end{bmatrix}, \mathbf \Sigma \end{pmatrix} \\ \mathbf \Sigma & = \begin{bmatrix} \sigma_{\alpha^\text{1}} & 0 & 0 & 0 & 0 & 0 \\ 0 & \sigma_{\alpha^\text{2}} & 0 & 0 & 0 & 0 \\ 0 & 0 & \sigma_{\alpha^\text{3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \sigma_{\beta_{2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \sigma_{\beta_{3}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \sigma_{\beta_{4}} \\ \end{bmatrix} \mathbf R \begin{bmatrix} \sigma_{\alpha^\text{1}} & 0 & 0 & 0 & 0 & 0 \\ 0 & \sigma_{\alpha^\text{2}} & 0 & 0 & 0 & 0 \\ 0 & 0 & \sigma_{\alpha^\text{3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \sigma_{\beta_{2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \sigma_{\beta_{3}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \sigma_{\beta_{4}} \\ \end{bmatrix} \\ \mathbf{R} &\sim \text{LKJcorr}(1) \end{aligned}

Any feedback would be greatly appreciated!

2

I think perhaps this is more accurate?

\begin{aligned} \text{Y1}_i & \sim \operatorname{Bernoulli} (p_i) \\ \operatorname{logit} (p_i) & = \alpha^\text{1}_{Subject_i} + \beta_{1} \text{X1}_{i} + \beta_{2} \text{X2}_{i} + \beta_{3} \text{X3}_{i} + \beta_{4} \text{X4}_{i} \\ \alpha_{Subject}^\text{1} & \sim \operatorname{Normal}(\overline{\alpha}^\text{1}_{Subject},\sigma_{Subject}) \\ \overline{\alpha}^\text{1}_{Subject} & \sim \operatorname{Normal}(0, 1) \\ \beta^\text{1}_{1,2,3,4} & \sim \operatorname{Normal}(0, 1) \\ \sigma_{Subject} & \sim \operatorname{Normal}(0, 1) \\ \text{Y2}_i & \sim \operatorname{Bernoulli} (p_i) \\ \operatorname{logit} (p_i) & = \alpha^\text{2}_{Subject_i} + \beta_{1} \text{X1}_{i} + \beta_{2} \text{X2}_{i} + \beta_{3} \text{X3}_{i} + \beta_{4} \text{X4}_{i} \\ \alpha_{Subject}^\text{2} & \sim \operatorname{Normal}(\overline{\alpha}^\text{2}_{Subject},\sigma_{Subject}) \\ \overline{\alpha}^\text{2}_{Subject} & \sim \operatorname{Normal}(0, 1) \\ \beta^\text{2}_{1,2,3,4} & \sim \operatorname{Normal}(0, 1) \\ \sigma_{Subject} & \sim \operatorname{Normal}(0, 1) \\ \text{Y3}_i & \sim \operatorname{Bernoulli} (p_i) \\ \operatorname{logit} (p_i) & = \alpha^\text{3}_{Subject_i} + \beta_{1} \text{X1}_{i} + \beta_{2} \text{X2}_{i} + \beta_{3} \text{X3}_{i} + \beta_{4} \text{X4}_{i} \\ \alpha_{Subject}^\text{3} & \sim \operatorname{Normal}(\overline{\alpha}^\text{3}_{Subject},\sigma_{Subject}) \\ \overline{\alpha}^\text{3}_{Subject} & \sim \operatorname{Normal}(0, 1) \\ \beta^\text{3}_{1,2,3,4} & \sim \operatorname{Normal}(0, 1) \\ \sigma_{Subject} & \sim \operatorname{Normal}(0, 1) \\ \begin{bmatrix} \alpha_{\text{Subject}}^\text{1} \\ \alpha_{\text{Subject}}^\text{2} \\ \alpha_{\text{Subject}}^\text{3} \\ \beta^\text{1}_{2} \\ \beta^\text{1}_{3} \\ \beta^\text{1}_{4} \\ \beta^\text{2}_{2} \\ \beta^\text{2}_{3} \\ \beta^\text{2}_{4} \\ \beta^\text{3}_{2} \\ \beta^\text{3}_{3} \\ \beta^\text{3}_{4} \end{bmatrix} & \sim \operatorname{MVNormal} \begin{pmatrix} \begin{bmatrix} \alpha^\text{1} \\ \alpha^\text{2} \\ \alpha^\text{3} \\ \beta^\text{1}_{2} \\ \beta^\text{1}_{3} \\ \beta^\text{1}_{4} \\ \beta^\text{2}_{2} \\ \beta^\text{2}_{3} \\ \beta^\text{2}_{4} \\ \beta^\text{3}_{2} \\ \beta^\text{3}_{3} \\ \beta^\text{3}_{4} \end{bmatrix}, \mathbf \Sigma \end{pmatrix} \\ \mathbf \Sigma & = \begin{bmatrix} \sigma_{\alpha^\text{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \sigma_{\alpha^\text{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \sigma_{\alpha^\text{3}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sigma_{\beta^\text{1}_{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \sigma_{\beta^\text{1}_{3}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{1}_{4}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{2}_{2}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{2}_{3}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{2}_{4}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{3}_{2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{3}_{3}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{3}_{4}} \\ \end{bmatrix} \mathbf R \begin{bmatrix} \sigma_{\alpha^\text{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \sigma_{\alpha^\text{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \sigma_{\alpha^\text{3}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sigma_{\beta^\text{1}_{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \sigma_{\beta^\text{1}_{3}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{1}_{4}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{2}_{2}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{2}_{3}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{2}_{4}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{3}_{2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{3}_{3}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sigma_{\beta^\text{3}_{4}} \\ \end{bmatrix} \\ \mathbf{R} &\sim \text{LKJcorr}(1) \end{aligned}

This gets cut off a bit on desktop and almost entirely on mobile. Is there any way I can fix that?

3

Hello,

One thing that can help folks answer your question here is which part of the model or notation are you looking for feedback on? Is the model behaving well? Throwing errors?

thanks!

4

The model works well and gives good estimates.

I’m just trying to figure out the notation for publication/dissemination.