Unexpected results of `emtrends()` with meausurement error model

[Kenji Hayashi]

I've encountered (what I think might be) unexpected behavior when using emmeans (specifically emtrends() here) with brms measurement error models. I'm not totally sure if this is a brms or emmeans issue, but I'm opening it here since it seems to be associated with multivariate brms models with mi() terms.

The issue arises when trying to calculate the effect of some predictor variable that has measurement error, e.g. emtrends(fit_mi, ~ xobs, var = "xobs", resp = "y"). Some observations are that (1) the result of this function call changes between calls, (2) the estimated trend is often quite off from the corresponding slope coefficient, and (3) the reported lower/upper HPD have very large magnitudes relative to the estimated trend. Perhaps the first point has to do with the measurement error associated with xobs, but I'm not sure if/how this can be controlled when calculating emmeans/emtrends, and I'm really not sure what's going on with the other points.

Here's a reproducible example:

library(brms)
library(emmeans)

set.seed(253)

# Simulate data
n <- 100 # number of observations
a <- 0 # intercept
b <- 0.5 # slope
sigma <- 1 # response scale
xtrue <- runif(n, -3, 3) # true x
xsd <- abs(rnorm(n, 0, 0.5)) # measurement error
xobs <- rnorm(n, xtrue, xsd) # observed x
y <- rnorm(n, a + b*xtrue, sigma) # response
df <- data.frame(y, xobs, xsd)

# Specify formula for model with measurement error
form_mi <- 
  bf(y ~ mi(xobs)) +
  bf(xobs | mi(xsd) ~ 1) +
  gaussian() +
  set_rescor(FALSE)

# Specify formula for model without measurement error (for comparison)
form <- bf(y ~ xobs) + gaussian()

# Fit models
fit_mi <- brm(form_mi, df)
fit <- brm(form, df)

# emtrends for model with measurement error
emtrends(fit_mi, ~ xobs, var = "xobs", resp = "y") # results change per call

   xobs xobs.trend lower.HPD upper.HPD
 -0.013       1.41      -154       147

Point estimate displayed: median 
HPD interval probability: 0.95 

# emtrends for model without measurement error
emtrends(fit, ~ xobs, var = "xobs")

   xobs xobs.trend lower.HPD upper.HPD
 -0.013      0.441     0.326     0.555

Point estimate displayed: median 
HPD interval probability: 0.95 

Session info:

R version 4.3.1 (2023-06-16)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Ventura 13.4

Matrix products: default
BLAS:   /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

time zone: America/Los_Angeles
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] emmeans_1.10.0 brms_2.20.4    Rcpp_1.0.11   

loaded via a namespace (and not attached):
 [1] tidyselect_1.2.0     dplyr_1.1.3          farver_2.1.1         loo_2.7.0            fastmap_1.1.1       
 [6] tensorA_0.36.2       shinystan_2.6.0      promises_1.2.1       shinyjs_2.1.0        digest_0.6.33       
[11] estimability_1.5     mime_0.12            lifecycle_1.0.4      StanHeaders_2.32.6   ellipsis_0.3.2      
[16] processx_3.8.2       magrittr_2.0.3       posterior_1.5.0      compiler_4.3.1       rlang_1.1.3         
[21] tools_4.3.1          igraph_1.5.1         utf8_1.2.4           labeling_0.4.3       prettyunits_1.2.0   
[26] bridgesampling_1.1-2 htmlwidgets_1.6.2    curl_5.1.0           pkgbuild_1.4.2       plyr_1.8.9          
[31] dygraphs_1.1.1.6     abind_1.4-5          miniUI_0.1.1.1       withr_2.5.1          grid_4.3.1          
[36] stats4_4.3.1         fansi_1.0.6          xts_0.13.2           xtable_1.8-4         colorspace_2.1-0    
[41] inline_0.3.19        ggplot2_3.5.0        scales_1.3.0         gtools_3.9.5         cli_3.6.2           
[46] mvtnorm_1.2-4        crayon_1.5.2         generics_0.1.3       RcppParallel_5.1.7   rstudioapi_0.15.0   
[51] reshape2_1.4.4       rstan_2.32.6         stringr_1.5.0        shinythemes_1.2.0    bayesplot_1.11.1    
[56] parallel_4.3.1       matrixStats_1.0.0    base64enc_0.1-3      vctrs_0.6.5          V8_4.3.3            
[61] Matrix_1.5-4.1       jsonlite_1.8.7       callr_3.7.3          crosstalk_1.2.1      glue_1.7.0          
[66] codetools_0.2-19     ps_1.7.5             DT_0.32              distributional_0.3.2 stringi_1.7.12      
[71] gtable_0.3.4         later_1.3.1          QuickJSR_1.1.3       munsell_0.5.0        tibble_3.2.1        
[76] colourpicker_1.3.0   pillar_1.9.0         htmltools_0.5.6      Brobdingnag_1.2-9    R6_2.5.1            
[81] shiny_1.7.5          lattice_0.21-8       markdown_1.9         backports_1.4.1      threejs_0.3.3       
[86] httpuv_1.6.11        rstantools_2.4.0     coda_0.19-4.1        gridExtra_2.3        nlme_3.1-162        
[91] checkmate_2.2.0      zoo_1.8-12           pkgconfig_2.0.3   

[Paul-Christian Bürkner]

I think this question is better asked on stan discourse.

[Kenji Hayashi]

Thank you for the suggestion - I've now posted this question on the Stan Discourse: https://discourse.mc-stan.org/t/unexpected-behavior-of-emtrends-with-brms-models-with-me-vs-mi/34525