[R-sig-ME] How to obtain monotonic effect of ordered factor predictor in lme4 package

[Cristobal Moya]

Dear list members,

I have a question regarding monotonic effects with lme4. I also posted this
in StackOverflow (https://stackoverflow.com/q/68546489/6832021), which is
what I reproduce below.

I want to obtain a monotonic effect for an ordinal predictor with the
function `lmer()` from the package lme4. My reference is the estimate that
can be obtained with `mo()` in the brms package.

Below a reprex with the desired estimate (leaving aside the different
statistical approach behind both packages) in `m1`, and what happens by
default (`m2`) when an ordered factor is used in `lmer()`

library(brms)
library(lme4)
sleepstudy$Days <- factor(sleepstudy$Days, ordered = T)
m1 <- brm(Reaction ~ mo(Days) + (1 | Subject), data = sleepstudy, chains =
2)
#> Compiling Stan program...
...
summary(m1)
#>  Family: gaussian
#>   Links: mu = identity; sigma = identity
#> Formula: Reaction ~ mo(Days) + (1 | Subject)
#>    Data: sleepstudy (Number of observations: 180)
#> Samples: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
#>          total post-warmup samples = 2000
#>
#> Group-Level Effects:
#> ~Subject (Number of levels: 18)
#>               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sd(Intercept)    39.73      7.85    27.73    58.76 1.01      538      751
#>
#> Population-Level Effects:
#>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept   257.11     10.87   235.14   277.72 1.00      468      774
#> moDays       10.12      1.01     8.16    12.09 1.00     1603     1315
#>
#> Simplex Parameters:
#>            Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> moDays1[1]     0.07      0.06     0.00     0.20 1.00     1343      747
#> moDays1[2]     0.07      0.05     0.00     0.20 1.00     1275      524
#> moDays1[3]     0.13      0.07     0.01     0.29 1.00     1337      591
#> moDays1[4]     0.10      0.07     0.00     0.28 1.00     1600      850
#> moDays1[5]     0.16      0.09     0.01     0.34 1.00     1285      658
#> moDays1[6]     0.09      0.06     0.00     0.24 1.00     1543      840
#> moDays1[7]     0.09      0.07     0.00     0.25 1.00     1534      992
#> moDays1[8]     0.16      0.08     0.02     0.32 1.00     1897      906
#> moDays1[9]     0.13      0.08     0.01     0.31 1.00     1839      936
#>
#> Family Specific Parameters:
#>       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma    31.25      1.81    27.93    35.19 1.00     1726     1341
#>
#> Samples were drawn 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).

m2 <- lmer(Reaction ~ Days + (1 | Subject), data = sleepstudy)
summary(m2)
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: Reaction ~ Days + (1 | Subject)
#>    Data: sleepstudy
#>
#> REML criterion at convergence: 1731.8
#>
#> Scaled residuals:
#>     Min      1Q  Median      3Q     Max
#> -3.3473 -0.5293  0.0317  0.5328  4.2570
#>
#> Random effects:
#>  Groups   Name        Variance Std.Dev.
#>  Subject  (Intercept) 1375.5   37.09
#>  Residual              987.6   31.43
#> Number of obs: 180, groups:  Subject, 18
#>
#> Fixed effects:
#>             Estimate Std. Error t value
#> (Intercept)  298.508      9.050  32.985
#> Days.L        95.074      7.407  12.835
#> Days.Q         7.744      7.407   1.045
#> Days.C        -0.705      7.407  -0.095
#> Days^4         5.889      7.407   0.795
#> Days^5         1.754      7.407   0.237
#> Days^6        -6.036      7.407  -0.815
#> Days^7        -1.695      7.407  -0.229
#> Days^8        -4.161      7.407  -0.562
#> Days^9         6.435      7.407   0.869
...

How could an equivalent monotonic effect of `moDays` in `m1` can be
obtained with lme4?

I'm grateful for anyone who can provide some orientation. Kind regards,
--

Cristóbal Moya

Research Fellow

Chair for Social Structure Analysis of Social Inequalities

Faculty of Sociology

Bielefeld University

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