线性混合 Model（NLME 或 LMER） - 为系数/估计设置界限

Question

I am trying to build put bounds on Mixed effect model. I usually use LMER function for my models but I am not able to find any way to put bounds on the coefficients. I tried using LME but even this was not helpful. Can anyone help here?

library(nlme)
library(lmerTest)


myDat = structure(list(Score = c(1.62, 2.18, 2.3, 3.46, 3.85, 4.7, 1.41, 
                                 2.21, 3.32, 2.73, 3.34, 3.27, 2.14, 2.73, 2.74, 3.39, 3.59, 4.01, 
                                 1.81, 1.83, 3.22, 3.64, 3.51, 4.26), Subject = structure(c(1L, 
                                                                                            1L, 1L, 2L, 2L, 2L, 3L, 3L, 3L, 4L, 4L, 4L, 5L, 5L, 5L, 6L, 6L, 
                                                                                            6L, 7L, 7L, 7L, 8L, 8L, 8L), .Label = c("A", "B", "C", "D", "E", 
                                                                                                                                    "F", "G", "H"), class = "factor"), Condition = structure(c(1L, 
                                                                                                                                                                                               1L, 1L, 2L, 2L, 2L, 1L, 1L, 1L, 2L, 2L, 2L, 1L, 1L, 1L, 2L, 2L, 
                                                                                                                                                                                               2L, 1L, 1L, 1L, 2L, 2L, 2L), .Label = c("No", "Yes"), class = "factor"), 
                       Time = structure(c(1L, 2L, 3L, 1L, 2L, 3L, 1L, 2L, 3L, 1L, 
                                          2L, 3L, 1L, 2L, 3L, 1L, 2L, 3L, 1L, 2L, 3L, 1L, 2L, 3L), .Label = c("1PM", 
                                                                                                              "2PM", "3PM"), class = "factor")), .Names = c("Score", "Subject", 
                                                                                                                                                            "Condition", "Time"), class = "data.frame", row.names = c(NA, 
                                                                                                                                                                                                                      -24L))

m1 <- lmer(Score ~ Condition + Time + Condition*Time + (1 | Subject),
          data = myDat)
fixef(m1)
ranef(m1)


# This is a lame attempt to get positive coeffiecients but I am not sure what objective function, I need to spcify
m2 <- lme(Score ~ Condition + Time + Condition*Time,
          data = myDat, random=c(~1 | Subject)
          # ,
          # nlminb(start = c(0,0,0,0,0),lower = c(0,0,0,0,0))
          )

fixef(m2)
ranef(m2)

Answer 1

If you only want to entertain the possibility of positive parameters, I do think Ben Bolker has written about implementing this in lme4. Although, in a Bayesian methodology you can place priors on the parameters with the support you desire. The example below uses the lower bound argument, lb , from the brms package (which has similar formula notation) to encode a folded normal prior. Of course, there are many possible priors with the correct support.

library(lme4)
library(brms)

myDat <-  structure(list(Score = c(1.62, 2.18, 2.3, 3.46, 3.85, 4.7, 1.41, 
                                 2.21, 3.32, 2.73, 3.34, 3.27, 2.14, 2.73, 2.74, 3.39, 3.59, 4.01, 
                                 1.81, 1.83, 3.22, 3.64, 3.51, 4.26), Subject = structure(c(1L, 
                                                                                            1L, 1L, 2L, 2L, 2L, 3L, 3L, 3L, 4L, 4L, 4L, 5L, 5L, 5L, 6L, 6L, 
                                                                                            6L, 7L, 7L, 7L, 8L, 8L, 8L), .Label = c("A", "B", "C", "D", "E", 
                                                                                                                                    "F", "G", "H"), class = "factor"), Condition = structure(c(1L, 
                                                                                                                                                                                               1L, 1L, 2L, 2L, 2L, 1L, 1L, 1L, 2L, 2L, 2L, 1L, 1L, 1L, 2L, 2L, 
                                                                                                                                                                                               2L, 1L, 1L, 1L, 2L, 2L, 2L), .Label = c("No", "Yes"), class = "factor"), 
                       Time = structure(c(1L, 2L, 3L, 1L, 2L, 3L, 1L, 2L, 3L, 1L, 
                                          2L, 3L, 1L, 2L, 3L, 1L, 2L, 3L, 1L, 2L, 3L, 1L, 2L, 3L), .Label = c("1PM", 
                                                                                                              "2PM", "3PM"), class = "factor")), .Names = c("Score", "Subject", 
                                                                                                                                                            "Condition", "Time"), class = "data.frame", row.names = c(NA, 
## your first try                                                                                                                                                                                                                       -24L))

m1 <- lmer(Score ~ Condition + Time + Condition*Time + (1 | Subject),
           data = myDat)
summary(m1)
# 
# Linear mixed model fit by REML ['lmerMod']
# Formula: Score ~ Condition + Time + Condition * Time + (1 | Subject)
# Data: myDat
# 
# REML criterion at convergence: 25.7
# 
# Scaled residuals: 
#   Min      1Q  Median      3Q     Max 
# -1.4009 -0.5397  0.1392  0.4146  1.2662 

# Random effects:
#   Groups   Name        Variance Std.Dev.
# Subject  (Intercept) 0.05619  0.2370  
# Residual             0.11329  0.3366  
# Number of obs: 24, groups:  Subject, 8
# 
# Fixed effects:
#   Estimate Std. Error t value
# (Intercept)            1.7450     0.2058   8.477
# ConditionYes           1.5600     0.2911   5.359
# Time2PM                0.4925     0.2380   2.069
# Time3PM                1.1500     0.2380   4.832
# ConditionYes:Time2PM  -0.2250     0.3366  -0.668
# ConditionYes:Time3PM  -0.3950     0.3366  -1.174
# 
# Correlation of Fixed Effects:
#   (Intr) CndtnY Tim2PM Tim3PM CY:T2P
# ConditionYs -0.707                            
# Time2PM     -0.578  0.409                     
# Time3PM     -0.578  0.409  0.500              
# CndtnY:T2PM  0.409 -0.578 -0.707 -0.354       
# CndtnY:T3PM  0.409 -0.578 -0.354 -0.707  0.500

## brms - get priors

get_prior(Score ~ Condition + Time + Condition*Time + (1 | Subject),
          data = myDat, family= gaussian())

prior_positive <-  c(
    prior(normal(0, 1), class = Intercept),
    prior(normal(0, 1), class = b, lb = 0), # specify lower bound with "lb"
    #prior(cauchy(0,1), class = shape),
    prior(exponential(1), class = sd),
    prior(exponential(1), class = sigma))

  
mod1_positive <- brm(Score ~ Condition + Time + Condition*Time + (1 | Subject),
        data = myDat, family= gaussian(), prior = prior_positive)
  

summary(mod1_positive)

# Family: gaussian 
# Links: mu = identity; sigma = identity 
# Formula: Score ~ Condition + Time + Condition * Time + (1 | Subject) 
# Data: myDat (Number of observations: 24) 
# Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
# total post-warmup samples = 4000
# 
# Group-Level Effects: 
#   ~Subject (Number of levels: 8) 
# Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
# sd(Intercept)     0.26      0.16     0.02     0.65 1.00      954     1653
# 
# Population-Level Effects: 
#   Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
# Intercept                1.91      0.21     1.48     2.32 1.00     1645     2280
# ConditionYes             1.11      0.27     0.55     1.65 1.00     1431     1742
# Time2PM                  0.27      0.17     0.02     0.65 1.00     1861     1176
# Time3PM                  0.82      0.22     0.37     1.23 1.00     1793     1129
# ConditionYes:Time2PM     0.28      0.21     0.01     0.76 1.00     2183     1490
# ConditionYes:Time3PM     0.24      0.20     0.01     0.75 1.00     1624     1475
# 
# Family Specific Parameters: 
#   Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
# sigma     0.40      0.08     0.27     0.58 1.00     1379     2100
# 
# 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).