Question

我正在使用多级模型来尝试描述纵向变化中的不同模式。 Dingemanse et al (2010)描述了一个＆＃39;扇出＆＃39;当随机效应完全相关时的模式。然而，我发现当随机效应之间的关系是非线性但在观察到的间隔内单调增加时会出现类似的模式。在这种情况下，随机效应不是完全相关的，而是由函数描述的。请参阅下面的示例以获取此示例。该示例仍然具有高的截距 - 斜率相关性（> .9），但是可以获得低于.7的相关性，同时仍然保持完美的截距 - 斜率关系。

我的问题：有没有办法在使用nlme或其他R包的多级模型中包含随机效果之间的完美（非线性）关系？ MLwiN有一种约束斜率 - 截距协方差的方法，这将是一个开始，但不足以包括非线性关系。到目前为止，我还没有找到nlme的解决方案，但也许您知道其他一些可以在模型中包含此功能的软件包。

道歉的编码道歉。我希望我的问题足够清楚，但如果有任何事情需要澄清，请告诉我。非常感谢任何帮助或替代解决方案。

set.seed(123456)

# Change function, quadratic
# Yit = B0ij + B1ij*time + B2ij*time^2
chn <- function(int, slp, slp2, time){
  score<-int + slp * time+ slp2 * time^2 
  return(score)
}


# Set N, random intercept, time and ID
N<-100
start<-rnorm(N,100,15) # Random intercept
time<- matrix(1:15,ncol = 15, nrow = 100,byrow = T) # Time, balanced panel data
ID<-1:N # ID variable

# Random intercept, linear slope: exp(intercept/25)/75, quadratic slope: exp(intercept/25)/250 
score3<-matrix(NA,ncol = ncol(time), nrow = N)
for(x in ID){
  score3[x,]<-chn(start[x],exp(start[x]/25)/75,exp(start[x]/25)/250,time[x,])
}

#Create dataframe
df<- data.frame(ID,score3)
df2<- melt(df,id = 'ID')
df2$variable<-as.vector(time)


# plot
ggplot(df2, aes(x= variable, y= value)) + geom_line(aes(group = ID)) +
  geom_smooth(method = "lm", formula = y ~ x + I(x^2), se =F, size = 2, col ="red" )


# Add noise and estimate model
df2$value2<-df2$value + rnorm(N*ncol(time),0,2)

# Random intercept
mod1<-lme(value2 ~ variable + I(variable^2),
          random= list(ID = ~1),
          data=df2,method="ML",na.action=na.exclude)
summary(mod1)

# Random slopes
mod2<-update(mod1,.~.,random= list(ID = ~variable + I(variable^2)))
summary(mod2)


pairs(ranef(mod2))

Answer 1

根据@MattTyers的建议，我使用rjags进行贝叶斯方法。它是对随机效应之间已知关系的模拟数据的第一次尝试，但似乎产生了准确的估计（比nlme模型更好）。我仍然有点担心Gelman收敛诊断以及如何将此解决方案应用于实际数据。但是，我认为如果有人正在解决同样的问题，我会发布我的答案。

# BAYESIAN ESTIMATE
library(ggplot2); library(reshape2)
# Set new dataset
set.seed(12345)

# New dataset to separate random and fixed
N<-100              # Number of respondents
int<-100            # Fixed effect intercept
U0<-rnorm(N,0,15)   # Random effect intercept
slp_lin<-1          # Fixed effect linear slope
slp_qua<-.25        # Fixed effect quadratic slope
ID<- 1:100          # ID numbers
U1<-exp(U0/25)/7.5  # Random effect linear slope
U2<-exp(U0/25)/25   # Random effect quadratic slope
times<-15           # Max age
err <- matrix(rnorm(N*times,0,2),ncol = times, nrow = N) # Residual term
age <- 1:15         # Ages

# Create matrix of 'math' scores using model
math<-matrix(NA,ncol = times, nrow = N)

for(i in ID){

  for(j in age){

math[i,j] <- (int + U0[i]) + 
  (slp_lin + U1[i])*age[j] + 
  (slp_qua + U2[i])*(age[j]^2) + 
  err[i,j] 

}}

# Melt dataframe and plot scores
e.long<-melt(math)
names(e.long) <- c("ID","age","math")
ggplot(e.long,aes(x= age, y= math)) + geom_line(aes(group = ID))

# Create dataframe for rjags
dat<-list(math=as.numeric(e.long$math),
          age=as.numeric(e.long$age), 
          childnum=e.long$ID,
          n=length(e.long$math), 
          nkids=length(unique(e.long$ID)))
lapply(dat , summary)


library(rjags)

# Model with uninformative priors
model_rnk<-"
model{

#Model, fixed effect age and random intercept-slope connected
for( i in 1:n)
{
  math[i]~dnorm(mu[i], sigm.inv)
  mu[i]<-(b[1] + u[childnum[i],1]) + (b[2]+ u[childnum[i],2]) * age[i] + 
  (b[3]+ u[childnum[i],3]) * (age[i]^2) 
}

#Random slopes
for (j in 1:nkids)
{
  u[j, 1] ~ dnorm(0, tau.a)
  u[j, 2] <- exp(u[j,1]/25)/7.5
  u[j, 3] <- exp(u[j,1]/25)/25
}

#Priors on fixed intercept and slope priors
b[1] ~ dnorm(0.0, 1.0E-5)
b[2] ~ dnorm(0.0, 1.0E-5)
b[3] ~ dnorm(0.0, 1.0E-5)

# Residual variance priors
sigm.inv ~ dgamma(1.5, 0.001)# precision of math[i]
sigm<- pow(sigm.inv, -1/2)   # standard deviation


# Varying intercepts, varying slopes priors
tau.a ~ dgamma(1.5, 0.001)
sigma.a<-pow(tau.a, -1/2)
}"

#Initialize the model
mod_rnk<-jags.model(file=textConnection(model_rnk), data=dat, n.chains=2 )

#burn in 
update(mod_rnk, 5000)

#collect samples of the parameters
samps_rnk<-coda.samples(mod_rnk, variable.names=c( "b","sigma.a", "sigm"), n.iter=5000, n.thin=1)

#Numerical summary of each parameter:
summary(samps_rnk)

gelman.diag(samps_rnk,  multivariate = F)

# nlme model
library(nlme)
Stab_rnk2<-lme(math ~ age + I(age^2),
               random= list(ID = ~age + I(age^2)),
               data=e.long,method="ML",na.action=na.exclude)
summary(Stab_rnk2)

结果与人口价值非常接近

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

                Mean       SD  Naive SE Time-series SE
    b[1]    100.7409 0.575414 5.754e-03      0.1065523
    b[2]      0.9843 0.052248 5.225e-04      0.0064052
    b[3]      0.2512 0.003144 3.144e-05      0.0003500
    sigm      1.9963 0.037548 3.755e-04      0.0004056
    sigma.a  16.9322 1.183173 1.183e-02      0.0121340

nlme估计值更差（除随机截距外）

Random effects:
 Formula: ~age + I(age^2) | ID
 Structure: General positive-definite, Log-Cholesky parametrization
            StdDev      Corr        
(Intercept) 16.73626521 (Intr) age  
age          0.13152688 0.890       
I(age^2)     0.03752701 0.924  0.918
Residual     1.99346015             

Fixed effects: math ~ age + I(age^2) 
                Value Std.Error   DF  t-value p-value
(Intercept) 103.85896 1.6847051 1398 61.64816       0
age           1.15741 0.0527874 1398 21.92586       0
I(age^2)      0.30809 0.0048747 1398 63.20204       0

指定R中随机效应之间的非线性关系

1 个答案: