# Statistical Significance Issue in Mixed Model

A multilevel model, with one explanatory variable at the individual level (X) and one explanatory variable at the group level (Z):

$$Y_{ij}=\gamma_{00}+\gamma_{10}X_{ij}+\gamma_{01}Z_{j}+\gamma_{11}X_{ij}Z_{j}+u_{0j}+u_{1j}X_{ij}+e_{ij}$$

correlation between $u_{0j}$ and $u_{1j}$ is $0$ . In This pdf , it is written in p.90 that

The interaction effect of these simulated characteristics are presented in table 4. Tested with a blockwise Bonferroni correction, none of the interactions were statistically significant .

But I found all fixed effect $(\gamma_{00},\gamma_{10},\gamma_{01},\gamma_{11})$ and all random effects $(u_{0j},u_{1j})$ statistically significant except individuals-level residual$(e_{ij})$ .

Now my question is if all of them are insignificant according to the mentioned paper , how can the model be valid ? By indicating all of those insignificant , what do they imply ?

Any help is appreciated. Thanks .

### EDIT :

I found all fixed effect $(\gamma_{00},\gamma_{10},\gamma_{01},\gamma_{11})$ and all random effects $(u_{0j},u_{1j})$ statistically significant according to this post and the codes are copied here :

simfun <- function(J,n_j,g00,g10,g01,g11,sig2_0,sig01,sig2_1){
N <- sum(rep(n_j,J))
x <- rnorm(N)
z <- rnorm(J)
mu <- c(0,0)
sig <- matrix(c(sig2_0,sig01,sig01,sig2_1),ncol=2)
u   <- MASS::mvrnorm(J,mu=mu,Sigma=sig)

b_0j <- g00 + g01*z + u[,1]
b_1j <- g10 + g11*z + u[,2]
y <- rep(b_0j,each=n_j)+rep(b_1j,each=n_j)*x + rnorm(N,0,sqrt(0.5))
sim_data <- data.frame(Y=y,X=x,Z=rep(z,each=n_j),group=rep(1:J,each=n_j))
}

fit <- function(J,n_j,g00,g10,g01,g11,sig2_0,sig01,sig2_1){
dat <- simfun(J,n_j,g00,g10,g01,g11,sig2_0,sig01,sig2_1)
full <- lmer(Y~X+Z+X:Z+(X||group),data=dat,control=lmerControl(optCtrl=list(maxfun=20000)))

#Testing significance of random intercept
null.U0 <- update(full, .~.-(1 | group))
dev.U0 <- as.numeric(2*(logLik(full)-logLik(null.U0)))
p.U0 <- 0.5*(1-pchisq(dev.U0,1))*(1/6) #multiplied by (1/6) since there are 6 parameters to be tested, that is there are 6 null hypothesis.

#Testing significance of random slope
null.U1 <- update(full, .~.-(0 + X | group))
dev.U1 <- as.numeric(2*(logLik(full)-logLik(null.U1)))
p.U1 <- 0.5*(1-pchisq(dev.U1,1))*(1/6)

#Testing significance of intercept of fixed part
null.int <- update(full, .~.-1)
dev.int <- as.numeric(2*(logLik(full)-logLik(null.int)))
p.int <- (1-pchisq(dev.U1,1))*(1/6)

#Testing significance of X of fixed part
null.x <- update(full, .~.-X)
dev.x <- as.numeric(2*(logLik(full)-logLik(null.x)))
p.x <- (1-pchisq(dev.x,1))*(1/6)

#Testing significance of Z of fixed part
null.z <- update(full, .~.-X)
dev.z <- as.numeric(2*(logLik(full)-logLik(null.z)))
p.z <- (1-pchisq(dev.z,1))*(1/6)

#Testing significance of interaction part
null.xz <- update(full, .~.-X:Z)
dev.xz <- as.numeric(2*(logLik(full)-logLik(null.xz)))
p.xz <- (1-pchisq(dev.xz,1))*(1/6)

pvals <- data.frame(p.U0=p.U0,p.U1=p.U1,p.int=p.int,p.x=p.x,p.z=p.z,p.xz=p.xz)
}

c1 <- replicate(1000,fit(30,5,1,.3,.3,.3,(1/18),0,(1/18)))
colMeans(apply(c1,2,unlist))

• "Bu I found all fixed effect (γ 00 ,γ 10 ,γ 01 ,γ 11 ) and all random effects (u 0j ,u 1j ) statistically significant." Maybe you can post how you find this. Aug 9, 2015 at 10:03
• I found them statistically significant according to this post stats.stackexchange.com/questions/166298/… Aug 10, 2015 at 14:59