What would be an illustrative picture for linear mixed models? Say that you are in the library of your department of statistics, and that you come across a book with the following picture in the front page. 

You will probably think that this is a book about linear regression things.
What would be the picture that would make you think about linear mixed models?
 A: So something not "extremely elegant" but showing random intercepts and slopes too with R. (I guess it would be even cooler if if showed the actual equations also)

N =100; set.seed(123);


x1 = runif(N)*3; readings1 <- 2*x1 + 1.0 + rnorm(N)*.99;
x2 = runif(N)*3; readings2 <- 3*x2 + 1.5 + rnorm(N)*.99;
x3 = runif(N)*3; readings3 <- 4*x3 + 2.0 + rnorm(N)*.99;
x4 = runif(N)*3; readings4 <- 5*x4 + 2.5 + rnorm(N)*.99;
x5 = runif(N)*3; readings5 <- 6*x5 + 3.0 + rnorm(N)*.99;

X = c(x1,x2,x3,x4,x5);
Y = c(readings1,readings2,readings3,readings4,readings5)
Grouping  = c(rep(1,N),rep(2,N),rep(3,N),rep(4,N),rep(5,N))

library(lme4);
LMERFIT <- lmer(Y ~ 1+ X+ (X|Grouping))

RIaS <-unlist( ranef(LMERFIT)) #Random Intercepts and Slopes
FixedEff <- fixef(LMERFIT)    # Fixed Intercept and Slope

png('SampleLMERFIT_withRandomSlopes_and_Intercepts.png', width=800,height=450,units="px" )
par(mfrow=c(1,2))
plot(X,Y,xlab="x",ylab="readings")
plot(x1,readings1, xlim=c(0,3), ylim=c(min(Y)-1,max(Y)+1), pch=16,xlab="x",ylab="readings" )
points(x2,readings2, col='red', pch=16)
points(x3,readings3, col='green', pch=16)
points(x4,readings4, col='blue', pch=16)
points(x5,readings5, col='orange', pch=16)
abline(v=(seq(-1,4 ,1)), col="lightgray", lty="dotted");        
abline(h=(seq( -1,25 ,1)), col="lightgray", lty="dotted")   

lines(x1,FixedEff[1]+ (RIaS[6] + FixedEff[2])* x1+ RIaS[1], col='black')
lines(x2,FixedEff[1]+ (RIaS[7] + FixedEff[2])* x2+ RIaS[2], col='red')
lines(x3,FixedEff[1]+ (RIaS[8] + FixedEff[2])* x3+ RIaS[3], col='green')
lines(x4,FixedEff[1]+ (RIaS[9] + FixedEff[2])* x4+ RIaS[4], col='blue')
lines(x5,FixedEff[1]+ (RIaS[10]+ FixedEff[2])* x5+ RIaS[5], col='orange') 
legend(0, 24, c("Group1","Group2","Group3","Group4","Group5" ), lty=c(1,1), col=c('black','red', 'green','blue','orange'))
dev.off()

A: 
This graph taken from the Matlab documentation of nlmefit strikes me as one really exemplifying the concept of random intercepts and slopes quite obviously. Probably something showing groups of heteroskedasticity in the residuals of an OLS plot would be also pretty standard but I wouldn't give a "solution".
A: For a talk, I've used the following picture which is based on the sleepstudy dataset from the lme4 package. The idea was to illustrate the difference between independent regression fits from subject-specific data (gray) versus predictions from random-effects models, especially that (1) predicted values from random-effects model are shrinkage estimators and that (2) individuals trajectories share a common slope with a random-intercept only model (orange). The distributions of subject intercepts are shown as kernel density estimates on the y-axis (R code).

(The density curves extend beyond the range of observed values because there are relatively few observations.)
A more 'conventional' graphic might be the next one, which is from Doug Bates (available on R-forge site for lme4, e.g. 4Longitudinal.R), where we could add individual data in each panel.

