nlmer longitudinal data

Question

I've been working with the R Orthodont dataset in the "nlme" package. Just use install.packages("nlme");library(nlme);head(Orthodont) to take a look. The dataset is comprised of distance between the pituitary and the pterygomaxillary fissure measured in 27 children over time. Using the lme4 package I can fit a nonlinear mixed effects model using a logistic curve as my functional form. I can choose to have the asymptote and midpoint entered as random effects

nm1 <- nlmer(distance ~ SSlogis(age,Asym, xmid, scal) ~ (Asym | Subject) + (xmid | Subject), Orthodont, start = c(Asym =25,xmid = 11, scal = 3), corr = FALSE,verb=1)

What I would really like to know is if the gender changes these parameters. Unfortunately online examples do not include both subject and group examples. Is this even possible with the lme4 package?

Answer 1

I believe it is possible to do such a thing by creating a function for custom model formulation and its gradient. The standard SSlogis function uses a logistic function of the following form:

f(input) = Asym/(1+exp((xmid-input)/scal)) # as in ?SSlogis

Instead of calling the SSlogis, you could modify the above statement to suit your needs. I believe you would wish to see if the gender has effect on a fixed effect. Here is example code for modifying a gender specific Asym subpopulation effect in Asym2:

# Just for loading the data, we will use lme4 for model fitting, not nlme
library(nlme)
library(lme4)
# Careful when loading both nlme and lme4 as they have overlap, strange behaviour may occur

# A more generalized form could be taken e.g. from http://en.wikipedia.org/wiki/Generalised_logistic_curve
# A custom model structure:
Model <- function(age, Asym, Asym2, xmid, scal, Gender) 
{
    # Taken from ?SSlogis, standard form:
    #Asym/(1+exp((xmid-input)/scal))
    # Add gender-specific term to Asym2
    (Asym+Asym2*Gender)/(1+exp((xmid-age)/scal))
    # Evaluation of above form is returned by this function
}

# Model gradient, notice that we include all 
# estimated fixed effects like 'Asym', 'Asym2', 'xmid' and 'scal' here,
# but not covariates from the data: 'age' and 'Gender'
ModelGradient <- deriv(
    body(Model)[[2]], 
    namevec = c("Asym", "Asym2", "xmid", "scal"), 
    function.arg=Model
)

A rather typical way of introducing a gender effect is with binary coding. I will transform the Sex-variable to a binary coded Gender:

# Binary coding for the gender
Orthodont2 <- data.frame(Orthodont, Gender = as.numeric(Orthodont[,"Sex"])-1)
#> table(Orthodont2[,"Gender"])
# 0  1 
#64 44 
# Ordering data based on factor levels so they don't mix up paneling in lattice later on
Orthodont2 <- Orthodont2[order(Orthodont2[,"Subject"]),]

I can then fit the customized model:

# Fit the non-linear mixed effects model
fit <- nlmer(
    # Response
    distance ~ 
    # Fixed effects
    ModelGradient(age = age, Asym, Asym2, xmid, scal, Gender = Gender) ~ 
    # replaces: SSlogis(age,Asym, xmid, scal) ~ 
    # Random effects
    (Asym | Subject) + (xmid | Subject), 
    # Data
    data = Orthodont2, 
    start = c(Asym = 25, Asym2 = 15, xmid = 11, scal = 3))

What happens is that when Gender==0 (Male), the model achieves values:

(Asym+Asym2*0)/(1+exp((xmid-age)/scal)) = (Asym)/(1+exp((xmid-age)/scal))

which is actually the standard SSlogis-function form. However, there is now the binary switch that if Gender==1 (Female):

(Asym+Asym2)/(1+exp((xmid-age)/scal))

so that the asymptotic level that we achieve as age increases is actually Asym + Asym2, not just Asym, for female individuals.

Notice also that I do not specify a new random effect for Asym2. Because Asym is non-specific to the gender, female individuals can also have variance in their individuals asymptotic levels due to Asym-term. Model fit:

> summary(fit)
Nonlinear mixed model fit by the Laplace approximation 
Formula: distance ~ ModelGradient(age = age, Asym, Asym2, xmid, scal,      Gender = Gender) ~ (Asym | Subject) + (xmid | Subject) 
   Data: Orthodont2 
   AIC   BIC logLik deviance
 268.7 287.5 -127.4    254.7
Random effects:
 Groups   Name Variance Std.Dev.
 Subject  Asym 7.0499   2.6552  
 Subject  xmid 4.4285   2.1044  
 Residual      1.5354   1.2391  
Number of obs: 108, groups: Subject, 27

Fixed effects:
      Estimate Std. Error t value
Asym    29.882      1.947  15.350
Asym2   -3.493      1.222  -2.859
xmid     1.240      1.068   1.161
scal     5.532      1.782   3.104

Correlation of Fixed Effects:
      Asym   Asym2  xmid  
Asym2 -0.471              
xmid  -0.584  0.167       
scal   0.901 -0.239 -0.773

Looks like there might be a gender specific effect (t -2.859), so that female patients seem to reach a bit lower values of 'distance' as 'age' increases: 29.882 - 3.493 = 26.389

I'm not necessarily suggesting that this is a good/best model, just showing how you could go on with customizing the non-linear models in lme4. Visualizations for the model require a bit of tinkering if you want to extract the non-linear fixed effects (in similar fashion to the visualization for linear models in How do I extract lmer fixed effects by observation? ):

# Extracting fixed effects components by calling the model function, a bit messy but it works
# I like to do this for visualizing the model fit
fixefmat <- matrix(rep(fixef(fit), times=dim(Orthodont2)[1]), ncol=length(fixef(fit)), byrow=TRUE)
colnames(fixefmat) <- names(fixef(fit))
Orthtemp <- data.frame(fixefmat, Orthodont2)
attach(Orthtemp)
# see str(Orthtemp)
# Evaluate the function for rows of the attached data.frame to extract fixed effects corresponding to observations
fix = as.vector(as.formula(body(Model)[[2]]))
detach(Orthtemp)

nobs <- 4 # 4 observations per subject
legend = list(text=list(c("y", "Xb + Zu", "Xb")), lines = list(col=c("blue", "red", "black"), pch=c(1,1,1), lwd=c(1,1,1), type=c("b","b","b")))
require(lattice)
xyplot(
    distance ~ age | Subject, 
    data = Orthodont2,
    panel = function(x, y, ...){
        panel.points(x, y, type='b', col='blue')
        panel.points(x, fix[(1+nobs*(panel.number()-1)):(nobs*(panel.number()))], type='b', col='black')
        panel.points(x, fitted(fit)[(1+nobs*(panel.number()-1)):(nobs*(panel.number()))], type='b', col='red')
    },
    key = legend
)

# Residuals
plot(Orthodont2[,"distance"], resid(fit), xlab="y", ylab="e")

# Distribution of random effects
par(mfrow=c(1,2))
hist(ranef(fit)[[1]][,1], xlab="Random 'Asym'", main="")
hist(ranef(fit)[[1]][,2], xlab="Random 'xmid'", main="")
# Random 'xmid' seems a bit skewed to the right and may violate normal distribution assumption
# This is due to M13 having a bit abnormal growth curve (random effects):
#           Asym       xmid
#M13  3.07301310  3.9077583

Graphics output:

Notice how in above figure the Female (F##) individuals are slightly lower than their Male (M##) counterparts (black lines). E.g. M10 <-> F10 difference in the middle area panels.

Residuals and random effects for observing some characteristics of the specified model. Individual M13 seems a bit tricky.