knitr::opts_chunk$set(echo = TRUE)
In general, the broken stick model smoothes the observed growth trajectory. What happens of all observations are already aligned to the break ages? Does the model perfectly represent the data? Is the covariance matrix of the random effects ($\Omega)$ equal to the covariance between the measurements? Is $\sigma^2$ equal to zero?
We adapt code from http://www.davekleinschmidt.com/sst-mixed-effects-simulation/simulations_slides.pdf to generate test data:
library("plyr") library("mvtnorm") make_data_generator <- function(resid_var = 1, ranef_covar = diag(c(1, 1)), n = 100 ) { ni <- nrow(ranef_covar) generate_data <- function() { # sample data set under mixed effects model with random slope/intercepts simulated_data <- rdply(n, { b <- t(rmvnorm(n = 1, sigma = ranef_covar)) epsilon <- rnorm(n = length(b), mean = 0, sd = sqrt(resid_var)) b + epsilon }) data.frame( subject = rep(1:n, each = ni), age = rep(1:ni, n), simulated_data) } }
Let us first model the perfect situation where $\sigma^2 = 0$ (so we set resid_var
to zero) and where the ages align perfectly.
set.seed(77711) covar <- matrix(c(1, 0.7, 0.5, 0.3, 0.7, 1, 0.8, 0.5, 0.5, 0.8, 1, 0.6, 0.3, 0.5, 0.6, 1), nrow = 4) gen_dat <- make_data_generator(n = 10000, ranef_covar = covar, resid_var = 1) data <- gen_dat() head(data)
Check the correlation matrix of the $y$'s.
library("tidyr") library("dplyr") d <- as_tibble(data[,-3]) broad <- t(spread(d, subject, X1))[-1,] cor(broad)
Fit broken stick model, with knots specified at ages 1:4
.
library("brokenstick") knots <- 1:3 boundary <- c(1, 4) fit <- brokenstick(X1 ~ age | subject, data, knots = knots, boundary = boundary) omega <- fit$omega beta <- fit$beta sigma2 <- fit$sigma2 round(beta, 2) round(sigma2, 4) # correlation random effects round(covar, 3) round(omega, 2) # covariances measured data round(omega + diag(sigma2, 4), 3) round(cov(broad), 3) # convert to time-to-time correlation matrix round(cov2cor(omega + diag(sigma2, 4)), 3) round(cor(broad), 3)
cov2cor()
to the time-to-time correlation matrix.Broken Stick Model for Irregular Longitudinal Data
lmer
and kr
methodsbrokenstick()
for model fittingpredict()
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