old <- options(digits = 3) 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) } }
We choose between the perfect situation where $\sigma^2 = 0$ and the noisy case $\sigma^2 = 1$ and where the ages align perfectly.
resid_var <- 0 resid_var <- 1 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 = resid_var) data <- gen_dat() head(data)
We wish to reproduce the correlation matrix among the $y$'s from the mixed model estimates. The target correlation matrix is:
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, method = "lmer") omega <- get_omega(fit, hide = "no") beta <- coef(fit, hide = "no") sigma2 <- fit$sigma2 round(beta, 2) round(sigma2, 4) # correlation random effects round(covar, 3) round(omega, 2) # covariances measured data round(omega + diag(sigma2, 4), 3) round(cov(broad), 3) # convert to time-to-time correlation matrix round(cov2cor(omega + diag(sigma2, 4)), 3) round(cor(broad), 3) z <- predict(fit, x = "knots", include_data = FALSE, shape = "wide")[, -1] # off-diagonal elements of covariance of broken stick estimates approach correlation # not enough variance in the diagonal because of smoothing cov(z) # correlations of broken stick estimates are inflated because of smoothing cor(z)
cov2cor(hatC)
reproduces the sample time-to-time correlation matrix.cov <- get_omega(fit) chat <- cov + diag(fit$sigma2, nrow(cov)) r <- cov2cor(chat) r
options(old)
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