bivrp {bivrp}

Bivariate Residual Plots with Simulation Polygons

Description

Produces a bivariate residual plot with simulation polygons to assess goodness-of-fit of bivariate statistical models, provided the user supplies three functions: one to obtain model diagnostics, one to simulate data from a fitted model object, and one to refit the model to simulated data.

Usage

bivrp(obj, sim = 99, conf = .95, diagfun, simfun, fitfun, verb = FALSE,
      sort.res = TRUE, closest.angle = TRUE, angle.ref = - pi,
      counter.clockwise = TRUE, xlab, ylab, main,
      clear.device = FALSE, point.col, point.pch, ...)
      
## S3 method for class 'bivrp'
print(x, ...)

Arguments

obj

fitted model object

sim

number of simulations used to compute envelope. Default is 99

conf

confidence level of the simulated polygons. Default is 0.95

diagfun

user-defined function used to obtain the diagnostic measures from the fitted model object

simfun

user-defined function used to simulate a random sample from the model estimated parameters

fitfun

user-defined function used to re-fit the model to simulated data

verb

logical. If TRUE, prints each step of the simulation procedure

sort.res

logical. If TRUE, points will be sorted using angles formed with the origin (type of ordering can be fine-tuned with arguments closest.angle, angle.ref and counter.clockwise).

closest.angle

logical. If FALSE, points will be sorted starting from the angle defined in angle.ref, if TRUE, points will be sorted starting from the closest angle to the observed bivariate sample ranked as first

angle.ref

the reference angle from which points will be sorted starting from the closest angle to the input (in radians). Defaults to - pi

counter.clockwise

logical. Should the points be ordered counter-clockwise or clockwise from the reference angle?

xlab

argument passed to par

ylab

argument passed to par

main

argument passed to par

clear.device

logical. If TRUE, clears the plotting device after producing the bivariate residual plot with simulation polygons

point.col

a vector of length 2 with the colors of the points that are inside and outside of the simulated polygons

point.pch

a vector of length 2 with the point characters of the points that are inside and outside of the simulated polygons

...

further arguments passed to plot.bivrp

x

an object of class bivrp

Details

This approach relies on the same strategy used for producing half-normal plots with simulation envelopes. Given a vector of bivariate model diagnostics, the angle each point makes with the origin is calculated to order them. This can be fine-tuned using the logical arguments closest.angle, angle.ref, and counter.clockwise, see the Arguments section above.

Then, sim bivariate response variables are simulated from the fitted model, using the same model matrices, error distribution and fitted parameters, using the function defined as simfun. The model is refitted to each simulated sample, using the function defined as fitfun. Next, we obtain the same type of model diagnostics, using diagfun, again ordered the same way the original bivariate sample was. We have, for each bivariate diagnostic, sim simulated bivariate diagnostics forming the whole cloud of simulated diagnostics.

By default, we then obtain the convex hulls of each set of the $s$ sets of points and obtain a reduced polygon whose area is (conf * 100)% of the original convex hull's area, forming the simulated polygon. This is equivalent to passing the argument reduce.polygon = "proportional" to plot.bivrp. The argument reduce.polygon = "bag" can be used to obtain a (conf * 100)% bagplot as the simulated polygon instead of a convex hull. The points are then connected to the centroids of their respective simulated polygons and, if they lie outside the polygons, they are drawn in red. For the final display, the polygons are erased so as to ease visualization.

There is no automatic implementation of a bivariate model in this function, and hence users must provide three functions for bivrp. The first function, diagfun, must extract the desired model diagnostics from a model fit object. The second function, simfun, must return the response variable, simulated using the same error distributions and estimated parameters from the fitted model. The third and final function, fitfun, must return a fitted model object. See the Examples section.

This function produces a plot by passing the computed object to plot.bivrp. The print method returns a data.frame containing all ordered simulated bivariate diagnostics.

Value

The function returns an object of class "bivrp", which is a list containing the following components:

reslist.ord

list of ordered diagnostics from model refitting to each simulated dataset

res.original.ord

original model diagnostics

res1

diagnostics from variable 1

res2

diagnostics from variable 2

res.original1

original model diagnostics for variable 1

res.original2

original model diagnostics for variable 2

conf

confidence level of the simulated polygons

Author(s)

Rafael A. Moral <[e-mail]> and John Hinde

See Also

plot.bivrp

Examples

## simulating a bivariate normal response variable

require(mvtnorm)

n <- 40
beta1 <- c(2, .4)
beta2 <- c(.2, .2)
x <- seq(1, 10, length = n)
X <- model.matrix(~ x)
mu1 <- X%*%beta1
mu2 <- X%*%beta2
sig1 <- 2
sig2 <- 3
sig12 <- -1.7
Sig1 <- diag(rep(sig1), n)
Sig2 <- diag(rep(sig2), n)
Sig12 <- diag(rep(sig12), n)
V <- rbind(cbind(Sig1, Sig12),
           cbind(Sig12, Sig2))

set.seed(2016)
Y <- as.numeric(rmvnorm(1, c(mu1, mu2), V))

## code for fitting the model estimating covariance or not
bivnormfit <- function(Y, X, covariance) {
  n <- nrow(X)
  p <- ncol(X)
  y <- cbind(Y[1:n],Y[(n+1):(2*n)])
  XtXinv <- solve(crossprod(X, X))
  beta.hat <- XtXinv %*% crossprod(X, y)
  mu.hat <- X%*%beta.hat
  sigma.hat <- 1/n * t(y - mu.hat) %*% (y - mu.hat)
  if(!covariance) sigma.hat <- diag(diag(sigma.hat))
  cov.betas <- sigma.hat %x% XtXinv
  se.s1 <- sqrt(2*sigma.hat[1]^2/(n-p+1))
  se.s2 <- sqrt(2*sigma.hat[4]^2/(n-p+1))
  if(!covariance) se.s12 <- NA else {
    rho <- sigma.hat[2]/sqrt(sigma.hat[1]*sigma.hat[4])
    se.s12 <- sqrt((1+rho^2)*sigma.hat[1]*sigma.hat[4]/(n-p+1))
  }
  se.betas <- sqrt(diag(cov.betas))
  se.sigma <- c(se.s1, se.s2, se.s12)
  coefs <- c(beta.hat, sigma.hat[1], sigma.hat[4], sigma.hat[2])
  names(coefs) <- c("beta1.0", "beta1.1", "beta2.0", "beta2.1", "sig1", "sig2", "sig12")
  fitted <- c(mu.hat)
  resid <- Y - fitted
  Sig1 <- diag(rep(sigma.hat[1]), n)
  Sig2 <- diag(rep(sigma.hat[4]), n)
  Sig12 <- diag(rep(sigma.hat[2]), n)
  V <- rbind(cbind(Sig1, Sig12),
             cbind(Sig12, Sig2))
  llik <- dmvnorm(Y, c(mu.hat), V, log = TRUE)
  ret <- list("coefs" = coefs, "covariance" = covariance, "n" = n, 
              "X" = X, "fitted" = fitted, "resid" = resid, "loglik" = llik,
              "Y" = Y, "se" = c(se.betas, se.sigma))
  class(ret) <- "bivnormfit"
  return(ret)
}

## fitting bivariate models with and without estimating covariance
fit0 <- bivnormfit(Y, X, covariance=FALSE)
fit1 <- bivnormfit(Y, X, covariance=TRUE)
## likelihood-ratio test
2*(fit0$loglik - fit1$loglik)
pchisq(54.24, 1, lower=FALSE)

## function for extracting diagnostics (raw residuals)
dfun <- function(obj) {
  r <- obj$resid
  n <- obj$n
  return(list(r[1:n], r[(n+1):(2*n)]))
}

## function for simulating new response variables
sfun <- function(obj) {
  n <- obj$n
  fitted <- obj$fitted
  sig1 <- obj$coefs[5]
  sig2 <- obj$coefs[6]
  if(obj$covariance) sig12 <- obj$coefs[7] else sig12 <- 0
  Sig1 <- diag(rep(sig1), n)
  Sig2 <- diag(rep(sig2), n)
  Sig12 <- diag(rep(sig12), n)
  V <- rbind(cbind(Sig1, Sig12),
             cbind(Sig12, Sig2))
  Y <- as.numeric(rmvnorm(1, c(mu1, mu2), V))
  return(list(Y[1:n], Y[(n+1):(2*n)], "X" = obj$X, 
              "covariance" = obj$covariance))
}

## function for refitting the model to simulated data
ffun <- function(new.obj) {
  Ynew <- c(new.obj[[1]], new.obj[[2]])
  bivnormfit(Ynew, new.obj$X, new.obj$covariance)
}

## Bivariate residual plot for model 1 (without estimating covariance)
plot1 <- bivrp(fit0, diagfun=dfun, simfun=sfun, fitfun=ffun, verb=TRUE)
## without polygon area reduction
plot(plot1, conf=1)
## drawing polygons
plot(plot1, add.polygon=TRUE)
## without ordering
plot(plot1, theta.sort=FALSE, kernel=TRUE, add.dplots=TRUE, superpose=TRUE)

## Bivariate residual plot for model 2 (estimating covariance)
plot2 <- bivrp(fit1, diagfun=dfun, simfun=sfun, fitfun=ffun, verb=TRUE)
## without polygon area reduction
plot(plot2, conf=1)
## drawing polygons
plot(plot2, add.polygon=TRUE, conf=1)
## without ordering
plot(plot2, theta.sort=FALSE, kernel=TRUE, add.dplots=TRUE, superpose=TRUE)