Measurement error distributions

Truncated Normal and Uniform distributions, where the response is also subject to a Normally distributed measurement error.

Usage

dmenorm(
  x,
  mean = 0,
  sd = 1,
  lower = -Inf,
  upper = Inf,
  sderr = 0,
  meanerr = 0,
  log = FALSE
)

pmenorm(
  q,
  mean = 0,
  sd = 1,
  lower = -Inf,
  upper = Inf,
  sderr = 0,
  meanerr = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

qmenorm(
  p,
  mean = 0,
  sd = 1,
  lower = -Inf,
  upper = Inf,
  sderr = 0,
  meanerr = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

rmenorm(n, mean = 0, sd = 1, lower = -Inf, upper = Inf, sderr = 0, meanerr = 0)

dmeunif(x, lower = 0, upper = 1, sderr = 0, meanerr = 0, log = FALSE)

pmeunif(
  q,
  lower = 0,
  upper = 1,
  sderr = 0,
  meanerr = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

qmeunif(
  p,
  lower = 0,
  upper = 1,
  sderr = 0,
  meanerr = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

rmeunif(n, lower = 0, upper = 1, sderr = 0, meanerr = 0)

Arguments

x, q: vector of quantiles.
mean: vector of means.
sd: vector of standard deviations.
lower: lower truncation point.
upper: upper truncation point.
sderr: Standard deviation of measurement error distribution.
meanerr: Optional shift for the measurement error distribution.
log, log.p: logical; if TRUE, probabilities $p$ are given as $\log(p)$, or log density is returned.
lower.tail: logical; if TRUE (default), probabilities are $P[X <= x]$, otherwise, $P[X > x]$.
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.

Value

dmenorm, dmeunif give the density, pmenorm, pmeunif give the distribution function, qmenorm, qmeunif give the quantile function, and rmenorm, rmeunif generate random deviates, for the Normal and Uniform versions respectively.

Details

The normal distribution with measurement error has density

$$ \frac{\Phi(u, \mu_2, \sigma_3) - \Phi(l, \mu_2, \sigma_3)}{\Phi(u, \mu_2, \sigma_3) - \Phi(l, \mu_2, \sigma_3)} \phi(x, \mu_0 + \mu_\epsilon, \sigma_2)$$

where

$$\sigma_2^2 = \sigma_0^2 + \sigma_\epsilon^2,$$

$$\sigma_3 = \sigma_0 \sigma_\epsilon / \sigma_2,$$

$$\mu_2 = (x - \mu_\epsilon) \sigma_0^2 + \mu_0 \sigma_\epsilon^2, $$

$\mu_0$ is the mean of the original Normal distribution before truncation,
$\sigma_0$ is the corresponding standard deviation,
$u$ is the upper truncation point,
$l$ is the lower truncation point,
$\sigma_\epsilon$ is the standard deviation of the additional measurement error,
$\mu_\epsilon$ is the mean of the measurement error (usually 0).
$\phi(x)$ is the density of the corresponding normal distribution, and
$\Phi(x)$ is the distribution function of the corresponding normal distribution.

The uniform distribution with measurement error has density

$$(\Phi(x, \mu_\epsilon+l, \sigma_\epsilon) - \Phi(x, \mu_\epsilon+u, \sigma_\epsilon)) $$$$ / (u - l)$$

These are calculated from the original truncated Normal or Uniform density functions $f(. | \mu, \sigma, l, u)$ as

$$ \int f(y | \mu, \sigma, l, u) \phi(x, y + \mu_\epsilon, \sigma_\epsilon) dy $$

If sderr and meanerr are not specified they assume the default values of 0, representing no measurement error variance, and no constant shift in the measurement error, respectively.

Therefore, for example with no other arguments, dmenorm(x), is simply equivalent to dtnorm(x), which in turn is equivalent to dnorm(x).

These distributions were used by Satten and Longini (1996) for CD4 cell counts conditionally on hidden Markov states of HIV infection, and later by Jackson and Sharples (2002) for FEV1 measurements conditionally on states of chronic lung transplant rejection.

These distribution functions are just provided for convenience, and are not optimised for numerical accuracy or speed. To fit a hidden Markov model with these response distributions, use a hmmMETNorm or hmmMEUnif constructor. See the hmm-dists help page for further details.

References

Satten, G.A. and Longini, I.M. Markov chains with measurement error: estimating the 'true' course of a marker of the progression of human immunodeficiency virus disease (with discussion) Applied Statistics 45(3): 275-309 (1996)

Jackson, C.H. and Sharples, L.D. Hidden Markov models for the onset and progression of bronchiolitis obliterans syndrome in lung transplant recipients Statistics in Medicine, 21(1): 113–128 (2002).

Author

C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk

Examples


## what does the distribution look like?
x <- seq(50, 90, by=1)
plot(x, dnorm(x, 70, 10), type="l", ylim=c(0,0.06)) ## standard Normal
lines(x, dtnorm(x, 70, 10, 60, 80), type="l")       ## truncated Normal
## truncated Normal with small measurement error
lines(x, dmenorm(x, 70, 10, 60, 80, sderr=3), type="l")