medists: Measurement error distributions In msm: Multi-State Markov and Hidden Markov Models in Continuous Time

Description

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

Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ``` 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. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `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].

Details

The normal distribution with measurement error has density

(Phi(upper, mu2, sigma3) - Phi(lower, mu2, sigma3)) / (Phi(upper, mean, sd) - Phi(lower, mean, sd)) * phi(x, mean + meanerr, sigma2)

where

sigma2*sigma2 = sd*sd + sderr*sderr,

sigma3 = sd*sderr / sigma2,

mu2 = (x - meanerr)*sd*sd + mean*sderr*sderr,

mean is the mean of the original Normal distribution before truncation,
sd is the corresponding standard deviation,
u is the upper truncation point,
l is the lower truncation point,
sderr is the standard deviation of the additional measurement error,
meanerr 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, meanerr+l, sderr) - Phi(x, meanerr+u, sderr)) / (upper - lower)

These are calculated from the original truncated Normal or Uniform density functions f(. | mu, sd) as

integral f(y | mu, sd, l, u) phi(x, y + meanerr, sderr) 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.

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.

Author(s)

C. H. Jackson [email protected]

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).

`dnorm`, `dunif`, `dtnorm`
 ```1 2 3 4 5 6``` ```## 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") ```