merror (version 3.0)

omx: Compute full information maximum likelihood estimates of accuracy and precision for the nonconstant bias measurement error model using 'OpenMx'.

Description

Compute full information maximum likelihood (FIML) estimates of accuracy (bias) and precision (imprecision) for the nonconstant bias measurement error model. 'OpenMx' functions are used to construct and fit a one latent factor model. Likelihood-based confidence intervals and bootstrapped confidence interval are can be determined for all model parameters and relevant functions of the model parameters.

Usage

omx(data, rvEst=rep(1,ncol(data)), mubarEst=mean(data[,1]), interval=0.95, 
  reps=500,bs.q=c(0.025,0.975), bs=TRUE)

Value

fit

'OpenMx' fit object containing the results (FIML parameter estimates, etc)

ci

Likelihood-based confidence intervals for all parameters and certain useful functions of parameters.

boot

Object created by 'mxBootstrap' 'OpenMx' function. Not returned if bs=FALSE.

q.boot

data.frame containing the standard error and quantile estimates based on bootstrapped samples. Not returned if bs=FALSE.

abs

A \(3 \times N\) matrix of the estimated alphas, betas, and the raw imprecision standard deviations for each of the \(N\) methods. The results can be passed to the merror cplot to produce a calibration plot.

bsReps

Number of bootstrapped samples. Default is 500. Not returned if bs=FALSE.

model

The 'OpenMx' one-factor model.

Arguments

data

\(n\) (no. of items) \(\times N\) (no. of methods) matrix or data.frame containing the measurements. \(N\) must be \(\ge 3\). Missing values are allowed.

rvEst

A vector of \(N\) residual variance starting values, one for each corresponding method.

mubarEst

A scalar starting value for estimating the the true mean value.

interval

Confidence level for likelihood-based confidence intervals. Should be a scalar value greater than 0 and less than 1.

reps

Number of bootstrap samples. Ignored if bs=FALSE.

bs.q

A vector of desired quantiles for bootstrapped samples. Default is ci.q=c(0.025,0.975).

bs

A boolean indicating whether bootstapped samples are to be generated. Default is TRUE.

Author

Richard A. Bilonick

Details

Measurement Error Model:

$$x_{ik}=\alpha_i+\beta_i\mu_k+\epsilon_{ik}$$

where \(x_{ik}\) is the measurement by the ith of \(N\) methods for the kth of \(n\) items, \(i = 1\) to \(N\ge 3\), \(k = 1\) to \(n\), \(\mu_k\) is the true value for the kth item, \(\epsilon_{ik}\) is the normally distributed random error with variance \(\sigma_i^2\) for the ith method and the kth item, and \(\alpha_i\) and \(\beta_i\) are the accuracy parameters for the ith method. The beta for the first column of data) is set to one. The corresponding alpha is set to 0. These constraints or similar are required for model identification.

The imprecision for the ith method is \(\sigma_i\). If all alphas are zeroes and all betas are ones, there is no bias. If all betas equal 1, then there is a constant bias. If some of the betas differ from one there is a nonconstant bias. Note that the individual betas are not unique - only ratios of the betas are unique. If you divide all the betas by \(\beta_i\), then the betas represent the scale bias of the other devices/methods relative to device/method \(i\). Also, when the betas differ from one, the sigmas are not directly comparable because the measurement scales (size of the units) differ. To make the sigmas comparable, divide them by their corresponding beta.

Technically, the alphas and betas describe the measurements in terms of the unknown true values (i.e., the unknown true values can be thought of as a latent variable). The "true values" are ALWAYS unknown (unless you have a real, highly accurate reference method/device). The real goal is to calibrate one device/method in terms of another. This is easily accomplished because each measurement is a linear function of the same unknown true values. For methods 1 and 2, the calibration curve is given by:

$$E[x_{1k}]=\left(\alpha_1-\alpha_2\beta_1/\beta_2\right)+\left(\beta_1/\beta_2\right)E[x_{2k}]$$

or equivalently

$$E[x_{2k}]=\left(\alpha_2-\alpha_1\beta_2/\beta_1\right)+\left(\beta_2/\beta_1\right)E[x_{1k}]$$.

Use cplot, with the alpha.beta.sigma argument specified, to display this calibration curve, calibration equation, and the corresponding scale-bias adjusted imprecision standard deviations.

Note that likelihood confidence intervals and bootstrapped confidence intervals can be returned. Wald-type intervals based on the standard errors are alos available by using the confint function on the returned fit object. See examples.

See Also

ncb.od, alpha.beta.sigma

Examples

if (FALSE) {
library(OpenMx)
library(merror)

data(pm2.5)

pm <- pm2.5

# OpenMx does not like periods in data column names
names(pm) <- c('ms_conc_1','ws_conc_1','ms_conc_2','ws_conc_2','frm')

# Fit model with FRM sampler as reference
omxfit <- omx(data=pm[,c(5,1:4)],bs.q=c(0.025,0.5,0.975),reps=100)

# Look at results
summary(omxfit$fit)$parameters[,c(1,5,6)] # Parameter estimates and standard errors
round(omxfit$ci[,1:3],3) # Likelihood-based intervals

# Estimated standard errors and quantiles based on bootstrapped samples
round(omxfit$q.boot,3)

# Wald-type intervals
#   - note variances not standard deviations and different ordering
confint(omxfit$fit) 

omxfit$abs # Use with cplot

# Make a calibration plot using the results from omx instead of the default ncb.od
cplot(pm[,c(5,1:4)],1,2,alpha.beta.sigma=omxfit$abs)
}