RMark (version 3.0.0)

mata.wald: Model-Averaged Tail Area Wald (MATA-Wald) confidence intervals

Description

A generic function to compute model averaged estimates and their standard errors or variance-covariance matrix model-averaged tail area (MATA) construction.

Usage

mata.wald(theta.hats, se.theta.hats, model.weights, normal.lm=FALSE, 
                          residual.dfs=0, alpha=0.025)
 
       tailarea.z(theta, theta.hats, se.theta.hats, model.weights, alpha)
       tailarea.t(theta, theta.hats, se.theta.hats, model.weights, alpha, residual.dfs)

Arguments

theta.hats

A vector containing the estimates of theta under each candidate model.

se.theta.hats

A vector containing the estimated standard error of each estimate in 'theta.hats'.

model.weights

A vector containing the model weights for each candidate model. Calculated from an information criterion, such as AIC. See Turek and Fletcher (2012) for details of calculation. All model weights must be non-negative, and sum to one.

normal.lm

Specify normal.lm=TRUE for the normal linear model case, and normal.lm=FALSE otherwise. When normal.lm=TRUE, the argument 'residual.dfs' must also be supplied. See USAGE section, and Turek and Fletcher (2012) for additional details.

residual.dfs

A vector containing the residual (error) degrees of freedom under each candidate model. This argument must be provided when the argument normal.lm=TRUE.

alpha

The desired lower and upper error rate. Specifying alpha=0.025 corresponds to a 95 alpha=0.05 to a 90 Default value is alpha=0.025.

theta

value for root finding in tailarea.z and tailarea.t

Author

Daniel Turek<[e-mail]>

Details

The main function, mata.wald(...), may be used to construct model-averaged confidence intervals, using the model-averaged tail area (MATA) construction. The idea underlying this construction is similar to that of a model-averaged Bayesian credible interval. This function returns the lower and upper confidence limits of a MATA-Wald interval.

Two usages are supported. For the normal linear model case, and quantity of interest theta, > mata.wald(theta.hats, se.theta.hats, model.weights, alpha, normal.lm=TRUE, residual.dfs) returns a (1-2*alpha)100 Corresponds to the solutions of equations (2) and (3) of Turek and Fletcher (2012). The argument 'residual.dfs' is required for this usage.

When the sampling distribution for the estimate of theta is asymptotically normal (e.g. MLEs), possibly after a transformation, > mata.wald(theta.hats, se.theta.hats, model.weights, alpha, normal.lm=FALSE) returns a (1-2*alpha)100 on a transformed scale. Back-transformation of both confidence limits may be necessary. Corresponds to solutions to the equations in Section 3.2 of Turek and Fletcher (2012).

References

Turek, D. and Fletcher, D. (2012). Model-Averaged Wald Confidence Intervals. Computational Statistics and Data Analysis, 56(9), p.2809-2815.

Examples

# The example code below, uncommented, generates single-model Wald 
# and model-averaged MATA-Wald 95% confidence intervals for theta.
#
#    EXAMPLE: Normal linear prediction
#    =================================
#
# Data 'y', covariates 'x1' and 'x2', all vectors of length 'n'.
# 'y' taken to have a normal distribution.
# 'x1' specifies treatment/group (factor).
# 'x2' a continuous covariate.
#
# Take the quantity of interest (theta) as the predicted response 
# (expectation of y) when x1=1 (second group/treatment), and x2=15.

n = 20                              # 'n' is assumed to be even
x1 = c(rep(0,n/2), rep(1,n/2))      # two groups: x1=0, and x1=1
x2 = rnorm(n, mean=10, sd=3)
y = rnorm(n, mean = 3*x1 + 0.1*x2)  # data generation

x1 = factor(x1)
m1 = glm(y ~ x1)                    # using 'glm' provides AIC values.
m2 = glm(y ~ x1 + x2)               # using 'lm' doesn't.
aic = c(m1$aic, m2$aic)
delta.aic = aic - min(aic)
model.weights = exp(-0.5*delta.aic) / sum(exp(-0.5*delta.aic))
residual.dfs = c(m1$df.residual, m2$df.residual)

p1 = predict(m1, se=TRUE, newdata=list(x1=factor(1), x2=15))
p2 = predict(m2, se=TRUE, newdata=list(x1=factor(1), x2=15))
theta.hats = c(p1$fit, p2$fit)
se.theta.hats = c(p1$se.fit, p2$se.fit)

#  AIC model weights
model.weights

#  95% Wald confidence interval for theta (under Model 1)
theta.hats[1] + c(-1,1)*qt(0.975, residual.dfs[1])*se.theta.hats[1]

#  95% Wald confidence interval for theta (under Model 2)
theta.hats[2] + c(-1,1)*qt(0.975, residual.dfs[2])*se.theta.hats[2]

#  95% MATA-Wald confidence interval for theta (model-averaging)
mata.wald(theta.hats=theta.hats, se.theta.hats=se.theta.hats, 
        model.weights=model.weights, normal.lm=TRUE, residual.dfs=residual.dfs)