AIC- or AICc-weighted average of estimated `real' or `beta' parameters from multiple fitted openCR models.
The modelAverage generic is imported from secr (>= 4.5.0).
# S3 method for openCR
modelAverage(object, ..., realnames = NULL, betanames = NULL,
newdata = NULL, alpha = 0.05, dmax = 10, covar = FALSE, average = c("link",
"real"), criterion = c("AIC","AICc"), CImethod = c("Wald", "MATA"))
# S3 method for openCRlist
modelAverage(object, ..., realnames = NULL, betanames = NULL,
newdata = NULL, alpha = 0.05, dmax = 10, covar = FALSE, average = c("link",
"real"), criterion = c("AIC","AICc"), CImethod = c("Wald", "MATA"))
A list (one component per parameter) of model-averaged estimates, their
standard errors, and a \(100(1-\alpha)\)% confidence
interval. The interval for real parameters is backtransformed from the
link scale. If there is only one row in newdata or beta
parameters are averaged or averaging is requested for only one parameter
then the array is collapsed to a matrix. If covar = TRUE then a
list is returned with separate components for the estimates and the
variance-covariance matrices.
openCR or openCRlist objects
other openCR objects (modelAverage.openCR() only)
character vector of real parameter names
character vector of beta parameter names
optional dataframe of values at which to evaluate models
alpha level for confidence intervals
numeric, the maximum AIC or AICc difference for inclusion in confidence set
logical, if TRUE then return variance-covariance matrix
character string for scale on which to average real parameters
character, information criterion to use for model weights
character, type of confidence interval (see Details)
Models to be compared must have been fitted to the same data and use the
same likelihood method (full vs conditional). If realnames =
NULL and betanames = NULL then all real parameters will be
averaged; in this case all models must use the same real parameters. To
average beta parameters, specify betanames (this is ignored if a
value is provided for realnames). See predict.openCR
for an explanation of the optional argument newdata;
newdata is ignored when averaging beta parameters.
Model-averaged estimates for parameter \(\theta\) are given by
$$\hat{\theta} = \sum\limits _k w_k \hat{\theta}_k$$ where the subscript \(k\) refers to a specific
model and the \(w_k\) are AIC or AICc weights (see
AIC.openCR for details). Averaging of real parameters may be
done on the link scale before back-transformation
(average="link") or after back-transformation
(average="real").
Models for which dAIC > dmax (or dAICc > dmax) are given a
weight of zero and effectively are excluded from averaging.
Also, $$\mbox{var} (\hat{\theta}) = \sum\limits _{k} { w_{k} ( \mbox{var}(\hat{\theta}_{k} | \beta _k) + \beta _k ^2)} $$
where \(\hat{\beta} _k = \hat{\theta}_k - \hat{\theta}\) and the variances are asymptotic estimates from fitting each model \(k\). This follows Burnham and Anderson (2004) rather than Buckland et al. (1997).
Two methods are offered for confidence intervals. The default `Wald'
uses the above estimate of variance. The alternative `MATA'
(model-averaged tail area) avoids estimating a weighted variance and
is thought to provide better coverage at little cost in increased
interval length (Turek and Fletcher 2012). Turek and Fletcher (2012)
also found averaging with AIC weights (here criterion = 'AIC')
preferable to using AICc weights, even for small
samples. CImethod does not affect the reported standard errors.
Buckland S. T., Burnham K. P. and Augustin, N. H. (1997) Model selection: an integral part of inference. Biometrics 53, 603--618.
Burnham, K. P. and Anderson, D. R. (2002) Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. Second edition. New York: Springer-Verlag.
Burnham, K. P. and Anderson, D. R. (2004) Multimodel inference - understanding AIC and BIC in model selection. Sociological Methods & Research 33, 261--304.
Turek, D. and Fletcher, D. (2012) Model-averaged Wald confidence intervals. Computational statistics and data analysis 56, 2809--2815.
AIC.openCR,
make.table,
openCR.fit,
openCRlist
## Compare two models fitted previously
cjs1 <- openCR.fit(dipperCH, model=p~1)
cjs2 <- openCR.fit(dipperCH, model=p~session)
AIC(cjs1, cjs2)
modelAverage(cjs1, cjs2)
## or
cjs12 <- openCRlist(cjs1, cjs2)
modelAverage(cjs12)
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