post is a generic method for computing posterior probabilities of a fitted model.
post.moc computes the posterior mixture probabilities for each
subject of a fitted moc model.
residuals.moc computes response, deviance, gradient and mixture
residuals. The residuals are optionally weighted by the posterior
mixture probabilities, globally (with post) or within each
group (in that case post is divided by its mean for each group).
npmle.gradient computes the components and average of the
mixture gradient function at some specified parameters values
$$\frac{1}{n}\sum_{i=1}^n\Bigl{(}\frac{\hat{h}_j(y_i|\,x_i)}
{\sum_k\hat{\pi}_k(z_i,x_i)\,\hat{h}_k(y_i|\,x_i)} - 1\Bigr{)}$$
or the empirical mixture gradient which is
$$\frac{1}{n}\sum_{i=1}^n\Bigl{(}\frac{d\hat{F}(y_i)}{\sum_k\hat{\pi}_k(z_i,x_i)\,
\hat{h}_k(y_i|\,x_i)} - 1\Bigr{)}$$
where \(d\hat{F}()\) is the empirical measure.
post(object, …)
# S3 method for moc
post(object, …)# S3 method for moc
residuals(object, …,
type = c("deviance","response","mixture","gradient"),
post.weight = TRUE, within = FALSE)
npmle.gradient(object, parm = object$coef, gradient = TRUE,
average = FALSE)
Object of class moc.
Type of residuals: either "deviance" (the default), "response", "gradient" and "mixture" (see description and details).
Specify if the residuals must be weighted by the posterior mixture probabilities. Weighting is preferable, it is the default.
Boolean that specify if the posterior weights are rescaled within each mixture group.
Parameters values at which evaluation of the gradient takes place.
Boolean specifying if you require the density for each
mixture group (TRUE) or the empirical measure (FALSE) in
the numerator of npmle.gradient (see the description above).
Boolean that specify if npmle.gradient should return
the individual components or the average.
Unused.
residuals.moc returns an array of class
residuals.moc and residuals with attributes type,
post.weight and within. All these methods return their
values invisibly.
Posterior probabilities are the conditional probabilities of mixture groups given the subject response observations and are computed using the formula: $$\hat{\tau}_{i,k} = \frac{\hat{\pi}_k(z_i,x_i)\, \hat{h}_k( y_i | x_i)}{\sum_k \hat{\pi}_k(z_i, x_i)\, \hat{h}_k( y_i | x_i)}$$
Response residuals are simply the difference between the observed and expected values, $$response = y - expected$$
Deviance residuals are defined as properly scaled difference in the log likelihood at the observed and fitted value. $$deviance = \Bigl{\lbrace}2\cdot wt\cdot \log{\Bigl{(}\frac{\mathrm{density}(y,y,shape,extra)} {\mathrm{density}(y,mu,shape,extra)}\Bigr{)}}\Bigr{\rbrace}^{1/2} \cdot \mathrm{sign}(response)$$
The npmle.gradient function is primarily intended to compute the
components which are used to define the mixture and gradient residuals in
residuals.moc. Mixture residuals uses the empirical gradient
components while gradient residuals uses the mixture gradient
components defined in the description above. The average of the first
form above which we call the gradient can be used to check some
minimal requirement about a proposed solution given with parm: if it
is a maximum likelihood (not necessarily a non-parametric maximum
likelihood) the average should be \(0\).
See Lindsay, B.G. (1983) for details about the use of the gradient
function for finite mixture and non-parametric mixture and Lindsay,
B.G. and Roeder, K. (1992) for residuals diagnostics specific to mixture
models.
Globally weighted residuals are preferable to detect influential data, wrong number of groups and differences between groups. Rescaled weight residuals are more useful when plotted against some variables or variable index to detect misspecified regression function or density.
McLachlan, G. and Peel, D. (2000) Finite mixture models, Wiley-Interscience, New York.
Lindsay, B. G. and Roeder, K. (1992) Residual diagnostics for mixture models, Journal of the American Statistical Association, 87, pp. 785--794.