aster2-package {aster2}

Aster Models

Description

Aster models are exponential family graphical models that combine aspects of generalized linear models and survival analysis.

This package is still under development, only about half finished. However, it does do maximum likelihood for unconditional aster models with dependence groups, which the old package aster does not.

The main differences between this package and the old package are as follows.

1. The old package had triple indices for model matrices. The first index ran over individuals, the second index over nodes of the graph for an individual, and the third index over regression coefficients. Consequently the model matrix was represented (sometimes, but not consistently) as a three-dimensional array rather than a matrix, which was very confusing, even to the package author. This package ignores individuals, one index runs over all nodes of the combined graph for all individuals. Thus model matrices are always matrices.

2. The old package did not implement dependence groups, although they were described in Geyer, Wagenius and Shaw (2007). This package does. Consequently, this package requires a data frame, a vector pred that indicates predecessors, a vector group that indicates individuals in the same dependence group, and a vector fam that indicates families to specify a saturated aster model (the old package required only the data frame, pred, and fam). To facilitate the old style model specification, there is a new function asterdata that constructs objects of class "asterdata" given an old style data frame, pred, and fam. All other functions of the package take objects of class "asterdata" as model specifications.

3. The function predict.aster in the old package did some parameter transformations, but not all, and the returned value, when a list, had a component gradient, that was undocumented but useful in applying the delta method. The functions transformSaturated, transformConditional, and transformUnconditional in this package transform between any of the following parameter vectors: the conditional canonical parameter \theta, the unconditional canonical parameter \varphi, the conditional mean value parameter \xi, the unconditional mean value parameter \mu, the canonical affine submodel canonical parameter \beta, and (unconditional aster models only) the canonical affine submodel mean value parameter \tau (this last parameter is new, not discussed in the cited papers below, it is \tau = M^T \mu, where M is the model matrix). The change of parameter from \tau to \beta is equivalent to maximum likelihood estimation for an unconditional aster model when the value \tau = M^T y is used, where y is the response vector. All of these transformation functions also compute derivatives, if requested. See examples.

Bugs

Functions analogous to aster, anova, and predict in the old package are missing, thus model fitting, hypothesis tests, and confidence intervals are more cumbersome. In fact, since there is no function to calculate log likelihoods (like mlogl in the old package), there is no way to do likelihood ratio tests (but Rao or Wald tests could be done, for unconditional aster models, since the derivative of the log likelihood is observed minus expected M^T (y - \mu).

References

Geyer, C. J., Wagenius, S., and Shaw, R. G. (2007) Aster Models for Life History Analysis. Biometrika 94 415–426.

Shaw, R. G., Geyer, C. J., Wagenius, S., Hangelbroek, H. H. and Etterson, J. R. (2008) Unifying Life History Analyses for Inference of Fitness and Population Growth. American Naturalist, 172, E35–E47.

See Also

asterdata, transformSaturated, families

Examples

## Not run: # perfectly good example but takes longer to run than CRAN allows
data(echinacea)
#### estimate MLE (simpler model than in Biometrika paper cited, not as good)
hdct <- as.numeric(grepl("hdct", as.character(echinacea$redata$varb)))
modmat <- model.matrix(resp ~ varb + nsloc + ewloc + pop * hdct - pop,
    data = echinacea$redata)
tau.hat <- as.numeric(t(modmat) %*% echinacea$redata$resp)
beta.hat <- transformUnconditional(tau.hat, modmat, echinacea,
    from = "tau", to = "beta")
inverse.fisher <- jacobian(tau.hat, echinacea, transform = "unconditional",
    from = "tau", to = "beta", modmat = modmat)
#### now have MLE (beta.hat) and pseudo-inverse of Fisher information
#### (inverse.fisher), pseudo-inverse because modmat is not full rank
foo <- cbind(beta.hat, sqrt(diag(inverse.fisher)))
foo <- cbind(foo, foo[ , 1]/foo[ , 2])
foo <- cbind(foo, 2 * pnorm(- abs(foo[ , 3])))
dimnames(foo) <- list(colnames(modmat),
    c("Estimate", "Std. Error", "z value", "Pr(>|z|)"))
printCoefmat(foo)
#### coefficients constrained to be zero because parameterization is not
#### identifiable have estimate zero and std. error zero (and rest NA)

#### estimate fitness in populations
#### generate new data with one individual in each pop at location (0, 0)
pop.names <- levels(echinacea$redata$pop)
pop.idx <- match(pop.names, as.character(echinacea$redata$pop))
pop.id <- echinacea$redata$id[pop.idx]
newdata <- subset(echinacea, echinacea$redata$id %in% pop.id)
newdata$redata[ , "nsloc"] <- 0
newdata$redata[ , "ewloc"] <- 0
hdct <- as.integer(grepl("hdct", as.character(newdata$redata$varb)))
#### modmat for new data
newmodmat <- model.matrix(resp ~ varb + nsloc + ewloc + pop * hdct - pop,
    data = newdata$redata)
#### matrix that when multiplied mean value parameter vector gives fitness
#### in each pop
amat <- matrix(NA, nrow = length(pop.id), ncol = nrow(newmodmat))
for (i in 1:nrow(amat))
    amat[i, ] <- as.numeric(grepl(paste("^", pop.id[i], ".hdct", sep = ""),
        rownames(newmodmat)))
#### transform to expected fitness parameters
efit <- transformUnconditional(beta.hat, newmodmat, newdata,
    from = "beta", to = "mu")
efit <- as.numeric(amat %*% efit)
#### jacobian matrix of this transformation
jack <- jacobian(beta.hat, newdata, transform = "unconditional",
    from = "beta", to = "mu", modmat = newmodmat)
#### delta method standard errors
sefit <- sqrt(diag(amat %*% jack %*% inverse.fisher %*% t(jack) %*% t(amat)))
foo <- cbind(efit, sefit)
dimnames(foo) <- list(pop.names, c("Est. fitness", "Std. Error"))
print(foo)

## End(Not run)