Variance Functions

Source: R/mc_variance_function.R

Compute the variance function and its derivatives with respect to regression, dispersion and power parameters.

mc_variance_function(mu, power, Ntrial, variance, inverse,
                           derivative_power, derivative_mu)

mc_power(mu, power, inverse, derivative_power, derivative_mu)

mc_binomialP(mu, power, inverse, Ntrial,
                    derivative_power, derivative_mu)

mc_binomialPQ(mu, power, inverse, Ntrial,
                     derivative_power, derivative_mu)

Arguments

mu

a numeric vector. In general the output from mc_link_function.
power

a numeric value (power and binomialP) or a vector (binomialPQ) of the power parameters.
Ntrial

number of trials, useful only when dealing with binomial response variables.
variance

a string specifying the name (power, binomialP or binomialPQ) of the variance function.
inverse

logical. Compute the inverse or not.
derivative_power

logical if compute (TRUE) or not (FALSE) the derivatives with respect to the power parameter.
derivative_mu

logical if compute (TRUE) or not (FALSE) the derivative with respect to the mu parameter.

Source

Bonat, W. H. and Jorgensen, B. (2016) Multivariate covariance generalized linear models. Journal of Royal Statistical Society - Series C 65:649--675.

Value

A list with from one to four elements depends on the arguments.

Details

The function mc_variance_function computes three features related with the variance function. Depending on the logical arguments, the function returns \(V^{1/2}\) and its derivatives with respect to the parameters power and mu, respectivelly. The output is a named list, completely informative about what the function has been computed. For example, if inverse = FALSE, derivative_power = TRUE and derivative_mu = TRUE. The output will be a list, with three elements: V_sqrt, D_V_sqrt_power and D_V_sqrt_mu.

See also

mc_link_function.

Examples

x1 <- seq(-1, 1, l = 5)
X <- model.matrix(~x1)
mu <- mc_link_function(beta = c(1, 0.5), X = X, offset = NULL,
                       link = "logit")
mc_variance_function(mu = mu$mu, power = c(2, 1), Ntrial = 1,
                     variance = "binomialPQ", inverse = FALSE,
                     derivative_power = TRUE, derivative_mu = TRUE)
#> $V_sqrt
#> 5 x 5 diagonal matrix of class "ddiMatrix"
#>      [,1]     [,2]      [,3]      [,4]      [,5]     
#> [1,] 0.382466         .         .         .         .
#> [2,]        . 0.3846942         .         .         .
#> [3,]        .         . 0.3791238         .         .
#> [4,]        .         .         . 0.3668165         .
#> [5,]        .         .         .         . 0.3491967
#> 
#> $D_V_sqrt_p
#> 5 x 5 diagonal matrix of class "ddiMatrix"
#>      [,1]        [,2]        [,3]        [,4]        [,5]       
#> [1,] -0.09065917           .           .           .           .
#> [2,]           . -0.07441352           .           .           .
#> [3,]           .           . -0.05938248           .           .
#> [4,]           .           .           . -0.04620587           .
#> [5,]           .           .           .           . -0.03516643
#> 
#> $D_V_sqrt_q
#> 5 x 5 diagonal matrix of class "ddiMatrix"
#>      [,1]       [,2]       [,3]       [,4]       [,5]      
#> [1,] -0.1862757          .          .          .          .
#> [2,]          . -0.2186738          .          .          .
#> [3,]          .          . -0.2489444          .          .
#> [4,]          .          .          . -0.2754662          .
#> [5,]          .          .          .          . -0.2970639
#> 
#> $D_V_sqrt_mu
#> [1]  0.10792045 -0.03313504 -0.18624891 -0.35165473 -0.52998085
#>