Variance Functions

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

Usage

mt_variance_function(mu, power, Ntrial, variance,
                           derivative_power, derivative_mu)

mt_tweedie(mu, power, Ntrial, derivative_power, derivative_mu)

mt_binomialP(mu, power, Ntrial,
                    derivative_power, derivative_mu)

mt_binomialPQ(mu, power, Ntrial,
                     derivative_power, derivative_mu)

mt_constant(mu, power, Ntrial, derivative_power, derivative_mu)

Source

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

Arguments

mu: a numeric vector. In general the output from mt_link_function.
power: a numeric value (tweedie 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 (constant, tweedie, binomialP or binomialPQ) of the variance function.
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.

Value

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

Details

The function mt_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 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.

Author

Wagner Hugo Bonat, wbonat@ufpr.br

Examples

x1 <- seq(-1, 1, l = 5)
X <- model.matrix(~x1)
mu <- mt_link_function(beta = c(1, 0.5), X = X, offset = NULL,
                       link = "logit")
mt_variance_function(mu = mu$mu, power = c(2, 1), Ntrial = 1,
                     variance = "binomialPQ",
                     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_inv_sqrt_mu
#> [1] -0.7377650  0.2239011  1.2957812  2.6134777  4.3463022
#>