calcStandardErrors: Standard errors for predictions

In LMMsolver: Linear Mixed Model Solver

Standard errors for predictions

Description

Calculates the standard errors for predictions D \hat{u}, see Welham et al. 2004 and Gilmour et al. 2004 for details.

Usage

calcStandardErrors(C, D)

Arguments

C	a symmetric matrix of class spam
D	a matrix of class spam

Details

The prediction error variance is given by D C^{-1} D', where C is the mixed model coefficient matrix, and D defines linear combinations of fixed and random effects. The standard errors are given by the the square root of the diagonal. To calculate the standard errors in an efficient way we use that

\frac{\partial log|C + \xi_i d_i d_i'|}{\partial \xi_i} |_{\xi_i=0} = trace(C^{-1} d_i d_i') = trace(d_i' C^{-1} d_i) = d_i' C^{-1} d_i,

where d_i is row i of matrix D. The values of d_i' C^{-1} d_i can be calculated more efficient, avoiding the calculation of the inverse of C, by using Automated Differentiation of the Choleksy algorithm, see section 2.3 in Smith (1995) for details.

Value

a vector with standard errors for predictions D \hat{u}.

References

Welham, S., Cullis, B., Gogel, B., Gilmour, A., & Thompson, R. (2004). Prediction in linear mixed models. Australian & New Zealand Journal of Statistics, 46(3), 325-347.

Smith, S. P. (1995). Differentiation of the Cholesky algorithm. Journal of Computational and Graphical Statistics, 4(2), 134-147.

Gilmour, A., Cullis, B., Welham, S., Gogel, B., & Thompson, R. (2004). An efficient computing strategy for prediction in mixed linear models. Computational statistics & data analysis, 44(4), 571-586.

LMMsolver documentation built on Sept. 11, 2024, 7:31 p.m.