multistagegain.each {selectiongain}

Function for calculating the selection gain in each stage

Description

In some situations, the user wants to know the increase of \Delta G in each stage so that it is possible to determine the stage which contributes most to \Delta G. This function calculates \Delta G stepwise for each stage.

Usage

multistagegain.each(corr, Q, alg, Vg)

Arguments

corr

is the correlation matrix of y and X, which is introduced in the function multistagecorr. The correlation matrix must be symmetric and positive-definite. If the estimated correlation matrix is negative-definite, it must be adjusted before using this function. Before starting the calculations, it is recommended to check the correlation matrix.

Q

are the coordinates of the truncation points, which are the output of the function multistagetp that we are going to introduce.

Vg

correspond to the genetic variance or variance of the GCA effects. The default value is 1

alg

is used to switch between two algorithms. If alg = GenzBretz(), which is by default, the quasi-Monte Carlo algorithm from Genz et al. (2009, 2013), will be used. If alg = Miwa(), the program will use the Miwa algorithm (Mi et al., 2009), which is an analytical solution of the MVN integral. Miwa's algorithm has higher accuracy (7 digits) than quasi-Monte Carlo algorithm (5 digits). However, its computational speed is slower. We recommend to use the Miwa algorithm.

Details

This function calculates the well-known selection gain \Delta G, which is described by Cochran (1951), for each stage.

Value

The output is given as (\Delta G_1(y), \Delta G_2(y)-\Delta G_1(y), \Delta G_3(y)-\Delta G_2(y), ...) where \Delta G_i(y) refers to the total selection gain after the first i stages of selection.

Author(s)

Xuefei Mi

References

A. Genz and F. Bretz. Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics, Vol. 195, Springer-Verlag, Heidelberg, 2009.

A. Genz, F. Bretz, T. Miwa, X. Mi, F. Leisch, F. Scheipl and T. Hothorn. mvtnorm: Multivariate normal and t distributions. R package version 0.9-9995, 2013.

G.M. Tallis. Moment generating function of truncated multi-normal distribution. J. Royal Stat. Soc., Ser. B, 23(1):223-229, 1961.

H.F. Utz. Mehrstufenselektion in der Pflanzenzuechtung (in German). Doctor thesis, University Hohenheim, 1969.

W.G. Cochran. Improvement by means of selection. In J. Neyman (ed.) Proc. 2nd Berkeley Symp. on Mathematical Statistics and Probability. University of California Press, Berkeley, 1951.

X. Mi, T. Miwa and T. Hothorn. Implement of Miwa's analytical algorithm of multi-normal distribution. R Journal, 1:37-39, 2009.

See Also

selectiongain()

Examples

# example 1

corr=matrix( c(1,      0.3508,0.3508,0.4979,
               0.3508, 1,     0.3016,0.5630,
               0.3508, 0.3016,1,     0.5630,
               0.4979, 0.5630,0.5630,1),
              nrow=4
)

multistagegain.each(Q=c(0.4308,0.9804,1.8603),corr=corr)

# examples 2

 alpha1<- 1/24
 alpha2<- 1
 Q=multistagetp(alpha=c(alpha1,alpha2),corr=corr[2:3,2:3])


corr=matrix( c(1,        0.7071068,0.9354143,
               0.7071068,1,        0.7559289,
               0.9354143,0.7559289,1),
              nrow=3
)

multistagegain.each(Q=Q,corr=corr)