selectiongain (version 2.0.22)

multistagegain.each: Function for calculating the selection gain in each stage

Description

In some situations, the user wants to know the increase of $\Delta G$ for 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, Vg, alg)

Arguments

corr

is the correlation matrix of y and X, which is introduced in function multistagecorr. The correlation matrix must be symmetric and positive-definite. Before starting the calculations, the user is recommended to check the correlation matrix.

Q

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

Vg

is true breeding value. 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 (2009) will be used. If alg = Miwa(), the program will use the Miwa algorithm (Mi et. al., 2009), whic

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.

Details

This function calculates the well-known selection gain $\Delta G$, which is described by Cochran (1951). For one-stage selection the gain is defined as $\Delta G = i V_g \rho_{1}$, where $i$ is the selection intensity, $\rho_{1}$ is the correlation between the true breeding value (its variance is $V_g$) and the selection index (Utz 1969).

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. Journal of the Royal Statistical Society, Series B, 23(1):223-229, 1961. H.F. Utz. Mehrstufenselektion in der Pflanzenzuechtung. Doctor thesis, University Hohenheim, 1969. W.G. Cochran. Improvent by means of selection. In: Proceedings Second Berkeley Symposium on Math Stat Prof, pp449-470, 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

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)

# further examples 3 for the JSS paper

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

k=c(-200,Q)

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

alphaofx=pmvnorm(lower=k,corr=corr)

multistagegain(Q=Q,corr=corr)

multistagegain(Q=Q,corr=corr,partial=TRUE)

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