selectiongain (version 2.0.1)

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

Description

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

Usage

multistagegain.each(Q, corr, alg, lim.y)

Arguments

Q

(length n) refers to the coordinates of the truncation points Q, which is the output of the next function (multistagetp) that we are going to introduced.

corr

(n+1-dimensional matrix) is the correlation matrix of y and X. The correlation matrix must be symmetric and positive-definite. Before starting the calculations, the user is recommended to check the correlation matrix, which is usually obtained by analysis

alg

is used to switch between two algorithms. If alg = GenzBretz(), which is by default, the quasi-Monte Carlo algorithm from Genz(1999) will be used. If alg = Miwa(), the program will use the Miwa algorithm (Mi2009), which an analytical solution of the MVN i

lim.y

is the lower limit of y and is set to -200 as default, which is on the safe side.

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 $\Delta G(y)$, which is described by Cochran (1951). For one-stage selection the gain is defined as $\Delta G (y) = i \rho_{y} \rho_{1}$, where $i$ is the selection intensity, $\rho_{1}$ is the correlation between the true breeding value and the selection index $y$ (Utz 1969). More details are in the JSS paper section 3.2.

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 0.9-9, 2010. 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. Mehrstufenselecktion 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. X. Mi, H.F. Utz. and A.E. Melchinger. R package selectiongain: A tool for efficient calculation and optimization of the expected gain from multi-stage selection. J Stat Softw. (submitted)

See Also

selectiongain

Examples

k=c(-200,0.4308,0.9804,1.8603)

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,stages=TRUE)

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