selectiongain (version 2.0.50.1)

multistagegain: Function for calculating the expected multi-stage selection gain

Description

This is the main function of the package and uses the following equation given by Tallis (1961) for y, which the true genotypic value is: $\frac{\partial m(\textbf{t})}{\partial t_0}|_{\textbf{t}=\textbf{0}}= E(X_0=y) =\frac{1}{\alpha} \sum_{k=0}^{n} \rho_{0,k} \, \phi_1(q_k) \, \Phi_{n} (A_{k,s};R_k)$ to calculate the expected selection gain defined by Cochran (1951) for given correlation matrix and coordinates of the truncation points.

Usage

multistagegain(corr, Q, alg,parallel)

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 funct

Q

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

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.

parallel

is a logical variable to desided if the multiple cores can be used for computing, by default is FALSE. The users have to notice that assign cores also cost time. So this procedure can only be efficient if the dim >5.

Value

• The returned value is the expected gain of selection.

Details

This function calculates the well-known selection gain $\Delta G$, which is described by Cochran (1951), for multi-stage selection. For one-stage selection the gain is defined as $\Delta G = i \delta_y \rho_{1}$, where $i$ is the selection intensity, $\rho_{1}$ is the correlation between the true breeding value, which has variance $\delta_y^2$, 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. 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

No link

Examples

Q=c(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(corr=corr,Q=Q)