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.
multistagegain(corr, Q, alg, parallel, Vg)The returned value is the expected gain of selection.
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.
are the coordinates of the truncation points, which are the output of the function multistagetp that we are going to introduce.
correspond to the genetic variance or variance of the GCA effects. The value entered here is only used during the last multiplication of the expected selection gain times the squared root of the genetic variance or the variance of the GCA effects. The default value is 1, and in this case the breeder is adviced to make the multiplication outside the function, as showed in the example by Mi et al 2014 page 1415
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.
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.
Xuefei Mi
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).
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.
X. Mi, F. Utz, F. Technow and A. E. Melchinger. Optimizing Resource Allocation for Multistage Selection in Plant Breeding with R package selectiongain. Crop Science 54:1413-1418. 2014
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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, alg=Miwa())
# value 1.227475
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