selectiongain (version 2.0.6)

multistagevariance: Expected gain for k-stages selection

Description

This function uses the algorithm described by Tallis (1961) to calculate the variance (second moment) after multi-stage selection.

Usage

multistagevariance(Q, corr, alg, lim.y)

Arguments

Q

Vector with length n. It refers to the coordinates of the truncation points Q, which is the output of the function multistagetp that we are going to introduced.

corr

(n+1)-dimensional matrix. It is the correlation matrix of true value y and selection indices X. More detail see multistagegain.

alg

An object used to switch between two algorithms. More detail see multistagegain.

lim.y

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

Value

• The output is the value of $\psi_n(y;\textbf{S}_{Q})=E(y^2;\textbf{S}_{Q})-[E(y;\textbf{S}_{Q})]^2$.

Details

The theory for multi-stage selection is based on the first moment (mathematical expectation) of the true value $y$ in the selected area Cochran (1951). The $\Delta G(y)$ is defined as the difference of the mathematical expectation of $y$ after selection and the corresponding value before selection. For simplicity, the variance of $y$ is set to $1$, i.e., $\sigma_{y}^2=1$. If $\sigma_{y}^2 \neq1$, the selection gain and the variance among the selected candidates have to be multiplied with $\sigma_{y}$ or $\sigma_{y}^2$, respectively. In order to calculate the expectation of $y$ after $n$ stages of selection, we have to determine the one-sided integral of $y$ over the right-sided area $\textbf{S}_{Q}={x_1>q_1,\ldots,x_n>q_n}$ defined by the truncation point $\textbf{Q}={q_1,q_2,\ldots,q_n}$. The value of $\Delta G_n(y)$ is noted as $E(Y;\textbf{S}_{Q})$. We have to distinguish it with $E(Y)$, which is the integral of $y$ over the whole area $\textbf{S}={x_1>-\infty,\ldots,x_n>-\infty}$ with $\textbf{Q}={-\infty,\ldots,-\infty}$ as the truncation point. Thus, the expectation of $y$ after $n$ stages of selection is, $$\Delta G_n(y, \textbf{S}_{Q}) = E(Y;\textbf{S}_{Q})= \alpha^{-1} \int_{-\infty} ^\infty \int_{q_{1}}^\infty\ldots\int_{q_{n}}^\infty y \, \phi_{n+1}(\textbf{x}^{*};\textbf{U}^{*}; \bm{\Sigma}^{*}) \, d \textbf{x}^*,$$ where $$\alpha = \Phi_n (\textbf{Q},\bm{\Sigma})= \int_{q_{1}}^\infty\ldots\int_{q_{n}}^\infty \phi_{n}(\textbf{x};\textbf{U}; \bm{\Sigma}) \, d \textbf{x}, \label{equ-alpha0}$$ and $\Phi_n$ is the distribution function of the MVN, $\phi_{n}$ is the density function of MVN, and $\bm{\Sigma}$ is the correlation matrix of $\textbf{X}$. $\bm{\Sigma}^{*}$ is the correlation matrix of $\textbf{X}^*$. It comprises $\bm{\Sigma}$, but has one dimension more pertaining to the correlations between $Y=X_0$ and the selection indices $\textbf{X}$. The mean vector $\textbf{U}^{*}={u_0,u_1,\ldots,u_n}$ of $\phi_{n+1}$ is omitted, assuming $\textbf{U}^{*}={0,0,\ldots,0}$ without loss of generality, and consequently, we write shortly $\phi_{n+1} (\textbf{x}^{*};\bm{\Sigma}^{*})$ and $\phi_{n} (\textbf{x};\bm{\Sigma})$. The selection gain is the first moment, while the selected fraction $\alpha$ over all $n$ stages of selection corresponds to the zero-th moment of the one-sided truncated MVN distribution of $\textbf{X}$. The variance among candidates of y in the selected area $\textbf{S}_{Q}$ is defined as the second central moment, $\psi_n(y)=E(Y^2;\textbf{S}_{Q}) - \left[E(Y;\textbf{S}_{Q})\right]^2$, where $$E(Y^2;\textbf{S}_{Q}) = \alpha^{-1} \int_{-\infty} ^\infty \int_{q_{1}}^\infty...\int_{q_{n}}^\infty y^2\, \phi_{n+1}(\textbf{x}^{*}; \bm{\Sigma}^{*}) \, d \textbf{x}^*$$

See Also

multistagegain

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  
)


multistagevariance(Q=Q,corr=corr,alg=Miwa)

var.time.miwa=system.time (var.miwa<-multistagevariance(Q=Q,corr=corr,alg=Miwa))

var.time.bretz=system.time (var.bretz<-multistagevariance(Q=Q,corr=corr))


var.time.miwa
var.miwa[1]
var.time.bretz
var.bretz[1]

# further examples 1 


Q= c(0.9674216, 1.6185430)
corr=matrix( c(1,      0.7071068, 0.9354143,
               0.7071068, 1,      0.7559289,
               0.9354143, 0.7559289, 1    
             ), 
              nrow=3  
)


multistagevariance(Q=Q,corr=corr,alg=Miwa)

var.time.miwa=system.time (var.miwa<-multistagevariance(Q=Q,corr=corr,alg=Miwa))

var.time.bretz=system.time (var.bretz<-multistagevariance(Q=Q,corr=corr))


var.time.miwa
var.miwa[1]
var.time.bretz
var.bretz[1]


# further examples 

 alpha1<- 1/(24)^0.5
 alpha2<- 1/(24)^0.5
 Q=multistagetp(alpha=c(alpha1,alpha2),corx=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  
)

multistagevariance(Q=Q,corr=corr,alg=Miwa)