Calculates correlations from variances and covariances, together with their variances and covariances (S.A. Gezan).

### Options

`PRINT` = string token |
Output required (`summary` ); default `summ` |
---|---|

`IVARIANCES` = variate |
Indexes of the two variances in the `ESTIMATES` variate; no default – must be set |

`ICOVARIANCE` = scalar |
Index of the covariance in the `ESTIMATES` variate; no default – must be set |

### Parameters

`ESTIMATES` = variates |
Estimated values of the variances and covariances |
---|---|

`VCOVARIANCE` = symmetric matrices |
Variance-covariance matrix of the variances and covariances |

`FUNCTIONESTIMATE` = scalars |
Saves the estimated value of the function |

`SE` = scalars |
Saves the standard error of the function estimate |

`NEWESTIMATES` = variates |
Saves new vectors of estimates, including the estimated value of the function |

`NEWVCOVARIANCE` = symmetric matrices |
Saves variance-covariance matrices for the new vectors (including the function estimate) |

### Description

`FNCORRELATION`

estimates correlations from variances and covariances. The estimated values of the variances and covariances, are contained in a variate supplied by the `ESTIMATES`

parameter. The positions of the two variances in the `ESTIMATES`

variate are specified (in a variate of length two) by the `IVARIANCES`

option, and the position of the covariance is specified (in a scalar) by the `ICOVARIANCES`

option. The variances and covariances of the `ESTIMATES`

are supplied (in a symmetric matrix) by the `VCOVARIANCE`

parameter.

The estimated correlation can be saved by the `FUNCTIONESTIMATE`

parameter, and its standard error can be saved by the `SE`

option (both in scalars). The `NEWESTIMATES`

parameter can save a new variate of estimates, containing first the original `ESTIMATES`

variate and then the function estimate. The corresponding variance-covariance matrix can be saved (in a symmetric matrix) by the `NEWVCOVARIANCE`

parameter.

Options: `PRINT`

, `IVARIANCES`

, `ICOVARIANCE`

.

Parameters: `ESTIMATES`

, `VCOVARIANCE`

, `FUNCTIONESTIMATE`

, `SE`

, `NEWESTIMATES`

, `NEWVCOVARIANCE`

.

### Method

The correlation function *w* is calculated from the random variances *f* and *g*, and covariance *h* by the expression:

*w* = *h* / √( *f* × *g* )

The variance of the estimated correlation is approximated using a first-order Taylor series expansion (i.e. the *delta* method); see Holland (2006).

var(*w*) = var( *h* / sqrt( *f* × g ) )

= E(*w*)^{2} × { var(*f*) / (4 × E(*f*)^{2}) + var(*g*) / (4 × E(*g*)^{2}) + var(*h*) / E(*h*)^{2}

+ cov(*f*,*g*) / (2 × E(*f*) × E(*g*)) – cov(*f*,*h*) / (E(*f*) × E(*h*)) – cov(*h*,*g*) / (E(*h*) × E(*g*))}

### Reference

Holland, J.B. (2006). Estimating genotypic correlations and their standard errors using multivariate restricted maximum likelihood estimation with SAS Proc MIXED. *Crop Sci.*, 46, 642-654.

### See also

Procedures: `FNLINEAR`

, `FNPOWER`

.

Commands for: Calculations and manipulation.

### Example

CAPTION 'FNCORRELATION example'; STYLE=meta VARIATE [VALUES=4.01,19.63,13.65] est0 SYMMETRICMATRIX [ROWS=3; VALUES=150.40,-31.85,161.13,0.93,-9.32,23.31] vcov0 FNCORRELATION [PRINT=summary; IVARIANCES=!(2,3); ICOVARIANCE=!(1)]\ ESTIMATES=est0; VCOVARIANCE=vcov0; FUNCTIONESTIMATE=corr; SE=se;\ NEWESTIMATES=newest; NEWVCOVARIANCE=newvcov PRINT corr,se & newest,newvcov