For single trait model, multiple traits model and solving mixed model equation, HIBLUP can calculate the reliability and prediction error variance (PEV) of all random effects. The mathematical formula for reliability calculation is as follows:

\begin{equation} \begin{aligned} r^{2} &= 1-\frac{PEV}{\sigma^{2}} \end{aligned} \end{equation}

where \sigma^{2} is the estimated variance from REML analysis for the random effect of interest. The prediction error variance PEV can be obtained from the following equation:

\begin{equation} \begin{aligned} PEV &= C_{ii}*\sigma_{e}^{2} \end{aligned} \end{equation}

where \sigma_e^{2} is the estimated residual variance. C_{ii} is the i_{th} diagonal element of the inverse matrix of left hand side (LHS) of mixed model equation.

By default, HIBLUP does not calculate and output those statistics, as the reliability estimation usually takes longer than whole REML process, thus please add flag `--r2`

only if you really want to get the reliability.

Taking single trait model for an example, the script to obtain the reliability is as follows:

./hiblup --single-trait --pheno demo.phe --pheno-pos 8 --dcovar 2,3 #fixed effect --qcovar 4,5 #covariates --rand 7 #non-genetic (environmental) random effect --bfile demo #genetic random effect --r2 #to calculate the reliability --threads 32 --out demo

Then each random effect, including residual, has several additional columns listing its corresponding standard error, PEV and reliability, overview of the file:

ID dam dam_SE dam_PEV dam_R2 GA GA_SE GA_PEV GA_R2 residuals residuals_SE residuals_PEV residuals_R2 IND1001 0.496645 3.75992 14.137 0.255863 6.66127 4.27597 18.2839 0.774767 -1.45194 3.18313 10.1323 0.55248 IND1002 0.496645 3.75992 14.137 0.255863 6.66127 4.27597 18.2839 0.774767 2.04383 3.17906 10.1064 0.553624 IND1003 0.674833 3.53678 12.5088 0.341566 2.08201 4.62091 21.3528 0.736963 -0.104053 4.14567 17.1866 0.24091 IND1004 0.674833 3.53678 12.5088 0.341566 -0.682836 4.70199 22.1087 0.727651 0.9083 4.17824 17.4577 0.228936