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*To*: jces@kvl.dk*Subject*: Re: About !GP bending of US matrix*From*: Arthur Gilmour <gilmoua@ornsun.agric.nsw.gov.au>*Date*: Tue, 30 Mar 1999 09:19:02 +1000 (EST)*Cc*: asreml@ram.chiswick.anprod.csiro.au*Reply-To*: Arthur Gilmour <gilmoua@ornsun.agric.nsw.gov.au>*Sender*: asreml-owner@ram.chiswick.anprod.csiro.au

Dear Joao, > > > I am troubled with an analysis which failed to provide a positive definite > result, and I would be pleased if you could give me some feedback to > some questions regarding this matter. > > The analysis concerns one trait measured in two sites and on the same > set of 37 half-sib families. Results for variance and covariance > parameters were as follows, for additive ("tid" denotes tree) and > residual terms in single and multi-site analysis of (co)variance > components: > > SINGLE-SITE ANALYSES OF BOTH TRIALS > > Site1 > > LogL=-276.982 S2= 1.8357 265 df 0.51574 1.0000 > Final parameter values 0.51581 1.0000 > > Source Model terms Gamma Component Compnt/StndErr %C > tid 893 893 0.515812 0.946862 1.70 0 P > Variance 271 265 1.00000 1.83567 3.51 0 P > > Site2 > > LogL=-397.776 S2= 0.91613 622 df 0.41763 1.0000 > Final parameter values 0.41766 1.0000 > > Source Model terms Gamma Component Compnt/StndErr %C > tid 905 905 0.417664 0.382633 2.42 0 P > Variance 628 622 1.00000 0.916125 6.35 0 P > > > MULTI-SITE ANALYSIS WITHOUT CONSTRAINTS > > LogL=-665.899 S2= 1. 887 df 1.7970 0.92075 0.99003 0.78443 0.37704 > Final parameter values 1.7971 0.92071 0.98993 0.78452 0.37709 > > Source Model terms Gamma Component Compnt/StndErr %C > Residual 1724 887 > Variance 0 0 1.79714 1.79714 3.43 0 P > Variance 0 0 0.920710 0.920710 6.42 0 P > site.tid US=UnStr 1 0.989931 0.989931 1.76 0 U > site.tid US=UnStr 1 0.784520 0.784520 3.18 0 U > site.tid US=UnStr 2 0.377092 0.377092 2.40 0 U > Covariance/Variance/Correlation Matrix US=UnStructu > 0.9899 1.284 > 0.7845 0.3771 > > > MULTI-SITE ANALYSIS WITH CONSTRAINTS > > LogL=-667.075 S2= 1. 887 df 1.9190 0.89997 0.78643 0.55299 0.39326 > Final parameter values 1.9190 0.89998 0.78643 0.55299 0.39326 > > Source Model terms Gamma Component Compnt/StndErr %C > Residual 1724 887 > Variance 0 0 1.91895 1.91895 8.61 0 P > Variance 0 0 0.899979 0.899979 13.02 0 P > site.tid US=UnStr 1 0.786427 0.786427 0.00 0 B > site.tid US=UnStr 1 0.552995 0.552995 0.00 0 B > site.tid US=UnStr 2 0.393264 0.393264 0.00 0 B > Covariance/Variance/Correlation Matrix US=UnStructu > 0.7864 0.9944 > 0.5530 0.3933 > > > Under the multi-site analysis without constraints, the additive genetic > correlation obtained across sites was 1.28, suggesting that the US > (co)variance matrix is non-positive definite. In spite of this, the job > converged well and the variance estimates are similar to those obtained > in the single-site analyses (NOTE: according to the likelihood ratio test, > the term "tid" was significant in both single-site analyses). > Constraints were imposed on the (co)variance parameters, to keep the > additive genetic correlation within the -1 to +1 range. Following this, the > additive covariance was substantially reduced and, particularly for site > 1, the variance estimates were altered when compared with the results > obtained for both multi-site analysis without constraints and single-site > analysis. > > > Regarding the results above, I have the following doubts which I would > be pleased if you could clarify: > > 1) Is the non-positive result reflecting the estimation of the additive > covariance from an analysis on small data sets (i.e. 271 observations in > site 1)? > From the unconstrained multivariate analysis, the phenotypic variance matrix is 1.797+.990=2.787 0.000+.784=0.784 0.921+0.377=1.298 implying a phenotypic correlation of 0.412 Is it possible there could be come nongenetic correlation between trees [e.g. due to nursury effects]. The model fitted attributes all the phenotypic covariance to genetics. You would need greater depth of pedigree or some other information to sort this out. You only have 37 half sib families so there is considerable variance associated with the variance components which is probably a strong component of the explanation. In the constrained model, ASREML has 'bent' the matrix which involves shrinking the variances towards their mean. Hence the Trait 1 variance is reduced but the trait 2 variance is increased and the covariance is decreased. As it needed to BEND the matrix in two successive iterations, it then fixed the values. Otherwise it would infact not converge because the AI updates would continually [in this case] make the matrix negative definite. What we really need [an it is on the drawing board] is a reduced rank formulation which implicitly [in the two variate case] would make the genetic correlation 1. The model can infact be fitted in the current version of ASREML using the and() function I think and manually iterating the variance ratio between the traits. A simpler approach is to use the CORR structure rather than the US structure which can constrain the correlation to 0.999 and this should give a fit better than the constrained US one. > 2) Particularly for site 1, the estimation of the heritability change from a > value of 0.29 (multisite analysis with constraints) to 0.34 or 0.355 > (single-site and unconstrained multi-site analyses). So, which of these > estimates is less affected by possible bias of the variance parameters? > The unconstrained [and single site] analysis is the least biased. By constraining the solution to the parameter space, you are introducing bias. I would use the CONSTRAINED CORR model as the best compromise. > 3) What are the limitations in terms of using the results above (i.e. > unconstrained versus constrained multi-site analysis) to infer about the > importance of GxE interactions for the trait? > If we assume the correlation is 1, there is no GxE interaction, just a scale effect. > 4) In the unconstrained multi-site analysis, the calculated standard error > of the additive genetic correlation estimate was 0.253. Does the standard > error tend to be smaller when the estimate of the correlation parameter > approaches (or exceeds) the limits of -1 or +1? In this case, is the > estimated standard error a meaningful measure of the precision of the > additive genetic correlation parameter? > The formula used in ASREML for the variance of the genetic correlation is given in section 5.2.3 of the manual and is a general formula for the variance of a ratio. I do not expect it will have any odd behaviour with respect to the value of 1. The se is proportional to the correlation. The estimate shows that the correlation is not different from 1. Arthur <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><> Arthur Gilmour PhD mailto:Arthur.Gilmour@agric.nsw.gov.au Senior Research Scientist (Biometrics) fax: <61> 2 6391 3899 NSW Agriculture <61> 2 6391 3922 Orange Agricultural Institute telephone work: <61> 2 6391 3815 Forest Rd, ORANGE, 2800, AUSTRALIA home: <61> 2 6362 0046 ASREML is still free by anonymous ftp from pub/aar on ftp.res.bbsrc.ac.uk or point your web browser at ftp://ftp.res.bbsrc.ac.uk/pub/aar/ To join the asreml discussion list, send the message subscribe mailto:asreml-request@chiswick.anprod.CSIRO.au To send messages to the list, mailto:asreml@chiswick.anprod.CSIRO.au Asreml mailinglist archive: http://www.chiswick.anprod.csiro.au/lists/asreml <> <> <> <> <> <> <> "You (Father) have given Him (Your Son) authority over all flesh, that He should give eternal life to as many as You have given Him. And this is eternal life, that they may know You, the only true God, and Jesus Christ whom You have sent." John 17: 2-3. <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><> -- Asreml mailinglist archive: http://www.chiswick.anprod.csiro.au/lists/asreml

**Follow-Ups**:**Re: About !GP bending of US matrix***From:*Kim Bunter <kbunter@metz.une.edu.au>

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