Contents - Index
Include any interaction terms in the model
Applying the "Find Associated Variables" function to the genotype-by-(marker+trait) table, three QTL for barley yield are identified:
The (Harrington allele of) QTL represented by mwg665c and mwg502 have positive effects on yield, whereas that represented by mwg838 has a negative effect. They together explained 25% of the mean yield variation. Note that these QTL are identified based on the "genotype main effect" in this eample, and any genotype-by-environment interactions are excluded by averaging across environments.
Detect and include epistasis terms in the regression and biplot
GGEbiplot provides two options for adding epistasis terms in the linear regression model and the biplot. The first is possible epistasis among selected markers, and the second is possible epistasis among all markers. This function is activated only when multiple regression is performed.
Epistasis among selected markers
Under Association, click Multivariate Analysis, then Include Interaction (Epistasis) terms, and select Between Selected Markers. GGEbiplot will first calculate the residues from the current multiple linear model; it will then construct a new set of "variables", which are all combinations among the selected markers, and then select among these new variables that are significantly associated with the residues. If any such variable is selected, GGEbiplot will put the selected variables together with the originally selected markers (i.e., the three markers on the above biplot) in the multiple regression, and re-conduct biplot analysis. If not, the following message will appear:
In our case, no significant interaction among selected markers were found, which is quite common in the Harrington * Tr306 barley mapping dataset.
Epistasis among all markers
Under Association, click Multivariate Analysis, click Include Interaction (Epistasis) terms, and select Between All Markers. GGEbiplot will first calculate the residues from the current multiple linear model; it will then construct a new set of "variables", which are possible marker*marker combinations among all markers, and then select among these new variables that are significantly associated with the residues. If any such variable is selected, GGEbiplot will put the newly selected variables together with the originally selected markers (i.e., the three markers on the above biplot), and re-conduct the biplot analysis. Before showing the new biplot, however, the following message will show:
Unwanted variables include any variables that are not a marker or marker combination. For example, some traits may be associated with the target traits, which should be removed because the purpose is to identify QTL. This can be done following this example. No such variables were selected in our example.
The search for epistasis among all markers identified one interaction that had significant effect on yield: abg613-abg609a * abg378, with abg613 and abg609a closely linked on chromosome 2:
Re-conduct multiple linear regression using yield as the dependent variable reveals that this interaction explained additional 8% of the yield variation:
Missing image: yieldepistasis.gif
This finding has not been reported by original researcher of the barley mapping project. It wouldn't it be fascinating to follow up this finding and find out why two markers from different chromosomes interacted to influence barley yield.
Apply the log ratio method to remove redundant or indirectly associated variables leads to the following final results:
To summarize, we found three QTL, represented by markers mwg665c, mwg502, and mwg838, plus one epistasis, abg613 * abg378, which altogether explained 33.5% of yield variation.