Generalized Linear Mixed Models for Mapping Quantitative Trait Loci
Xiaohong Che and Shizhong Xu
Supplemental Material for the Wheat Data Analysis
Responding to a reviewer's suggestion, we analyzed both the binomial and binary traits of
the wheat experiment using the linear mixed model (LMM) by ignoring the discrete
nature of the traits. The correct model should be the generalized linear mixed model
(GLMM), but we used the LMM as an ad hoc model to analyze the discrete traits.
Binomial trait analysis
Figure S1 shows the estimated QTL effects and LOD scores of the LMM analysis for the
binomial trait. Only one large QTL was detected using this ad hoc model. Comparing
Figure S1 here with Figure 5 of the main text, we can see that many small to median
sized QTL detected by the GLMM were missed. Results of leave-one-out cross validation
are shown in Table S1. The Pearson correlation coefficients between the observed and
predicted trait values were dropped from 0.517 (Expectation) and 0.529 (Overdispersion)
to 0.495 (Expectation) and 0.497 (Overdispersion). This means that the median sized
QTL detected by GLMM do contribute to the binomial trait variation, and ignoring the
discrete nature of the trait has decreased the predictability of the model.
Binary trait analysis
Figure S2 shows the estimated QTL effects and LOD scores of the LMM analysis for the
binary trait. Two QTL were detected on chromosome 2. The two QTL happen to be
located in two consecutive markers. Due to linkage, they are most likely the same QTL
split into two by the LMM. Comparing Figure S2 here with Figure 7 of the main text, we
are confident that they are indeed one QTL. Results of leave-one-out cross validation are
given in Table S2. The Pearson correlation coefficients between the observed and
predicted trait values were dropped from 0.472 (Expectation) and 0.473 (Overdispersion)
to 0.3888 (Expectation) and 0.3891 (Overdispersion). The substantial drops of the
predictabilities further support the one QTL hypothesis.
The conclusion of this analysis was that discrete traits should be analyzed using the
correct GLMM rather than the ad hoc LMM.
1
Figure S1. Binomial trait analyzed as normal trait of the wheat experiment using the
expectation method (blue) and the overdispersion method (red). The top panel shows the
estimated QTL effects and the bottom panel shows the LOD scores. Chromosomes are
separated by the dotted vertical lines. Positions of true markers are indicated by the
inward ticks on the horizontal axis.
2
Figure S2. Binary trait analyzed as normal trait of the wheat experiment using the
expectation method (blue) and the overdispersion method (red). The top panel shows the
estimated QTL effects and the bottom panel shows the LOD scores. Chromosomes are
separated by the dotted vertical lines. Positions of true markers are indicated by the
inward ticks on the horizontal axis.
3
Table S1. Comparison of predictability (via cross validation) for the binomial trait of the
wheat experiment using different methods (Expectation and Overdispersion) under
different models (GLMM and LMM).
Method
Model1
Pearson correlation ( ryyˆ )
Prediction error (PE)
Expectation
Overdispersion
GLMM
GLMM
0.516640
0.529022
0.100563
0.098658
Expectation
Overdispersion
LMM
LMM
0.495165
0.496879
0.101786
0.101515
1
GLMM - Generalized linear mixed model is the correct model; LMM - Linear mixed
model is the ad hoc model by ignoring the binomial nature of the trait.
4
Table S2. Comparison of predictability (via cross validation) for the binary trait of the
wheat experiment using different methods (Expectation and Overdispersion) under
different models (GLMM and LMM).
Method
Model1
Pearson correlation ( ryyˆ )
Prediction error (PE)
Expectation
Overdispersion
GLMM
GLMM
0.471551
0.472990
0.104914
0.104721
Expectation
Overdispersion
LMM
LMM
0.388857
0.389190
0.115758
0.115717
1
GLMM - Generalized linear mixed model is the correct model; LMM - Linear mixed
model is the ad hoc model by ignoring the binary nature of the trait.
5