Some important traits observed in forestry are dichotomous
(0 or 1). They have only two categories; such as survival (alive or
dead), disease incidence (infected or not infected), forkness (forked
or not forked), etc. When a randomize complete block field design
is used, the analysis of binomial traits can be carried out using
arcsin transformed plot percentages. However, this analysis method
may not be applicable when other plot configurations are used such
as single-tree plots.
Measuring a phenotypic expression on the 0-1 scale
produces a large amount of measurement error (environmental variance).
The error is the least when the proportion of the incidence is p=0.5
and becomes larger when p increases or decreases (see Falconer and
Mackay 1996).
where m= population mean, x= the distance
of the threshold value from m, or deviation of the threshold value
from the mean in standard deviation, T= threshold value,
i= mean of the individuals affected and p= proportion of
individuals that are affected.
Given a p value, x and i can
be obtained from a standardized normal distribution. For example,
if p=24% of the individuals are affected, then x=2.820 and i=3.117.
Heritability estimated on the observed scale (0-1)
needs to be transformed to continuous scale in order to remove the
scale effect by the following equation.
Lynch and Walsh sugegsted below equation for estimation
of heritability on the observed scale.
where, SigmaA(0,1) is additive genetic variance, Phi(1-Phi)
is the phenotypic variance on the observed scale. Phi_p is the frequency
of incidence in the population. The relationship between observed
and underlying scale heritabilities is given as follows.
where the numerator [p(xp)]^2 is the mean of affected
individuals of the population.
See Falconer and Mackay 1996, page 299-311, and Lynch
and Walsh 1998, page 738-744 for details.
Example 1 (fitting the model with ASReml)
Fusiform rust incidence in a clonally replicated trial of loblolly
pine was observed as infected (1) and not infected (0). The probability
of disease (p) was modeled with the generalized linear mixed model
(GLMM).
log[p/(1-p)]=µ+r+f+c.f+e
Since the disease incidence was observed on the 0-1
scale, incidence frequency was not used in the analysis but 0-1 values.
The GLMM was fitted using the logit function and fitting the model
with ASReml program.
!part rust #univariate model
for rust at age 3
rust3 !BIN !LOGIT ~ mu rep !r family clone.family
0
The proportion of infected individuals is p=(538/2369)=
0.23. Using this proportion, we can easily obtain deviation of threshold
value from the mean as a standard deviation (x=2.860) and the mean
of individuals affected (i=3.125). Variance components from the output:
| Source |
Estimate |
| Family |
0.58 |
| Clone.Family |
2.71 |
| Error |
1.00 |
Example 2 (fitting the model with SAS GLIMMIX)
Analysis of binomial traits can be carried out by a SAS macro code
called GLIMMIX.SAS. The code can be downloaded from the SAS
web site. Just type 'glimmix' in the search window and you will
find several macro programs for different versions of SAS.
*Call the
macro program ;
%inc 'c:\mysasfiles\glmmix macro.sas' ;
run;
* run the macro ;
%glimmix(data=a, procopt=method=reml,
stmts=%str( class rep family clone;
model y = block / s;
random family clone(family)/s ;
),
error=binomial,
link=logit );
run;
You may change the class variables, fixed and random effects according
to your model. If you are not interested in the Best Linear Unbiased
Predictors of the random factors in your model, you may remove the
/ and 's'.
The output includes model information, variance components as well
as solutions for fixed effects (BLUE) and solutions for random effects
(BLUPs). Click here to see an output.
Be aware that linear predictors for rust incidence were computed on
logit scale. To calculate predicted probability (p) of clones
or families, one needs to apply the inverse link function. For example,
probability of a genotype being infected by the disease can be calculated
as follows:
p = exp[BLUP(genotype)]/[1+exp(BLUP(genotype)]
Predicted probability values range between 0-1 and they are more
meaningful to interpret. For example, a p=0.16 value indicates that
probability of a genotype being infected is 16%.
Caution: SAS GLMMIX codes are still experimental. ASReml and
GLIMMIX analysis of the same threshold trait gave somewhat different
error variances. ASReml by default sets the environmenatl error to
'1.00'.