Tadpole Survival
Description of problem
A common problem in quantitative genetics is the estimation of the variation in the various genetic and environmental components that contribute to the variation in a trait. The most important component is usually the additive component, which is related to the strength of natural selection. This variation can be estimated from controlled mating experiments. Traditionally, the additive variance is estimated from the Male effect but here we are able to include the contributions from all of the levels of the experiment, which is clearly a more efficient use of the data
The data comes from part of a data set on survival of tadpoles, which used a "North Carolina I" design. 30 males were used in the experiment, and each one was mated to two females. The females were only mated to a single male. The eggs from each cross were split up into groups of between 15 and 51 eggs, and placed in several treatments and blocks. The data here comes from the high pH treatment, and two of the blocks (for simplicity, the block effect is not modelled here). The trait of interest is whether a tadpole survives to Gosner Growth Stage 25 (whatever that is). We also use the average egg size for a female as a covariate. The number of tadpoles in a batch of size N_{i} that survive to GS25 is n_{i}. We then use mixed model logistic regression:
n_{i} ~ Bin(N_{i}, p_{i})
logit(p_{i}) ~N( μ_{i}, σ^{2}_{e})
where
μ_{i} = μ_{0} + (α_{s} + α_{d})/2 + β_{d}, + δ_{d}/4
α_{s}, α_{d} ~ N(0, V_{Add})
β_{d} ~ N(0, V_{Mat})
δ_{d} ~ N(0, V_{Dom})
where μ_{0} is the overall mean, α_{s} and α_{d} are the sire's and dam's breeding values, β_{d} is the maternal effect and δ_{d} is the dominance effect. σ^{2}_{e} is the residual variance, which is made up of contributions from the additive and dominance variation, due to segregation. Rather than writing this as the sum of the variance contributions, it is quicker to simulate the errors directly, i.e. write
μ_{i} = μ_{0} + (α_{s} + α_{d})/2 + β_{d}, + δ_{d}/4
+ ε_{i}^{a}/2^{1/2} + (3/4)^{1/2} ε^{d}_{i}
where ε^{a;}_{i} ~ N (0, V_{Add}) and + ε^{d}_{i} ~ N (0, V_{Dom}). Now σ^{2}_{e} is V_{EnvW}, the within environment variance.
Because of the design of the experiemnt, there are identifiability problems with V_{Dom}, V_{EnvW} and V_{Mat}. However, our main interest is in the proportion of the total variance explained by V_{Add}, i.e. h^{2} = V_{Add}/(V_{Add} + V_{Dom} + V_{Mat} + V_{EnvW})
V_{Add}, V_{Dom}, V_{Mat} and V_{EnvW} are all given uninformative uniform priors with an upper limit that is larger than possible (i.e. much larger than the variance of the data). This gives better frequentist properties than the inverse gamma distribution.
BUGS Code
model; { for(i in 1 : N) { add.err[i] ~ dnorm(0, TauAdd) dom.err[i] ~ dnorm(0, TauDom) error[i] ~ dnorm( 0.0,TauEnvW) logit(pp[i]) < Mu0 + (alphaMale[Male[i]] + alphaFemale[Male[i], Female[i]])/2 + maternal[Male[i], Female[i]] + Delta[Male[i], Female[i]]/2 + add.err[i]/sqrt(2) + sqrt(3/4)*dom.err[i] + error[i] Alive[i] ~ dbin(pp[i],Total[i]) } for(m in 1 : MaleTot) { alphaMale[m] ~ dnorm( 0,TauAdd) for(f in 1:2) { alphaFemale[m,f] ~ dnorm( 0,TauAdd) maternal[m,f] ~ dnorm( 0,TauMat) Delta[m,f] ~ dnorm( 0,TauDom) } } Mu0 ~ dnorm( 0,0.1) VarAdd ~ dunif( 0,1000) VarDom ~ dunif( 0,1000) VarEnvW ~ dunif( 0,1000) VarMat ~ dunif( 0,1000) TauAdd < 1/VarAdd TauDom < 1/VarDom TauEnvW < 1/VarEnvW TauMat < 1/VarMat TotVar < VarAdd + VarDom + VarEnvW + VarMat Heritability < 1/ (TauAdd*TotVar) }
Data
Inits
list(VarAdd =1, VarDom =1, VarEnvW =1, VarMat =1)
Results
Two chains were run, and after a burnin of 1000 iterations, a further 10000 iterations per chain were run, to give 2x10000 samples. This took 130 seconds on a 2GHz processor.

mean 
sd 
MC_error 
val2.5pc 
median 
val97.5pc 
start 
sample 
Heritability 
0.1295 
0.1086 
0.006233 
0.009633 
0.1043 
0.4065 
1001 
20000 
Mu0 
1.818 
0.2898 
0.01345 
1.27 
1.806 
2.427 
1001 
20000 
VarAdd 
0.5855 
0.4927 
0.02826 
0.04388 
0.4738 
1.837 
1001 
20000 
VarDom 
0.3026 
0.2525 
0.0138 
0.02443 
0.241 
0.9459 
1001 
20000 
VarEnvW 
0.2269 
0.1914 
0.009831 
0.01516 
0.1799 
0.7208 
1001 
20000 
VarMat 
3.544 
1.045 
0.03034 
1.8 
3.445 
5.907 
1001 
20000 
The posterior for the heritability is low compared to the typical value of around 0.2. The Maternal component has the largest variance, suggesting that there is a large variation in maternal effect on early survival.