Quantitative Genetics Practical 11: A capstone on gain, diversity and resources
Learning objectives
Specify a complete programme comparison; separate observable selection information from genetic truth; evaluate gain, pedigree inbreeding and marker diversity on fixed bases; count resource use; and summarize paired outcomes while retaining the limits of a small simulation.
1. Declare the question and hold other decisions fixed
Does collecting a second independent phenotype record per candidate change genetic gain enough to justify its extra recording cost under this teaching model?
The objective is increased additive merit for one trait, with weight one. Scores are averages of observed phenotypes, not fitted breeding values. Compare these programmes:
Decision
One record
Two records
Candidates per cohort
24: 12 per role
Same
Observations per candidate
1
2
Selected parents
3 sires and 3 dams
Same
Selection score
Observed phenotype
Mean of two phenotypes
Mating
Uniform independent parental draws
Same
New offspring
24 per generation
Same
Horizon
2 generations, assumed 2 years
Same
The second observation has an independent residual in this model. Both observations are available before selection; neither extends the assumed generation interval. There are no fixed effects, permanent environment, dominance, mutation or migration. These assumptions make the effect of repeated measurement transparent. They do not describe every real recording programme.
2. Create one synthetic founder population
Use 24 unrelated, noninbred pedigree founders and six phased markers on one chromosome. Independent Bernoulli draws supply founder alleles; this is a toy panel, not a reconstruction of historical haplotypes. G is VCF ALT and the counted allele. Cumulative map positions are in cM; offspring meiosis uses the existing gsim Poisson no-interference model.
Marker effects and dosage centers stay fixed for all programmes and runs. The founder mean is the genetic-gain reference. Pedigree inbreeding is relative to the declared unrelated-founder base. Do not re-center a selected cohort when reporting gain.
3. Define records, time and accounting assumptions
A record is \(y_{iv}=g_i+e_{iv}\), with independent residual variance 0.04 trait units squared. For \(v\) independent records, the mean has residual variance \(0.04/v\). Averaging does not eliminate an environmental effect shared across visits; our example deliberately omits that component.
Costs are hypothetical accounting units: 5 per record, 20 per selected parent per generation, and 10 per offspring. They are resource charges, not trait economic weights or estimated farm profit. Each generation lasts one assumed year, including the recording stage.
The founder cohorts are already available. Accounting starts with their selection-stage records and includes two offspring cohorts. The terminal offspring cohort is evaluated through simulation truth and genotypes; it is not phenotyped for another selection cycle, so we charge no terminal recording stage.
4. Run a small paired comparison
Three runs share this same founder panel. For each run, the two policies share numbered record, mating and meiosis seed schedules. Visit 1 has the same record and observation-unit identity under both policies, so it retains its innovation when visit 2 is added. Between runs, seeds differ. Later candidate genotypes can differ across policies because selection changes parentage.
This pairing supports a difference within each run. It does not make the three runs independent founder-population experiments, nor remove all stochastic variability from their differences.
Show R code
run_programme <-function(visits,replicate) { policy <- policies$policy[match(visits,policies$visits)] current <- phase0; bed <- bed0; pedigree <- pedigree0 animals <- ids; sex <- parent_table$sex history <-list(); record_count <-0Lfor(t inseq_len(generations)) { seed_base <-10000*replicate+100*t g <-gsim_genetic_values(bed,state) record_table <-expand.grid(animal=animals,visit=seq_len(visits),stringsAsFactors=FALSE) record_table$trait <-"trait" record_table$record <-paste(record_table$animal,record_table$visit,sep=":") record_table$observation_unit <- record_table$record records <-gsim_records(g,records=record_table,residual_covariance=.04,seed=seed_base+1)# Observable records alone supply the decision score. scores <-tapply(records$records$value,records$records$animal,mean) selection <-gsim_select(animals,c(M=3,F=3),scores=scores,method="score",strata=sex) sires <- selection$animal[selection$selected & selection$stratum=="M"] dams <- selection$animal[selection$selected & selection$stratum=="F"] children <-sprintf("g%d-%02d",t,1:24) mating <-gsim_mate(pedigree,sires,dams,children,seed=seed_base+2,sex=setNames(rep(c("M","F"),12),children)) next_cohort <-gsim_simulate_cohort(current,mating,seed=seed_base+3,output=file.path(work,paste0("visits",visits,"-run",replicate,"-g",t))) plan <- mating$offspring offspring_g <-gsim_genetic_values(next_cohort,state) d <-gsim_genetic_values(next_cohort,dosage_state) p <-colMeans(d)/2 full <- mating$pedigree$pedigree tab <-data.frame(id=full$animal,sire=full$sire,dam=full$dam) tab <- tab[match(mating$pedigree$canonical_order,tab$id),] A <-mouse_relationship(tab) offspring_F <-diag(A)[children]-1 F_from_parents <- A[cbind(match(plan$sire,rownames(A)),match(plan$dam,colnames(A)))]/2stopifnot(max(abs(offspring_F-F_from_parents))<1e-12,identical(attr(offspring_g,"trait_state_checksum"),state$checksum),abs(mean(offspring_g)-sum((2*p-1)*effects[,"trait"]))<1e-12) parent_expectation <-mean((g[plan$sire,"trait"]+g[plan$dam,"trait"])/2) record_count <- record_count+nrow(records$records) cost <- record_count*charges["record"]+t*6*charges["parent"]+ t*24*charges["offspring"] history[[t]] <-data.frame(policy=policy,replicate=replicate,generation=t,years=t*years_per_generation,gain=mean(offspring_g)-base_mean,mean_F=mean(offspring_F),mean_HE=mean(2*p*(1-p)),records=record_count,cost=unname(cost),parents_used=length(mating$parents_used),mendelian_mean=mean(offspring_g)-parent_expectation) pedigree <- mating$pedigree; current <- next_cohort; bed <- next_cohort animals <- children; sex <- plan$sex }do.call(rbind,history)}history <-do.call(rbind,lapply(seq_len(replicates),function(r)do.call(rbind,lapply(policies$visits,function(v) run_programme(v,r)))))stopifnot(nrow(history)==12,all(history$mean_F>=0& history$mean_F<=1),all(history$mean_HE>=0& history$mean_HE<=.5),all(history$parents_used<=6))knitr::kable(history,digits=4,row.names=FALSE)
policy
replicate
generation
years
gain
mean_F
mean_HE
records
cost
parents_used
mendelian_mean
One record
1
1
1
0.1750
0.0000
0.4460
24
480
6
0.0042
One record
1
2
2
0.3479
0.0938
0.3591
48
960
6
-0.0312
Two records
1
1
1
0.2354
0.0000
0.4077
48
600
6
-0.0083
Two records
1
2
2
0.2938
0.0104
0.3484
96
1200
6
-0.0219
One record
2
1
1
0.2729
0.0000
0.4122
24
480
6
0.0260
One record
2
2
2
0.3333
0.0781
0.3711
48
960
6
-0.0427
Two records
2
1
1
0.2854
0.0000
0.4087
48
600
6
-0.0052
Two records
2
2
2
0.3125
0.0677
0.3825
96
1200
6
-0.0365
One record
3
1
1
0.2042
0.0000
0.4524
24
480
6
-0.0156
One record
3
2
2
0.3375
0.0990
0.3219
48
960
6
-0.0406
Two records
3
1
1
0.3000
0.0000
0.3640
48
600
6
0.0427
Two records
3
2
2
0.4312
0.0990
0.2571
96
1200
6
-0.0083
Selected-parent counts are fixed, but random mating can leave some selected parents unused. Offspring inbreeding depends on actual pairings and previous ancestry. Marker \(H_E\) is the mean \(2p(1-p)\) over the fixed six-marker panel, including invariant markers. It is not an estimate of genome-wide diversity or effective population size.
The Mendelian-mean column separates realized offspring mean from the true parentage-weighted additive expectation. Truth evaluates the decision after selection; it did not choose parents. These small dense pedigree and dosage summaries are for teaching, not a large-population workflow.
5. Evaluate terminal outcomes and paired differences
Compare both programmes at the same two-year horizon. Divide gain from the original base by elapsed years to report a realized average gain rate over this horizon. This is not a stationary long-term response rate.
The differences use the same run as a pair. Positive gain difference favors two records for terminal merit in that run; negative \(F\) difference favors it for pedigree inbreeding. More records cost 240 extra accounting units over the horizon in every run. No single scalar defines a winner unless the programme supplies a resource constraint or values these outcomes on a justified common scale.
The Monte Carlo standard error illustrates uncertainty of an average over independent seeded simulation runs conditional on this founder panel and model. Three runs are far too few for a dependable policy-performance assessment. It excludes uncertainty about founder populations, trait architecture, residual variance, costs and real deployment. Do not interpret it as a real-world confidence interval.
Figure 1: Paired terminal outcomes from three teaching runs. Lines join the same run under one and two records; all runs share one synthetic founder panel.
Question 1. Does reducing record noise guarantee more gain and less inbreeding in every run?
Suggested answer
No. Recording precision changes selection information; finite selection, mating and segregation still vary. More accurate selection does not automatically preserve diversity. With connected pedigrees, selecting more similar high-merit relatives can also affect future relationships. These outcomes need separate evaluation.
6. Make the decision conditional on resources and assumptions
For a hypothetical horizon budget of 1,000 accounting units, only the one-record programme is feasible under these charges. A budget of 1,200 makes both feasible. Feasibility and expected performance are different questions: an infeasible strategy is excluded by the budget, while an uncertain performance difference calls for more evidence within an agreed experimental design.
The trait objective is additive merit. Recording cost is paid by the programme. To translate gain into economic benefit, specify trait units, production volume, marginal values, timing and discounting. Subtracting these accounting costs directly from a mean genetic value would mix units.
Exercises
Verify the additional cost from record counts without using the accumulated cost column.
Suppose the second record delays each generation from one to 1.5 years, without changing genotypes or scores in these already simulated runs. Recalculate elapsed years and the realized average gain rate for that scenario. Explain what this post-processing does not model.
Suppose records have permanent-environment variance 0.03 and independent residual variance 0.04. Calculate environmental variance in the one- and two-record mean.
Suggested solutions
Show R code
extra <- (two$records-one$records)*charges["record"]stopifnot(all(extra==240),all(extra==paired$extra_cost))delayed_years <- generations*1.5delayed_gain_rate <- two$gain/delayed_yearsknitr::kable(data.frame(replicate=two$replicate,original_gain_per_year=two$gain_per_year,delayed_gain_per_year=delayed_gain_rate),digits=4,row.names=FALSE)
The delay calculation changes only the calendar-time denominator. It does not simulate age effects, changed survival, extra maintenance cost or additional cycles in a fixed calendar horizon. Permanent environment persists across visits, so only the independent residual part is divided by the number of records.
Capstone assignment
Write a short programme specification before extending the simulation:
State the trait objective, units, genetic base and deployment population.
Define records available at selection, any fitted evaluation and its validation target.
Declare eligibility, parent quotas, contribution limits and mating rules.
Specify horizon, generation intervals, resource charges and feasible budgets.
Choose gain, diversity and inbreeding outcomes and a planned paired replication design.
Then explain which conclusion your results support and which assumptions remain untested. Do not use simulated truth as a decision score unless deliberately labelling an oracle comparison.
Package connection and next topic
gsel owns objectives and programme policies; gsim supplies existing records, selection, mating and inheritance; gpop owns population-statistic definitions and their assumptions. Estimation tools fit genetic evaluations. This capstone composes those responsibilities without adding package interfaces.
Use the course map to connect both practical sequences, choose a teaching route and identify slide topics that need further worked examples.
