Last updated: 2021-08-19

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# 1) start a screen shell 
screen; # or screen -r if re-attaching...

# cbsurobbins SLURM 
salloc -n 20 --mem=60G --time=06:00:00;

# 2) start the singularity Linux shell inside that
singularity shell ~/rocker2.sif; 
# Project directory, so R will use as working dir.
cd /home/mw489/IITA_2021GS/;
# 3) Start R
R

Estimate selection error

Concept

Determine error-variance vs. plot-size/rep-number scaling for input to simulations.

Our downstream objective is to simulate a baseline and alternative breeding pipelines (variety development pipelines; VDPs) based on proposed differences in plot-size (number of plants), number of reps, locations and overall trial-stage size.

We can’t compute the same selection index for each trial because of variation in the traits scored.

Below, I outline an approach that integrates into the existing GS pipeline and deals with heterogeneity in the traits observed from trial-to-trial.

Approach:

Use the SELECTION INDEX GETGV from genomic prediction using all entire available training population and all of the latest available data as a best estimate of “true” net merit
- The predictions documented here is the most up-to-date. See the summary of genomic prediction results here.
- Alternative best estimate of “true” net merit: actual national performance trial data (e.g. from NCRPs in Nigeria?)
For each trial, analyze the cleaned plot-basis data:
1. Fit a univariate mixed-model to each trait scored
2. Extract trial-specific BLUPs for whatever clones were present
3. Compute the SELIND for the current trial using BLUPs for whatever component traits were scored (\(SI_{TrialBLUP}\)).
4. Regress \(SI_{GETGV}\) on the \(SI_{TrialBLUP}\)
5. Extract the \(\hat{\sigma}^2_e\) of the regression as the trial-specific estimate of the selection error
Using the results from all trials: Regress \(\hat{\sigma}^2_e\) on plot-size. Weight by number of clones available to measure \(\hat{\sigma}^2_e\). Extract coefficients to generate relative scaling profiles of error variances to input for simulating VDPs.

Analysis

library(genomicMateSelectR); 
library(tidyverse)

# SELIND GETGVS (for input to estimateSelectionError func below)
gpreds<-readRDS(file = here::here("output","genomicPredictions_full_set_2021Aug09.rds"))
getgvs<-gpreds$gblups[[1]] %>% 
  filter(predOf=="GETGV") %>% 
  select(GID,SELIND)

# CLEANED PLOT-LEVEL TRIAL DATA
dbdata<-readRDS(here::here("output","IITA_ExptDesignsDetected_2021Aug08.rds"))
### Restrict consideration to >2012 
### to measure the selection error during the current "era" at IITA.
trials2keep<-dbdata %>% 
  filter(studyYear>=2013) %>% 
  distinct(studyYear,locationName,studyName,TrialType,CompleteBlocks,IncompleteBlocks,MaxNOHAV) %>% 
  filter(!is.na(MaxNOHAV)) %$% 
  unique(studyName)
length(trials2keep) %>% paste0(.," trials")

[1] "641 trials"

# [1] 641 trials
dbdata %>% filter(studyName %in% trials2keep) %>% nrow %>% paste0(.," plots")

[1] "182195 plots"

# [1] 182195 plots
trialdata<-dbdata %>% 
  filter(studyYear>=2013,
         studyName %in% trials2keep) %>% 
  nest(TrialData=-c(studyYear,locationName,studyName,TrialType,CompleteBlocks,IncompleteBlocks,MaxNOHAV))
trialdata %<>% 
  mutate(propGenotyped=map_dbl(TrialData,
                              ~length(which(!is.na(unique(.$FullSampleName))))/length(unique(.$GID))))

# SELECTION INDEX WEIGHTS
## from IYR+IK
## note that not ALL predicted traits are on index
SIwts<-c(logFYLD=20,
         HI=10,
         DM=15,
         MCMDS=-10,
         logRTNO=12,
         logDYLD=20,
         logTOPYLD=15,
         PLTHT=10) 

# SOURCE FUNCTION estimateSelectionError() 
source(here::here("code","estimateSelectionError.R"))

trialdata %>% unnest(TrialData) %>% 
  summarise(Nplots=nrow(.),
            across(c(locationName,studyYear,studyName,TrialType,GID), ~length(unique(.)),.names = "N_{.col}")) %>% 
  rmarkdown::paged_table()

Summary of available plot-level data: See here for details about the cleaned trial data most recently downloaded from IITA/Cassavabase and used below. Number of unique plots, locations, years, etc. in the cleaned plot-basis data.

trialdata %>% unnest(TrialData) %>% 
  count(TrialType,CompleteBlocks,IncompleteBlocks) %>% 
  spread(TrialType,n) %>% 
  rmarkdown::paged_table()

Count the trial designs by TrialType:

trialdata %>% unnest(TrialData) %>% 
  distinct(studyYear,locationName,studyName,TrialType,CompleteBlocks,IncompleteBlocks,MaxNOHAV) %>% 
  filter(!is.na(MaxNOHAV)) %>% 
  mutate(TrialType=factor(TrialType,levels=c("CrossingBlock","GeneticGain","CET","ExpCET","PYT","AYT","UYT","NCRP"))) %>% 
  ggplot(.,aes(x=TrialType,y=MaxNOHAV,fill=TrialType)) + 
  geom_boxplot(notch = T) + 
  theme_bw() + theme(axis.text.x = element_text(angle=45,vjust=.5)) +
  labs(title = "Max number harvested as a proxy for planned plot size",
       subtitle="MaxNOHAV = The maximum number stands harvested per trial. studyYear>=2013")

Version	Author	Date
182b83c	wolfemd	2021-08-19
8a6c166	wolfemd	2021-08-11
3a1faa0	wolfemd	2021-07-14

###### unit test inputs for estimateSelectionError
# TrialData<-trialdata$TrialData[[1]]
# CompleteBlocks<-trialdata$CompleteBlocks[[1]]
# IncompleteBlocks<-trialdata$IncompleteBlocks[[1]]
# TrialData<-trialdata %>% filter(propGenotyped>0.75) %>% slice(4) %$% TrialData[[1]]
# CompleteBlocks<-trialdata %>% filter(propGenotyped>0.75) %>% slice(4) %$% CompleteBlocks[[1]]
# IncompleteBlocks<-trialdata %>% filter(propGenotyped>0.75) %>% slice(4) %$% IncompleteBlocks[[1]]
# ncores=4
# rm(TrialData,CompleteBlocks,IncompleteBlocks)

Run function estimateSelectionError() across trials to estimation selection errors.

require(furrr); plan(multisession, workers = 20)
options(future.globals.maxSize=+Inf); options(future.rng.onMisuse="ignore")
trialdata %<>% 
  mutate(SelectionError=future_pmap(.,estimateSelectionError,
                                    SIwts=SIwts,getgvs=getgvs))
plan(sequential)
saveRDS(trialdata,here::here("output","estimateSelectionError.rds"))

Result

Out of 633 trials, 516 produced successful model fits and subsequent estimates of TrialMSE (selection index error).

library(genomicMateSelectR)
estSelError<-readRDS(here::here("output","estimateSelectionError.rds"))
estSelError %<>% 
  select(-TrialData) %>% 
  unnest(SelectionError) %>% 
  select(-SI_BLUPs,-BLUPs,-SelectionError) %>% 
  filter(!is.na(TrialMSE))

Here’s the str() of the estimates I made.

estSelError %>% str

tibble [516 × 12] (S3: tbl_df/tbl/data.frame)
 $ studyYear       : int [1:516] 2013 2013 2013 2013 2013 2013 2013 2013 2013 2013 ...
 $ locationName    : chr [1:516] "Ibadan" "Ibadan" "Ibadan" "Ibadan" ...
 $ studyName       : chr [1:516] "13Ayt16ICTIB" "13ayt20pdIB" "13ayt20pdwhtrtIB" "13ayt20pdyrtIB" ...
 $ TrialType       : chr [1:516] "AYT" "AYT" "AYT" "AYT" ...
 $ MaxNOHAV        : num [1:516] 10 10 10 10 11 20 10 10 10 10 ...
 $ CompleteBlocks  : logi [1:516] TRUE TRUE TRUE TRUE TRUE TRUE ...
 $ IncompleteBlocks: logi [1:516] FALSE FALSE FALSE FALSE FALSE FALSE ...
 $ propGenotyped   : num [1:516] 0.983 0.516 0.575 0.75 0.6 ...
 $ cor2si          : num [1:516] -0.3344 0.2644 0.573 -0.0843 0.5199 ...
 $ r2_si           : num [1:516] 0.1118 0.0699 0.3284 0.0071 0.2703 ...
 $ TrialMSE        : num [1:516] 22982 15691 1933 7259 17167 ...
 $ NcloneForReg    : int [1:516] 30 10 23 25 11 11 11 11 9 28 ...

cor2si = correlation between SI computed from each trial’s BLUPs and the SI computed from GETGV (all training data and traits used)
r2_si = r-squared, regression of SI_GETGV on SI_TrialBLUP
TrialMSE = mean squared error from that regression
NcloneForReg = the number of clones with estimates of both SI_TrialBLUP and SI_GETGV for a given trial. Avoid considering trials with too few data points.

I made two plots below, one of the cor2si and one of the TrialMSE vs. MaxNOHAV (the proxy for plot size). Trial-results are scaled by NcloneForReg and a smooth spline is fit.

estSelError %>% 
  ggplot(.,aes(x=MaxNOHAV,y=cor2si,color=NcloneForReg,size=NcloneForReg)) + 
  geom_point() + geom_smooth() + theme_bw() + theme(panel.grid = element_blank()) +
  labs(title = "The cor(SI_GETGV,SI_TrialBLUPs) vs. MaxNOHAV as a proxy for plot size")

Version	Author	Date
182b83c	wolfemd	2021-08-19
8a6c166	wolfemd	2021-08-11
3a1faa0	wolfemd	2021-07-14

estSelError %>% 
  ggplot(.,aes(x=MaxNOHAV,y=TrialMSE,color=NcloneForReg,size=NcloneForReg)) + 
  geom_point() + geom_smooth() + theme_bw() + theme(panel.grid = element_blank()) +
  labs(title = "Trial-specific selection error ests. vs. MaxNOHAV")

Version	Author	Date
182b83c	wolfemd	2021-08-19

In my opinion, there is no clear pattern. I am biased to seeing a hint of a trend.

Look at a correlation matrix and do some regression analysis below.

Try to measure an effect size of increasing the number of stands per plot.

estSelError<-readRDS(here::here("output","estimateSelectionError.rds"))
# extract some covariates which might be predictive of cor2si, 
# which I think is on a uniform scale

# Nclone in trial: want est. of plot-size effect NOT dependent on Nclones
# estSelError$TrialData[[1]] %$% length(unique(GID)) 
# MeanPropNOHAV: measure trial quality by how "full" the plots were at harvest
# estSelError$TrialData[[1]] %$% mean(PropNOHAV) 
estSelError %<>%
  mutate(Nclones=map_dbl(TrialData,~length(unique(.$GID))),
         MeanPropNOHAV=map_dbl(TrialData,~mean(.$PropNOHAV,na.rm=T)),
         Nobs=map_dbl(TrialData,~nrow(.)),
         TrialDesign=paste0("Complete",CompleteBlocks,"Incomplete",IncompleteBlocks)) %>% 
  select(-TrialData) %>% 
  unnest(SelectionError) %>% 
  select(-SI_BLUPs,-BLUPs,-SelectionError) %>% 
  filter(!is.na(TrialMSE)) %>% 
  mutate(propAvailForReg=NcloneForReg/Nclones)
estSelError %>% str

tibble [516 × 17] (S3: tbl_df/tbl/data.frame)
 $ studyYear       : int [1:516] 2013 2013 2013 2013 2013 2013 2013 2013 2013 2013 ...
 $ locationName    : chr [1:516] "Ibadan" "Ibadan" "Ibadan" "Ibadan" ...
 $ studyName       : chr [1:516] "13Ayt16ICTIB" "13ayt20pdIB" "13ayt20pdwhtrtIB" "13ayt20pdyrtIB" ...
 $ TrialType       : chr [1:516] "AYT" "AYT" "AYT" "AYT" ...
 $ MaxNOHAV        : num [1:516] 10 10 10 10 11 20 10 10 10 10 ...
 $ CompleteBlocks  : logi [1:516] TRUE TRUE TRUE TRUE TRUE TRUE ...
 $ IncompleteBlocks: logi [1:516] FALSE FALSE FALSE FALSE FALSE FALSE ...
 $ propGenotyped   : num [1:516] 0.983 0.516 0.575 0.75 0.6 ...
 $ cor2si          : num [1:516] -0.3344 0.2644 0.573 -0.0843 0.5199 ...
 $ r2_si           : num [1:516] 0.1118 0.0699 0.3284 0.0071 0.2703 ...
 $ TrialMSE        : num [1:516] 22982 15691 1933 7259 17167 ...
 $ NcloneForReg    : int [1:516] 30 10 23 25 11 11 11 11 9 28 ...
 $ Nclones         : num [1:516] 60 31 40 60 30 65 30 30 30 81 ...
 $ MeanPropNOHAV   : num [1:516] 0.835 0.738 0.773 0.713 0.651 ...
 $ Nobs            : num [1:516] 240 124 160 240 120 260 120 120 120 275 ...
 $ TrialDesign     : chr [1:516] "CompleteTRUEIncompleteFALSE" "CompleteTRUEIncompleteFALSE" "CompleteTRUEIncompleteFALSE" "CompleteTRUEIncompleteFALSE" ...
 $ propAvailForReg : num [1:516] 0.5 0.323 0.575 0.417 0.367 ...

estSelError %>% 
  select(cor2si,TrialMSE,MaxNOHAV,NcloneForReg,Nclones,MeanPropNOHAV,Nobs,propAvailForReg) %>% 
  cor(., use='pairwise.complete.obs') %>% 
  round(.,2) %>% 
  corrplot::corrplot(type="lower")

Version	Author	Date
182b83c	wolfemd	2021-08-19

MeanPropNOHAV: Avg. proportion of stands harvested per plot. Measure of trial quality / success.
Nclones and Nobs: Measures of trial size
propAvailForReg = NcloneForReg/Nclones: a weight for the regressions below. The maximum weight is to be on trials with as many clones as intended scored on the selection index.

Linear model below: include all feasible co-variables and an interaction b/t TrialDesign and MaxNOHAV (plot size)

lm(cor2si~MaxNOHAV+TrialDesign+TrialDesign*MaxNOHAV+studyYear+Nclones+MeanPropNOHAV,
   data = estSelError,
   weights = propAvailForReg) %>% summary


Call:
lm(formula = cor2si ~ MaxNOHAV + TrialDesign + TrialDesign * 
    MaxNOHAV + studyYear + Nclones + MeanPropNOHAV, data = estSelError, 
    weights = propAvailForReg)

Weighted Residuals:
     Min       1Q   Median       3Q      Max 
-0.93878 -0.21431  0.01064  0.22382  0.57224 

Coefficients:
                                                  Estimate Std. Error t value
(Intercept)                                     -5.222e+01  2.674e+01  -1.953
MaxNOHAV                                         5.222e-03  1.598e-02   0.327
TrialDesignCompleteFALSEIncompleteTRUE          -3.082e-02  2.382e-01  -0.129
TrialDesignCompleteTRUEIncompleteFALSE           2.480e-02  2.255e-01   0.110
TrialDesignCompleteTRUEIncompleteTRUE           -1.013e-01  2.284e-01  -0.443
studyYear                                        2.580e-02  1.330e-02   1.940
Nclones                                          5.049e-04  1.668e-04   3.027
MeanPropNOHAV                                    3.762e-01  1.515e-01   2.484
MaxNOHAV:TrialDesignCompleteFALSEIncompleteTRUE  5.469e-03  1.913e-02   0.286
MaxNOHAV:TrialDesignCompleteTRUEIncompleteFALSE -7.924e-04  1.676e-02  -0.047
MaxNOHAV:TrialDesignCompleteTRUEIncompleteTRUE   7.955e-03  1.714e-02   0.464
                                                Pr(>|t|)   
(Intercept)                                      0.05134 . 
MaxNOHAV                                         0.74393   
TrialDesignCompleteFALSEIncompleteTRUE           0.89711   
TrialDesignCompleteTRUEIncompleteFALSE           0.91248   
TrialDesignCompleteTRUEIncompleteTRUE            0.65766   
studyYear                                        0.05293 . 
Nclones                                          0.00259 **
MeanPropNOHAV                                    0.01332 * 
MaxNOHAV:TrialDesignCompleteFALSEIncompleteTRUE  0.77513   
MaxNOHAV:TrialDesignCompleteTRUEIncompleteFALSE  0.96231   
MaxNOHAV:TrialDesignCompleteTRUEIncompleteTRUE   0.64279   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.3097 on 501 degrees of freedom
  (4 observations deleted due to missingness)
Multiple R-squared:  0.05904,   Adjusted R-squared:  0.04026 
F-statistic: 3.144 on 10 and 501 DF,  p-value: 0.0006501

lm(TrialMSE~MaxNOHAV+TrialDesign+TrialDesign*MaxNOHAV+studyYear+Nclones+MeanPropNOHAV,
   data = estSelError,
   weights = propAvailForReg) %>% summary


Call:
lm(formula = TrialMSE ~ MaxNOHAV + TrialDesign + TrialDesign * 
    MaxNOHAV + studyYear + Nclones + MeanPropNOHAV, data = estSelError, 
    weights = propAvailForReg)

Weighted Residuals:
    Min      1Q  Median      3Q     Max 
-9940.9 -2731.2  -549.3  2480.9 12845.6 

Coefficients:
                                                  Estimate Std. Error t value
(Intercept)                                     914475.297 356860.012   2.563
MaxNOHAV                                          -232.808    213.554  -1.090
TrialDesignCompleteFALSEIncompleteTRUE           -5230.826   3182.386  -1.644
TrialDesignCompleteTRUEIncompleteFALSE           -7568.083   3012.140  -2.513
TrialDesignCompleteTRUEIncompleteTRUE            -2341.172   3041.104  -0.770
studyYear                                         -446.189    177.483  -2.514
Nclones                                              4.471      2.225   2.010
MeanPropNOHAV                                      891.391   2023.441   0.441
MaxNOHAV:TrialDesignCompleteFALSEIncompleteTRUE    370.713    255.684   1.450
MaxNOHAV:TrialDesignCompleteTRUEIncompleteFALSE    396.631    223.945   1.771
MaxNOHAV:TrialDesignCompleteTRUEIncompleteTRUE     290.578    228.212   1.273
                                                Pr(>|t|)  
(Intercept)                                       0.0107 *
MaxNOHAV                                          0.2762  
TrialDesignCompleteFALSEIncompleteTRUE            0.1009  
TrialDesignCompleteTRUEIncompleteFALSE            0.0123 *
TrialDesignCompleteTRUEIncompleteTRUE             0.4418  
studyYear                                         0.0122 *
Nclones                                           0.0450 *
MeanPropNOHAV                                     0.6597  
MaxNOHAV:TrialDesignCompleteFALSEIncompleteTRUE   0.1477  
MaxNOHAV:TrialDesignCompleteTRUEIncompleteFALSE   0.0771 .
MaxNOHAV:TrialDesignCompleteTRUEIncompleteTRUE    0.2035  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4139 on 505 degrees of freedom
Multiple R-squared:  0.1047,    Adjusted R-squared:  0.08696 
F-statistic: 5.905 on 10 and 505 DF,  p-value: 1.819e-08

<6% variance explained (cor2si) and 10% for TrialMSE.

Next simplify a bit, remove the interactions and TrialDesign effect, but keep the covariates.

Simplify it a bit, e.g.: cor2si~MaxNOHAV+studyYear+Nclones+MeanPropNOHAV

lm_cor2si<-lm(cor2si~MaxNOHAV+studyYear+Nclones+MeanPropNOHAV,
   data = estSelError,
   weights = propAvailForReg)
lm_cor2si %>% summary


Call:
lm(formula = cor2si ~ MaxNOHAV + studyYear + Nclones + MeanPropNOHAV, 
    data = estSelError, weights = propAvailForReg)

Weighted Residuals:
     Min       1Q   Median       3Q      Max 
-0.95664 -0.21012  0.00948  0.22580  0.58414 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)   -5.633e+01  2.363e+01  -2.384   0.0175 *  
MaxNOHAV       7.920e-03  3.894e-03   2.034   0.0425 *  
studyYear      2.782e-02  1.174e-02   2.369   0.0182 *  
Nclones        5.196e-04  1.219e-04   4.264 2.39e-05 ***
MeanPropNOHAV  3.718e-01  1.487e-01   2.500   0.0127 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.3082 on 507 degrees of freedom
  (4 observations deleted due to missingness)
Multiple R-squared:  0.05678,   Adjusted R-squared:  0.04934 
F-statistic: 7.631 on 4 and 507 DF,  p-value: 5.642e-06

lm(TrialMSE~MaxNOHAV+studyYear+Nclones+MeanPropNOHAV,
   data = estSelError,
   weights = propAvailForReg) %>% summary


Call:
lm(formula = TrialMSE ~ MaxNOHAV + studyYear + Nclones + MeanPropNOHAV, 
    data = estSelError, weights = propAvailForReg)

Weighted Residuals:
    Min      1Q  Median      3Q     Max 
-9411.3 -2973.2  -403.7  2316.7 13128.4 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)   447088.115 324383.326   1.378   0.1687    
MaxNOHAV         110.772     53.086   2.087   0.0374 *  
studyYear       -217.692    161.210  -1.350   0.1775    
Nclones            7.450      1.678   4.440  1.1e-05 ***
MeanPropNOHAV   2125.966   2049.063   1.038   0.3000    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4248 on 511 degrees of freedom
Multiple R-squared:  0.04559,   Adjusted R-squared:  0.03812 
F-statistic: 6.102 on 4 and 511 DF,  p-value: 8.407e-05

# VarT=100
# h2=0.2
# VarE=(1-h2)*VarT
# tibble(plotSize=1:50) %>% 
#   mutate(expCorToSI=lm_cor2si$coefficients[["MaxNOHAV"]]*plotSize,
#          sqCorToSI=round(expCorToSI^2,2),
#          VarE=VarE*(1-sqCorToSI))

Conclusions

The analysis seems far from conclusive. Do we even expect the relation b/t plot-size and error variance to be linearly decreasing?

Certainly this will vary by breeding, location, and much more.

There are also possibly many alternative methods to analyze this problem, and alternative datasets to choose.

NEW APPROACH: While inconclusive, this exercise emphasizes a key concern for conducting simulations that alter the VDP: that the cost-benefit balance could depend on the relative information value/selection accuracy/error variance of different plot sizes and trial configurations. As a result, I propose to simulate a range of error-vs-plot size as part of the baseline simulation of each breeding program. If we observe a shift-point in the cost-benefit analysis we can then work with breeding programs to determine where their data indicate they lie and what changes are subsequently recommended.

sessionInfo()

R version 4.1.0 (2021-05-18)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur 10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] forcats_0.5.1            stringr_1.4.0            readr_2.0.1             
 [4] ggplot2_3.3.5            tidyverse_1.3.1          genomicMateSelectR_0.2.0
 [7] purrr_0.3.4              tidyr_1.1.3              dplyr_1.0.7             
[10] tibble_3.1.3             workflowr_1.6.2         

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.7        lattice_0.20-44   lubridate_1.7.10  here_1.0.1       
 [5] assertthat_0.2.1  rprojroot_2.0.2   digest_0.6.27     utf8_1.2.2       
 [9] R6_2.5.0          cellranger_1.1.0  backports_1.2.1   reprex_2.0.1     
[13] evaluate_0.14     highr_0.9         httr_1.4.2        pillar_1.6.2     
[17] rlang_0.4.11      readxl_1.3.1      rstudioapi_0.13   whisker_0.4      
[21] jquerylib_0.1.4   Matrix_1.3-4      rmarkdown_2.10    splines_4.1.0    
[25] labeling_0.4.2    munsell_0.5.0     broom_0.7.9       compiler_4.1.0   
[29] httpuv_1.6.1      modelr_0.1.8      xfun_0.25         pkgconfig_2.0.3  
[33] mgcv_1.8-36       htmltools_0.5.1.1 tidyselect_1.1.1  fansi_0.5.0      
[37] crayon_1.4.1      tzdb_0.1.2        dbplyr_2.1.1      withr_2.4.2      
[41] later_1.2.0       grid_4.1.0        nlme_3.1-152      jsonlite_1.7.2   
[45] gtable_0.3.0      lifecycle_1.0.0   DBI_1.1.1         git2r_0.28.0     
[49] magrittr_2.0.1    scales_1.1.1      cli_3.0.1         stringi_1.7.3    
[53] farver_2.1.0      fs_1.5.0          promises_1.2.0.1  xml2_1.3.2       
[57] bslib_0.2.5.1     ellipsis_0.3.2    generics_0.1.0    vctrs_0.3.8      
[61] tools_4.1.0       glue_1.4.2        hms_1.1.0         yaml_2.2.1       
[65] colorspace_2.0-2  corrplot_0.90     rvest_1.0.1       knitr_1.33       
[69] haven_2.4.3       sass_0.4.0

Empirical inputs for simulations - IITA

2021-Aug-13

Estimate selection error

Concept

Analysis

Result

Conclusions