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Advanced Normal Data Example
Aimee Lee Houde
2024-02-04
In this vignette, the fullfact package is explored using
advanced (functions designated by the number 2) for the standard model,
i.e. containing random effects for dam, sire, and dam by sire, for
normal data type or error structure, with the options of including
additional random effects for one position (e.g. tank) and/or one block
effect (e.g. several blocks of 2 \(\times\) 2 factorial matings).
Simple (functions designated by no number) for the standard model
only is explored in the vignette Simple Normal Data
Example.
Expert (functions designated by the number 3) for the standard model
with the ability of the user to include additional fixed and/or random
effects, such as a model including environment treatments and their
interactions is explored in the vignette Expert Normal Data
Example.
Non-normal error structures (e.g. binary, proportion, and/or count
data types) are explored in another three vignettes: (1) Simple
Non-normal Data Example, (2) Advanced Non-normal Data Example, and (3)
Expert Non-normal Data Example.
Load the package and example data
The example data set is an 11 \(\times\) 11 full factorial mating: 11 dams
and 11 sires with all combinations resulting in 121 families. There are
10 observations per family or 5 observations for each of two replicates
per family.
library("fullfact")
data(chinook_length)
head(chinook_length)
#> family repli dam sire tray cell length egg_size
#> 1 f1 r1 d1 s1 t7 1A 22.7 7.27
#> 2 f1 r2 d1 s1 t8 1A 22.0 7.27
#> 3 f1 r1 d1 s1 t7 1A 23.8 7.27
#> 4 f1 r2 d1 s1 t8 1A 21.8 7.27
#> 5 f1 r1 d1 s1 t7 1A 22.6 7.27
#> 6 f1 r2 d1 s1 t8 1A 22.4 7.27
Displayed are columns for family identities (ID), replicate ID, dam
ID, sire ID, incubation tray ID, incubation cell ID (within tray),
Chinook salmon length (mm) at hatch, and dam egg size (mm).
Observed variance components
Model random effects are dam, sire, and dam by sire. Options to
include one random position and/or one random block effect(s). Extracts
the dam, sire, dam, dam by sire, and residual variance components.
Calculates the total variance component. Calculates the additive
genetic, non-additive genetic, and maternal variance components.
Extracts optional position and block variance components.
Assuming the effects of epistasis are of negligible importance, the
additive genetic variance (VA) component is calculated as
four times the sire (VS), the non-additive genetic variance
(VN) component as four times the dam by sire interaction
(VD\(\times\)S), and the
maternal variance component (VM) as the dam (VD) –
sire (VS) (Lynch and Walsh 1998, p. 603). When there is
epistasis, those variance components will be overestimated and this may
explain why the percentage of phenotypic variance explained by the
components can add up to more than 100% in certain cases.
position is the column name containing position factor
information.
block is the column name containing block factor
information.
Significance values for the random effects are determined using
likelihood ratio tests (Bolker et al. 2009).
Lynch M, Walsh B. 1998. Genetics and Analysis of Quantitative Traits.
Sinauer Associates, Massachusetts.
Bolker BM, Brooks ME, Clark CJ, Geange SW, Poulsen JR, Stevens MHH,
White J-SS. 2009. Generalized linear mixed models: a practical guide for
ecology and evolution. Trends in Ecology and Evolution 24(3): 127-135.
DOI: 10.1016/j.tree.2008.10.008
For this example, we explore position (tray) effects for Chinook
salmon length.
length_mod2<- observLmer2(observ=chinook_length,dam="dam",sire="sire",response="length",
position="tray")
#> [1] "2024-02-04 15:48:48 PST"
#> Time difference of 0.3131449 secs
length_mod2
#> $random
#> effect variance percent d.AIC d.BIC Chi.sq p.value
#> 1 dam:sire 0.1798015 17.28121 141.49865 136.400272 143.49865 4.573043e-33
#> 2 tray 0.1077362 10.35482 121.50329 116.404911 123.50329 1.082069e-28
#> 3 sire 0.0000000 0.00000 -2.00000 -7.098376 0.00000 1.000000e+00
#> 4 dam 0.2029810 19.50905 29.98924 24.890867 31.98924 1.550287e-08
#>
#> $other
#> component variance percent
#> 1 Residual 0.5499266 52.85493
#> 2 Total 1.0404453 100.00000
#>
#> $calculation
#> component variance percent
#> 1 additive 0.0000000 0.00000
#> 2 nonadd 0.7192061 69.12483
#> 3 maternal 0.2029810 19.50905
Produces a list object containing three data frames. Each data frame
contains the raw variance components and the variance components as a
percentage of the total variance component. The first data frame also
contains the difference in AIC and BIC, and likelihood ratio test
Chi-square and p-value for all random effects.
Note
Default is Restricted maximum likelihood (REML) as
ml = F. Option for maximum likelihood (ML) is
ml = T.
Maximum likelihood (ML) estimates the parameters that maximize the
likelihood of the observed data and has the advantage of using all the
data and accounting for non-independence (Lynch and Walsh 1998, p. 779;
Bolker et al. 2009). On the other hand, ML has the disadvantage of
assuming that all fixed effects are known without error, producing a
downward bias in the estimation of the residual variance component. This
bias can be large if there are lots of fixed effects, especially if
sample sizes are small. Restricted maximum likelihood (REML) has the
advantage of not assuming the fixed effects are known and averages over
the uncertainty, so there can be less bias in the estimation of the
residual variance component. However, REML only maximizes a portion of
the likelihood to estimate the effect parameters, but is the preferred
method for analyzing large data sets with complex structure.
Statistical Power analysis
Power values are calculated by stochastically simulating data for a
number of iterations and then calculating the proportion of P-values
less than \(\alpha\) (e.g. 0.05) for
each component (Bolker 2008). Simulated data are specified by inputs for
variance component values and the sample sizes.
Bolker BM. 2008. Ecological Models and Data in R. Princeton
University Press, Princeton.
Defaults are alpha = 0.05 for 5%,
nsim = 100 for 100 simulations, and ml = F for
REML.
varcomp is a vector of dam, sire, dam by sire, residual,
position and/or block variance components, i.e. c(dam,sire,dam \(\times\) sire,residual,position/block). If
there is a position and a block, c(dam,sire,dam \(\times\) sire,residual,position,block).
nval is a vector of dam, sire, offspring per family, and
offspring per position or number of block sample sizes,
i.e. c(dam,sire,offspring,position/block). If there is a position and a
block, c(dam,sire,offspring,position,block).
position is optional number of positions.
block is optional vector of dams and sires per block,
e.g. c(2,2).
For this example, the variance components of observLmer2
above are used (i.e. dam= 0.2030, sire= 0, dam \(\times\) sire= 0.1798, residual= 0.5499,
tray= 0.1077) and the sample size of the Chinook salmon data set
(i.e. dam= 11, sire= 11, offspring= 10, offspring per position= 110).
Position was represented by 11 trays. The actual design was composed of
16 trays with 55–80 offspring each. However, powerLmer2
uses an equal number of offspring per position, so the number of trays
was decreased from 16 to 11.
Full analysis is 100 simulations. Example has 25 simulations.
#>powerLmer2(varcomp=c(0.2030,0,0.1798,0.5499,0.1077),nval=c(11,11,10,110),position=11) #full
#25 simulations
powerLmer2(varcomp=c(0.2030,0,0.1798,0.5499,0.1077),nval=c(11,11,10,110),position=11,nsim=25)
#> [1] "2024-02-04 15:48:49 PST"
#> [1] "Starting simulation: 1"
#> [1] "Starting simulation: 2"
#> [1] "Starting simulation: 3"
#> [1] "Starting simulation: 4"
#> [1] "Starting simulation: 5"
#> [1] "Starting simulation: 6"
#> [1] "Starting simulation: 7"
#> [1] "Starting simulation: 8"
#> [1] "Starting simulation: 9"
#> [1] "Starting simulation: 10"
#> [1] "Starting simulation: 11"
#> [1] "Starting simulation: 12"
#> [1] "Starting simulation: 13"
#> [1] "Starting simulation: 14"
#> [1] "Starting simulation: 15"
#> [1] "Starting simulation: 16"
#> [1] "Starting simulation: 17"
#> [1] "Starting simulation: 18"
#> [1] "Starting simulation: 19"
#> [1] "Starting simulation: 20"
#> [1] "Starting simulation: 21"
#> [1] "Starting simulation: 22"
#> [1] "Starting simulation: 23"
#> [1] "Starting simulation: 24"
#> [1] "Starting simulation: 25"
#> Time difference of 8.156208 secs
#> term n var_in var_out power
#> 1 dam 11 0.2030 0.17589809 1.00
#> 2 sire 11 0.0000 0.00631872 0.08
#> 3 dam.sire 121 0.1798 0.16449326 1.00
#> 4 position 11 0.1077 0.09609158 1.00
#> 5 residual NA 0.5499 0.54673647 NA
#Block examples using 8 dams, 8 sires (as four 2x2 blocks), and 20 offspring per family
#>powerLmer2(varcomp=c(0.2030,0,0.1798,0.5499,0.1077),nval=c(8,8,20,4),block=c(2,2))
#>powerLmer2(varcomp=c(0.2030,0,0.1798,0.5499,0.1077,0.1077),nval=c(8,8,20,40,4),position=8,
#>block=c(2,2)) #with position
There is sufficient power (\(\ge\)
0.8) for dam and dam by sire variance components, whereas there is
insufficient power (< 0.8) for the sire variance component. Albeit,
the sire component is near zero, so the low power may be an artifact.
There was also sufficient power for position (tray) variance components.
In the cases of insufficient power, the sample size of dam, sire, and/or
offspring can be increased until there is sufficient power.
Taking the reverse approach (can the sample size of dam, sire, or
offspring be reduced while maintaining sufficient power?) using the same
variance components and offspring sample size, dam and sire sample sizes
could be reduced from 11 to 7. The position number and its sample size
was reduced accordingly, i.e. 7 dams \(\times\) 7 sires \(\times\) 10 offspring = 490, divided by 10
trays for 49 offspring each.
#>powerLmer2(varcomp=c(0.2030,0,0.1798,0.5499,0.1077),nval=c(7,7,10,49),position=10) #full
#25 simulations
powerLmer2(varcomp=c(0.2030,0,0.1798,0.5499,0.1077),nval=c(7,7,10,49),position=10,nsim=25)
#> [1] "2024-02-04 15:48:57 PST"
#> [1] "Starting simulation: 1"
#> [1] "Starting simulation: 2"
#> [1] "Starting simulation: 3"
#> [1] "Starting simulation: 4"
#> [1] "Starting simulation: 5"
#> [1] "Starting simulation: 6"
#> [1] "Starting simulation: 7"
#> [1] "Starting simulation: 8"
#> [1] "Starting simulation: 9"
#> [1] "Starting simulation: 10"
#> [1] "Starting simulation: 11"
#> [1] "Starting simulation: 12"
#> [1] "Starting simulation: 13"
#> [1] "Starting simulation: 14"
#> [1] "Starting simulation: 15"
#> [1] "Starting simulation: 16"
#> [1] "Starting simulation: 17"
#> [1] "Starting simulation: 18"
#> [1] "Starting simulation: 19"
#> [1] "Starting simulation: 20"
#> [1] "Starting simulation: 21"
#> [1] "Starting simulation: 22"
#> [1] "Starting simulation: 23"
#> [1] "Starting simulation: 24"
#> [1] "Starting simulation: 25"
#> Time difference of 5.511813 secs
#> term n var_in var_out power
#> 1 dam 7 0.2030 0.22885061 0.84
#> 2 sire 7 0.0000 0.00587355 0.00
#> 3 dam.sire 49 0.1798 0.17430060 1.00
#> 4 position 10 0.1077 0.09689070 1.00
#> 5 residual NA 0.5499 0.55339969 NA
Bootstrap confidence intervals
Confidence intervals for the additive genetic, non-additive genetic,
and maternal variance components can be produced using the bootstrap-t
resampling method described by Efron and Tibshirani (1993, p. 160‒162).
Observations are resampled with replacement until the original sample
size is reproduced. The resampled data are then used in the model and
the additive genetic, non-additive genetic, and maternal variance
components are extracted. The process is repeated for a number of
iterations, typically 1,000 times, to produce a distribution for each
component. The confidence interval lower and upper limits and median are
extracted from the distribution.
Efron B, Tibshirani R. 1993. An Introduction to the Bootstrap.
Chapman and Hall, New York.
Resample observations
The resampRepli function is used to bootstrap resample
observations grouped by replicate ID within family ID for a specified
number of iterations to create the resampled data set. A similar
resampFamily function is able to resample observations
grouped by family ID only.
copy is a vector of column numbers (to copy the
contents). Does not need to contain the family and/or replicate
columns.
Full analysis is 1000 iterations. Example has 5 iterations.
#>resampRepli(dat=chinook_length,copy=c(3:8),family="family",replicate="repli",iter=1000) #full
#>resampFamily(dat=chinook_length,copy=c(3:8),family="family",iter=1000) #family only
resampRepli(dat=chinook_length,copy=c(3:8),family="family",replicate="repli",iter=5) #5 iterations
Because of the large file sizes that can be produced, the resampling
of each replicate Y per family X is saved separately as a common
separated (X_Y_resampR.csv) file in the working directory. These files
are merged to create the final resampled data set (resamp_datR.csv).
If using resampFamily, the file names are X_resampF.csv
per family and resamp_datF.csv for the final resampled data set.
Iteration variance components
The equivalent to observLmer2 is available for the final
bootstrap resampled data set, i.e. resampLmer2.
Default is ml = F for REML. The starting model number
start = and ending model number end = need to
be specified.
Full analysis is 1000 iterations. Example has 5 iterations.
#>length_datR<- read.csv("resamp_datR.csv") #1000 iterations
#>length_rcomp2<- resampLmer2(resamp=length_datR,dam="dam",sire="sire",response="length",
#>position="tray",start=1,end=1000) #full
data(chinook_resampL) #5 iterations
head(chinook_resampL)
#> dam1 sire1 tray1 cell1 length1 egg_size1 dam2 sire2 tray2 cell2 length2
#> 1 d1 s1 t7 1A 23.1 7.27 d1 s1 t7 1A 23.1
#> 2 d1 s1 t7 1A 22.7 7.27 d1 s1 t7 1A 22.7
#> 3 d1 s1 t7 1A 22.9 7.27 d1 s1 t7 1A 22.9
#> 4 d1 s1 t7 1A 23.1 7.27 d1 s1 t7 1A 22.7
#> 5 d1 s1 t7 1A 23.1 7.27 d1 s1 t7 1A 22.9
#> 6 d1 s1 t8 1A 22.4 7.27 d1 s1 t8 1A 22.3
#> egg_size2 dam3 sire3 tray3 cell3 length3 egg_size3 dam4 sire4 tray4 cell4
#> 1 7.27 d1 s1 t7 1A 23.8 7.27 d1 s1 t7 1A
#> 2 7.27 d1 s1 t7 1A 23.8 7.27 d1 s1 t7 1A
#> 3 7.27 d1 s1 t7 1A 23.8 7.27 d1 s1 t7 1A
#> 4 7.27 d1 s1 t7 1A 23.8 7.27 d1 s1 t7 1A
#> 5 7.27 d1 s1 t7 1A 23.8 7.27 d1 s1 t7 1A
#> 6 7.27 d1 s1 t8 1A 22.4 7.27 d1 s1 t8 1A
#> length4 egg_size4 dam5 sire5 tray5 cell5 length5 egg_size5
#> 1 22.6 7.27 d1 s1 t7 1A 22.9 7.27
#> 2 23.8 7.27 d1 s1 t7 1A 22.6 7.27
#> 3 23.1 7.27 d1 s1 t7 1A 22.6 7.27
#> 4 22.9 7.27 d1 s1 t7 1A 22.9 7.27
#> 5 23.8 7.27 d1 s1 t7 1A 22.7 7.27
#> 6 22.0 7.27 d1 s1 t8 1A 22.4 7.27
length_rcomp2<- resampLmer2(resamp=chinook_resampL,dam="dam",sire="sire",response="length",
position="tray",start=1,end=5)
#> [1] "2024-02-04 15:49:03 PST"
#> [1] "Working on model: 1"
#> [1] "Working on model: 2"
#> [1] "Working on model: 3"
#> [1] "Working on model: 4"
#> [1] "Working on model: 5"
#> Time difference of 0.4106691 secs
length_rcomp2[1:5,]
#> dam:sire tray sire dam Residual Total additive
#> 1 0.2156343 0.1010264 9.658700e-09 0.1933548 0.4996255 1.0096410 3.863480e-08
#> 2 0.2423708 0.1084802 3.551509e-10 0.1954621 0.5204227 1.0667358 1.420604e-09
#> 3 0.2204124 0.1072239 3.857906e-10 0.1713985 0.4844660 0.9835008 1.543162e-09
#> 4 0.1959125 0.1204563 1.996910e-03 0.1931546 0.4919967 1.0035171 7.987638e-03
#> 5 0.2263470 0.1308195 0.000000e+00 0.2085843 0.4787757 1.0445264 0.000000e+00
#> nonadd maternal
#> 1 0.8625374 0.1933548
#> 2 0.9694834 0.1954621
#> 3 0.8816494 0.1713985
#> 4 0.7836500 0.1911577
#> 5 0.9053880 0.2085843
The function provides a data frame with columns containing the raw
variance components for dam, sire, dam by sire, residual, total,
additive genetic, non-additive genetic, and maternal. Also columns
containing the raw variance components for the options of position
and/or block. The number of rows in the data frame matches the number of
iterations in the resampled data set and each row represents a model
number.
Plotting confidence intervals
The barMANA and boxMANA functions are
simple plotting functions for the confidence intervals or all values,
respectively, from the bootstrap and jackknife approaches. Default is to
display the percentage values as type = perc. Raw values
can be displayed as type = raw.
Within the functions, there are simple plot modifications available.
For the y-axis, min and max values can be species as ymin
and ymax, as well as the increment as yunit.
Also, magnification of the axis unit as cex_yaxis and label
as cex_ylab. The position of the legend can be specified as
leg. Default is “topright”.
Bar plot
The barMANA function produces bar graphs with the
bootstrap-t median (ciMANA2) or jackknife pseudo-value mean
(ciJack2) as the top of the shaded bar and error bars
covering the range of the confidence interval for each of the additive
genetic, non-additive genetic, and maternal values of a phenotypic
trait.
The length of the error bar can be specified in inches as
bar_len.
length_ci<- ciJack2(comp=chinook_jackL,position="tray",full=c(0,0.7192,0.2030,1.0404,0.1077))
oldpar<- par(mfrow=c(2,1))
barMANA(ci_dat=length_ci) #basic, top
barMANA(ci_dat=length_ci,bar_len=0.3,yunit=20,ymax=100,cex_ylab=1.3) #modified, bottom
Different traits can also be combined on the same bar plot using
trait specified in ciMANA2 or
ciJack2. The information is combined into a list object.
For the example, the jackknife CI is duplicated to simulate ‘different
traits’.
length_ci1<- ciJack2(comp=chinook_jackL,position="tray",full=c(0,0.7192,0.2030,1.0404,0.1077),
trait="length_1")
length_ci2<- ciJack2(comp=chinook_jackL,position="tray",full=c(0,0.7192,0.2030,1.0404,0.1077),
trait="length_2")
comb_bar<- list(raw=rbind(length_ci1$raw,length_ci2$raw),
percentage=rbind(length_ci1$percentage,length_ci2$percentage))
barMANA(ci_dat=comb_bar,bar_len=0.3,yunit=20,ymax=100,cex_ylab=1.3)
The legend is slightly off in the presented html version but is fine
with the R plotting device.
Box plot
The boxMANA function produces box plots using all values
for the bootstrap-t resampling data set (resampLmer2) or
jackknife resampling data set (JackLmer2).
oldpar<- par(mfrow=c(2,1))
boxMANA(comp=chinook_bootL) #from resampLmer2, basic, top
boxMANA(comp=chinook_bootL,yunit=20,ymax=100,cex_ylab=1.3,leg="topleft") #modified, bottom
Different traits can also be combined on the same box plot by adding
a “trait” column to the resampling data set. For the example, the
bootstrap-t data frame is duplicated to simulate ‘different traits’.
chinook_bootL1<- chinook_bootL; chinook_bootL2<- chinook_bootL #from resampLmer2
chinook_bootL1$trait<- "length_1"; chinook_bootL2$trait<- "length_2"
comb_boot<- rbind(chinook_bootL1,chinook_bootL2)
comb_boot$trait<- as.factor(comb_boot$trait)
boxMANA(comb_boot,yunit=20,ymax=100,cex_ylab=1.3,leg="topleft")
The recommended follow-up vignette is the Expert Normal Data
Example, covering the standard model with the ability of the
user to include additional fixed and/or random effects, such as a model
including environment treatments and their interactions.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.