Theoretical background
IBD Calculations
The calculations of the IBDs are based on Hidden Markov Models (HMM)
and inheritance vectors and are performed using the statgenIBD R package.
For details of the theory see Lander and Green
(1987) and
Huang et al. (2011). It is also possible to import IBD
probabilities computed using the RABBIT software (Zheng,
Boer, and Van Eeuwijk 2014, 2015, 2018). Using the statgenIBD
package, IBD probabilities can be calculated for many different types of
populations. See the
statgenIBD vignette for an overview of these populations. Using
RABBIT more complex populations such as MAGIC can also be analyzed.
QTL Mapping
We follow the methods described in Li et al.
(2021) for QTL
mapping. In the first step a model without markers is fitted:
\[y = X\beta + \varepsilon, \quad
\varepsilon \sim { }N\left( {0, \oplus_{k = 1}^{F} I_{{n_{k} }}
\sigma_{{\varepsilon_{k} }}^{2} } \right),\] where \(y\) is a vector with phenotypes, \(X\) the design matrix for the cross
structure, \(\beta\) a vector of fixed
cross intercepts, and the residual term \(\varepsilon\) has a cross-specific variance
covariance structure \(\oplus_{k = 1}^{F}
I_{{n_{k} \sigma_{\varepsilon_{k}}^{2}}}\) in which \(\sigma_{\varepsilon_{k}}^{2}\) is the
residual variance of the \(k^{th}\)
cross with family size \(n_{k}\), and
\(F\) is the number of families. If
\(F=1\), for example for a single
biparental cross or a MAGIC population, the variance covariance
structure of the residual term is homogeneous.
Then for each of the evaluation points \(q\) in the map a second model is
fitted:
\[y = X\beta + M_{q} a_{q} + \varepsilon,
a_{q} \sim { }N\left( {0,I_{P} \sigma_{q}^{2} } \right), \varepsilon
\sim { }N\left( {0, \oplus_{k = 1}^{F} I_{{n_{k} }}
\sigma_{{\varepsilon_{k} }}^{2} } \right),\] where \(M_{q}\) is the design matrix containing the
expected number of parental alleles as a function of IBD probabilities,
\(a_{q}\) is the vector of random
parental effects with variance covariance structure \(I_{P} \sigma_{q}^{2}\), where \(\sigma_{q}^{2}\) is the genetic variance of
the QTL effects at P parents.
Variance components of a putative QTL from the two models are
evaluated to perform the likelihood ratio test (LRT). The p-value is
calculated using a \(\chi^2\) mixture
approximation for the Likelihood Ratio Test.
For Multi-QTL Mapping (MQM) the first steps are the same. However
after the first round of scanning the QTL with the highest \(-log10(p)\) value is added as a cofactor in
the model and a new round of scanning is done. This is repeated until no
new QTLs are found with a \(-log10(p)\)
value above a predefined threshold, or until the predefined maximum
number of cofactors is added to the model. In a region of specified
width around a cofactor no new cofactors can be detected.
It is possible to include a polygenic \(g\) term in the models to control for
background genetic information. The models fitted then become:
\[y = X\beta + g + \varepsilon, \quad g
\sim { }N\left( {0,K \sigma_{g}^{2} } \right), \varepsilon \sim {
}N\left( {0, \oplus_{k = 1}^{F} I_{{n_{k} }} \sigma_{{\varepsilon_{k}
}}^{2} } \right),\]
\[y = X\beta + M_{q} a_{q} + g +
\varepsilon, a_{q} \sim { }N\left( {0,I_{P} \sigma_{q}^{2} } \right), g
\sim { }N\left( {0,K \sigma_{g}^{2} } \right), \varepsilon \sim {
}N\left( {0, \oplus_{k = 1}^{F} I_{{n_{k} }} \sigma_{{\varepsilon_{k}
}}^{2} } \right),\]
where the VCOV structure of the polygenic term \(g\) is \(K
\sigma_{g}^{2}\) with \(\sigma_{g}^{2}\) the variance of the
polygenic effect, and \(K\) a kinship
matrix. To avoid proximal contamination a chromosome specific kinship
matrix is used, computed using the markers on the other chromosomes.
The symmetric kinship matrix \(K\)
can have zero eigenvalues, which leads to problems solving the mixed
model equations. For that reason we use a spectral decomposition of
\(K\), \(K=U
D U^{-1}\), and only use the eigenvectors \(U_+\), with corresponding positive
eigenvalues in diagonal matrix \(D_+\).
Using the transformation \(g=Zu\), with
\(Z=U_+ D_+^{1/2}\) we obtain
\[y = X\beta + Zu + \varepsilon, \quad u
\sim { }N\left( {0,I \sigma_{u}^{2} } \right), \varepsilon \sim {
}N\left( {0, \oplus_{k = 1}^{F} I_{{n_{k} }} \sigma_{{\varepsilon_{k}
}}^{2} } \right),\]
\[y = X\beta + M_{q} a_{q} + Zu +
\varepsilon, a_{q} \sim { }N\left( {0,I_{P} \sigma_{q}^{2} } \right), g
\sim { }N\left( {0,I \sigma_{u}^{2} } \right), \varepsilon \sim {
}N\left( {0, \oplus_{k = 1}^{F} I_{{n_{k} }} \sigma_{{\varepsilon_{k}
}}^{2} } \right),\]
Example simulated data
As a first example for performing IBD calculations and QTL mapping
for a multi-parent population we use a relatively simple simulated data
set. The example contains simulated data for two F4DH populations. The
population type F4DH is cross between two parents, followed by 3
generations of selfings, followed by a DH generation, see statgenIBD for
details.
For the first population the parents where A and B, for the second
the parents where A and C. The first population consists of 100
individuals, the second of 80 individuals. This is a simple example of a
NAM population, having parent A as central parent. The data is simulated
with three QTLs, on chromosome 1, 2, and 3. All necessary data for this
population is available in the package.
Before doing QTL detection we compute IBD probabilities on a grid of
positions along the genome. This can be done using the
calcIBDMPP function in the statgenMPP package.
To perform IBD calculations a marker file is required for each of the
populations. These files should be a tab-delimited file with first
column ID identifying the genotype. The following columns should contain
marker information. The first rows should contain the parents used in
the cross. As an example, the file for the first cross, AxB, starts like
this:
| A |
1 |
2 |
2 |
2 |
| B |
2 |
2 |
2 |
1 |
| AxB0001 |
1 |
2 |
2 |
1 |
| AxB0002 |
2 |
2 |
2 |
2 |
In this example markers M1_1 and M1_4 are segregating in the AxB
cross, M1_2 and M1_3 not.
A map file is also required. This should also be a tab-delimited file
with three columns, “marker name”, “chromosome” and “position”. The map
file cannot contain a header and has to be identical for all
crosses.
Phenotypic data can be added as a data.frame when
computing IBD probabilities. Such a data.frame should have
a first column “genotype” and all other columns have to be numerical.
For the simulated NAM population the data.frame with
phenotypic data starts like this:
| AxB0001 |
9.89 |
| AxB0002 |
6.55 |
| AxB0003 |
7.90 |
| AxB0004 |
4.46 |
| AxB0005 |
5.21 |
| AxB0006 |
5.27 |
The phenotypic data only contains a single trait, yield. When
performing IBD calculations and specifying phenotypic data, the
phenotypic data will be combined with computed IBD probabilities. For
this, the genotype specified in the ID column in the marker file(s) will
be matched with the genotype in the genotype column in the
data.frame with phenotypic information. Phenotypic data for
all crosses can either be added from a single data.frame
containing phenotypic data for all crosses, or a list of
data.frames each containing phenotypic data for a single
cross.
## Specify files containing markers.
# One file for each of the two crosses.
markerFiles <- c(system.file("extdata/multipop", "AxB.txt",
package = "statgenMPP"),
system.file("extdata/multipop", "AxC.txt",
package = "statgenMPP"))
## Specify file containing map.
# Both crosses use the same map file.
mapFile <- system.file("extdata/multipop", "mapfile.txt",
package = "statgenMPP")
## Read phenotypic data
phenoDat <- read.delim(system.file("extdata/multipop", "AxBxCpheno.txt",
package = "statgenMPP"))
# Check contents.
head(phenoDat)
#> genotype yield
#> 1 AxB0001 9.89
#> 2 AxB0002 6.55
#> 3 AxB0003 7.90
#> 4 AxB0004 4.46
#> 5 AxB0005 5.21
#> 6 AxB0006 5.27
## Perform IBD calculations.
ABCMPP <- calcIBDMPP(crossNames = c("AxB", "AxC"),
markerFiles = markerFiles,
pheno = phenoDat,
popType = "F4DH",
mapFile = mapFile,
evalDist = 5)
With calcIBDMPP IBD probabilities are computed on a grid
for each of the crosses separately and then combined into a single
output object. The value of evalDist can be used to specify
the maximum distance (in cM) between two evaluation points on the grid.
The exact distance depends on the length of the chromosomes. The output
is stored in an object of class gDataMPP (genomic Data for
Multi Parent Populations) in which information about map, markers, and
phenotypic data is combined. With the summary function we
can get some insight in this information.
## Print summary
summary(ABCMPP)
#> map
#> Number of markers: 95
#> Number of chromosomes: 5
#>
#> markers
#> Number of markers: 95
#> Number of genotypes: 180
#> Parents: A, B, C
#> pheno
#> Number of traits: 1
#> Traitnames: yield
#> Number of genotypes: 180
#>
#> crosses
#> AxB:100
#> AxC: 80
To get a further idea about the population and the computed IBD
probabilities we can visualize the results. First we have a look at
structure of the pedigree of the population using
plotType = "pedigree" to get a general idea of what the
design looks like.
## Plot structure of the pedigree.
plot(ABCMPP, plotType = "pedigree")
Next we look at the genetic map using
plotType = "genMap". This will display the genetic map of
the population showing the length of each of the chromosomes and
indicating the positions where the IBD probabilities were calculated.
Optionally it is possible to highlight one or more markers using
highlight argument.
## Plot genetic map.
# Highlight marker on chromosome 3 at position 40.
plot(ABCMPP, plotType = "genMap", highlight = "EXT_3_40")
Finally we visualize the computed IBD probabilities across the genome
for a selected genotype using plotType = "singleGeno". This
plot will show the IBD probabilities for all parents for all positions
on the genome for the selected genotype.
## Plot IBD probabilities for genotype AxB0001.
plot(ABCMPP, plotType = "singleGeno", genotype = "AxB0001")
Using the computed IBD probabilities we can now do the actual QTL
Mapping using the selQTLMPP function. First we perform a
SQM by setting maxCofactors = 0. Our trait of interest,
yield, is specified in the trait argument of the
function.
## Perform Single-QTL Mapping.
ABCSQM <- selQTLMPP(MPPobj = ABCMPP,
trait = "yield",
maxCofactors = 0)
The results of the SQM can be plotted using the plot
function with plotType = "QTLProfile".
## Plot QTL Profile for ABC SQM.
plot(ABCSQM, plotType = "QTLProfile")
Already from the SQM the position of the three simulated QTLs is
quite clear. We can get an even better result using MQM. For that, again
we use the selQTLMPP function. We don’t specify the
maxCofactors to let the algorithm determine the number of
cofactors based on the threshold and QTLwindow. As long as new markers
are found with a \(-log10(p)\) value
above threshold and the maximum number of cofactors is not
reached, a new round of scanning is done. In the new round of scanning
the marker with the highest \(-log10(p)\) value from the previous round
is added to the cofactors. Based on the profile plot for the SQM the
threshold is set to 3. The QTLwindow is not
specified and therefore left at its default value of 10cM.
## Perform Multi-QTL Mapping.
ABCMQM <- selQTLMPP(MPPobj = ABCMPP,
trait = "yield",
threshold = 3)
It is possible to include a polygenic term in the model to control
for background genetic information. To do this a list of chromosome
specific kinship matrices can either be computed by the
selQTLMPP function or specified by the user. In the former
case it is enough to specify computeKin = TRUE. When doing
so the kinship matrix is computed by averaging \(Z Z^t\) over all markers, where \(Z\) is the genotype \(\times\) parents matrix with IBD
probabilities for the marker. A list of precomputed chromosome specific
kinship matrices can be specified in K. Note that adding a
kinship matrix to the model increases the computation time a lot,
especially for large populations. It is advisable to use parallel
computing when doing so (see Parallel
computing).
## Perform Multi-QTL Mapping.
# Compute kinship matrices.
ABCMQM_kin <- selQTLMPP(MPPobj = ABCMPP,
trait = "yield",
threshold = 3,
computeKin = TRUE)
Next we can visualize the results. First we plot the positions of the
QTLs found on the genetic map. This will produce a plot similar the the
genetic map plot we have seen before, but now the QTLs will be
highlighted. This plot can be made by specifying
plotType = "QTLRegion"
## Plot QTL Profile for ABC MQM.
plot(ABCMQM, plotType = "QTLRegion")
As for the SQM we can also plot the QTL profile. The QTLs found will
be highlighted in the profile in red.
## Plot QTL Profile for ABC MQM.
plot(ABCMQM, plotType = "QTLProfile")
It is also possible to plot the size of the parental effects for each
of the QTLs found. Positive effects of a parent on the trait will be
indicated by shades of red, negative effects by shades of blue. The
stronger the color, the larger the effect for the specific parent is.
This plot can be made using plotType = "parEffs".
## Plot QTL Profile for ABC MQM.
plot(ABCMQM, plotType = "parEffs")
Finally a combined plot of the QTL profile and the parental can be
made. In this plot the two previous plots are plotted above each other
with the chromosomes and positions aligned to allow for easily getting
an overview of which effect belongs to which QTL in the QTL profile.
This plot can be made using plotType = "QTLProfileExt".
## Plot QTL Profile for ABC MQM.
plot(ABCMQM, plotType = "QTLProfileExt")
A summary of the QTL-analyis gives a short overview containing the
total number of markers and the number of QTLs found. Also for all QTL
their position on the chromosome is shown as well as the nearest marker
on the original map, the explained variance and the effects of all
parents.
## Print summary
summary(ABCMQM)
#> Trait analysed: yield
#>
#> Data are available for 95 markers.
#> Threshold: 3
#>
#> Number of QTLs: 3
#>
#> evalPos chr pos mrkNear minlog10p varExpl eff_A eff_B eff_C
#> EXT_1_25 1 25 M1_3 16.45 0.242 -1.417 0.859 0.558
#> EXT_2_75 2 75 M2_8 8.59 0.217 0.833 -1.327 0.495
#> EXT_3_65 3 65 M3_7 11.69 0.287 0.984 0.560 -1.544
From the output of selQTLMPP the p-Values and effects
for all markers can be extracted. They are stored in a
data.table within the output object. The example below
shows how to extract them. The output will contain columns evalPos, chr
and pos with name, chromosome number and evaluation position and columns
pValue and eff_par1, eff_par2, eff_par.. with the effects of all parents
for that marker.
## Extract results of QTL mapping.
ABCMQMres <- ABCMQM$GWAResult$yield
head(ABCMQMres[, 1:8])
#> NULL
It is also possible to only extract the markers that are either QTLs
or within the window of one of the selected QTLs, as specified when
calling the selQTLMPP function. As for the full results,
this information in stored in the output as a data.table
that can be extracted as shown below. The columns in this
data.table are identical to those in the full results
except for an additional column at the end, snpStatus, that shows
whether a marker is a QTL or within the window of a QTL.
## Extract QTLs and markers within QTL windows.
ABCMQMQTL <- ABCMQM$signSnp$yield
head(ABCMQMQTL[, c(2:8, 10)])
#> NULL
Example maize
As a second example we use data from a maize NAM population described
in Giraud et al. (2014). The NAM population consists of 10
bi-parental doubled haploid (DH) crosses with central parent F353. The
total population consists of 841 individuals. Several traits were
measured in four locations across Europe. We calculated the best linear
unbiased estimations (BLUEs) of those traits using the R package statgenSTA. As an
example, we perform QTL mapping for only the mean value of the number of
days to silking (“mean_DtSILK”) across all locations. The data for this
population is available from the package in zipped format.
As for the simulated data, before doing QTL detection we first
compute IBD probabilities on a grid of positions along the genome.
## Define names of crosses.
crosses <- paste0("F353x", c("B73", "D06", "D09", "EC169", "F252", "F618",
"Mo17", "UH250", "UH304", "W117"))
head(crosses)
#> [1] "F353xB73" "F353xD06" "F353xD09" "F353xEC169" "F353xF252" "F353xF618"
## Specify files containing crosses.
## Extract them in a temporary directory.
tempDir <- tempdir()
crossFiles <- unzip(system.file("extdata/maize/maize.zip", package = "statgenMPP"),
files = paste0(crosses, ".txt"), exdir = tempDir)
## Specify file containing map.
mapFile <- unzip(system.file("extdata/maize/maize.zip", package = "statgenMPP"),
files = "map.txt", exdir = tempDir)
## Read phenotypic data.
phenoFile <- unzip(system.file("extdata/maize/maize.zip", package = "statgenMPP"),
files = "EUmaizePheno.txt", exdir = tempDir)
phenoDat <- read.delim(phenoFile)
head(phenoDat[, 1:5])
#> genotype INR_DMY KWS_DMY Syngenta_DMY TUM_DMY
#> 1 CFD02-003 182 NA 173 NA
#> 2 CFD02-006 172 201 159 243
#> 3 CFD02-010 208 237 159 228
#> 4 CFD02-024 185 219 161 214
#> 5 CFD02-027 206 226 185 221
#> 6 CFD02-036 180 223 163 216
## Perform IBD calculations.
maizeMPP <- calcIBDMPP(crossNames = crosses,
markerFiles = crossFiles,
pheno = phenoDat,
popType = "DH",
mapFile = mapFile,
evalDist = 5)
We then have a look at the summary and some of the plots to get an
overview of the pedigree and the computed probabilities.
## Print summary
summary(maizeMPP)
#> map
#> Number of markers: 262
#> Number of chromosomes: 10
#>
#> markers
#> Number of markers: 262
#> Number of genotypes: 841
#> Parents: F353, B73, D06, D09, EC169, F252, F618, Mo17, UH250, UH304, W117
#> pheno
#> Number of traits: 30
#> Traitnames: INR_DMY, KWS_DMY, Syngenta_DMY, TUM_DMY, ..., mean_NBPL
#> Number of genotypes: 841
#>
#> crosses
#> F353xB73 : 64
#> F353xD06 : 99
#> F353xD09 :100
#> F353xEC169: 66
#> F353xF252 : 96
#> F353xF618 :104
#> F353xMo17 : 53
#> F353xUH250: 94
#> F353xUH304: 81
#> F353xW117 : 84
## Plot structure of the pedigree.
plot(maizeMPP, plotType = "pedigree")
Now we can use the computed IBD probabilities to perform SQM.
## Perform Single-QTL Mapping.
maizeSQM <- selQTLMPP(MPPobj = maizeMPP,
trait = "mean_DtSILK",
maxCofactors = 0)
We plot the QTL profile for the SQM to get an idea of reasonable
values to use for the threshold and QTLwindow
in the MQM that we do next.
## Plot QTL Profile for maize SQM.
plot(maizeSQM, plotType = "QTLProfile")
Based on the plot for the SQM the threshold is set to 5
to restrict a bit the number of QTLs that will be detected in the MQM.
The QTLwindow is not specified and therefore left at its
default value of 10cM.
## Perform Multi-QTL Mapping.
maizeMQM <- selQTLMPP(MPPobj = maizeMPP,
trait = "mean_DtSILK",
threshold = 5)
We only look at the combined plot of the QTL Profile and the parental
effects. This should give us the most direct insight in the QTLs found
and the effects the different parents in the crosses have.
## Plot QTL Profile for maize MQM.
plot(maizeMQM, plotType = "QTLProfileExt")
Example barley
Instead of performing IBD calculations directly with the package, it
is also possible to import IBD probabilities computed using RABBIT
software (Zheng, Boer, and Van Eeuwijk 2014, 2015, 2018). The main
advantage of using RABBIT for IBD calculations is that it can handle
complex pedigree populations and therefore can also be used in cases
where the population structure is more complex than those that can be
computed using statgenIBD, e.g. in the maize NAM population
described before.
As an example we use a barley population described in Liller et al. (2017). This MPP design consists of 5
parents. Four wild parents were crossed with the cultivar Morex and then
backcrossed with Morex once. Individuals from the four families from the
backcrosses were then crossed with each other as in a full diallel
design, which generated six F6 families through five generations of
selfing. The trait of interest for this population is awn length
(“Awn_length”). As for the maize NAM population, the data for this
population is available in zipped format in the package.
RABBIT output can be read using the readRABBITMPP
function in statgenMPP. This has as input the standard
RABBIT output summary file and the pedigree file that needs to be
provided to RABBIT as well. This pedigree file is an optional input and
is only used for plotting the pedigree structure of the population.
Without it QTL mapping can still be performed. As for
calcIBDMPP the phenotypic data has to be provided as a
data.frame. This data.frame has been included
in the package.
## Specify files containing RABBIT output.
## Extract in a temporary directory.
tempDir <- tempdir()
inFile <- unzip(system.file("extdata/barley/barley_magicReconstruct.zip",
package = "statgenMPP"), exdir = tempDir)
## Specify pedigree file.
pedFile <- system.file("extdata/barley/barley_pedInfo.csv",
package = "statgenMPP")
## Read phenotypic data.
data("barleyPheno")
## read RABBIT output.
barleyMPP <- readRABBITMPP(infile = inFile,
pedFile = pedFile,
pheno = barleyPheno)
As for the maize example we can summarize and plot the imported data
to get a first idea of its content.
## Summary.
summary(barleyMPP)
#> map
#> Number of markers: 355
#> Number of chromosomes: 7
#>
#> markers
#> Number of markers: 355
#> Number of genotypes: 916
#> Parents: Morex, HID4, HID64, HID369, HID382
#> pheno
#> Number of traits: 1
#> Traitnames: Awn_length
#> Number of genotypes: 916
#>
#> crosses
#> 50:149
#> 51:152
#> 52:157
#> 53:130
#> 54:167
#> 55:161
## Plot structure of the pedigree.
plot(barleyMPP, plotType = "pedigree")
Performing SQM and MQM for imported RABBIT output works in the same
way as for IBD probabilities computed directly in the package. Since a
full scan would take long we precomputed the results and included them
in the package.
## Perform Multi-QTL Mapping with threshold 4.
barleyMQM <- selQTLMPP(MPPobj = barleyMPP,
trait = "Awn_length",
threshold = 4)
There is a very large QTL on chromosome 7. To be able to more clearly
distinguish the differences between the other QTLs they are plotted
separately.
## Plot QTL Profile for barley MQM - chromosome 1-6.
plot(barleyMQM, plotType = "QTLProfileExt", chr = 1:6)
## Plot QTL Profile for barley MQM - chromosome 7.
plot(barleyMQM, plotType = "QTLProfileExt", chr = 7)
The QTLs found are very similar in both position, size and effects as
described in Liller et al. (2017). This
can also be clearly seen by comparing the summary of the MQM with the
table of effects in this paper.
## Summary.
summary(barleyMQM)
#> Trait analysed: Awn_length
#>
#> Data are available for 355 markers.
#> Threshold: 4
#>
#> Number of QTLs: 12
#>
#> evalPos chr pos mrkNear minlog10p varExpl eff_Morex eff_HID4 eff_HID64 eff_HID369
#> 2_1053 1 47.06 2_1053 10.05 0.0212 -4.4815 1.72 3.563 -1.425
#> 2_1187 2 27.21 2_1187 4.86 0.0121 3.7109 -1.01 -0.200 -0.815
#> 1_1522 2 46.06 1_1522 4.73 0.0459 -0.0572 7.25 -2.554 -5.226
#> 2_0340 2 68.99 2_0340 6.14 0.0208 -3.8297 1.96 -0.811 -0.437
#> 1_1501 3 42.17 1_1501 13.42 0.0526 4.6614 -2.88 -1.654 -5.951
#> 1_1070 4 51.72 1_1070 5.83 0.0554 -4.7880 -1.89 8.331 0.844
#> 1_0611 4 102.75 1_0611 4.05 0.0299 -0.9099 -2.47 3.531 -4.200
#> 2_0645 5 56.84 2_0645 6.86 0.0268 5.2156 -2.56 -1.824 1.227
#> 1_0094 5 94.26 1_0094 14.48 0.0939 -8.4484 1.70 -3.081 9.928
#> 1_0120 6 1.67 1_0120 4.41 0.0372 2.6349 -1.39 -5.899 1.992
#> 2_0687 6 88.03 2_0687 5.70 0.0344 -4.3277 2.90 3.666 1.144
#> 1_1012 7 121.04 1_1012 65.39 0.2648 11.3779 5.77 -9.658 7.651
#> eff_HID382
#> 0.629
#> -1.681
#> 0.583
#> 3.119
#> 5.827
#> -2.502
#> 4.050
#> -2.061
#> -0.102
#> 2.665
#> -3.387
#> -15.136