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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>The logitr R Package: Obtaining Willingness to Pay Estimates from Preference Space and Willingness to Pay Space Utility Models</title>
<meta charset="utf-8" />
<meta name="author" content="John Paul Helveston" />
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layout: true
<!-- this adds the link footer to all slides, depends on my-footer class in css-->
<div class="footer-small">
<span>
https://github.com/jhelvy/2021-sawtooth-conf
</span>
</div>
---
name: title-slide
class: inverse, middle
background-image: url(images/blue.jpg)
# The `logitr` <svg style="height:0.8em;top:.04em;position:relative;fill:white;" viewBox="0 0 581 512"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg> Package: <a href='https://jhelvy.github.io/logitr/'><img src='images/logitr-hex.png' align="right" height="250"/></a>
## Obtaining Willingness to Pay Estimates from Preference Space and Willingness to Pay<br>Space Utility Models
### by John Paul Helveston
Sawtooth Software Conference: Turbo Choice Modeling Seminar
April 20, 2021
---
class: center
# Which would you choose?
.cols4[
## .center[$2.49]
<center>
<img src="images/dannon.png" width=300>
</center>
]
.cols4[
## .center[$2.99]
<center>
<img src="images/yoplait.png" width=300>
</center>
]
.cols4[
## .center[$1.99]
<center>
<img src="images/hiland.png" width=300>
</center>
]
.cols4[
## .center[$3.99]
<center>
<img src="images/weight.png" width=300>
</center>
]
---
# .center[Estimate marginal utilities]
</br>
<center>
<img src="images/utilityPreference.png" width="800">
</center>
--
.code100[
```
#> Estimate StdError tStat pVal signif
#> price -0.388627 0.024269 -16.0131 0 ***
#> brandhiland -3.116712 0.144968 -21.4993 0 ***
#> brandyoplait 1.446362 0.088698 16.3066 0 ***
#> branddannon 0.644086 0.054360 11.8486 0 ***
```
]
---
# .center[Convert marginal _utilities_ to marginal _WTPs_]
</br>
<center>
<img src="images/wtpHatComputed.png" width=250>
</center>
--
.code100[
```
#> Estimate StdError tStat pVal signif
#> brandhiland -8.019803 0.460815 -17.4035 0 ***
#> brandyoplait 3.721723 0.440084 8.4568 0 ***
#> branddannon 1.657337 0.183712 9.0214 0 ***
```
]
---
class: center
## Alternative approach: **Estimate a WTP-Space Model**
.leftcol30[
## Substitutions:
<center>
<img src="images/wtpComputed.png" width=200></br>
<img src="images/lambda.png" width=160>
</center>
]
.rightcol70[
## "Preference Space"
<center>
<img src="images/utilityPreference2.png" width="370">
</center>
## "WTP Space"
<center>
<img src="images/utilityWtp2.png" width=400>
</center>
]
---
class: center, middle, inverse
# What's the difference?
.leftcol[
## Preference Space
<center>
<img src="images/utilityPreference2Inverse.png" width=500>
</center>
.font200[&darr;]
<center>
<img src="images/wtpHatComputedInverse.png" width=200></br>
</center>
]
.rightcol[
## WTP Space
</br></br>
<center>
<img src="images/utilityWtp2Inverse.png" width=520>
</center>
]
---
## .center[Oops...WTP has undefined mean and variance]
</br>
.leftcol35[
### .center[Preference Space]
<center>
<img src="images/utilityPreference2.png" width=500>
</center>
.center[.font200[&darr;]]
<center>
<img src="images/wtpHatComputed.png" width=200></br>
</center>
]
.rightcol60[
</br>
`\(\hat{\alpha}\)` & `\(\hat{\beta}\)` are assumed to have asymptotically normal distributions .gray[(Bockstael and Strand, 1987)]
Implies error around `\(\hat{\omega}\)` follows a</br>[Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution)
]
---
class: center
## **Mixed logit**:
## Unreasonably large WTP variance across population
</br>
.leftcol40[
<center>
<img src="images/utilityPreference2.png" width=500>
</center>
.center[.font200[&darr;]]
<center>
<img src="images/wtpHatComputed.png" width=200></br>
</center>
]
.rightcol60[
<center>
<img src="images/betaNormal.png" width=380>
</br>
<img src="images/alphaNormal.png" width=380>
</center>
]
---
background-color: #fff
class: center
### Preference space model produces unreasonably large variance in WTP
.cols4[&zwj;]
.cols4[
<center>
<b>Preference Space</b>
<img src="images/betaNormal.png" width=200>
</center>
]
.cols4[
<center>
<b>WTP Space</b>
<img src="images/omegaNormal.png" width=200>
</center>
</br>
]
.cols4[&zwj;]
--
<center>
<img src="images/wtpCompare.png" width=900>
</center>
---
# .center[I'm not the first to identify these issues]
--
.leftcol[
.center[WTP computed from preference space model has undefined moments]
.font70[
- **Carson and Czajkowski (2019)**. "A new baseline model for estimating willingness to pay from discrete choice models." _Journal of Environmental Economics and Management_, 95:57–61.
]]
--
.rightcol[
.center[Preference space model produces unreasonably large variance in WTP]
.font70[
- **Train and Weeks (2005)**. "Discrete Choice Models in Preference and Willingness-to-Pay Space". In _Appl. Simul. Methods Environ. Resour. Econ._, Chapter 1, pages 1–16.
- **Sonnier, Ainslie, & Otter (2007)**. "Heterogeneity distributions of willingness-to-pay in choice models." _Quant. Mark. Econ._, 5(3):313–331.
]]
---
class: center, middle, inverse
# .fancy[Practical Considerations]
---
class: center
## .fancy[Practical Considerations]
> ### WTP space models produce immediately interpretable results</br>(with correct standard errors)
--
.leftcol[
Unit: "Utility" (relative)
<center>
<img src="images/utilityPreference2.png" width="250">
</center>
.code50[
```
#> Estimate StdError tStat pVal signif
#> price -0.388627 0.024269 -16.0131 0 ***
#> brandhiland -3.116712 0.144968 -21.4993 0 ***
#> brandyoplait 1.446362 0.088698 16.3066 0 ***
#> branddannon 0.644086 0.054360 11.8486 0 ***
```
]]
--
.rightcol[
Units: $ (absolute)
<center>
<img src="images/utilityWtp2.png" width="300">
</center>
.code50[
```
#> Estimate StdError tStat pVal signif
#> lambda 0.388627 0.024341 15.9658 0 ***
#> brandhiland -8.019803 0.460815 -17.4035 0 ***
#> brandyoplait 3.721723 0.440084 8.4568 0 ***
#> branddannon 1.657337 0.183712 9.0214 0 ***
```
]]
---
class: center
## .fancy[Practical Considerations]
> ### WTPs can be directly compared across different models</br>(even estimates from different data sets)
--
<center>
<img src="images/utility.png" width=700>
</center>
--
.leftcol[
**Preference Space**</br>Parameters proportional to `\(\sigma\)`
<center>
<img src="images/utilityPreferenceScaled2.png" width=420>
<img src="images/utilityPreference2.png" width=300>
</center>
]
--
.rightcol[
**WTP Space**</br>Parameters independent of `\(\sigma\)`
<center>
<img src="images/utilityWtpScaled2.png" width=450>
<img src="images/utilityWtp2.png" width=350>
</center>
]
---
class: center
## .fancy[Practical Considerations]
> ### No theoretical basis for believing that marginal _utilities_ versus marginal _WTPs_ should follow standard distributions
.leftcol[
<center>
<img src="images/betaNormal.png" width=350>
</center>
]
.rightcol[
<center>
<img src="images/omegaNormal.png" width=350>
</center>
]
---
## .center[.fancy[Practical Considerations]]
> ### .center[Neither space systematically predicts choice better]
.leftcol70[
- **Train and Weeks (2005)** and **Sonnier et al. (2007)** found preference space model fit data better.
- **Das et al. (2009)** found nearly identical model fit on out-of-sample predictions with each model specification.
]
---
class: center, middle, inverse
# ...but most software is built for
<center>
<img src="images/utilityPreference2Inverse.png" width="370">
</center>
# not
<center>
<img src="images/utilityWtp2Inverse.png" width=400>
</center>
---
class: center, middle, inverse
# `logitr` to the rescue!
<center>
<img src="images/logitr-hex.png" width=250>
</center>
---
# The logitr <svg style="height:0.8em;top:.04em;position:relative;fill:#165CAA;" viewBox="0 0 581 512"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg> Package <a href='https://jhelvy.github.io/logitr/'><img src='images/logitr-hex.png' align="right" height="240"/></a>
### Estimation of multinomial and mixed logit models in with "Preference" space or "Willingness-to-pay" (WTP) space utility parameterizations.
- Homogeneous multinomial logit models.
- Heterogeneous mixed logit models (normal and log-normal parameter distributions).
- Preference & WTP space utility parameterizations.
- Optional multistart optimization loop.
- Computing and comparing WTP from preference space and WTP space models.
- Simulating expected shares.
Source code: https://github.com/jhelvy/logitr
---
# .center[Installation]
.leftcol70[
Version 0.1.0 can be installed from the CRAN:
```r
install.packages("logitr")
```
The development version can be installed from GitHub:
```r
# install.packages("remotes")
remotes::install_github("jhelvy/logitr")
```
]
---
# .center[Data format]
Data must be arranged in a "long" format:
- Each row is an alternative from a choice observation.
- Choice observations do _not_ have to be symmetric.
Required variables:
- `choiceName`: A dummy variable for the chosen alternative (`1` or `0`).
- `obsIDName`: A sequence of repeated numbers identifying each unique choice observation, e.g. `1, 1, 2, 2, 3, 3`.
- `parNames`: Any other variables to use as model covariates.
---
# .center[Data format]
.leftcol[
```r
head(yogurt, 10)
```
```
#> choice obsID alt price brand
#> 1 0 1 1 8.1 dannon
#> 2 0 1 2 6.1 hiland
#> 3 1 1 3 7.9 weight
#> 4 0 1 4 10.8 yoplait
#> 5 1 2 1 9.8 dannon
#> 6 0 2 2 6.4 hiland
#> 7 0 2 3 7.5 weight
#> 8 0 2 4 10.8 yoplait
#> 9 1 3 1 9.8 dannon
#> 10 0 3 2 6.1 hiland
```
]
.rightcol[
- `choiceName = "choice"`
- `obsIDName = "obsID"`
- `parNames = c("price", "brand")`
]
---
# .center[Multinomial logit in **Preference Space**]
.leftcol45[
```r
library(logitr)
mnl_pref <- logitr(
data = yogurt,
choiceName = "choice",
obsIDName = "obsID",
parNames = c("price", "brand"))
)
*summary(mnl_pref)
```
<center>
<img src="images/utilityPreference2.png" width=300>
</center>
]
--
.rightcol55[.code50[
```
#> =================================================
#> MODEL SUMMARY:
#>
#> Model Space: Preference
#> Model Run: 1 of 1
#> Iterations: 20
#> Elapsed Time: 0h:0m:0.12s
#> Exit Status: 3
#> Weights Used?: FALSE
#>
#> Model Coefficients:
#> Estimate StdError tStat pVal signif
#> price -0.388627 0.024269 -16.0131 0 ***
#> brandhiland -3.116712 0.144968 -21.4993 0 ***
#> brandyoplait 1.446362 0.088698 16.3066 0 ***
#> branddannon 0.644086 0.054360 11.8486 0 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Model Fit Values:
#>
#> Log.Likelihood. -2665.1101915
#> Null.Log.Likelihood. -3343.7419990
#> AIC. 5338.2204000
#> BIC. 5361.3732000
#> McFadden.R2. 0.2029558
#> Adj..McFadden.R2 0.2017595
#> Number.of.Observations. 2412.0000000
```
]]
---
# .center[Multinomial logit in **WTP Space**]
.leftcol45[
```r
library(logitr)
mnl_wtp <- logitr(
data = yogurt,
choiceName = "choice",
obsIDName = "obsID",
* parNames = "brand",
* priceName = "price",
* modelSpace = "wtp")
)
summary(mnl_wtp)
```
<center>
<img src="images/utilityWtp2.png" width=300>
</center>
]
--
.rightcol55[.code50[
```
#> =================================================
#> MODEL SUMMARY:
#>
#> Model Space: Willingness-to-Pay
#> Model Run: 6 of 10
#> Iterations: 36
#> Elapsed Time: 0h:0m:0.23s
#> Exit Status: 3
#> Weights Used?: FALSE
#>
#> Model Coefficients:
#> Estimate StdError tStat pVal signif
#> lambda 0.388631 0.024269 16.0132 0 ***
#> brandhiland -8.019752 0.455555 -17.6044 0 ***
#> brandyoplait 3.721705 0.157656 23.6065 0 ***
#> branddannon 1.657322 0.165715 10.0010 0 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Model Fit Values:
#>
#> Log.Likelihood. -2665.1101915
#> Null.Log.Likelihood. -3343.7419990
#> AIC. 5338.2204000
#> BIC. 5361.3732000
#> McFadden.R2. 0.2029558
#> Adj..McFadden.R2 0.2017595
#> Number.of.Observations. 2412.0000000
```
]]
---
class: center, middle
## **.red[Caution]**</br>
## Log-likelihood function for WTP space models is</br>**non-convex** 😔
---
# .center[Use a Multistart]
.leftcol45[
```r
library(logitr)
mnl_wtp <- logitr(
data = yogurt,
choiceName = "choice",
obsIDName = "obsID",
parNames = "brand",
priceName = "price",
modelSpace = "wtp",
* options = list(numMultiStarts = 10)
)
summary(mnl_wtp)
```
<center>
<img src="images/utilityWtp2.png" width=300>
</center>
]
--
.rightcol55[.code50[
```
#> =================================================
#> SUMMARY OF ALL MULTISTART RUNS:
#>
#> run logLik iterations status
#> 1 1 -2833.932 80 4
#> 2 2 -2665.110 34 3
#> 3 3 -2665.110 34 3
#> 4 4 -2665.110 32 3
#> 5 5 -2834.285 76 3
#> 6 6 -2665.110 36 3
#> 7 7 -2665.110 30 3
#> 8 8 -2834.989 69 3
#> 9 9 -2834.567 70 3
#> 10 10 -2834.183 75 4
#> ---
#> Use statusCodes() to view the meaning of the status codes
#>
#> Below is the summary of run 6 of 10 multistart runs
#> (the run with the largest log-likelihood value)
#> =================================================
#> MODEL SUMMARY:
#>
#> Model Space: Willingness-to-Pay
#> Model Run: 6 of 10
#> Iterations: 36
#> Elapsed Time: 0h:0m:0.23s
#> Exit Status: 3
#> Weights Used?: FALSE
#>
#> Model Coefficients:
#> Estimate StdError tStat pVal signif
#> lambda 0.388631 0.024269 16.0132 0 ***
#> brandhiland -8.019752 0.455555 -17.6044 0 ***
#> brandyoplait 3.721705 0.157656 23.6065 0 ***
#> branddannon 1.657322 0.165715 10.0010 0 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Model Fit Values:
#>
#> Log.Likelihood. -2665.1101915
#> Null.Log.Likelihood. -3343.7419990
#> AIC. 5338.2204000
#> BIC. 5361.3732000
#> McFadden.R2. 0.2029558
#> Adj..McFadden.R2 0.2017595
#> Number.of.Observations. 2412.0000000
```
]]
---
# .center[Mixed logit in **Preference Space**]
.leftcol45[
```r
library(logitr)
mxl_pref <- logitr(
data = yogurt,
choiceName = "choice",
obsIDName = "obsID",
parNames = c("price", "brand"),
* randPars = c(brand = "n"),
options = list(numMultiStarts = 10)
)
summary(mxl_pref)
```
<center>
<img src="images/utilityPreference2.png" width=300>
<img src="images/betaNormal.png" width=200>
</center>
]
--
.rightcol55[.code50[
```
#> =================================================
#> MODEL SUMMARY:
#>
#> Model Space: Preference
#> Model Run: 5 of 10
#> Iterations: 43
#> Elapsed Time: 0h:0m:17s
#> Exit Status: 3
#> Weights Used?: FALSE
#>
#> Model Coefficients:
#> Estimate StdError tStat pVal signif
#> price -0.389810 0.044362 -8.7870 0.0000 ***
#> brandhiland_mu -3.124798 0.243803 -12.8169 0.0000 ***
#> brandyoplait_mu 1.449109 0.140574 10.3085 0.0000 ***
#> branddannon_mu 0.655452 0.138290 4.7397 0.0000 ***
#> brandhiland_sigma 0.056181 1.177854 0.0477 0.9620
#> brandyoplait_sigma 0.011127 0.388879 0.0286 0.9772
#> branddannon_sigma 0.172808 2.458086 0.0703 0.9440
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Model Fit Values:
#>
#> Log.Likelihood. -2665.1077011
#> Null.Log.Likelihood. -3343.7419990
#> AIC. 5344.2154000
#> BIC. 5384.7329000
#> McFadden.R2. 0.2029565
#> Adj..McFadden.R2 0.2008631
#> Number.of.Observations. 2412.0000000
#>
#> Summary of 10k Draws for Random Coefficients:
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> brandhiland (normal) -3.343157864 -3.1627301 -3.1248413 -3.1248770 -3.0869654 -2.921920
#> brandyoplait (normal) 1.406487337 1.4415994 1.4491051 1.4490984 1.4566085 1.488621
#> branddannon (normal) -0.009543227 0.5388061 0.6553104 0.6551855 0.7718474 1.259228
```
]]
---
# .center[Mixed logit in **WTP Space**]
.leftcol45[
```r
library(logitr)
mxl_wtp <- logitr(
data = yogurt,
choiceName = "choice",
obsIDName = "obsID",
parNames = "brand",
priceName = "price",
* randPars = c(brand = "n"),
* modelSpace = "wtp",
options = list(numMultiStarts = 10)
)
summary(mxl_wtp)
```
<center>
<img src="images/utilityWtp2.png" width=300>
<img src="images/omegaNormal.png" width=200>
</center>
]
--
.rightcol55[.code50[
```
#> =================================================
#> MODEL SUMMARY:
#>
#> Model Space: Willingness-to-Pay
#> Model Run: 6 of 10
#> Iterations: 51
#> Elapsed Time: 0h:0m:27s
#> Exit Status: 3
#> Weights Used?: FALSE
#>
#> Model Coefficients:
#> Estimate StdError tStat pVal signif
#> lambda 0.391114 0.037629 10.3938 0.0000 ***
#> brandhiland_mu -8.001475 0.579751 -13.8016 0.0000 ***
#> brandyoplait_mu 3.715322 0.181690 20.4487 0.0000 ***
#> branddannon_mu 1.682783 0.169619 9.9210 0.0000 ***
#> brandhiland_sigma 0.182098 2.623297 0.0694 0.9447
#> brandyoplait_sigma 0.034227 0.975524 0.0351 0.9720
#> branddannon_sigma 0.598862 3.586766 0.1670 0.8674
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Model Fit Values:
#>
#> Log.Likelihood. -2665.1066726
#> Null.Log.Likelihood. -3343.7419990
#> AIC. 5344.2133000
#> BIC. 5384.7308000
#> McFadden.R2. 0.2029568
#> Adj..McFadden.R2 0.2008634
#> Number.of.Observations. 2412.0000000
#>
#> Summary of 10k Draws for Random Coefficients:
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> brandhiland (normal) -8.7092358 -8.124422 -8.001614 -8.001730 -7.878848 -7.343892
#> brandyoplait (normal) 3.5842248 3.692225 3.715311 3.715291 3.738391 3.836856
#> branddannon (normal) -0.6217451 1.278549 1.682292 1.681859 2.086149 3.775156
```
]]
---
class: center, middle, inverse
# .fancy[Convenient helper functions]
---
## .center[`wtp()`: Compute WTP from Preference Space model]
</br>
.leftcol30[
<center>
<img src="images/wtpHatComputed.png" width=300>
</center>
]
.rightcol65[
```r
wtp(mnl_pref, priceName = "price")
```
```
#> Estimate StdError tStat pVal signif
#> lambda 0.388627 0.024204 16.0565 0 ***
#> brandhiland -8.019803 0.459432 -17.4559 0 ***
#> brandyoplait 3.721723 0.437562 8.5056 0 ***
#> branddannon 1.657337 0.183184 9.0474 0 ***
```
]
---
### .center[`wtpCompare()`: Compare WTP from Preference & WTP space models]
</br>
.leftcol70[
```r
wtpCompare(mnl_pref, mnl_wtp, priceName = "price")
```
```
#> pref wtp difference
#> lambda 0.388627 0.3886309 3.940e-06
#> brandhiland -8.019803 -8.0197522 5.083e-05
#> brandyoplait 3.721723 3.7217045 -1.848e-05
#> branddannon 1.657337 1.6573215 -1.547e-05
#> logLik -2665.110192 -2665.1101915 2.000e-08
```
]
---
### .center[`simulateShares()`: Expected shares for a set of alternatives]
</br>
.leftcol[
Define a set of alternatives
```r
alts <- subset(
yogurt, obsID == 42,
select = c('price', 'brand'))
alts
```
```
#> price brand
#> 1 6.3 dannon
#> 2 6.1 hiland
#> 3 7.9 weight
#> 4 11.5 yoplait
```
]
--
.rightcol[
Compute expected shares
```r
simulateShares(
model = mnl_pref,
alts = alts,
alpha = 0.025
)
```
```
#> share_mean share_low share_high
#> Alt: 1 0.62391502 0.56509154 0.67356217
#> Alt: 2 0.01568871 0.01205067 0.02013178
#> Alt: 3 0.17593678 0.16175071 0.19017909
#> Alt: 4 0.18445949 0.13613464 0.24552081
```
]