Estimating Models with {logitr}

---

## .center[Estimating Models with {logitr}]

]

### <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:white;overflow:visible;position:relative;"><path d="M272 304h-96C78.8 304 0 382.8 0 480c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32C448 382.8 369.2 304 272 304zM48.99 464C56.89 400.9 110.8 352 176 352h96c65.16 0 119.1 48.95 127 112H48.99zM224 256c70.69 0 128-57.31 128-128c0-70.69-57.31-128-128-128S96 57.31 96 128C96 198.7 153.3 256 224 256zM224 48c44.11 0 80 35.89 80 80c0 44.11-35.89 80-80 80S144 172.1 144 128C144 83.89 179.9 48 224 48z"/></svg> John Paul Helveston
### <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:white;overflow:visible;position:relative;"><path d="M243.4 2.6l-224 96c-14 6-21.8 21-18.7 35.8S16.8 160 32 160v8c0 13.3 10.7 24 24 24H456c13.3 0 24-10.7 24-24v-8c15.2 0 28.3-10.7 31.3-25.6s-4.8-29.9-18.7-35.8l-224-96c-8.1-3.4-17.2-3.4-25.2 0zM128 224H64V420.3c-.6 .3-1.2 .7-1.8 1.1l-48 32c-11.7 7.8-17 22.4-12.9 35.9S17.9 512 32 512H480c14.1 0 26.5-9.2 30.6-22.7s-1.1-28.1-12.9-35.9l-48-32c-.6-.4-1.2-.7-1.8-1.1V224H384V416H344V224H280V416H232V224H168V416H128V224zm128-96c-17.7 0-32-14.3-32-32s14.3-32 32-32s32 14.3 32 32s-14.3 32-32 32z"/></svg> The George Washington University | Dept. of Engineering Management and Systems Engineering
### <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:white;overflow:visible;position:relative;"><path d="M152 64H296V24C296 10.75 306.7 0 320 0C333.3 0 344 10.75 344 24V64H384C419.3 64 448 92.65 448 128V448C448 483.3 419.3 512 384 512H64C28.65 512 0 483.3 0 448V128C0 92.65 28.65 64 64 64H104V24C104 10.75 114.7 0 128 0C141.3 0 152 10.75 152 24V64zM48 248H128V192H48V248zM48 296V360H128V296H48zM176 296V360H272V296H176zM320 296V360H400V296H320zM400 192H320V248H400V192zM400 408H320V464H384C392.8 464 400 456.8 400 448V408zM272 408H176V464H272V408zM128 408H48V448C48 456.8 55.16 464 64 464H128V408zM272 192H176V248H272V192z"/></svg> June 15, 2023

]

---

# Many FOSS options model estimation

R packages:

- [{logitr}](https://github.com/jhelvy/logitr): Fastest, mixed logit, WTP space.
- [{apollo}](http://www.apollochoicemodelling.com/): Most flexible, great documentation.
- [{mlogit}](https://www.jstatsoft.org/article/view/v095i11): The OG R package.
- [{gmnl}](https://www.jstatsoft.org/article/view/v079i02): Generalized logit model (though slow).
- [{mixl}](https://github.com/joemolloy/fast-mixed-mnl): Good for big datasets (uses C for speed).

Python packages:

- [{xlogit}](https://xlogit.readthedocs.io/en/latest/): Basically Python version of {logitr}.

[Stan](https://www.inwt-statistics.com/blog/understand-customer-decision-making-discrete-choice-models-with-rstan): For the Bayesians.

---

# Many FOSS options model estimation

R packages:

- [{logitr}](https://github.com/jhelvy/logitr): Fastest, mixed logit, WTP space. .red[<- I wrote this one, so I'm showcasing it!]
- [{apollo}](http://www.apollochoicemodelling.com/): Most flexible, great documentation.
- [{mlogit}](https://www.jstatsoft.org/article/view/v095i11): The OG R package.
- [{gmnl}](https://www.jstatsoft.org/article/view/v079i02): Generalized logit model (though slow).
- [{mixl}](https://github.com/joemolloy/fast-mixed-mnl): Good for big datasets (uses C for speed).

Python packages:

- [{xlogit}](https://xlogit.readthedocs.io/en/latest/): Basically Python version of {logitr}.

[Stan](https://www.inwt-statistics.com/blog/understand-customer-decision-making-discrete-choice-models-with-rstan): For the Bayesians.

---

## {logitr} is fast!

---

## {logitr} supports two common forms of utility models

## Preference Space

]

## WTP Space

]

---

## .center[{logitr} has a similar UI with {cbcTools}]

## .center[{cbcTools}]

```r
power <- cbc_power(
    nbreaks = 10,
    n_q     = 6,
    data    = data,
    obsID   = "obsID",
    outcome = "choice",
    pars    = c("price", "type", "freshness")
)
```

]

## .center[{logitr}]

```r
model <- logitr(
    data    = data,
    obsID   = "obsID",
    outcome = "choice",
    pars    = c("price", "type", "freshness")
)
```

]

---

# Utility model refresher

---

# Which would you choose?

## .center[$2.49]

]

## .center[$2.99]

]

## .center[$1.99]

]

## .center[$3.99]

]

---

# .center[Estimate marginal utilities]

</br>

```
#>                Estimate Std. Error   z-value Pr(>|z|)
#> price        -0.3886257 0.02426923 -16.01311        0
#> brandhiland  -3.1167063 0.14496806 -21.49926        0
#> brandyoplait  1.4463603 0.08869767  16.30663        0
#> branddannon   0.6440868 0.05435965  11.84862        0
```

]

---

# .center[Convert marginal _utilities_ to marginal _WTPs_]

</br>

```
#>              Estimate Std. Error  z-value  Pr(>|z|)    
#> brandhiland  -8.01982    0.46096 -17.3980 < 2.2e-16 ***
#> brandyoplait  3.72173    0.15890  23.4214 < 2.2e-16 ***
#> branddannon   1.65734    0.16832   9.8463 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

]

---

## Alternative approach: **Estimate a WTP-Space Model**

## Substitutions:

]

## "Preference Space"

## "WTP Space"

]

---

# What's the difference?

## Preference Space

]

## WTP Space

</br></br>
<center>
<img src="images/utilityWtp2Inverse.png" width=520>
</center>

]

---

## **Mixed logit**:
## Unreasonably large WTP variance across population

</br>

]

]

---

background-color: #fff
class: center

### Preference space model produces unreasonably large variance in WTP

<center>
<b>Preference Space</b>
<img src="images/betaNormal.png" width=200>
</center>

]

<center>
<b>WTP Space</b>
<img src="images/omegaNormal.png" width=200>
</center>
</br>

]

---

# .fancy[Practical Considerations]

---

## .fancy[Practical Considerations]

> ### WTP space models produce immediately interpretable results

Unit: "Utility" (relative)

]]

Units: $ (absolute)

```
#>               Estimate Std. Error  z-value  Pr(>|z|)    
#> scalePar      0.388626   0.024399  15.9280 < 2.2e-16 ***
#> brandhiland  -8.019815   0.460961 -17.3980 < 2.2e-16 ***
#> brandyoplait  3.721731   0.158903  23.4214 < 2.2e-16 ***
#> branddannon   1.657345   0.168321   9.8463 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

]]

---

## .fancy[Practical Considerations]

> ### WTPs can be directly compared across different models</br>(even estimates from different data sets)

**Preference Space**</br>Parameters proportional to `$\sigma$`

]

**WTP Space**</br>Parameters independent of `$\sigma$`

]

---

## .center[.fancy[Practical Considerations]]

> ### .center[Neither space systematically predicts choice better]

-  **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.

]

---

# ...but most software is built for

# not

---

# `logitr` to the rescue!

---

# The logitr <svg aria-hidden="true" role="img" viewBox="0 0 581 512" style="height:1em;width:1.13em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:#165CAA;overflow:visible;position:relative;"><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.

- Multinomial logit (MNL) models
- Mixed logit (MXL) models with normal and log-normal parameter
  distributions.
- Preference space and WTP space utility parameterizations.
- Weighted models to differentially weight individual observations.
- Uncorrelated or correlated heterogeneity covariances for mixed logit
  models.
- Functions for computing WTP from preference space models.
- Functions for predicting expected probabilities and outcomes for sets
  of alternatives based on an estimated model.
- A parallelized multistart optimization loop that uses different random
  starting points in each iteration to search for different local minima
  (useful for non-convex problems like MXL models or models with WTP
  space parameterizations).

---

# .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:

- `outcome`: A dummy variable for the chosen alternative (`1` or `0`).
- `obsID`: A sequence of repeated numbers identifying each unique choice observation, e.g. `1, 1, 2, 2, 3, 3`.
- `pars`: Any other variables to use as model covariates.

---

# .center[Data format]

```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
```

]

- `outcome = "choice"`
- `obsID = "obsID"`
- `pars = c("price", "brand")`

]

---

# .center[Multinomial logit in **Preference Space**]

```r
mnl_pref <- logitr(
  data    = yogurt,
  outcome = "choice",
  obsID   = "obsID",
  pars    = c("price", "brand")
)

*summary(mnl_pref)
```

]

```
#> =================================================
#> 
#> Model estimated on: Fri Jun 09 10:12:27 2023 
#> 
#> Using logitr version: 1.1.0 
#> 
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("price", 
#>     "brand"))
#> 
#> Frequencies of alternatives:
#>        1        2        3        4 
#> 0.402156 0.029436 0.229270 0.339138 
#> 
#> Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
#>                                 
#> Model Type:    Multinomial Logit
#> Model Space:          Preference
#> Model Run:                1 of 1
#> Iterations:                   20
#> Elapsed Time:        0h:0m:0.01s
#> Algorithm:        NLOPT_LD_LBFGS
#> Weights Used?:             FALSE
#> Robust?                    FALSE
#> 
#> Model Coefficients: 
#>               Estimate Std. Error z-value  Pr(>|z|)    
#> price        -0.388626   0.024269 -16.013 < 2.2e-16 ***
#> brandhiland  -3.116706   0.144968 -21.499 < 2.2e-16 ***
#> brandyoplait  1.446360   0.088698  16.307 < 2.2e-16 ***
#> branddannon   0.644087   0.054360  11.849 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>                                      
#> Log-Likelihood:         -2665.1101915
#> Null Log-Likelihood:    -3343.7419990
#> AIC:                     5338.2203830
#> BIC:                     5361.3732000
#> McFadden R2:                0.2029558
#> Adj McFadden R2:            0.2017595
#> Number of Observations:  2412.0000000
```

]]

---

# .center[Multinomial logit in **WTP Space**]

```r
library(logitr)

mnl_wtp <- logitr(
  data     = yogurt,
  outcome  = "choice",
  obsID    = "obsID",
* pars     = "brand",
* scalePar = "price"
)

summary(mnl_wtp)
```

]

```
#> =================================================
#> 
#> Model estimated on: Fri Jun 09 10:12:41 2023 
#> 
#> Using logitr version: 1.1.0 
#> 
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = "brand", 
#>     scalePar = "price")
#> 
#> Frequencies of alternatives:
#>        1        2        3        4 
#> 0.402156 0.029436 0.229270 0.339138 
#> 
#> Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
#>                                  
#> Model Type:     Multinomial Logit
#> Model Space:   Willingness-to-Pay
#> Model Run:                 1 of 1
#> Iterations:                    40
#> Elapsed Time:         0h:0m:0.02s
#> Algorithm:         NLOPT_LD_LBFGS
#> Weights Used?:              FALSE
#> Robust?                     FALSE
#> 
#> Model Coefficients: 
#>               Estimate Std. Error z-value  Pr(>|z|)    
#> scalePar      0.388633   0.024269  16.013 < 2.2e-16 ***
#> brandhiland  -8.019717   0.455549 -17.605 < 2.2e-16 ***
#> brandyoplait  3.721711   0.157655  23.607 < 2.2e-16 ***
#> branddannon   1.657290   0.165712  10.001 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>                                      
#> Log-Likelihood:         -2665.1101915
#> Null Log-Likelihood:    -3343.7419990
#> AIC:                     5338.2203830
#> BIC:                     5361.3732000
#> McFadden R2:                0.2029558
#> Adj McFadden R2:            0.2017595
#> Number of Observations:  2412.0000000
```

]]

---

## **.red[Caution]**</br>

## Log-likelihood function for WTP space models is</br>**non-convex**  😔

---

# .center[Use a Multistart]

```r
mnl_wtp <- logitr(
  data     = yogurt,
  outcome  = "choice",
  obsID    = "obsID",
  pars     = "brand",
  scalePar = "price",
* numMultiStarts = 10
)

summary(mnl_wtp)
```

]

```
#> =================================================
#> 
#> Model estimated on: Fri Jun 09 10:13:43 2023 
#> 
#> Using logitr version: 1.1.0 
#> 
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = "brand", 
#>     scalePar = "price", numMultiStarts = 10)
#> 
#> Frequencies of alternatives:
#>        1        2        3        4 
#> 0.402156 0.029436 0.229270 0.339138 
#> 
#> Summary Of Multistart Runs:
#>    Log Likelihood Iterations Exit Status
#> 1        -2665.11         40           3
#> 2        -2665.11         39           3
#> 3        -2665.11         43           3
#> 4        -2665.11         47           3
#> 5        -2665.11         54           3
#> 6        -2665.11         42           3
#> 7        -2665.11         39           3
#> 8        -2665.11         44           3
#> 9        -2665.11         39           3
#> 10       -2665.11         38           3
#> 
#> Use statusCodes() to view the meaning of each status code
#> 
#> Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
#>                                  
#> Model Type:     Multinomial Logit
#> Model Space:   Willingness-to-Pay
#> Model Run:               10 of 10
#> Iterations:                    38
#> Elapsed Time:         0h:0m:0.02s
#> Algorithm:         NLOPT_LD_LBFGS
#> Weights Used?:              FALSE
#> Robust?                     FALSE
#> 
#> Model Coefficients: 
#>               Estimate Std. Error z-value  Pr(>|z|)    
#> scalePar      0.388631   0.024269  16.013 < 2.2e-16 ***
#> brandhiland  -8.019767   0.455557 -17.604 < 2.2e-16 ***
#> brandyoplait  3.721702   0.157655  23.607 < 2.2e-16 ***
#> branddannon   1.657316   0.165715  10.001 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>                                      
#> Log-Likelihood:         -2665.1101915
#> Null Log-Likelihood:    -3343.7419990
#> AIC:                     5338.2203830
#> BIC:                     5361.3732000
#> McFadden R2:                0.2029558
#> Adj McFadden R2:            0.2017595
#> Number of Observations:  2412.0000000
```

]]

---

## .center[Mixed logit in<br>**Preference Space**]

```r
mxl_pref <- logitr(
  data     = yogurt,
  outcome  = "choice",
  obsID    = "obsID",
  pars     = c("price", "brand"),
* randPars = c(brand = "n"),
  numMultiStarts = 10
)
```

]

## .center[Mixed logit in<br>**WTP Space**]

```r
mxl_wtp <- logitr(
  data     = yogurt,
  outcome  = "choice",
  obsID    = "obsID",
  pars     = "brand",
  scalePar = "price",
* randPars = c(brand = "n"),
* randScale = "ln",
  numMultiStarts = 10
)
```

]

---

# .fancy[Convenient helper functions]

---

### .center[`predict()`: Expected shares for a set of alternatives]

</br>

Define a set of alternatives

```r
data <- subset(
  yogurt, obsID == 42, 
  select = c('price', 'brand', 'obsID'))

data
```

```
#>   price   brand obsID
#> 1   6.3  dannon    42
#> 2   6.1  hiland    42
#> 3   7.9  weight    42
#> 4  11.5 yoplait    42
```

]

Predict probabilities

```r
predict(
  mnl_pref,
  newdata = data,
  obsID = "obsID", 
  returnData = TRUE
)
```

```
#>   obsID predicted_prob price   brand
#> 1    42     0.62391435   6.3  dannon
#> 2    42     0.01568877   6.1  hiland
#> 3    42     0.17593683   7.9  weight
#> 4    42     0.18446005  11.5 yoplait
```

]

---

# Your turn

- Be sure to have downloaded and unzipped the [practice code](https://jhelvy.github.io/2023-qux-conf-conjoint/practice/2023-qux-conf-conjoint.zip).
- Open the `2023-qux-conf-conjoint.Rproj` file to open RStudio.
- In RStudio, open the `estimating-models.R` file.
- Experiment with estimating different models (use either one of the example datasets included in the package, or simulate your own data!)

---

<br>

## .center[{logitr} documentation:<br>https://jhelvy.github.io/logitr/]

## .center[Back to workshop website:<br> https://jhelvy.github.io/2023-qux-conf-conjoint/]

@JohnHelveston <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:white;overflow:visible;position:relative;"><path d="M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z"/></svg><br>
@jhelvy <svg aria-hidden="true" role="img" viewBox="0 0 496 512" style="height:1em;width:0.97em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:white;overflow:visible;position:relative;"><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg><br>
@jhelvy <svg aria-hidden="true" role="img" viewBox="0 0 576 512" style="height:1em;width:1.12em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:white;overflow:visible;position:relative;"><path d="M385.2 167.6c6.4 0 12.6.3 18.8 1.1C387.4 90.3 303.3 32 207.7 32 100.5 32 13 104.8 13 197.4c0 53.4 29.3 97.5 77.9 131.6l-19.3 58.6 68-34.1c24.4 4.8 43.8 9.7 68.2 9.7 6.2 0 12.1-.3 18.3-.8-4-12.9-6.2-26.6-6.2-40.8-.1-84.9 72.9-154 165.3-154zm-104.5-52.9c14.5 0 24.2 9.7 24.2 24.4 0 14.5-9.7 24.2-24.2 24.2-14.8 0-29.3-9.7-29.3-24.2.1-14.7 14.6-24.4 29.3-24.4zm-136.4 48.6c-14.5 0-29.3-9.7-29.3-24.2 0-14.8 14.8-24.4 29.3-24.4 14.8 0 24.4 9.7 24.4 24.4 0 14.6-9.6 24.2-24.4 24.2zM563 319.4c0-77.9-77.9-141.3-165.4-141.3-92.7 0-165.4 63.4-165.4 141.3S305 460.7 397.6 460.7c19.3 0 38.9-5.1 58.6-9.9l53.4 29.3-14.8-48.6C534 402.1 563 363.2 563 319.4zm-219.1-24.5c-9.7 0-19.3-9.7-19.3-19.6 0-9.7 9.7-19.3 19.3-19.3 14.8 0 24.4 9.7 24.4 19.3 0 10-9.7 19.6-24.4 19.6zm107.1 0c-9.7 0-19.3-9.7-19.3-19.6 0-9.7 9.7-19.3 19.3-19.3 14.5 0 24.4 9.7 24.4 19.3.1 10-9.9 19.6-24.4 19.6z"/></svg><br>
jhelvy.com <svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:white;overflow:visible;position:relative;"><path d="M579.8 267.7c56.5-56.5 56.5-148 0-204.5c-50-50-128.8-56.5-186.3-15.4l-1.6 1.1c-14.4 10.3-17.7 30.3-7.4 44.6s30.3 17.7 44.6 7.4l1.6-1.1c32.1-22.9 76-19.3 103.8 8.6c31.5 31.5 31.5 82.5 0 114L422.3 334.8c-31.5 31.5-82.5 31.5-114 0c-27.9-27.9-31.5-71.8-8.6-103.8l1.1-1.6c10.3-14.4 6.9-34.4-7.4-44.6s-34.4-6.9-44.6 7.4l-1.1 1.6C206.5 251.2 213 330 263 380c56.5 56.5 148 56.5 204.5 0L579.8 267.7zM60.2 244.3c-56.5 56.5-56.5 148 0 204.5c50 50 128.8 56.5 186.3 15.4l1.6-1.1c14.4-10.3 17.7-30.3 7.4-44.6s-30.3-17.7-44.6-7.4l-1.6 1.1c-32.1 22.9-76 19.3-103.8-8.6C74 372 74 321 105.5 289.5L217.7 177.2c31.5-31.5 82.5-31.5 114 0c27.9 27.9 31.5 71.8 8.6 103.9l-1.1 1.6c-10.3 14.4-6.9 34.4 7.4 44.6s34.4 6.9 44.6-7.4l1.1-1.6C433.5 260.8 427 182 377 132c-56.5-56.5-148-56.5-204.5 0L60.2 244.3z"/></svg><br>
jph@gwu.edu <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:white;overflow:visible;position:relative;"><path d="M16.1 260.2c-22.6 12.9-20.5 47.3 3.6 57.3L160 376V479.3c0 18.1 14.6 32.7 32.7 32.7c9.7 0 18.9-4.3 25.1-11.8l62-74.3 123.9 51.6c18.9 7.9 40.8-4.5 43.9-24.7l64-416c1.9-12.1-3.4-24.3-13.5-31.2s-23.3-7.5-34-1.4l-448 256zm52.1 25.5L409.7 90.6 190.1 336l1.2 1L68.2 285.7zM403.3 425.4L236.7 355.9 450.8 116.6 403.3 425.4z"/></svg>

]]