The cbcTools Package: Tools for Designing and Testing Choice-Based Conjoint Surveys in R

<div class="footer-small">
<span>
https://github.com/jhelvy/2022-sawtooth-conf
</span>
</div>

---

# The `cbcTools` Package <a href='https://jhelvy.github.io/cbcTools/'><img src='images/logo.png' align="right" height="300"/></a>

## Tools for Designing and Testing<br>Choice-Based Conjoint Surveys in <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:white;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>

### by John Paul Helveston

Sawtooth Software Conference

May 06, 2022

---

### Designing a Choice-Based Conjoint Survey is Hard

---

### Designing a Choice-Based Conjoint Survey is Hard

---

### Designing a Choice-Based Conjoint Survey is Hard

---

# .center[A simple conjoint experiment about _cars_]

Attribute | Levels
----------|----------
Brand     | GM, BMW, Ferrari
Price     | $20k, $40k, $100k

```
Attribute counts:

brand:
  GM   BMW  Ferrari 
  10    11    6

price:

20k  40k 100k 
  9    9   9
```

]

```
Pairwise attribute counts:

brand & price:
         
          20k 40k 100k
  GM        3   0    7
  BMW       4   5    2
  Ferrari   2   4    0
```

]

---

# .center[A simple conjoint experiment about _cars_]

Attribute | Levels
----------|----------
Brand     | GM, BMW, Ferrari
Price     | $20k, $40k, $100k

```
Attribute counts:

brand:
  GM    BMW   Ferrari 
  92    80     98

price:

20k  40k 100k 
  91   84   95 
```

]

```
Pairwise attribute counts:

brand & price:
         
          20k 40k 100k
  GM      31  31  30
  BMW     25  25  30
  Ferrari 35  28  35
```

]

---

# .center[Bayesian D-efficient designs]

### .center[Maximize information on "Main Effects" according to priors]

Attribute | Levels | Prior
----------|-------------------|----------
Brand     | GM, BMW, Ferrari  | 0, 1, 2
Price     | $20k, $40k, $100k | 0, -1, -4

```
Attribute counts:

brand:
  GM    BMW   Ferrari 
  93    90     86

price:

20k  40k 100k 
  97   93   78
```

]

```
Pairwise attribute counts:

brand & price:
         
          20k 40k 100k
  GM      52  41  0
  BMW     30  30  30
  Ferrari 15  22  49
```

]

---

# .center[Bayesian D-efficient designs]

### .center[Attempts to maximize information on .red[Main Effects]]

---

### .center[...but .red[interaction effects] are confounded in D-efficient designs]

---

# .center[But what about other factors?]

<br>

- What if I add one more choice question to each respondent?

- What if I increase the number of alternatives per choice question?

- What if I use a labeled design (aka "alternative-specific design")?

- What if there are interaction effects?

---

# The `cbcTools` Package

---

---

---

---

---

# .center[Define the attributes and levels]

<br>

```r
levels <- list(
  price     = c(1.00, 1.50, 2.00, 2.50, 3.00, 3.50, 4.00), # $ per pound
  type      = c("Fuji", "Gala", "Honeycrisp"),
  freshness = c("Excellent", "Average", "Poor")
)
```

```r
levels
```

```
#> $price
#> [1] 1.0 1.5 2.0 2.5 3.0 3.5 4.0
#> 
#> $type
#> [1] "Fuji"       "Gala"       "Honeycrisp"
#> 
#> $freshness
#> [1] "Excellent" "Average"   "Poor"
```

---

---

# .center[Generate all possible profiles]

<br>

```r
profiles <- cbc_profiles(levels)
```

```r
head(profiles)
```

```
#>   profileID price type freshness
#> 1         1   1.0 Fuji Excellent
#> 2         2   1.5 Fuji Excellent
#> 3         3   2.0 Fuji Excellent
#> 4         4   2.5 Fuji Excellent
#> 5         5   3.0 Fuji Excellent
#> 6         6   3.5 Fuji Excellent
```

]

```r
tail(profiles)
```

```
#>    profileID price       type freshness
#> 58        58   1.5 Honeycrisp      Poor
#> 59        59   2.0 Honeycrisp      Poor
#> 60        60   2.5 Honeycrisp      Poor
#> 61        61   3.0 Honeycrisp      Poor
#> 62        62   3.5 Honeycrisp      Poor
#> 63        63   4.0 Honeycrisp      Poor
```

]

---

# .center[Attribute-specific levels]

<br>

```r
levels <- list(
  price = c(1.00, 1.50, 2.00, 2.50, 3.00, 3.50, 4.00),
  freshness = c("Excellent", "Average", "Poor"),
  type = list(
    "Fuji" = list(
        price = c(2.00, 2.50, 3.00)
    ),
    "Gala" = list(
        price = c(1.00, 1.50, 2.00)
    ),
    "Honeycrisp" = list(
        price = c(2.50, 3.00, 3.50, 4.00),
        freshness = c("Excellent", "Average")
    )
  )
)
```

---

# .center[Generate restricted set of profiles]

<br>

```r
profiles <- cbc_profiles(levels)
```

```r
head(profiles)
```

```
#>   profileID price freshness type
#> 1         1   2.0 Excellent Fuji
#> 2         2   2.5 Excellent Fuji
#> 3         3   3.0 Excellent Fuji
#> 4         4   2.0   Average Fuji
#> 5         5   2.5   Average Fuji
#> 6         6   3.0   Average Fuji
```

]

```r
tail(profiles)
```

```
#>    profileID price freshness       type
#> 21        21   3.5 Excellent Honeycrisp
#> 22        22   4.0 Excellent Honeycrisp
#> 23        23   2.5   Average Honeycrisp
#> 24        24   3.0   Average Honeycrisp
#> 25        25   3.5   Average Honeycrisp
#> 26        26   4.0   Average Honeycrisp
```

]

---

---

# .center[Generate a survey design]

```r
design <- cbc_design( 
  profiles = profiles,
  n_resp   = 300, # Number of respondents
  n_alts   = 3,   # Number of alternatives per question
  n_q      = 6    # Number of questions per respondent
)
```

```r
head(design)
```

```
#>   respID qID altID obsID profileID price       type freshness
#> 1      1   1     1     1        60   2.5 Honeycrisp      Poor
#> 2      1   1     2     1        39   2.5 Honeycrisp   Average
#> 3      1   1     3     1        37   1.5 Honeycrisp   Average
#> 4      1   2     1     2        58   1.5 Honeycrisp      Poor
#> 5      1   2     2     2         3   2.0       Fuji Excellent
#> 6      1   2     3     2        38   2.0 Honeycrisp   Average
```

---

# .center[Include a "no choice" option]

```r
design <- cbc_design(
  profiles  = profiles,
  n_resp    = 300, # Number of respondents
  n_alts    = 3,   # Number of alternatives per question
  n_q       = 6,    # Number of questions per respondent
* no_choice = TRUE
)
```

```r
head(design)
```

```
#>   respID qID altID obsID profileID price type_Fuji type_Gala type_Honeycrisp freshness_Excellent freshness_Average freshness_Poor no_choice
#> 1      1   1     1     1         3   2.0         1         0               0                   1                 0              0         0
#> 2      1   1     2     1        38   2.0         0         0               1                   0                 1              0         0
#> 3      1   1     3     1        19   3.0         0         0               1                   1                 0              0         0
#> 4      1   1     4     1         0   0.0         0         0               0                   0                 0              0         1
#> 5      1   2     1     2        30   1.5         0         1               0                   0                 1              0         0
#> 6      1   2     2     2        53   2.5         0         1               0                   0                 0              1         0
```

---

## .center[Make a labeled design]

```r
design <- cbc_design(
  profiles = profiles,
  n_resp   = 300, # Number of respondents
  n_alts   = 3,   # Number of alternatives per question
  n_q      = 6,    # Number of questions per respondent
* label    = "type"
)
```

```r
head(design)
```

```
#>   respID qID altID obsID profileID price       type freshness
#> 1      1   1     1     1         3   2.0       Fuji Excellent
#> 2      1   1     2     1        53   2.5       Gala      Poor
#> 3      1   1     3     1        38   2.0 Honeycrisp   Average
#> 4      1   2     1     2         3   2.0       Fuji Excellent
#> 5      1   2     2     2        50   1.0       Gala      Poor
#> 6      1   2     3     2        42   4.0 Honeycrisp   Average
```

---

# .center[Make a Bayesian D-efficient design]

### .center[(coming soon!)]

```r
design <- cbc_design(
  profiles = profiles,
  n_resp   = 300, # Number of respondents
  n_alts   = 3,   # Number of alternatives per question
  n_q      = 6,   # Number of questions per respondent
* priors = list(
*   price     = -0.1,
*   type      = c(0.1, 0.2),
*   freshness = c(0.1, -0.2)
* )
)
```

---

# .center[Make a Bayesian D-efficient design]

### .center[(coming soon!)]

<br>

## - Check out the [`idefix`](https://www.jstatsoft.org/article/view/v096i03) package

## - Import a design: .blue[Sawtooth] <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:blue;overflow:visible;position:relative;"><path d="M190.5 66.9l22.2-22.2c9.4-9.4 24.6-9.4 33.9 0L441 239c9.4 9.4 9.4 24.6 0 33.9L246.6 467.3c-9.4 9.4-24.6 9.4-33.9 0l-22.2-22.2c-9.5-9.5-9.3-25 .4-34.3L311.4 296H24c-13.3 0-24-10.7-24-24v-32c0-13.3 10.7-24 24-24h287.4L190.9 101.2c-9.8-9.3-10-24.8-.4-34.3z"/></svg> <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:1em;width:0.75em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:blue;overflow:visible;position:relative;"><path d="M224 136V0H24C10.7 0 0 10.7 0 24v464c0 13.3 10.7 24 24 24h336c13.3 0 24-10.7 24-24V160H248c-13.2 0-24-10.8-24-24zm-96 144c0 4.42-3.58 8-8 8h-8c-8.84 0-16 7.16-16 16v32c0 8.84 7.16 16 16 16h8c4.42 0 8 3.58 8 8v16c0 4.42-3.58 8-8 8h-8c-26.51 0-48-21.49-48-48v-32c0-26.51 21.49-48 48-48h8c4.42 0 8 3.58 8 8v16zm44.27 104H160c-4.42 0-8-3.58-8-8v-16c0-4.42 3.58-8 8-8h12.27c5.95 0 10.41-3.5 10.41-6.62 0-1.3-.75-2.66-2.12-3.84l-21.89-18.77c-8.47-7.22-13.33-17.48-13.33-28.14 0-21.3 19.02-38.62 42.41-38.62H200c4.42 0 8 3.58 8 8v16c0 4.42-3.58 8-8 8h-12.27c-5.95 0-10.41 3.5-10.41 6.62 0 1.3.75 2.66 2.12 3.84l21.89 18.77c8.47 7.22 13.33 17.48 13.33 28.14.01 21.29-19 38.62-42.39 38.62zM256 264v20.8c0 20.27 5.7 40.17 16 56.88 10.3-16.7 16-36.61 16-56.88V264c0-4.42 3.58-8 8-8h16c4.42 0 8 3.58 8 8v20.8c0 35.48-12.88 68.89-36.28 94.09-3.02 3.25-7.27 5.11-11.72 5.11s-8.7-1.86-11.72-5.11c-23.4-25.2-36.28-58.61-36.28-94.09V264c0-4.42 3.58-8 8-8h16c4.42 0 8 3.58 8 8zm121-159L279.1 7c-4.5-4.5-10.6-7-17-7H256v128h128v-6.1c0-6.3-2.5-12.4-7-16.9z"/></svg> <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:blue;overflow:visible;position:relative;"><path d="M190.5 66.9l22.2-22.2c9.4-9.4 24.6-9.4 33.9 0L441 239c9.4 9.4 9.4 24.6 0 33.9L246.6 467.3c-9.4 9.4-24.6 9.4-33.9 0l-22.2-22.2c-9.5-9.5-9.3-25 .4-34.3L311.4 296H24c-13.3 0-24-10.7-24-24v-32c0-13.3 10.7-24 24-24h287.4L190.9 101.2c-9.8-9.3-10-24.8-.4-34.3z"/></svg> <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:blue;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>

---

---

# .center[Check design **balance**]

```r
cbc_balance(design)
```

```
Attribute counts:

price:

1 1.5   2 2.5   3 3.5   4 
    825 797 743 743 767 779 746

type:

Fuji       Gala Honeycrisp 
    1842       1769       1789

freshness:
  
    Excellent   Average      Poor 
         1813      1775      1812 
```

]

```
Pairwise attribute counts:

price & type:
     
      Fuji Gala Honeycrisp
  1    304  252        269
  1.5  274  251        272
  2    257  254        232
  2.5  240  254        249
  3    249  263        255
  3.5  257  250        272
  4    261  245        240
```

]

---

# .center[Check design **overlap**]

```r
cbc_overlap(design)
```

```
Counts of attribute overlap:
(# of questions with N unique levels)

price:

1    2    3 
    31  630 1139

type:

1    2    3 
   156 1248  396

freshness:

1    2    3 
   175 1189  436 
```

]

---

---

# .center[Simulate random choices]

```r
data <- cbc_choices(
  design = design,
  obsID  = "obsID"
)
```

```r
head(data)
```

```
#>   respID qID altID obsID profileID price       type freshness choice
#> 1      1   1     1     1         3   2.0       Fuji Excellent      0
#> 2      1   1     2     1        53   2.5       Gala      Poor      0
#> 3      1   1     3     1        38   2.0 Honeycrisp   Average      1
#> 4      1   2     1     2         3   2.0       Fuji Excellent      1
#> 5      1   2     2     2        50   1.0       Gala      Poor      0
#> 6      1   2     3     2        42   4.0 Honeycrisp   Average      0
```

---

# .center[Simulate choices according to a prior]

```r
data <- cbc_choices(
  design = design,
  obsID = "obsID",
* priors = list(
*   price     = -0.1,
*   type      = c(0.1, 0.2),
*   freshness = c(0.1, -0.2)
* )
)
```

]

Attribute | Level | Utility 
----------|-----------
**Price** | Continuous | -0.1
**Type**  | Fuji | 0
          | Gala | 0.1
          | Honeycrisp  | 0.2
**Freshness**  | Average | 0
               | Excellent | 0.1
               | Poor  | -0.2

]

---

# .center[Simulate choices according to a prior]

```r
data <- cbc_choices(
  design = design,
  obsID = "obsID",
  priors = list(  
    price = -0.1, 
*   type = randN(
*     mu    = c(0.1, 0.2),
*     sigma = c(0.5, 1)
*   ),
    freshness = c(0.1, -0.2) 
  ) 
)
```

]

Attribute | Level | Utility 
----------|-----------
**Price** | Continuous | -0.1
**Type**  | Fuji | 0
          | Gala | N(0.1, 0.5)
          | Honeycrisp  | N(0.2, 1)
**Freshness**  | Average | 0
               | Excellent | 0.1
               | Poor  | -0.2

]

---

# .center[Simulate choices according to a prior]

```r
data <- cbc_choices(
  design = design,
  obsID = "obsID",
  priors = list( 
    price     = -0.1, 
    type      = c(0.1, 0.2), 
    freshness = c(0.1, -0.2),
*   "price*type" = c(0.1, 0.5)
  ) 
)
```

]

]

---

---

# .center[Conduct a power analysis]

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

```r
head(power)
```

```
#>   sampleSize             coef         est         se
#> 1         30            price -0.18558034 0.09627804
#> 2         30         typeGala -0.11287630 0.18806682
#> 3         30   typeHoneycrisp -0.00373311 0.18247312
#> 4         30 freshnessAverage -0.23740384 0.21995530
#> 5         30    freshnessPoor -0.58571733 0.23470664
#> 6         60            price -0.13680799 0.06768533
```

]

```r
tail(power)
```

```
#>    sampleSize             coef         est         se
#> 45        270    freshnessPoor -0.18341634 0.07544056
#> 46        300            price -0.11054677 0.02899401
#> 47        300         typeGala  0.09544003 0.05943437
#> 48        300   typeHoneycrisp  0.18369313 0.05829540
#> 49        300 freshnessAverage  0.13523452 0.07015579
#> 50        300    freshnessPoor -0.19427738 0.07161628
```

]

---

# .center[Conduct a power analysis]

```r
plot(power)
```

---

# .center[Conduct a power analysis]

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

]

```r
plot(power_int)
```

]

---

---

---

background-color: #fff
background-image: url("images/cbcTools-sawtooth.png")
background-position: center
background-size: contain

---

<br>

# .center[.font150[Thanks!]]

### `cbcTools` documentation: https://jhelvy.github.io/cbcTools/

### Slides: https://jhelvy.github.io/2022-sawtooth-conf

.footer-large[
.right[
@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 0 512 512" 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="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></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="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg>
]]

---

# Extra slides