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layout: true

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&lt;div class="footer-small"&gt;
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https://github.com/jhelvy/2022-sawtooth-conf
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---

name: title-slide
class: inverse, middle
background-image: url(images/blue.jpg)

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

## Tools for Designing and Testing&lt;br&gt;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

---

class: center

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

--

&lt;center&gt;
&lt;img src="images/tradeoffs1.png" width=90%&gt;
&lt;/center&gt;

---

class: center

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

&lt;center&gt;
&lt;img src="images/tradeoffs2.png" width=90%&gt;
&lt;/center&gt;

---

class: center

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

&lt;center&gt;
&lt;img src="images/tradeoffs3.png" width=90%&gt;
&lt;/center&gt;

---

# .center[A simple conjoint experiment about _cars_]

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

.center[**Design: .red[9] choice sets, .blue[3] alternatives each**]

--

.leftcol[

```
Attribute counts:

brand:
  GM   BMW  Ferrari 
  10    11    6 

price:

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

]

--

.rightcol[

```
Pairwise attribute counts:

brand &amp; 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

.center[**Design: .red[90] choice sets, .blue[3] alternatives each**]

--

.leftcol[

```
Attribute counts:

brand:
  GM    BMW   Ferrari 
  92    80     98

price:

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

]

--

.rightcol[

```
Pairwise attribute counts:

brand &amp; 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

--

.leftcol[

```
Attribute counts:

brand:
  GM    BMW   Ferrari 
  93    90     86

price:

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

]

--

.rightcol[

```
Pairwise attribute counts:

brand &amp; 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]]

--

&lt;center&gt;
&lt;img src="images/design_compare.png" width=100%&gt;
&lt;/center&gt;

---

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

--

&lt;center&gt;
&lt;img src="images/design_compare_int.png" width=120%&gt;
&lt;/center&gt;

---

# .center[But what about other factors?]

&lt;br&gt;

- 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?

---

class: middle, center, inverse 

# The `cbcTools` Package 

&lt;center&gt;
&lt;img src="images/logo.png" width=30%&gt;
&lt;/center&gt;

---

background-image: url("images/process.png")

---

background-image: url("images/process_labels.png")

---

class: center, middle 
background-image: url("images/process_labels.png")

.border[
&lt;center&gt;
&lt;img src="images/cbc_screenshot.png" width=80%&gt;
&lt;/center&gt;
]

---

background-image: url("images/process_levels.png")

---

# .center[Define the attributes and levels]

&lt;br&gt;


```r
levels &lt;- 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
```

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

---

background-image: url("images/process_profiles.png")

---

# .center[Generate all possible profiles]

&lt;br&gt;


```r
profiles &lt;- cbc_profiles(levels)
```

--

.leftcol[


```r
head(profiles)
```

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

]

.rightcol[


```r
tail(profiles)
```

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

]

---

# .center[Attribute-specific levels]

&lt;br&gt;


```r
levels &lt;- 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]

&lt;br&gt;


```r
profiles &lt;- cbc_profiles(levels)
```

--

.leftcol[


```r
head(profiles)
```


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

]

.rightcol[


```r
tail(profiles)
```


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

]

---

background-image: url("images/process_design.png")

---

# .center[Generate a survey design]


```r
design &lt;- 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)
```

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

---

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


```r
design &lt;- 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)
```

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

---

## .center[Make a labeled design]

.center[.font100[(aka "alternative-specific design")]]


```r
design &lt;- 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)
```

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

---

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

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


```r
design &lt;- 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!)]

&lt;br&gt;

## - 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>

---

background-image: url("images/process_inspect.png")

---

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


```r
cbc_balance(design)
```

--

.leftcol[

```
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 
```

]

--

.rightcol[

```
Pairwise attribute counts:

price &amp; 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)
```

--

.leftcol[

```
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 
```

]

---

background-image: url("images/process_choices.png")

---

# .center[Simulate random choices]


```r
data &lt;- cbc_choices(
  design = design,
  obsID  = "obsID"
)
```

--


```r
head(data)
```

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

---

# .center[Simulate choices according to a prior]

.leftcol[


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

]

.rightcol[

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]

.leftcol[


```r
data &lt;- 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) 
  ) 
)
```

]

.rightcol[

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]

.leftcol[


```r
data &lt;- 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)
  ) 
)
```

]

.rightcol[

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
**Price x Type**  | Fuji | 0
          | Gala | 0.1
          | Honeycrisp  | 0.5

]

---

background-image: url("images/process_power.png")

---

# .center[Conduct a power analysis]


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

--

.leftcol[


```r
head(power)
```

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

]

.rightcol[


```r
tail(power)
```

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

]

---

# .center[Conduct a power analysis]


```r
plot(power)
```

&lt;img src="figs/unnamed-chunk-30-1.png" width="576" /&gt;

---

# .center[Conduct a power analysis]

.leftcol[


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

]

.rightcol[


```r
plot(power_int)
```

&lt;img src="figs/unnamed-chunk-32-1.png" width="576" /&gt;

]

---

background-image: url("images/process_labels.png")

&lt;center&gt;
&lt;img src="images/logo.png" width=20%&gt;
&lt;/center&gt;

---

background-image: url("images/sawtooth-cbcTools.png")

---

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

---

class: inverse
background-image: url(images/blue.jpg)

&lt;br&gt;

# .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>&lt;br&gt;
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]]

---

class: center, middle, inverse 

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