AE 18: Pokemon Inference

Today we’ll explore the question “Are taller Pokémon speedier than shorter Pokémon?”

Packages

library(tidyverse)
library(tidymodels)

Data

The data for this application exercise was originally gathered using the Complete Pokémon Dataset from Kaggle.

First, let’s load the data:

pokemon <- read_csv("data/pokemon.csv")

To keep things simple, we’ll work with a subset of the data, the generation 4 Pokémon.

pokemon <- pokemon |>
  filter(generation == 4)

glimpse(pokemon)

Rows: 121
Columns: 17
$ name            <chr> "Turtwig", "Grotle", "Torterra", "Chimchar", "Monferno…
$ generation      <dbl> 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, …
$ type_1          <chr> "Grass", "Grass", "Grass", "Fire", "Fire", "Fire", "Wa…
$ type_2          <chr> NA, NA, "Ground", NA, "Fighting", "Fighting", NA, NA, …
$ height_m        <dbl> 0.4, 1.1, 2.2, 0.5, 0.9, 1.2, 0.4, 0.8, 1.7, 0.3, 0.6,…
$ weight_kg       <dbl> 10.2, 97.0, 310.0, 6.2, 22.0, 55.0, 5.2, 23.0, 84.5, 2…
$ total_points    <dbl> 318, 405, 525, 309, 405, 534, 314, 405, 530, 245, 340,…
$ hp              <dbl> 55, 75, 95, 44, 64, 76, 53, 64, 84, 40, 55, 85, 59, 79…
$ attack          <dbl> 68, 89, 109, 58, 78, 104, 51, 66, 86, 55, 75, 120, 45,…
$ defense         <dbl> 64, 85, 105, 44, 52, 71, 53, 68, 88, 30, 50, 70, 40, 6…
$ sp_attack       <dbl> 45, 55, 75, 58, 78, 104, 61, 81, 111, 30, 40, 50, 35, …
$ sp_defense      <dbl> 55, 65, 85, 44, 52, 71, 56, 76, 101, 30, 40, 60, 40, 6…
$ speed           <dbl> 31, 36, 56, 61, 81, 108, 40, 50, 60, 60, 80, 100, 31, …
$ catch_rate      <dbl> 45, 45, 45, 45, 45, 45, 45, 45, 45, 255, 120, 45, 255,…
$ base_friendship <dbl> 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70…
$ base_experience <dbl> 64, 142, 236, 62, 142, 240, 63, 142, 239, 49, 119, 218…
$ is_legendary    <lgl> FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE…

We are interested in whether a Pokémon’s log height (measured in meters) can predict its speed (how fast it is).

Visualize

1. Review! Create a new variable for the Pokémon’s log height.

pokemon <- pokemon |>
  mutate(log_height = log(height_m))

2. Your turn: Plot the data and the line of best fit.

# add code here

Point Estimation

1. Your turn: Fit the linear model to these data:

observed_fit <- pokemon |>
  specify(speed ~ log_height) |>
  fit()
observed_fit

# A tibble: 2 × 2
  term       estimate
  <chr>         <dbl>
1 intercept     72.4 
2 log_height     7.39

Note

This gives the exact same numbers that you get if you use linear_reg() |> fit(), but we need this new syntax because it plays nice with the tools we have for confidence intervals and hypothesis tests.

2. Your turn: Typeset the equation for the model fit:

\[ \widehat{\text{speed}} = \_\_\_ + \_\_\_ \times \log(\text{height}) \]

3. Your turn: Interpret the slope and the intercept estimates:

   • Add your answer here.
   • Add your answer here.

Hypothesis Testing

Let’s consider the hypotheses:

\[ H_0:\beta_1=0\quad vs\quad H_A: \beta_1\neq 0. \]

The null hypothesis corresponds to the claim that log_height and speed are uncorrelated.

1. Simulate and plot the null distribution for the slope:

set.seed(24601)
null_dist <- pokemon |>
  specify(speed ~ log_height) |>
  hypothesize(null = "independence") |>
  generate(reps = 500, type = "permute") |>
  fit()

# add code to plot here

2. Add a vertical line to your plot indicating the point estimate of the slope from your original data analysis:

# add code here

3. Compute the \(p\)-value for this test and interpret it:

# add code here

Add your interpretation here.

Interval Estimation

1. Demo: Using seed 24601, generate 500 bootstrap samples, and store them in a new data frame called bstrap_samples.

set.seed(24601)
bstrap_samples <- pokemon |>
  specify(speed ~ log_height) |>
  generate(reps = 500, type = "bootstrap")

2. Demo: Fit a linear model to each of these bootstrap samples and store the estimates in a new data frame called bstrap_fits.

bstrap_fits <- bstrap_samples |>
  fit()

3. Your turn: Use linear_reg() |> fit(...) to fit a linear model to bootstrap sample number 347, and verify that you get the same estimates as the ones contained in bstrap_fits.

# add code here

Note

The only point I’m making here is that this new bootstrap code is not performing a fundamentally new task. It’s performing an old task (fitting the linear model), but it’s repeating it A LOT.

4. Demo: Compute 95% confidence intervals for the slope and the intercept using the get_confidence_interval command.

ci_95 <- get_confidence_interval(
  bstrap_fits,
  point_estimate = observed_fit,
  level = 0.95,
  type = "percentile"
)
ci_95

# A tibble: 2 × 3
  term       lower_ci upper_ci
  <chr>         <dbl>    <dbl>
1 intercept    67.4       76.6
2 log_height    0.942     13.3

5. Your turn: Verify that you get the same numbers when you manually calculate the quantiles of the slope estimates using summarize and quantile. Pay attention to the grouping.

# add code here

Note

Same point as before. There’s no magic here. get_confidence_interval is just a convenient way of doing something that you already knew how to do.

6. BONUS: You can visualize the confidence interval:

visualize(bstrap_fits) +
  shade_confidence_interval(ci_95)