## Today’s data

The pokemon dataset contains data on all currently extant Pokemon (as of August 1, 2020), including alternate forms and mega evolutions. The dataset contains information including its name, baseline battle statistics (base stats), types, and whether they are “legendary” (broadly defined).

Pokemon are fantastic creatures with unique characteristics, and may be used to battle each other in combat. The battle-effectiveness of a Pokemon is in part determined by their “base stats,” which consist of HP or “hit points”, attack, defense, special attack, special defense, and speed. Some Pokemon are considered “legendary” if they are exceptionally rare (in some sense). Legendary Pokemon are often, but not always, more powerful than their non-legendary counterparts.

Pokemon and Pokemon names are trademarks of Nintendo.

Let’s read in the dataset and create a new variable legendary.

pokemon <- read_csv("pokemon.csv")

pokemon <- pokemon %>%
mutate(legendary = ifelse(leg_status %in% c("Legendary", "Mythical"), 1, 0))

Some important variables in the dataset are listed below:

• pokedex_number: a catalogue number of each Pokemon
• name: the name of the Pokemon
• hp, atk, def, spa, spd, spe: a Pokemon’s battle statistics (HP, attack, defense, special attack, special defense, and speed, respectively)
• bst: the total base stats of the Pokemon
• legendary: whether the Pokemon is legendary
• type_1: the primary type of the Pokemon

## Non-continuous outcomes

In previous tutorials, we’ve focused on using linear regression as a tool to

• Make predictions about new observations
• Describe relationships
• Perform statistical inference

These examples have all had continuous response variables. But can we do similar tasks for binary categorical response variables too? Today’s goals are to predict whether a Pokemon is a legendary based on its base stats and describe the relationship between stats and probability of being legendary.

Suppose we have some hypothetical Pokemon with the following base stats. Would we classify them as legendary Pokemon based on these characteristics?

Pokemon HP ATK DEF SPA SPD SPE
Crapistat 55 25 30 60 50 102
Mediocra 90 110 130 75 80 45
Literally Dragonite 91 134 95 100 100 80
Broaken 104 125 105 148 102 136

## Problems with linear models

Suppose we consider the following model for $$p$$, the probability of being a legendary Pokemon: ${p} = \beta_0 + \beta_1HP + \beta_2ATK + \cdots + \beta_6\times SPE$

Take a moment and think about what might go wrong when we use this model.

Let’s create this model and create some diagnostic plots. Click on Run Code in each of the code fields below to see the residual plot and a histogram of the residuals.

m2 <- lm(legendary ~ hp + atk + def + spa + spd + spe, data = pokemon)

ggplot(data = augment(m2), aes(x = .fitted, y = .resid)) +
geom_point() +
labs(x = "Predicted", y = "Residual", title = "Residual plot")
ggplot(data = augment(m2), aes(x = .fitted)) +
geom_histogram() +
labs(x = "Predicted Values", y = "Count")

What problems do you see?

We see that the linear model assumptions are definitely not satisfied. Also, in looking at the predicted values, many of the Pokemon are predicted to have negative probabilities of being legendary (which doesn’t make sense at all!). Similarly, it’s possible to get predicted probabilities greater than 1 in such a linear probability model.

## Introducing log-odds

Linear models can predict any value from $$-\infty$$ to $$\infty$$, but probabilities are restricted to lie between 0 and 1. Is there a way to create a model for a transformation of the probabilitiy in a principled way such that it’s not a problem if we make predictions that can take on any value?

Suppose the probability of an event is $$p$$ Then the odds that the event occurs is $$\frac{p}{1-p}$$. Taking the natural log of the odds, we have the logit (or log-odds) of $$p$$: $logit(p) = \log\left(\frac{p}{1-p}\right)$ Note that in statistics, when we say “log” we always mean a logarithm with the natural base $$e$$. By taking this logit transformation, although $$p$$ is constrained tolie within 0 and 1, $$logit(p)$$ can range from $$-\infty$$ to $$\infty$$.

## Logistic regression

Let’s instead consider the following linear model for the log-odds of $$p$$:

${logit(p)} = \beta_0 + \beta_1\times HP + \beta_2\times ATK + \cdots + \beta_6\times SPE$

Since there is a one-to-one relationship between probabilities and log-odds, we can undo the previous function. If we create a linear model on the log-odds, we can “work backwards” to obtain predicted probabilities that are guaranteed to lie between 0 and 1. To “work backwards,” we use the logistic function:

$f(x) = \frac{1}{1 + e^{-x}} = \frac{e^x}{1 + e^x}$

So, our linear model for $$logit(p)$$ is equivalent to

$p = \frac{e^{\beta_0 + \beta_1HP + \beta_2ATK + \cdots + \beta_6SPE}}{1 + e^{\beta_0 + \beta_1\times HP + \beta_2\times ATK + \cdots + \beta_6\times SPE}}$

## Model fitting

We fit the logistic regression model using the glm function, which stands for generalized linear model. We need to additionally tell R that we want a logistic model, and so we include the argument family = "binomial".

In general, the syntax is:

moderl <- glm(outcome ~ pred1 + pred2 + ..., data = ___, family = "binomial")

where outcome is a binary numeric outcome with values 0 and 1, and pred_ are the predictor variables. As well, we can use the tidy function to see a tidy output of the model object.

### Exercise 1

Fit a logistic regression model predicting legendary status using the predictors hp, atk, def, spa, spd, and spe using the pokemon dataset, and examine the tidy output of this model.

logit_mod <- glm(___ ~ ___, data = ___,
family = "binomial")
tidy(___)

We have a linear model on a transformation of the response. Thus, we can interpret estimated coefficients analogously to what we’ve learned before for linear models.

## Interpreting coefficients

Let’s take a look at the output from the model in the previous exercise:

Holding all other predictors constant, for each unit increase in base speed, we expect the log-odds of being legendary to increase by 0.0366.

If we exponentiate this, then we have the multiplicative effect on the odds scale (instead of the additive effect on the log-odds scale). Thus, an equivalent interpretation might be that a Pokemon that has a base speed one unit larger than another would have exp(0.0366) $$\approx$$ 1.0373 times the odds of being legendary (holding all other predictors constant).

Note: categorical variables are converted into binary dummy variables just as in regular linear models, and are interpreted as the difference in log-odds compared to the baseline value (holding all else constant). We can also exponentiate the coefficient estimate, which gives us the odds ratio comparing an observation satisfying a certain dummy condition vs. the baseline (holding all else constant).

## Predicting new values

Like in regular linear regression, we can predict outcomes given any set of predictor values. In this case, our outcome is the log-odds of being legendary given a Pokemon’s base stats.

We can the augment function (part of the broom package) to create predicted probabilities. The following code creates a new dataset that has the predictor values for our four hypothetical Pokemon from before. We will use the augment function that takes our new dataset and plugs it into an existing logistic regression model (in this case, one called logit_mod). Finally, we will pull the .fitted values as a vector.

new_pokemon <- tibble(hp = c(55, 90, 91, 104),
atk = c(25, 110, 134, 125),
def = c(30, 130, 95, 105),
spa = c(60, 75, 100, 148),
spd = c(50, 80, 100, 102),
spe = c(102, 45, 80, 136))

pred_log_odds <- augment(logit_mod, newdata = new_pokemon) %>%
pull(.fitted)

These are the predicted log-odds of being legendary, given the observed base stats of our four hypothetical Pokemon.

How might we go from log-odds back to probabilities? We can “undo” the earlier transformation! Letting $$y$$ be the log-odds and $$p$$ the probability, we have

$p = \frac{e^y}{1 + e^y}$

### Exercise 2

Using this transformation and your predicted log-odds, what are the predicted probabilitieS of being legendary? How might you classify these Pokemon given their predicted probabilities?

newdata <- tibble(___,
___,
...)

pred_probs <- augment(___, newdata = ___) %>%
mutate(p = exp(___)/(1 + exp(___)) %>%
pull(p)