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Introduction to Regression in R Cheat Sheet by

Building, assessing, and making predictions using basic regression models.

Simple Linear Regression in R

Regression models allow you explore relati­onships between a response and explan­atory variables.
You can use the model to make predic­tions.
It is always a good idea to visualize a dataset such as scatte­rplots.
The intercept is the y value when x is zero. In some cases, its interp­ret­ations might make sense. For instance, on average a house with zero conven­ience stores nearby had a price of 8.2242 TWD per square meter.
The slope is the amount y increases by if you increase x by one.
If your sole explan­atory variable is catego­rical, the intercept is the response variable's mean of the omitted category. The coeffi­cients of each category are means relative to the intercept. You can change this if you like so that the coeffi­cients are the means of each category.

Simple regression code

# Assume you have a real estate dataset and want to build a model predicting prices using n_convenience stores nearby within walking distance.

# Visualize the two variables. What is the relationship?
ggplot(taiwan_real_estate, aes(n_convenience, price_twd_msq)) +
  geom_point(alpha = 0.5) +
  geom_smooth(method='lm', se=FALSE)

# Run a linear regression of price_twd_msq vs. n_convenience
lm(price_twd_msq ~ n_convenience, data = taiwan_real_estate)

# Visualize prices v age_category using a histogram. What do you see?
ggplot(taiwan_real_estate, aes(price_twd_msq)) +
  # Make it a histogram with 10 bins
  geom_histogram(bins=10) +
  # Facet the plot so each house age group gets its own panel
  facet_wrap(vars(house_age_years))

# calculate means by each age category
taiwan_real_estate %>% 
  # Group by house age
  group_by(house_age_years) %>% 
  # Summarize to calculate the mean house price/area
  summarize(mean_by_group = mean(price_twd_msq))

# Run a linear regression of price_twd_msq vs. house_age_years
mdl_price_vs_age <- lm(data=taiwan_real_estate, price_twd_msq~house_age_years) #add +0 after house_age_years to get each category's mean.
 
# See the result
mdl_price_vs_age

Predic­tions and Model objects

Extrap­olating means making predic­tions outside the range of observed data. Even if you use nonsense explan­atory data to make predic­tions, the model won't throw an error and give you a predic­tion. Understand your data to determine if a prediction is nonsense or not.
It is useful to have values you want to use to make predic­tions (test data) in a tibble. Thus, you can store your predic­tions in the same tibble and make plots.
coeffi­cie­nts­(mo­del­_ob­ject) returns a named, numeric vector of the coeffi­cients.
fitted­_va­lue­s(m­ode­l_o­bject) returns predic­tions on the original data.
residu­als­(mo­del­_ob­ject) returns the difference between actual response values minus predic­tions. They are a measure of inaccu­racy.
broom:­:ti­dy(­mod­el_­object) returns coeffi­cients and its detail.
broom:­:au­gme­nt(­mod­el_­object) returns observ­ation level detail such as residuals, fitted values, etc.
broom:­:gl­anc­e(m­ode­l_o­bject) returns model-­level results (perfo­rmance metrics).
Residuals exist due to problems in the model and fundam­ental random­ness. And extreme cases are also often due to random­ness.
Eventu­ally, extreme cases will more look like average cases (b/c they don't presist over time). This is called regression to the mean.
Due to regression to the mean, a baseball player does not hit as many home runs this year as he did the year before.
If there is no straight line relati­onship between the response variable and the explan­atory variable, it is sometimes possible to create one by transf­orming one or both of the variables.
If you transf­ormed the response variable, you must "­bac­k-t­ran­sfo­rm" your predic­tions.

How to make predic­tions and view model objects?

# Model prices and n_convenience
mdl_price_vs_conv <- lm(formula = price_twd_msq ~ n_convenience, data = taiwan_real_estate)

# Create a tibble of integer values from 0 to 10.
explanatory_data <- tibble(n_convenience = 0:10)

# Make predictions and store them in prediction_data
prediction_data <- explanatory_data %>% 
  mutate(price_twd_msq = predict(mdl_price_vs_conv, explanatory_data))

# Plot the predictions along with all points
ggplot(taiwan_real_estate, aes(n_convenience, price_twd_msq)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  # Add a point layer of prediction data, colored yellow
  geom_point(color='yellow', data=prediction_data)

# -------------- Regression to the mean example-------------

# Suppose you have data on annual returns from investing in companies in the SP500 index and you're interested in knowing if the invested performance stays the same from 2018 to 2019. 

# Using sp500_yearly_returns, plot return_2019 vs. return_2018
ggplot(data=sp500_yearly_returns, aes(x=return_2018, y=return_2019)) +
  # Make it a scatter plot
  geom_point() +
  # Add a line at y = x, colored green, size 1
  geom_abline(slope=1, color='green', size=1) +
  # Add a linear regression trend line, no std. error ribbon
  geom_smooth(method='lm', se=FALSE) +
  # Fix the coordinate ratio
  coord_fixed()

# Transforming variables and back-transforming the response---------------

# Assume you've facebook advertising data; how many people see the adds and how many people click on them.

mdl_click_vs_impression <- lm(
  I(n_clicks^0.25) ~ I(n_impressions^0.25),
  data = ad_conversion
)
explanatory_data <- tibble(
  n_impressions = seq(0, 3e6, 5e5)
)
prediction_data <- explanatory_data %>% 
  mutate(
    n_clicks_025 = predict(mdl_click_vs_impression, explanatory_data),
    n_clicks = n_clicks_025 ^ 4
  )

ggplot(ad_conversion, aes(n_impressions  0.25, n_clicks  0.25)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  # Add points from prediction_data, colored green
  geom_point(data=prediction_data, color='green')

Quanti­fying Model Fit

Coeffi­cient of determ­ination is the proportion of variance in the response variable that is predic­table from the explan­atory variable. 1 means a perfect fit and 0 means the worst possible fit.
For simple linear regres­sion, coeff of determ­ination is correl­ation between the response and explan­atory squared.
Residual standard error (or sum of squared residuals) is a typical difference between prediction and an observed response. This is sigma in broom:­:gl­anc­e(m­odel)
Root mean squared error (RMSE) also works but the denomi­nator is number of observ­ations and not degrees of freedom.
If the linear regression model is a good fit, then the residuals are normally distri­buted and their mean is zero. This assumption can be checked using the residual v fitted values plot. The blue trend line should closely follow the y=0 line.
The Q-Q plot whether the residuals follow a normal distri­bution. If the points track along the diagonal line, they are normally distri­buted.
The scale-­loc­ation plot shows whether the size of residuals get bigger or smaller as the fitted values change.
Leverage quantifies how extreme your explan­atory variables are. These values are stored under .hat in augment().
Influence measures how much model the model would change if you left the observ­ation out of the dataset when modeling. Contained in .cooksd column in augment().

Code to quantify model's fit.

# Plot three diagnostics for mdl_price_vs_conv
library(ggplot2)
library(ggfortify)
autoplot(mdl_price_vs_conv, which=1:3, nrow=3, ncol=1)

# Plot the three outlier diagnostics for mdl_price_vs_conv
autoplot(mdl_price_vs_dist, which=4:6, nrow=3, ncol=1)

Simple Logistic regression in R

Build this model when the response is binary. Predic­tions are probab­ilities and not amounts.
The responses follow a logistic (s-shaped) curve. You can have the model return probab­ilties by specifying the response type in predict().
Odds ratio is the proability of something happening divided by the probab­ility that it doesn't.
This is sometimes easier to reason about than probab­ili­ties, partic­ularly when you want to make decisions about choices. For example, if a customer has a 20% chance of churning, it maybe more intuitive to say "the chance of them not churning is four times higher than the chance of them churni­ng".
One downside to probab­ilities and odds ratios for logistic regression predic­tions is that the prediction lines for each are curved.
A nice property of logistic regression odds ratio is that on a log-scale they change linearly with the explan­atory variable.
This makes it harder to reason about what happens to the prediction when you make a change to the explan­atory variable. The logarithm of the odds ratio (the "log odds ratio") does have a linear relati­onship between predicted response and explan­atory variable.
We use confusion matricies to quantify the fit of logistic regres­sion.
Accuracy is the proportion of correct predic­tions.
Sensit­ivity is the proportion of true positives. TP/(FN­+TP). Proportion of observ­ations where the actual response was true where the model also predicted were true.
Specificty is the proportion of true negatives. TN/(TN­+FP). Proportion of observ­ations where the actual response was false where the model also predicted that they were false.

Code for Logistic regression in R

plt_churn_vs_relationship ggplot(churn, aes(time_since_first_purchase, has_churned)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE, color = "red") +
  # Add a glm trend line, no std error ribbon, binomial family
  geom_smooth(method='glm', se=FALSE, method.args=list(family=binomial))

# Fit a logistic regression of churn vs. 
# length of relationship using the churn dataset
mdl_churn_vs_relationship <- glm(has_churned ~ time_since_first_purchase, data=churn, family='binomial')

# See the result
mdl_churn_vs_relationship

# Make predictions. "response" type returns probabilities of churning. 
prediction_data <- explanatory_data %>% 
  mutate(   
    has_churned = predict(mdl_churn_vs_relationship, explanatory_data, type = "response"),
    most_likely_outcome = round(has_churned) # easier to interpret. 
  )

# Update the plot
plt_churn_vs_relationship +
  # Add most likely outcome points from prediction_data, colored yellow, size 2
  geom_point(data=prediction_data, size=2, color='yellow', aes(y=most_likely_outcome))

# Odds ratio------------------------------------------

# From previous step
prediction_data <- explanatory_data %>% 
  mutate(   
    has_churned = predict(mdl_churn_vs_relationship, explanatory_data, type = "response"),
    odds_ratio = has_churned / (1 - has_churned), 
    log_odds_ratio = log(odds_ratio)
  )

# Using prediction_data, plot odds_ratio vs. time_since_first_purchase
ggplot(data=prediction_data, aes(x=time_since_first_purchase, y=odds_ratio)) +
  # Make it a line plot
  geom_line() +
  # Add a dotted horizontal line at y = 1. Indicates where churning is just as likely as not churning. 
  geom_hline(yintercept=1, linetype='dotted')

Quanti­fying Logistic's Model Fit

# Get the confusion matrix.
library(yardstick)

# Get the actual and most likely responses from the dataset
actual_response <- churn$has_churned
predicted_response <- round(fitted(mdl_churn_vs_relationship))

# Create a table of counts
outcomes <- table(predicted_response, actual_response)

# Convert outcomes to a yardstick confusion matrix
confusion <- conf_mat(outcomes)

# Get performance metrics for the confusion matrix
summary(confusion, event_level = 'second')
       
 

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