R Linear Regression: Linear Models (Introduction)

Confounding: Are BBs more Predictive?

• Association is not causation!
• Although it may appear that BB cause runs, it is actually the HR that cause most of these runs. We say that BB are confounded with HR.
• Regression can help us account for confounding.

Code: Confounding

# find regression line for predicting runs from BBs
library(tidyverse)
library(Lahman)
bb_slope <- Teams %>% 
  filter(yearID %in% 1961:2001 ) %>% 
  mutate(BB_per_game = BB/G, R_per_game = R/G) %>% 
  lm(R_per_game ~ BB_per_game, data = .) %>% 
  .$coef %>%
  .[2]
bb_slope

# compute regression line for predicting runs from singles
singles_slope <- Teams %>% 
  filter(yearID %in% 1961:2001 ) %>%
  mutate(Singles_per_game = (H-HR-X2B-X3B)/G, R_per_game = R/G) %>%
  lm(R_per_game ~ Singles_per_game, data = .) %>%
  .$coef  %>%
  .[2]
singles_slope

# calculate correlation between HR, BB and singles
Teams %>% 
  filter(yearID %in% 1961:2001 ) %>% 
  mutate(Singles = (H-HR-X2B-X3B)/G, BB = BB/G, HR = HR/G) %>%  
  summarize(cor(BB, HR), cor(Singles, HR), cor(BB,Singles))

Stratification and Multivariate Regression

• A first approach to check confounding is to keep HRs fixed at a certain value and then examine the relationship between BB and runs.
• The slopes of BB after stratifying on HR are reduced, but they are not 0, which indicates that BB are helpful for producing runs, just not as much as previously thought.

Code: Stratification & Multivariate Regression

# stratify HR per game to nearest 10, filter out strata with few points
dat <- Teams %>% filter(yearID %in% 1961:2001) %>%
  mutate(HR_strata = round(HR/G, 1), 
         BB_per_game = BB / G,
         R_per_game = R / G) %>%
  filter(HR_strata >= 0.4 & HR_strata <=1.2)
  
# scatterplot for each HR stratum
dat %>% 
  ggplot(aes(BB_per_game, R_per_game)) +  
  geom_point(alpha = 0.5) +
  geom_smooth(method = "lm") +
  facet_wrap( ~ HR_strata)
  
# calculate slope of regression line after stratifying by HR
dat %>%  
  group_by(HR_strata) %>%
  summarize(slope = cor(BB_per_game, R_per_game)*sd(R_per_game)/sd(BB_per_game))
  
# stratify by BB
dat <- Teams %>% filter(yearID %in% 1961:2001) %>%
  mutate(BB_strata = round(BB/G, 1), 
         HR_per_game = HR / G,
         R_per_game = R / G) %>%
  filter(BB_strata >= 2.8 & BB_strata <=3.9) 

# scatterplot for each BB stratum
dat %>% ggplot(aes(HR_per_game, R_per_game)) +  
  geom_point(alpha = 0.5) +
  geom_smooth(method = "lm") +
  facet_wrap( ~ BB_strata)
  
# slope of regression line after stratifying by BB
dat %>%  
  group_by(BB_strata) %>%
  summarize(slope = cor(HR_per_game, R_per_game)*sd(R_per_game)/sd(HR_per_game)) 

Linear Models

• “Linear” here does not refer to lines, but rather to the fact that the conditional expectation is a linear combination of known quantities.
• In Galton's model, we assume Y (son's height) is a linear combination of a constant and X (father's height) plus random noise. We further assume that ϵi are independent from each other, have expected value 0 and the standard deviation σ which does not depend on i.
• Note that if we further assume that ϵ is normally distributed, then the model is exactly the same one we derived earlier by assuming bivariate normal data.
• We can subtract the mean from X to make β0 more interpretable.