library(tidyverse)
library(easystats)
penguins <- read_csv("../../data/penguins.csv")
penguins
## # A tibble: 344 × 8
## species island bill_len bill_dep flipper_len body_mass sex year
## <chr> <chr> <dbl> <dbl> <dbl> <dbl> <chr> <dbl>
## 1 Adelie Torgersen 39.1 18.7 181 3750 male 2007
## 2 Adelie Torgersen 39.5 17.4 186 3800 female 2007
## 3 Adelie Torgersen 40.3 18 195 3250 female 2007
## 4 Adelie Torgersen NA NA NA NA <NA> 2007
## 5 Adelie Torgersen 36.7 19.3 193 3450 female 2007
## 6 Adelie Torgersen 39.3 20.6 190 3650 male 2007
## 7 Adelie Torgersen 38.9 17.8 181 3625 female 2007
## 8 Adelie Torgersen 39.2 19.6 195 4675 male 2007
## 9 Adelie Torgersen 34.1 18.1 193 3475 <NA> 2007
## 10 Adelie Torgersen 42 20.2 190 4250 <NA> 2007
## # ℹ 334 more rowsStatistical Methods with R
Multiple Regression
Developed by Jeffrey M. Girard
Venn diagram for simple regression
\[y_i = b_0 + b_1x_i + e_i\]
Each oval is one variable’s variance
Two oval’s overlap is their covariance
a is the variance in \(x\) unrelated to \(y\)
b is the variance shared by \(x\) and \(y\)
e is the variance in \(y\) unrelated to \(x\)
\(b_1\) represents b
\(R^2\) represents b / (b + e)
Venn diagram for two predictors
\[y_i = b_0 + b_1x_{1i} + b_2x_{2i} + e_i\]
a is the var in \(x_1\) unrelated to \(y\) or \(x_2\)
b is the var shared by \(x1\) and \(y\) (not \(x_2\))
c is the var shared by \(x_1\), \(x_2\), and \(y\)
d is the var shared by \(x_1\) and \(x_2\) (not \(y\))
e is the var in \(y\) unrelated to \(x_1\) or \(x_2\)
f is the var shared by \(x_2\) and \(y\) (not \(x_1\))
g is the var in \(x_2\) unrelated to \(x_1\) or \(y\)
Zero-order effect of \(x_1\)
\[y_i = b_0 + b_1x_{1i} + e_i\]
If we regress \(y\) on \(x_1\) only, we get the zero-order (i.e., total) effect of \(x_1\)
\(b_1\) represents b + c
\(e_i\) represents e + f
\(R^2\) represents (b + c) / (b + c + e + f)
Zero-order effect of \(x_2\)
\[y_i = b_0 + b_1x_{2i} + e_i\]
If we regress \(y\) on \(x_2\) only, we get the zero-order (i.e., total) effect of \(x_2\)
\(b_1\) represents c + f
\(e_i\) represents e + b
\(R^2\) represents (c + f) / (c + f + e + b)
Partial effects of \(x_1\) and \(x_2\)
\[y_i = b_0 + b_1x_{1i} + b_2x_{2i} + e_i\]
If we regress \(y\) on both \(x_1\) and \(x_2\), we get their partial (i.e., unique) effects
\(b_1\) represents b only
\(b_2\) represents f only
\(e_i\) represents e only
\(R^2\) represents (b + c + f) / (b + c + f + e)
c is included in \(R^2\) but neither slope
Collinearity
\[y_i = b_0 + b_1x_{1i} + b_2x_{2i} + e_i\]
If \(x_1\) and \(x_2\) are highly correlated, then c will be large and b and f may be small
Small slopes may lead you to conclude that \(x_1\) and \(x_2\) are not related to \(y\)
But this isn’t quite right… it’s just that they don’t uniquely explain \(y\) (rather, it is their overlap/shared variance that does)
So check zero-order and partial effects
Plotting marginal effects
library(modelbased)plot(estimate_relation(
model = fit3,
by = "bill_len"
))
plot(estimate_relation(
model = fit3,
by = "flipper_len"
))
Plotting marginal effects
plot(estimate_relation(
model = fit3,
by = c(
"bill_len",
"flipper_len=[sd]"
)
))
plot(estimate_relation(
model = fit3,
by = c(
"flipper_len",
"bill_len=[sd]"
)
))