Linear regression

A/B Testing in R

Lauryn Burleigh

Data Scientist

Least squares

Residual sum of squares
- Each point residual - squared, then summed
Line of best fit - smallest sum of squares
Error - mean square error
- Sum of squares / N

Positive correlation with line of best fit through enjoyment on the x-axis and time on the y-axis and purple lines indicating residuals, with the sum of squares formula indicated at the top left.

Linear regression model

linear <- lm(Time ~ Enjoy, 
                data = Pizza) 
summary(linear)

Call:
lm(formula = Time ~ Enjoy, data = pizza)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.89270 -0.59857  0.04758  0.67764  2.12600 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  5.31964    0.19886  26.750  < 2e-16 ***
Enjoy        0.07707    0.01672   4.608 7.26e-06 ***

Residual standard error: 0.8947 on 198 degrees of freedom
Multiple R-squared:  0.09687,    
Adjusted R-squared:  0.09231 
F-statistic: 21.24 on 1 and 198 DF,  p-value: 7.262e-06

Assessing assumptions

Homoscedasticity

Consistent variance

plot(fitted(linear), resid(linear));
abline(0,0)

A scatter plot of the fitted model data on the x-axis and residuals of the model on the y-axis with a horizontal line at 0 on the y-axis.

Normality

qqnorm(resid(linear));
qqline(resid(linear), col = "red")

A scatter plot with the residual theoretical values on the x-axis and sample values on the y-axis with a 45-degree line in red.

Making predictions

Enjoy  <- 12
topredict <-  data.frame(Enjoy) 
predict(linear, newdata = topredict)

       1 
6.244452

Enjoy  <- c(12, 14)
topredict <-  data.frame(Enjoy) 
predict(linear, newdata = topredict)

       1        2 
6.244452 6.398587

Including groups

grplinear <- lm(Time ~ Enjoy + Topping,
                data = Pizza) 
summary(grplinear)

Enjoy  <- c(12, 14)
Topping <- "Cheese"
topredict <-  data.frame(Enjoy, Topping) 
predict(grplinear, newdata = topredict)

       1        2 
6.136022 6.269139

Call:
lm(formula = Time ~ Enjoy + Topping, 
data = pizza)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.87771 -0.51529  0.03993  0.68685  2.19460 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)       5.53891    0.24823  22.314  < 2e-16 ***
Enjoy             0.06656    0.01815   3.668 0.000315 ***
ToppingCheese -0.20159    0.13729  -1.468 0.143606

Residual standard error: 0.8921 on 197 degrees of freedom
Multiple R-squared:  0.1066,    
Adjusted R-squared:  0.09758 
F-statistic: 11.76 on 2 and 197 DF,  p-value: 1.499e-05

Let's practice!

A/B Testing in R