library(datasets)
library(knitr)
cars.data <- cars

One factor regression is not commutative. To prove it I will be using cars dataset.

Since both columns, speed and dist are continuous variables a regression model can be built two different ways.

Linear Model for Case 1

case1.lm <- lm(cars.data$dist ~ cars.data$speed)
case1.lm
## 
## Call:
## lm(formula = cars.data$dist ~ cars.data$speed)
## 
## Coefficients:
##     (Intercept)  cars.data$speed  
##         -17.579            3.932
summary(case1.lm)
## 
## Call:
## lm(formula = cars.data$dist ~ cars.data$speed)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -29.069  -9.525  -2.272   9.215  43.201 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     -17.5791     6.7584  -2.601   0.0123 *  
## cars.data$speed   3.9324     0.4155   9.464 1.49e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.38 on 48 degrees of freedom
## Multiple R-squared:  0.6511, Adjusted R-squared:  0.6438 
## F-statistic: 89.57 on 1 and 48 DF,  p-value: 1.49e-12

Regression model for Case 1 is \(distance = -17.5791 + 3.9324 * speed\)

Linear Model for Case 2

case2.lm <- lm(cars.data$speed ~ cars.data$dist)
case2.lm
## 
## Call:
## lm(formula = cars.data$speed ~ cars.data$dist)
## 
## Coefficients:
##    (Intercept)  cars.data$dist  
##         8.2839          0.1656
summary(case2.lm)
## 
## Call:
## lm(formula = cars.data$speed ~ cars.data$dist)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.5293 -2.1550  0.3615  2.4377  6.4179 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     8.28391    0.87438   9.474 1.44e-12 ***
## cars.data$dist  0.16557    0.01749   9.464 1.49e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.156 on 48 degrees of freedom
## Multiple R-squared:  0.6511, Adjusted R-squared:  0.6438 
## F-statistic: 89.57 on 1 and 48 DF,  p-value: 1.49e-12

Regression model for Case 2 is \(speed = 8.2839 + 0.1656 * distance\)

case1.speed = seq(10, 30, by=1)
case1.distance <- -17.5791 + (3.9324 * case1.speed)
case1.df <- data.frame(case1.speed,case1.distance)
colnames(case1.df) <- c('Speed(mph)', 'Distance(feet)')
kable(case1.df, align='l', caption = "Regression model estimates - Speed As Predictor - Distance As Output")
Regression model estimates - Speed As Predictor - Distance As Output
Speed(mph) Distance(feet)
10 21.7449
11 25.6773
12 29.6097
13 33.5421
14 37.4745
15 41.4069
16 45.3393
17 49.2717
18 53.2041
19 57.1365
20 61.0689
21 65.0013
22 68.9337
23 72.8661
24 76.7985
25 80.7309
26 84.6633
27 88.5957
28 92.5281
29 96.4605
30 100.3929 
case2.distance <- case1.distance
case2.speed <- 8.2839 + (0.1656 * case2.distance)
case2.df <- data.frame(case2.distance,case2.speed)
colnames(case2.df) <- c('Distance(feet)', 'Speed(mph)')
kable(case2.df, align='l', caption = "Regression model estimates - Distance As Predictor - Speed As Output")
Regression model estimates - Distance As Predictor - Speed As Output
Distance(feet) Speed(mph)
21.7449 11.88486
25.6773 12.53606
29.6097 13.18727
33.5421 13.83847
37.4745 14.48968
41.4069 15.14088
45.3393 15.79209
49.2717 16.44329
53.2041 17.09450
57.1365 17.74570
61.0689 18.39691
65.0013 19.04812
68.9337 19.69932
72.8661 20.35053
76.7985 21.00173
80.7309 21.65294
84.6633 22.30414
88.5957 22.95535
92.5281 23.60655
96.4605 24.25776
100.3929 24.90896 
plot(case1.distance,case1.speed,type="l",col="red", xlab = 'Distance', ylab = 'Speed')
lines(case2.distance,case2.speed,col="green")

Red line plots Speed As Predictor - Distance As Output. Green line plots Distance As Predictor - Speed As Output. For the first case, I have used speeds between 10 mph to 30 mph to predict distance. In the second case, I have used the distance from the first instance as a predictor to predict speed. Output speed in latter case does not match to the first case.

Had both models followed commutative property they would have overlapped each other.

Conclusion