DATA_605_Assignment_11_DThonn

** DATA_605_Assignment_11_Thonn - Linear Regression-1 **

# install libraries if needed
#install.packages("permutations")
library(permutations)

## Warning: package 'permutations' was built under R version 3.3.3

## 
## Attaching package: 'permutations'

## The following object is masked from 'package:stats':
## 
##     cycle

#install.packages('gtools')
library(gtools)

#install.packages('gvlma')
library(gvlma)

#install.packages('lmtest')
library(lmtest)

## Warning: package 'lmtest' was built under R version 3.3.3

## Loading required package: zoo

## Warning: package 'zoo' was built under R version 3.3.3

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

** Assignment HW 11 **

Using the “cars” dataset in R, build a linear model for stopping distance as a function of speed and replicate the analysis of your textbook chapter 3 (visualization, quality evaluation of the model, and residual analysis.)

# load R dataset cars into a dataframe
cars1 <- cars
head(cars1)

##   speed dist
## 1     4    2
## 2     4   10
## 3     7    4
## 4     7   22
## 5     8   16
## 6     9   10

#print(cars1)

# examine the dataframe cars1 structure
str(cars1)

## 'data.frame':    50 obs. of  2 variables:
##  $ speed: num  4 4 7 7 8 9 10 10 10 11 ...
##  $ dist : num  2 10 4 22 16 10 18 26 34 17 ...

create a linear model and Visualize the data

# create a linear regression model for cars1
cars1.lm <- lm(cars1$dist ~ cars1$speed)

# plot the cars data
plot(cars1$speed,cars1$dist)
abline(cars1.lm)

# Note: the data looks fairly linear and the model fits reasonably well by appearance

Check a summary of the cars1.lm model

summary(cars1.lm)

## 
## Call:
## lm(formula = cars1$dist ~ cars1$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 *  
## cars1$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

# Findings:
# The pr of the model is less than .05 (.0123) so is significant
# The equation for the line is (y=stop_distance) y = -17.5791 + 3.9324 * x (x=speed)

Check the residuals

hist(cars1.lm$residuals)

# the histogram of residuals shows a partial shape of normal distribution though skewed to the right
# so not a perfect fit

plot(fitted(cars1.lm),resid(cars1.lm))

# the residuals are randomly distributedon on the residual chart which indicates good linearity

# the sample model fits the theoretical between -2 and +1.5, thought there is divergence after +1.5
# therefore, the model is a partial fit with some divergence
qqnorm(resid(cars1.lm))
qqline(resid(cars1.lm))

Check the gvlma function for overall criteria

gvlma(cars1.lm)

## 
## Call:
## lm(formula = cars1$dist ~ cars1$speed)
## 
## Coefficients:
## (Intercept)  cars1$speed  
##     -17.579        3.932  
## 
## 
## ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
## USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
## Level of Significance =  0.05 
## 
## Call:
##  gvlma(x = cars1.lm) 
## 
##                     Value  p-value                   Decision
## Global Stat        15.801 0.003298 Assumptions NOT satisfied!
## Skewness            6.528 0.010621 Assumptions NOT satisfied!
## Kurtosis            1.661 0.197449    Assumptions acceptable.
## Link Function       2.329 0.126998    Assumptions acceptable.
## Heteroscedasticity  5.283 0.021530 Assumptions NOT satisfied!

# Note: there are several issues found in the output of this test.  See below for details.

# Conclusion: 
# There are issues in the gvlma check as shown below with prob < .05 for (3) criteria.
# Heteroscedasticity, Skewness, and Global Stat fail
# Conclusion: the cars1.lm model is less than an ideal model for the cars data.

# Global Stat        15.801 0.003298 Assumptions NOT satisfied!
# Skewness            6.528 0.010621 Assumptions NOT satisfied!
# Kurtosis            1.661 0.197449    Assumptions acceptable.
# Link Function       2.329 0.126998    Assumptions acceptable.
# Heteroscedasticity  5.283 0.021530 Assumptions NOT satisfied!

** END **