Old Faithful Markdown Assignment

Step One: Load libraries and transformation functions

# ANOVA WORKFLOW - The Nuts and Bolts
library(ggplot2)
library(plyr)

# Data Transformation Section ----
Cube.Tns <- function (x) { x ^ 3 }
Square.Tns <- function (x) { x ^ 2 }
Raw.Tns <- function (x) { x }
Sqrt.Tns <- function (x) { sqrt(x) }
Log.Tns <- function (x) { log10(x + 0.00001) }
RecipRoot.Tns <- function (x) { -1 / sqrt(x) }
Recip.Tns <- function (x) { -1 / (x) }
InvSquare.Tns <- function (x) { -1 / (x ^ 2) }

Step Two: Import the data

data("faithful")  # data import
head(faithful)

##   eruptions waiting
## 1     3.600      79
## 2     1.800      54
## 3     3.333      74
## 4     2.283      62
## 5     4.533      85
## 6     2.883      55

summary(faithful)

##    eruptions        waiting    
##  Min.   :1.600   Min.   :43.0  
##  1st Qu.:2.163   1st Qu.:58.0  
##  Median :4.000   Median :76.0  
##  Mean   :3.488   Mean   :70.9  
##  3rd Qu.:4.454   3rd Qu.:82.0  
##  Max.   :5.100   Max.   :96.0

str(faithful)

## 'data.frame':    272 obs. of  2 variables:
##  $ eruptions: num  3.6 1.8 3.33 2.28 4.53 ...
##  $ waiting  : num  79 54 74 62 85 55 88 85 51 85 ...

Step Three: Inspect the data distributions

The following histograms provide insight into the normality of the data.

hist(faithful$eruptions) # Eruptions histogram

Figure: Histogram of Eruption Length and Waiting Times from faithful data set

hist(faithful$waiting) # Waiting Time histogram

Figure: Histogram of Eruption Length and Waiting Times from faithful data set

The following two QQ Plots will give a better assessment of the normality of the data.

qqnorm(faithful$eruptions)
qqline(faithful$eruptions)

Figure: QQ Plot of Eruption Length from the faithful data set

qqnorm(faithful$waiting)
qqline(faithful$waiting)

Figure: QQ Plot of Waiting Times from the faithful data set

The following two boxplots will also give an assessment of the data distribution.

boxplot(faithful$eruptions) # Eruptions boxplot

Figure: Boxplot of Eruption Length from faithful data set

boxplot(faithful$waiting) #Waiting Times boxplot

Figure: Boxplot of Waiting Times from faithful data set

Step Four: Transformation needed?

Transformations of these data do not improve normality of the data according to the QQ plots. Therefore, the following linear regression model will operate with the untransformed data.

Step Five: Run the linear regression model

LRerupt <- lm(faithful$eruptions ~ faithful$waiting) # Generates object
plot(LRerupt) # assumptions all met

Figure: Diagnostic plots for the untransformed Eruption data

Figure: Diagnostic plots for the untransformed Eruption data

Figure: Diagnostic plots for the untransformed Eruption data

Figure: Diagnostic plots for the untransformed Eruption data

summary(LRerupt) # significant p-value

## 
## Call:
## lm(formula = faithful$eruptions ~ faithful$waiting)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.29917 -0.37689  0.03508  0.34909  1.19329 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      -1.874016   0.160143  -11.70   <2e-16 ***
## faithful$waiting  0.075628   0.002219   34.09   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4965 on 270 degrees of freedom
## Multiple R-squared:  0.8115, Adjusted R-squared:  0.8108 
## F-statistic:  1162 on 1 and 270 DF,  p-value: < 2.2e-16

Step Six: Plot the Results

p <- ggplot(faithful, aes(eruptionsLOG, waitingLOG)) + geom_point()
p + geom_smooth(method = "lm", se = FALSE) + labs(x = "Eruptions") + labs(y = "Waiting Times")