Dat 605 - Discussion Week 11

require(ggthemes)
library(tidyverse)
library(magrittr)
library(tidyr)
library(dplyr)
library(lubridate)
library(ggplot2)
library(fpp2)   
library(forecast)
library(ggpubr)
library(boot)

A Simple Linear Regression Model

• Here is an example of a statistical relationship. The response variable y is the mortality due to skin cancer (number of deaths per 10 million people) and the predictor variable x is the latitude (degrees North) at the center of each of 49 states in the U.S

• The data were compiled in the 1950s, so Alaska and Hawaii were not yet states, and Washington, D.C. is included in the data set even though it is not technically a state.

#Read a txt file, named "mtcars.txt"
my_data <- read.table(file.choose(), stringsAsFactor = FALSE,sep=",",na.strings = c("NA"," ",""),strip.white = T,
                       header = TRUE, dec =".")
head(my_data)

##         State  Lat Mort Ocean  Long
## 1     Alabama 33.0  219     1  87.0
## 2     Arizona 34.5  160     0 112.0
## 3    Arkansas 35.0  170     0  92.5
## 4  California 37.5  182     1 119.5
## 5    Colorado 39.0  149     0 105.5
## 6 Connecticut 41.8  159     1  72.8

Including Plots

cor(my_data$Lat, my_data$Mort) #correlation between stopping distance vs. speed

## [1] -0.8245178

# Compute the analysis of variance
res.aov <- aov(Mort~ Lat, data = my_data)
# Summary of the analysis
summary(res.aov)

##             Df Sum Sq Mean Sq F value   Pr(>F)    
## Lat          1  36464   36464    99.8 3.31e-13 ***
## Residuals   47  17173     365                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# 1. Homogeneity of variances
plot(res.aov, 1)

# 2. Normality
plot(res.aov, 2)

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Summary and Conclusion:

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- The skin cancer data is an ideal candidate for a simple linear regression model.  The predictor (latitude) and the response variables are highly correlated at -82%.  The statistical diagnostics revealed that all the conditions of linearity, Homogeneity of variances (residuals) and signifcance of the slope meets the conditions for a simple linear regression. Further analysis, also showed that the residuals are truly normal and for the purist, do not need further data transforms.  But in this case, we can say that the linear regression formula is as follows:

- Therefore the linear equation: y = -6*x + 389.2 can be use for prediciton 

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