Intro to Data Science HW 7

Attribution statement: (choose only one and delete the rest)

# 1. I did this homework by myself, with help from the book and the professor.
library(tidyverse)
## -- Attaching packages --------------------------------------- tidyverse 1.3.1 --
## v ggplot2 3.3.5     v purrr   0.3.4
## v tibble  3.1.5     v dplyr   1.0.7
## v tidyr   1.1.4     v stringr 1.4.0
## v readr   2.0.2     v forcats 0.5.1
## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()
library(dplyr)
library(ggplot2)
library(imputeTS)
## Warning: package 'imputeTS' was built under R version 4.1.2
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo

The chapter on linear models (“Lining Up Our Models”) introduces linear predictive modeling using the tool known as multiple regression. The term “multiple regression” has an odd history, dating back to an early scientific observation of a phenomenon called “regression to the mean.” These days, multiple regression is just an interesting name for using linear modeling to assess the connection between one or more predictor variables and an outcome variable.


In this exercise, you will predict Ozone air levels from three predictors.

  1. We will be using the airquality data set available in R. Copy it into a dataframe called air and use the appropriate functions to summarize the data.
air <- airquality
glimpse(air)
## Rows: 153
## Columns: 6
## $ Ozone   <int> 41, 36, 12, 18, NA, 28, 23, 19, 8, NA, 7, 16, 11, 14, 18, 14, ~
## $ Solar.R <int> 190, 118, 149, 313, NA, NA, 299, 99, 19, 194, NA, 256, 290, 27~
## $ Wind    <dbl> 7.4, 8.0, 12.6, 11.5, 14.3, 14.9, 8.6, 13.8, 20.1, 8.6, 6.9, 9~
## $ Temp    <int> 67, 72, 74, 62, 56, 66, 65, 59, 61, 69, 74, 69, 66, 68, 58, 64~
## $ Month   <int> 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,~
## $ Day     <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,~
  1. In the analysis that follows, Ozone will be considered as the outcome variable, and Solar.R, Wind, and Temp as the predictors. Add a comment to briefly explain the outcome and predictor variables in the dataframe using ?airquality.
?airquality
## starting httpd help server ... done
# Ozone: Mean ozone in parts per billion from 1300 to 1500 hours at Roosevelt Island
# Solar.R: Solar radiation in Langleys in the frequency band 4000–7700 Angstroms from 0800 to 1200 hours at Central Park
# Wind: Average wind speed in miles per hour at 0700 and 1000 hours at LaGuardia Airport
# Temp: Maximum daily temperature in degrees Fahrenheit at La Guardia Airport.
  1. Inspect the outcome and predictor variables – are there any missing values? Show the code you used to check for that.
air %>% 
  summarise(ozone_na=sum(is.na(Ozone)), solar_na = sum(is.na(Solar.R)), wind_na = sum(is.na(Wind)), temp_na = sum(is.na(Temp)))
##   ozone_na solar_na wind_na temp_na
## 1       37        7       0       0
# 37 missing values in the Ozone field, 7 missing vlues in the Solar.R field, Wind and Temp both have zero missing values
  1. Use the na_interpolation() function from the imputeTS package (remember this was used in a previous HW) to fill in the missing values in each of the 4 columns. Make sure there are no more missing values using the commands from Step C.
air$Ozone <- na_interpolation(air$Ozone)
air$Solar.R <- na_interpolation(air$Solar.R) 
air$Wind <- na_interpolation(air$Wind) 
air$Temp <- na_interpolation(air$Temp)

air %>% 
  summarise(ozone_na = sum(is.na(Ozone)),
            solar_na = sum(is.na(Solar.R)),
            wind_na = sum(is.na(Wind)),
            temp_na = sum(is.na(Temp)))
##   ozone_na solar_na wind_na temp_na
## 1        0        0       0       0
  1. Create 3 bivariate scatterplots (X-Y) plots (using ggplot), for each of the predictors with the outcome. Hint: In each case, put Ozone on the Y-axis, and a predictor on the X-axis. Add a comment to each, describing the plot and explaining whether there appears to be a linear relationship between the outcome variable and the respective predictor.
ggplot(air) +
  geom_point(aes(x=Solar.R, y=Ozone)) # appears to be a slightly positive, but closer to a zero, correlation relationship between Solar.R and Ozone

ggplot(air) +
  geom_point(aes(x=Wind, y=Ozone)) # appears to be a negative correlation relationship between Wind and Ozone

ggplot(air) +
  geom_point(aes(x=Temp, y=Ozone)) # appears to be a positive correlation relationship between Temp and Ozone

  1. Next, create a simple regression model predicting Ozone based on Wind, using the lm( ) command. In a comment, report the coefficient (aka slope or beta weight) of Wind in the regression output and, if it is statistically significant, interpret it with respect to Ozone. Report the adjusted R-squared of the model and try to explain what it means.
srm <- lm(Ozone ~ Wind, data = air)
srm
## 
## Call:
## lm(formula = Ozone ~ Wind, data = air)
## 
## Coefficients:
## (Intercept)         Wind  
##      89.021       -4.592
summary(srm)
## 
## Call:
## lm(formula = Ozone ~ Wind, data = air)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -50.332 -18.332  -4.155  14.163  94.594 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  89.0205     6.6991  13.288  < 2e-16 ***
## Wind         -4.5925     0.6345  -7.238 2.15e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 27.56 on 151 degrees of freedom
## Multiple R-squared:  0.2576, Adjusted R-squared:  0.2527 
## F-statistic: 52.39 on 1 and 151 DF,  p-value: 2.148e-11
# Coefficient: -4.592
# p-value: 2.148e-11
# Adjusted R-squared: 0.2527
# For every one unit increase in the independent variable Wind, we can expect a -4.5925 unit change in Ozone.
  1. Create a multiple regression model predicting Ozone based on Solar.R, Wind, and Temp.
    Make sure to include all three predictors in one model – NOT three different models each with one predictor.
mrm <- lm(Ozone ~ Solar.R + Wind + Temp, data = air)
summary(mrm)
## 
## Call:
## lm(formula = Ozone ~ Solar.R + Wind + Temp, data = air)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -39.651 -15.622  -4.981  12.422 101.411 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -52.16596   21.90933  -2.381   0.0185 *  
## Solar.R       0.01654    0.02272   0.728   0.4678    
## Wind         -2.69669    0.63085  -4.275 3.40e-05 ***
## Temp          1.53072    0.24115   6.348 2.49e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 24.26 on 149 degrees of freedom
## Multiple R-squared:  0.4321, Adjusted R-squared:  0.4207 
## F-statistic: 37.79 on 3 and 149 DF,  p-value: < 2.2e-16
  1. Report the adjusted R-Squared in a comment – how does it compare to the adjusted R-squared from Step F? Is this better or worse? Which of the predictors are statistically significant in the model? In a comment, report the coefficient of each predictor that is statistically significant. Do not report the coefficients for predictors that are not significant.
# Adjusted R-squared:  0.4207
# The adjusted r-squared for the multiple regression model, predicting Ozone via the Solar.R, Wind, and Temp predictor variables, was 0.402 while the adjusted r-squared for the simple regression model, predicting Ozone via the Wind predictor variable, was 0.2527. Showing an improvement in the adjusted r-squared through adding the two additional preditor variables of Solar.R and Temp. 
# Of the three predictor variables included in the multiple regression model, only two showed statistical significance in predicting Ozone levels. Those two being the Wind and Temp predictors.
# The coefficient of the predictor variable Wind was -2.69669 
# The coefficient of the predictor variable Temp was 0.153072
  1. Create a one-row data frame like this:
predDF <- data.frame(Solar.R=290, Wind=13, Temp=61)

and use it with the predict( ) function to predict the expected value of Ozone:

predict(mrm, predDF)
##       1 
## 10.9464
# The predicted value of Ozone using the multiple regression model from part G is estimated at 10.9464 using the inputted dataframe, predDF, from the first portion of part I.
  1. Create an additional multiple regression model, with Temp as the outcome variable, and the other 3 variables as the predictors.

Review the quality of the model by commenting on its adjusted R-Squared.

mrm_j <- lm(Temp ~ Ozone + Solar.R + Wind, data = air)
mrm_j
## 
## Call:
## lm(formula = Temp ~ Ozone + Solar.R + Wind, data = air)
## 
## Coefficients:
## (Intercept)        Ozone      Solar.R         Wind  
##    74.69322      0.13905      0.01575     -0.58018
summary(mrm_j)
## 
## Call:
## lm(formula = Temp ~ Ozone + Solar.R + Wind, data = air)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -18.831  -4.802   1.174   4.880  18.004 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 74.693222   2.796787  26.707  < 2e-16 ***
## Ozone        0.139055   0.021907   6.348 2.49e-09 ***
## Solar.R      0.015751   0.006737   2.338  0.02072 *  
## Wind        -0.580176   0.195774  -2.963  0.00354 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.313 on 149 degrees of freedom
## Multiple R-squared:  0.4148, Adjusted R-squared:  0.403 
## F-statistic: 35.21 on 3 and 149 DF,  p-value: < 2.2e-16
# Adjusted R-squared:  0.403
# All three predictors used (Ozone, Solar.R, Wind) are statistically significant in predicting Temp
# 40.3% of the variation in Temp can be explained by the variation amongst the variables of Ozone, Solar.R, and Wind.