Notebook Instructions

About

  • In a given year, if it rains more, we may see that there might be an increase in crop production. This is because more water may lead to more plants.

  • This is a direct relationship; the number of fruits may be able to be predicted by amount of waterfall in a certain year.

  • This example represents simple linear regression, which is an extremely useful concept that allows us to predict values of a certain variable based off another variable.

  • This lab will explore the concepts of simple linear regression, multiple linear regression, and watson analytics.

Load Packages in R/RStudio

We are going to use tidyverse a collection of R packages designed for data science.

## Loading required package: tidyverse
## ── Attaching packages ───────────────────────────────────────────────────────────────────────────────── tidyverse 1.2.1 ──
## ✔ ggplot2 2.2.1     ✔ purrr   0.2.4
## ✔ tibble  1.4.2     ✔ dplyr   0.7.4
## ✔ tidyr   0.8.0     ✔ stringr 1.2.0
## ✔ readr   1.1.1     ✔ forcats 0.2.0
## ── Conflicts ──────────────────────────────────────────────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
## Loading required package: plotly
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout

Task 1: Correlation Analysis

1A) Read the csv file into R Studio and display the dataset.

  • Name your dataset ‘mydata’ so it easy to work with.

  • Commands: read_csv() rename() head()

Extract the assigned features (columns) to perform some analytics.

mydata = read.csv(file="data/Advertising.csv")
head(mydata)
##   X    TV radio newspaper sales
## 1 1 230.1  37.8      69.2  22.1
## 2 2  44.5  39.3      45.1  10.4
## 3 3  17.2  45.9      69.3   9.3
## 4 4 151.5  41.3      58.5  18.5
## 5 5 180.8  10.8      58.4  12.9
## 6 6   8.7  48.9      75.0   7.2

1B) Create a correlation table for your to compare the correlations between all variables. Remove any variables where correlation between variables is irrelevant or inaccurate

  • Commands: cor() mydata[ -c(“COLUMN_NAME OR COLUMN_NUMBER”) ]
corr = cor(mydata [-c(1)] )
corr
##                   TV      radio  newspaper     sales
## TV        1.00000000 0.05480866 0.05664787 0.7822244
## radio     0.05480866 1.00000000 0.35410375 0.5762226
## newspaper 0.05664787 0.35410375 1.00000000 0.2282990
## sales     0.78222442 0.57622257 0.22829903 1.0000000

1C) Why is the value “1.0” down the diagonal? Which pairs seem to have the strongest correlations, list the pairs.

The value is “1.0” down the diagonal because that is the correlation between itself (i.e. TV with TV). The pairs that seem to have the strongest correlation are Sales and TV with .78, Radio and Sales with .58

1D-a) Identifying the dependent variable (y) and one independent variable (x_i) using the correlation table to identify a variable with a coefficient greater than 0.20 and lower than 0.60. Use those two variables to create a scatterplot to visualize the data. Note any patterns or relation between the two variables

  • Commands: qplot( x = VARIABLE, y = VARIABLE, data = mydata)
qplot( x = mydata$radio, y = mydata$sales, data = mydata)

##The variables have a positive correlation

1D-b) Create a 3D scatterplot between the two of the strongest correlated variables to the dependent variable. Note any patterns and the coordinates of three points with the heights values (x,y,z)

p <- plot_ly(mydata, x = ~radio, y = ~TV, z = ~sales, marker = list(size = 5)) %>%
  add_markers() 
p

##The coordinates of three points with the highest values of Y are: X= 43, Y= 287.6, Z= 26.The highest points for all three variables is towards the left, bottom corner.

Task 2: Regression Analysis

2A) Create a linear regression model by identifying the dependent variable (y) and for independent variable (x_i) use the correlation table to identify a variable with a coefficient greater than 0.20 and lower than 0.60. (same variables as 1D-a)

  • Commands: lm( y ~ x )
#Simple Linear Regression Model

reg <- lm(mydata$sales ~ mydata$radio )

2B) Use the regression model to create a report. Note the R-Squared and Adjusted R-Squared values, determine if this is a good or bad fit for your data?

  • Commands: Use the summary() function to create a report for the linear model
#Summary of Simple Linear Regression Model

summary(reg)
## 
## Call:
## lm(formula = mydata$sales ~ mydata$radio)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -15.7305  -2.1324   0.7707   2.7775   8.1810 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   9.31164    0.56290  16.542   <2e-16 ***
## mydata$radio  0.20250    0.02041   9.921   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.275 on 198 degrees of freedom
## Multiple R-squared:  0.332,  Adjusted R-squared:  0.3287 
## F-statistic: 98.42 on 1 and 198 DF,  p-value: < 2.2e-16

sales_predicted = 9.3116 + 0.2025 (X)

The R-Squared Value is 0.332 and adjusted R-Squared value is 0.3287, which is not very accurate

2C) Create a plot for the dependent (y) and independent (x) variables Note any patterns or relation between the two variables describe the trend line.

  • The trend line will show how far the predictions are from the actual value
  • The distance from the actual versus the predicted is the residual
#p <- qplot( x = INDEPENDENT_VARIABLE, y = DEPENDENT_VARIABLE, data = mydata) + geom_point()

p <- qplot( x = radio, y = sales, data = mydata) + geom_point()

#Add a trend line plot using the a linear model
p + geom_smooth(method = "lm", formula = y ~ x)

This model is not very accurate. There are some outliers that are bringing the trend line down

2D-a) Create a Multiple linear regression model and summary report using the two strongest correlated variables and the dependent variable. Note the R-Squared and Adjusted R-Squared values, determine if this is a good or bad fit for your data? Compared this model to the previous model, which model is better?

  • Sometimes, one variable is very good at predicting another variable. But most times, there are more than one factors that affect the prediction of another variable.
  • While increased rainfall is a good predictor of increased crop supply, decreased herbivores can also result in an increase of crops.

  • This idea is a loose metaphor for multiple linear regression.

  • Multiple linear regression lm(y ~ x_0 + x_1 + x_2 + … x_i )

  • Where y is the predicted/dependent variable and the x variables are the predictors/independent variable

  • commands: lm( y ~ x_1 + x_2 ) summary( reg_model )

#Multiple Linear Regression Model
mlr1 <- lm( mydata$sales ~ mydata$radio + mydata$TV )

#Summary of Multiple Linear Regression Model
summary(mlr1)
## 
## Call:
## lm(formula = mydata$sales ~ mydata$radio + mydata$TV)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.7977 -0.8752  0.2422  1.1708  2.8328 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   2.92110    0.29449   9.919   <2e-16 ***
## mydata$radio  0.18799    0.00804  23.382   <2e-16 ***
## mydata$TV     0.04575    0.00139  32.909   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.681 on 197 degrees of freedom
## Multiple R-squared:  0.8972, Adjusted R-squared:  0.8962 
## F-statistic: 859.6 on 2 and 197 DF,  p-value: < 2.2e-16

2.9211 - .18799 (radio) - .04575 (TV)

The R-Squared and Adj R-Squared values went way up from the previous model. This model is more accurate than the previous model.

2D-b) Create a Multiple Linear Regression Model using all relevant independent variables and the dependent variable. Note the R-Squared and Adjusted R-Squared values, determine if this is a good or bad fit for your data?

mlr2 <- lm( mydata$sales ~ mydata$radio + mydata$TV + mydata$newspaper)
summary (mlr2)
## 
## Call:
## lm(formula = mydata$sales ~ mydata$radio + mydata$TV + mydata$newspaper)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.8277 -0.8908  0.2418  1.1893  2.8292 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       2.938889   0.311908   9.422   <2e-16 ***
## mydata$radio      0.188530   0.008611  21.893   <2e-16 ***
## mydata$TV         0.045765   0.001395  32.809   <2e-16 ***
## mydata$newspaper -0.001037   0.005871  -0.177     0.86    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.686 on 196 degrees of freedom
## Multiple R-squared:  0.8972, Adjusted R-squared:  0.8956 
## F-statistic: 570.3 on 3 and 196 DF,  p-value: < 2.2e-16

2.938889 + ( 0.188530radio) + (0.045765 TV) +(-0.001037 * Newspaper)

Based purely on the values for R-Squared and Adjusted R-Squared, which linear regression model is best in predicting the dependent variable? Explain why

The second linear regression model is the most accurate based on the adjusted R-squared value in predicting the depedent variable, while the second and third linear regression model have an equal R-squared value. This could be because the third model adds newspaper as an independent variable, which is not as strongly correlated with sales as the other two variables.

2E) Use the three different models to predicted the dependent variable for the given values of the independent variables.

  • Variable: Radio = 69
  • Variable: TV = 255
  • Variable: newspaper = 75

MODEL 1

model_1 = 9.31164 + 0.2025*69
model_1
## [1] 23.28414

MODEL 2

model_2 = 2.9211 + (.18799*69) + (.04575*255)
model_2
## [1] 27.55866

MODEL 3

model_3 = 2.938889 + ( 0.188530*69) + (0.045765*255) +(-0.001037*75)
model_3
## [1] 27.53976

Task 3: Watson Analysis

To complete the last task, follow the directions found below. Make sure to screenshot and attach any pictures of the results obtained or any questions asked.

3A) Use the Predictive module to analyze the given data. Note any interesting patterns add an screenshot of what you found.

knitr::include_graphics('Screen Shot 2018-02-28 at 2.35.38 PM.png')

##Radio and TV spending can strongly predict sales

3B) Note the predictive power strength of reported variables. Consider the one field predictive model only, describe your findings and add and screenshot

Tv or radio are not very strong predictors of sales on their own (59% and 32%), but combined they have a strength of 94%

knitr::include_graphics('Screen Shot 2018-02-28 at 4.03.05 PM.png')

### 3C) How do Watson results reconcile with your findings based on the R regression analysis in task 2? Explain how. ##Watson results are similiar to the results from the R regression analysis. The model in task two, which included both TV and radio was more accurate than the inidivual model and the model with all 3 variables.