• Relationships between two variables
• Linear Relationships: The equation of a straight line
• Relationships between two variables
• Linear regression models
• Estimating the coefficients: Least Squares
• Interpreting the slope with a continuous explanatory variable
• Prediction/Supervised learning using a linear regression model
• R² - Coefficient of Determination
• Introduction to Multiple Regression

• Suppose that we are statistical consultants hired by a client to provide advice on how to improve sales of a particular product.

• The Advertising data set consists of the sales of that product in 200 different markets, along with advertising budgets for the product in each of those markets for three different media: TV, radio, and newspaper.

glimpse(Advertising)

## Observations: 200
## Variables: 4
## $ TV        <dbl> 230.1, 44.5, 17.2, 151.5, 180.8, 8.7, 57.5, 120.2, 8...
## $ radio     <dbl> 37.8, 39.3, 45.9, 41.3, 10.8, 48.9, 32.8, 19.6, 2.1,...
## $ newspaper <dbl> 69.2, 45.1, 69.3, 58.5, 58.4, 75.0, 23.5, 11.6, 1.0,...
## $ sales     <dbl> 22.1, 10.4, 9.3, 18.5, 12.9, 7.2, 11.8, 13.2, 4.8, 1...

• It is not possible for our client to directly increase sales of the product, but they can control the advertising expenditure in each of the three media.

• Therefore, if we determine that there is an association between advertising and sales, then we can instruct our client to adjust advertising budgets, thereby indirectly increasing sales.

What is the relationship between sales and TV budget?

Advertising %>% ggplot(aes(x = TV, y = sales)) + geom_point() + theme_minimal()

• In general, as the budget for TV increases sales increases.
• Although, sometimes increasing the TV budget didn't increase sales.
• The relationship between these two variables is approximately linear.

A perfect linear relationship between an independent variable \(x\) and dependent variable \(y\) has the mathematical form:

\[y = \beta_0+\beta_1x.\]

iop\(\beta_0\) is called the \(y\)-intercept and \(\beta_1\) is called the slope.

If the relationship between \(y\) and \(x\) is perfectly linear then the scatter plot could look like:

What is the equation of straight line that fits these points?

First four observations:

## # A tibble: 4 x 2
##       x        y
##   <dbl>    <dbl>
## 1     0   0.0000
## 2     2 133.3333
## 3     4 266.6667
## 4     6 400.0000

Use analytic geometry to find the equation of the straight line: pick two any points \((x^{(1)},y^{(1)})\) and \((x^{(2)},y^{(2)})\) on the line.

The slope is:

\[m = \frac{y^{(1)} - y^{(2)}}{x^{(1)} - x^{(2)}}.\]

So the equation of the line with slope \(m\) passing through \((x^{(1)},y^{(1)})\) is

\[y - y^{(1)} = m(x - x^{(1)}) \Rightarrow y =mx +b,\]

where \(b=y^{(1)}-mx^{(1)}.\)

What is the equation of the 'best' straight line that fits these points?

## # A tibble: 4 x 2
##       x     y
##   <dbl> <dbl>
## 1    -4    16
## 2    -2     4
## 3     0     0
## 4     2     4

• Sometimes the relationship between two variables in non-linear.
  
• If the realtionship is non-linear then fitting a straight line to the data is not useful in describing the relationship.

• Let \(y\) be life expectancy of a component, and \(x\) the age of the component.
• There is a relationship between \(y\) and \(x\), but it is not linear.

p <- data_frame(x = age, y = life_exp) %>% 
  ggplot(aes(x = x, y = y)) + geom_point() + theme_minimal() 
p

p + geom_smooth(method = "lm", se = F)

• Each market is an observation, but each column is the amount spent on TV, radio, newspaper advertising.

## # A tibble: 3 x 4
##      TV radio newspaper sales
##   <dbl> <dbl>     <dbl> <dbl>
## 1 230.1  37.8      69.2  22.1
## 2  44.5  39.3      45.1  10.4
## 3  17.2  45.9      69.3   9.3

• The data are not tidy since each column corresponds to the values of advertising budget for different media.

• Tidy the data by creating a column for advertising budeget and another column for type of advertising.
• We can use the gather function in the tidyr library (part of the tidyverse library) to tidy the data.

Advertising_long <- Advertising %>% 
  select(TV, radio, newspaper, sales) %>% 
  gather(key = adtype, value = amount, TV, radio, newspaper)
head(Advertising_long)

## # A tibble: 6 x 3
##   sales adtype amount
##   <dbl> <chr>   <dbl>
## 1 22.1  TV     230   
## 2 10.4  TV      44.5 
## 3  9.30 TV      17.2 
## 4 18.5  TV     152   
## 5 12.9  TV     181   
## 6  7.20 TV       8.70

Advertising_long %>% 
  ggplot(aes(amount, sales)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE) +
  facet_grid(. ~ adtype)

• The advertising budgets (newspaper, radio, TV) are the input/independent/covariates and the dependent variable is sales.

The simple linear regression model can describe the relationship between sales and amont spent on radio advertising through the model

\[y_i =\beta_0 + \beta_1 x_i + \epsilon_i,\] where \(i=1,\ldots,n\) and \(n\) is the number of observations.

Advertising_long %>% 
  filter(adtype == "radio") %>%
  ggplot(aes(amount, sales)) + 
  geom_point()

The equation:

\[y_i =\beta_0 + \beta_1 x_i + \epsilon_i\] is called a regression model and since we have only one independent variable it is called a simple regression model.

• \(y_i\) is called the dependent or target variable.
• \(\beta_0\) is the intercept parameter.
• \(x_i\) is the independent variable, covariate, feature, or input.
• \(\beta_1\) is called the slope parameter.
• \(\epsilon_i\) is called the error parameter.

In general, models of the form

\[y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \cdots + \beta_k x_{ik} + \epsilon_{i},\] where \(i=1,\ldots,n\), with \(k>1\) independent variables are called multiple regression models.

• The \(\beta_j\)'s are called parameters and the \(\epsilon_i\)'s errors.
• The values of of neither \(\beta_j\)'s nor \(\epsilon_i\)'s can ever be known, but they can be estimated.
• The "linear" in Linear Regression means that the equation is linear in the parameters \(\beta_j\).
  
• This is a linear regression model: \(y_i = \beta_0 + \beta_1 \sqrt{x_{i1}} + \beta_2 x_{i2}^2 + \epsilon_{i}\)
• This is not a linear regression model (i.e., a nonlinear regression model): \(y_i = \beta_0 + \sin(\beta_1) x_{i1} + \beta_2 x_{i2} + \epsilon_{i}\)

## # A tibble: 6 x 2
##   sales amount
##   <dbl>  <dbl>
## 1  22.1   37.8
## 2  10.4   39.3
## 3   9.3   45.9
## 4  18.5   41.3
## 5  12.9   10.8
## 6   7.2   48.9

head(Advertising_long %>% 
  filter(adtype == "radio")) %>%
  select(sales,amount)

## # A tibble: 6 x 2
##   sales amount
##   <dbl>  <dbl>
## 1  22.1   37.8
## 2  10.4   39.3
## 3   9.3   45.9
## 4  18.5   41.3
## 5  12.9   10.8
## 6   7.2   48.9

\(m = \frac{22.1-10.4}{37.8-39.8}=\) -5.85, \(b = 22.1-\frac{22.1-10.4}{37.8-39.8}\times37.8=\) 243.23. So, the equation of the straight line is:

\[y=243.23-5.85x.\]

The equation \(y=243.23-5.85x\) is shown on the scatter plot.

• For a fixed value of amount spent on radio ads the corresponding sales has variation. It's neither strictly increasing nor decreasing.

• But, the overall pattern displayed in the scatterplot shows that on average sales increase as amount spent on radio ads increases.

The Least Squares approach is to find the y-intercept \(\beta_0\) and slope \(\beta_1\) of the straight line that is closest to as many of the points as possible.

To find the values of \(\beta_0\) and slope \(\beta_1\) that fit the data best we can minimize the sum of squared errors \(\sum_{i=1}^n \epsilon_i^2\):

\[\sum_{i=1}^n \epsilon_i^2 = \sum_{i=1}^n \left(y_i -\beta_0 - \beta_1 x_i\right)^2\]

So, we want to minimize a function of \(\beta_0, \beta_1\)

\[L(\beta_0,\beta_1) = \sum_{i=1}^n \left(y_i -\beta_0 - \beta_1 x_i\right)^2,\]

where \(x_i\)'s are numbers and therfore constants.

• The derivative of \(L(\beta_0,\beta_1)\) with respect to \(\beta_0\) treats \(\beta_1\) as a constant. This is also called the partial derivative and is denoted as \(\frac{\partial L}{\partial \beta_0}.\)

• To find the values of \(\beta_0\) and \(\beta_1\) that minimize \(L(\beta_0,\beta_1)\) we set the partial derivatives to zero and solve:

\[ \begin{aligned} \frac{\partial L}{\partial \beta_0} &= -2 \sum_{i=1}^n (y_i -\beta_0 - \beta_1 x_i) = 0, \\ \frac{\partial L}{\partial \beta_0} &= -2 \sum_{i=1}^n (y_i -\beta_0 - \beta_1 x_i)x_{i} =0. \end{aligned} \]

The values of \(\beta_0\) and \(\beta_1\) that are solutions to above equations are denoted \(\hat \beta_0\) and \(\hat \beta_1\) respectively.

It can be shown that

\[ \begin{aligned} \hat{\beta_0} &= \bar{y} - \hat{\beta_1} \bar{x} \\ \hat{\beta_1} &= \frac{(\sum_{i=1}^n y_ix_i) - n \bar{x}\bar{y}}{(\sum_{i=1}^n x_i^2) - n\bar{x}^2}, \end{aligned} \]

where, \(\bar{y} = \sum_{i=1}^n{y_i}/n\), and \(\bar{x} = \sum_{i=1}^n{x_i}/n.\)

\(\hat \beta_0\) and \(\hat \beta_1\) are called the least squares estimators of \(\beta_0\) and \(\beta_1\).

The R syntax for defining relationships between inputs such as amount spent on newspaper advertising and outputs such as sales is:

sales ~ newspaper

The tilde ~ is used to define the what the output variable (or outcome, on the left-hand side) is and what the input variables (or predictors, on the right-hand side) are.

A formula that has three inputs can be written as

sales ~ newspaper + TV + radio

mod_paper <- lm(sales ~ newspaper, data = Advertising)
mod_paper_summary <- summary(mod_paper)
mod_paper_summary$coefficients

##               Estimate Std. Error   t value     Pr(>|t|)
## (Intercept) 12.3514071 0.62142019 19.876096 4.713507e-49
## newspaper    0.0546931 0.01657572  3.299591 1.148196e-03

• (Intercept) is the estimate of \(\hat \beta_0\).
• newspaper is the estimate of \(\hat \beta_1\).

The estimated linear regression of sales on newspaper is:

\[y_i = 12.35 + 0.05x_i,\] where \(y_i\) is sales in the \(i^{th}\) market and \(x_i\) is the dollar amount spent on newspaper advertising in the \(i^{th}\) market.

• The slope \(\hat \beta_1\) is the amount of change in \(y\) for a unit change in \(x\).
• Sales increase by 0.05 for each dollar spent on advertising.
• The intercept \(\hat \beta_0\) is the average of \(y\) when \(x_i=0\).
• The average sales is 12.35 when the amount spent on advertising is zero.

After a linear regression model is estimated from data it can be used to calculate predicted values using the regression equation

\[\hat y_{i} = \hat \beta_0 + \hat \beta_1 x_{i}.\]

\(\hat y_{i}\) is the predicted value of the \(i^{th}\) response \(y_i\).

The \(i^{th}\) residual is \[e_i = y_i - \hat y_i.\]

The amount spent on newspaper advertising in the first market is:

Advertising %>% filter(row_number() == 1)

## # A tibble: 1 x 4
##      TV radio newspaper sales
##   <dbl> <dbl>     <dbl> <dbl>
## 1 230.1  37.8      69.2  22.1

• The predicted sales using the regression model is: \(12.35 + 0.05\times 69.2 =\) 16.14.
• The observed sales for region is 22.1.
• The error or residual is \(y_1-\hat {y_1}=\) 5.96.

The predicted and residual values from a regression model can be obtained using the predict() and residual() functions.

mod_paper <- lm(sales ~ newspaper, data = Advertising)
sales_pred <- predict(mod_paper)
head(sales_pred)

##        1        2        3        4        5        6 
## 16.13617 14.81807 16.14164 15.55095 15.54548 16.45339

sales_resid <- residuals(mod_paper)
head(sales_resid)

##         1         2         3         4         5         6 
##  5.963831 -4.418066 -6.841639  2.949047 -2.645484 -9.253389

• The regression model is a good fit when the residuals are small.
• Thus, we can measure the quality of fit by the sum of squares of the residuals \(\sum_{i=1}^n (y_{i}- \hat y_{i})^2\).
  
• This quantity depends on the units in which \(y_i\)'s are measured. A measure of fit that does not depend on the units is:

\[R^2 = 1- \frac{\sum_{i=1}^n e_i^2}{\sum_{i=1}^n (y_i-\bar y)^2}.\]

• \(R^2\) is often called the coeffcient of determination.
• \(0 \le R^2 \le 1,\) where 1 indicates a perfect match between the observed and predicted values and 0 indicates an poor match.

The summary() method calculates \(R^2\)

mod_paper <- lm(sales ~ newspaper, data = Advertising)
mod_paper_summ <- summary(mod_paper)
mod_paper_summ$r.squared

## [1] 0.05212045

• \(R^2=\) 0.0521204. This indicates a poor fit.

The diamonds data set contains the prices and other attributes of almost 54,000 diamonds. The variables are as follows:

## Observations: 53,940
## Variables: 10
## $ carat   <dbl> 0.23, 0.21, 0.23, 0.29, 0.31, 0.24, 0.24, 0.26, 0.22, ...
## $ cut     <ord> Ideal, Premium, Good, Premium, Good, Very Good, Very G...
## $ color   <ord> E, E, E, I, J, J, I, H, E, H, J, J, F, J, E, E, I, J, ...
## $ clarity <ord> SI2, SI1, VS1, VS2, SI2, VVS2, VVS1, SI1, VS2, VS1, SI...
## $ depth   <dbl> 61.5, 59.8, 56.9, 62.4, 63.3, 62.8, 62.3, 61.9, 65.1, ...
## $ table   <dbl> 55, 61, 65, 58, 58, 57, 57, 55, 61, 61, 55, 56, 61, 54...
## $ price   <int> 326, 326, 327, 334, 335, 336, 336, 337, 337, 338, 339,...
## $ x       <dbl> 3.95, 3.89, 4.05, 4.20, 4.34, 3.94, 3.95, 4.07, 3.87, ...
## $ y       <dbl> 3.98, 3.84, 4.07, 4.23, 4.35, 3.96, 3.98, 4.11, 3.78, ...
## $ z       <dbl> 2.43, 2.31, 2.31, 2.63, 2.75, 2.48, 2.47, 2.53, 2.49, ...

Question: Predict the price of diamonds based on carot size.

Let's select training and test sets.

set.seed(2)
diamonds_train <- diamonds %>% 
  mutate(id = row_number()) %>% 
  sample_frac(size = 0.8)

diamonds_test <- diamonds %>% 
  mutate(id = row_number()) %>% 
  # return all rows from diamonds where there are not 
  # matching values in diamonds_train, keeping just 
  # columns from diamonds.
  anti_join(diamonds_train, by = 'id') 

• Now fit a regression model on diamonds_train.

mod_train <- lm(price ~ carat, data = diamonds_train)
mod_train_summ <- summary(mod_train)
mod_train_summ$r.squared

## [1] 0.848017

• Evaluate the prediction error using root mean square error using the training model on diamonds_test.

\[\text {RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n (y_i - \hat y_i)^2}\]

• RMSE can be used to compare different sizes of data sets on an eual footing and the square root ensures that RMSE is on the same scale as \(y\).

• Calculate RMSE using test and training data.

y_test <- diamonds_test$price
yhat_test <- predict(mod_train, newdata = diamonds_test)
n_test <- length(diamonds_test$price)

# test RMSE
rmse <- sqrt(sum((y_test - yhat_test)^2) / n_test)
rmse

## [1] 1553.208

y_train <- diamonds_train$price
yhat_train <- predict(mod_train, newdata = diamonds_train)
n_train <- length(diamonds_train$price)

# train RMSE
sqrt(sum((y_train - yhat_train)^2) / n_train)

## [1] 1547.402

We will add other variables to the regression model to investigate if we can decrease the prediction error.

mrmod_train <- lm(price ~ carat + cut + color + clarity, data = diamonds_train)
mrmod_train_summ <- summary(mrmod_train)
mrmod_train_summ$r.squared

## [1] 0.9152898

y_test <- diamonds_test$price
yhat_test <- predict(mrmod_train, newdata = diamonds_test)
n_test <- length(diamonds_test$price)
mr_rmse <- sqrt(sum((y_test - yhat_test)^2) / n_test)
mr_rmse

## [1] 1149.881

• The simple linear regression model had \(R^2=\) 0.848017 and RMSE = 1553.2083635.