Simple Linear Regression Example in R

A Simple Linear Regression Example can help reinforce the intuition of simple linear regression models. Consider an example of salary vs. years of experience. This is a good example to start with because the results intuitively make sense.

  • The null hypothesis will be that years of experience has no impact on salary.
  • The significance level is set at 5%. This level is arbitrary, but is most often used.
  • If the p-value for the years of experience variable is less then the significance level then the null hypothesis is rejected.

The first two steps in R for this simple linear regression example are to import the dataset, and split the dataset into a training and test set. The syntax for this is as follows:

dataset = read.csv(‘Salary_Data.csv’)install.packages(‘caTools’)
library(caTools)
set.seed(123)
split = sample.split(dataset$Salary, SplitRatio = 2/3)
training_set = subset(dataset, split == TRUE)
test_set = subset(dataset, split == FALSE)

It is assumed that the csv data file is in the working directory. It contains thirty rows of observations. The caTools package is used to split the dataset, wherein Salary is the dependent variable. The split ratio was set at two-thirds. So, the training set will have twenty observations and the test set will have ten.

Once the previous code is executed, the next step is to fit the simple linear regression to the training set. In essence, this creates the line of best fit.

regressor = lm(formula = Salary ~ YearsExperience,
data = training_set)

In the above syntax, the lm function was used to build the regression model. The two essential parameters were passed which are the formula and the data.

Remember, this model was built with the twenty observations in the training set. This model has no idea of what the observations are in the test set. So the test set can be used to make predictions based on the model. And then, predictions from the test set can be compared with data from the test set. The predict function is used to make predictions.

y_pred = predict(regressor, newdata = test_set)

It may help to write the predictions to a new csv file:

write.csv(y_pred, file = “salary_predictions.csv)

With a bit of finesse, the ggplot2 library can be used to plot the test set data against the regression model that was built with the training set.

Simple Linear Regression Example
The red dots represent actual data from the test set.

It is easy to see that the regression model does a great job of predicting results from the test set. In some cases, the red data points sit very near the line itself, so some predictions are very accurate.

Using the summary(regressor) function outputs the data we need to definitively if the null hypothesis should be rejected.

In this simple linear regression example the summary tells us the p-value for the years of experience variable is 1.52e-14. This is much less than 1%, so it easily falls below the significance level of 5%. Thus, the null hypothesis is rejected. In other words, there is a correlation between years of experience vs. salary.

Furthermore, the summary indicates an R squared value of 0.9649. This is close to 1. It is safe to say the correlation is very strong.

Confusion Matrix in Data Mining Explained

The Confusion Matrix in data mining is used to explain Type I and a Type II errors from your results. These results are also referred to as false positives and false negatives.

confusion matrix in data mining

A false positive is when something is predicted to occur but does not occur. A false negative is when something is predicted to not occur, but it does occur.

The common notation is:

  • y for the actual values
  • y^ for predicted values

A confusion matrix in data mining can give a quick overview of how the prediction model has performed. It is used to see accuracy in Logistic Regression and K-Nearest Neighbor classification models, for example.

Conusion Matrix Accuracy
A confusion matrix makes it easy to calculate the accuracy and error rates .

accuracy rateIn the example above, the prediction model accurately predicted 35 events  that did not occur. And it accurately predicted 50 events that did not occur. The test set in this example has 100 events. From this, finding the accuracy or error rate is quite simple.

So, don’t let the name confuse you!

KNN Classifier Algorithm

The KNN Classifier Algorithm (K-Nearest Neighbor) is straight forward and not difficult to understand. Assume you have a dataset and have already identified two categories of data from the set. For the sake of simplicity, let’s say the dataset is represented by two columns of data, X1 and X2. The question is, how do you determine which category a new data point would belong to? Does it best fit the first category, or the second?

knn classifier algorithm

This is where the K-Nearest Neighbor algorithm comes in. It will help us classify the new data point.

 

 

 

KNN Classifier Algorithm Steps

knn steps

Typically, the number of neighbors chosen is 5. And the euclidean distance formula is mostly used. Other numbers of neighbors can be used, and a different distance formula can be used. It’s up the person to decide on how they want the model built.

euclidean distance formula
The Euclidan distance formula is most commonly used in the KNN Classifier Algorithm

As you can see in our example; the new data point is closer with two points in the green category, and with three points in the red category. We have exhausted our number of neighbors of five that we set for the algorithm, so we classify the new data point in the red category.

KNN Model
In this example, the K-Nearest Neighbor process dictates the new data point to belong in the red category

While the K-Nearest Neighbor Algorithm is based on a simple concept, it can can model some surprising accurate predictions.

Logistic Regression Model Intuition

A Logistic Regression Model is made from statistical analysis in which there are one or more independent variables that determine a binary outcome.

For example, a company sends out mailers to buy a product. The company has data that shows the age of the customer and if they bought it or not.

Logistic Regression Model
This Logistic Regression Model represents the age of the consumer and if they bought the product or not.

You can see the data implies that older people are more likely to buy the product.

Can this be modeled? A simple linear regression model will not work well. Moreover, a Linear Regression extends beyond the 1 value. It would be silly to say there is more than a 100% chance of anything to happen.

Linear Regression Model
This is a Linear Regression Model, but it does not fit the data well.

The key to remember for this example is you want to predict probability, and probability ranges from 0 to 1.

Logistic Regression Model Formula

Logistic Regression FormulaTo get the formula for a Logistic Regression Model, you apply the Sigmoid Function to a the Simple Linear Regression equation. Solve for y inside the Sigmoid Function, and substitute the value of y in theLinear Equation.

 

Use of the Logistic Regression formula transforms the look of a Linear Regression Model. With this formula you can predict probability.

Take four random variables for the independent variable x. Project the values on the curve. These projections are the fitted values.

predict probability

This information allows to give probability. It works slightly different if you want to make a binary prediction. In this case, you make a prediction if the customer will buy the product.

To make a prediction you choose an arbitrary, horizontal line. The 50% line is a fair line to choose. And then, any projected values on the Logistic Regression Model that shows below this line you would make a no precition. Any value above the line you would predict a yes value.

After predictions are made, a confusion matrix is used to give the accuracy of the predictions.

confusion matrix

The second column of the top row gives he number or false positives. This is an outcome predicted to happen but in reality did not happen. The second row of the first column show the number of false negatives.

R Squared Value Explained – For Regression Models

The R Squared value is a useful parameter for interpreting statistical results. However, it is often used without a clear understanding of its underlying principles.

The ordinary least squares method is used for finding the best fit line for a simple linear regression model. With this method, you find the sum of all the squared differences between the actual values and the predicted values on the regression line. This sum is found for all regression lines. The line with the least sum becomes the regression model, because it’s the best fitting line. The sum itself can be referred to as the sum of squares of residuals (SSres).

sum of squares of residuals
The sum of squares of residuals is sum of all the squared differences between the actual values and the predicted values on the regression line.

Now, consider the average line. For example, in the salary vs. experience example, the average line represents the average salary. If you take the squared some of differences between the actual observation points, and the corresponding points on the average line, then you have what is called the total sum of squares (SStot). Once you have this, you can find the R Squared value.

R Squared Value
Understanding how the R Squared value is calculated will give insight to the meaning of its value.

The R Squared Value Close to 1 is Good

Since the ordinary least squares method finds the minimum SSres value, then the smaller SSres value you have will result in R2 being closer to 1. The closer your R2 value is to 1 indicates a better regression line. And it could indicate that your regression model will make better predictions for test data.

To say the R Squared value in words; it is one minus the sum of squares of residuals, divided by the total sum of squares.