One of the most popular analytical methods for statistical analysis is regression analysis. In this article we explore the basics of regression analysis and how R can be used to effectively perform it.
Getting started with regression
Regression means the unbiased prediction of the conditional expected value, using independent variables, and the dependent variable. A dependent variable is the variable that we want to predict. Examples of a dependent variable could be a number such as price, sales, or weight. An independent variable is a characteristic, or feature, that helps to determine the dependent variable. So, for example, the independent variable of weight could help to determine the dependent variable of weight. Regression analysis can be used in forecasting, time series modeling, and cause and effect relationships.
Simple linear regression
R can help us to build prediction stories with Tableau. Linear regression is a great starting place when you want to predict a number, such as profit, cost, or sales. In simple linear regression, there is only one independent variable x, which predicts a dependent value, y. Simple linear regression is usually expressed with a line that identifies the slope that helps us to make predictions. So, if sales = x and profit = y, what is the slope that allows us to make the prediction? We will do this in R to create the calculation, and then we will repeat it in R. We can also color-code it so that we can see what is above and what is below the slope.
Using lm() function
What is linear regression? Linear regression has the objective of finding a model that fits a regression line through the data well, whilst reducing the discrepancy, or error, between the data and the regression line. We are trying here to predict the line of best fit between one or many variables from a scatter plot of points of data. To find the line of best fit, we need to calculate a couple of things about the line. We can use the lm() function to obtain the line, which we can call m:
- We need to calculate the slope of the line m
- We also need to calculate the intercept with the y axis c
So we begin with the equation of the line:
y = mx + c
To get the line, we use the concept of Ordinary Least Squares (OLS). This means that we sum the square of the y-distances between the points and the line. Furthermore, we can rearrange the formula to give us beta (or m) in terms of the number of points n, x, and y. This would assume that we can minimize the mean error with the line and the points. It will be the best predictor for all of the points in the training set and future feature vectors.
Example in R
Let’s start with a simple example in R, where we predict women’s weight from their height. If we were articulating this question per Microsoft’s Team Data Science Process, we would be stating this as a business question during the business understanding phase. How can we come up with a model that helps us to predict what the women’s weight is going to be, dependent on their height? Using this business question as a basis for further investigation, how do we come up with a model from the data, which we could then use for further analysis? Simple linear regression is about two variables, an independent and a dependent variable, which is also known as the predictor variable. With only one variable, and no other information, the best prediction is the mean of the sample itself. In other words, when all we have is one variable, the mean is the best predictor of any one amount.
The first step is to collect a random sample of data. In R, we are lucky to have sample data that we can use. To explore linear regression, we will use the women dataset, which is installed by default with R. The variability of the weight amount can only be explained by the weights themselves, because that is all we have. To conduct the regression, we will use the lm function, which appears as follows:
model <- lm(y ~ x, data=mydata)
To see the women dataset, open up RStudio. When we type in the variable name, we will get the contents of the variable. In this example, the variable name women will give us the data itself. The women’s height and weight are printed out to the console, and here is an example: > women When we type in this variable name, we get the actual data itself, which we can see next:
We can visualize the data quite simply in R, using the plot(women) command. The plot command provides a quick and easy way of visualizing the data. Our objective here is simply to see the relationship of the data. The results appear as follows:
Now that we can see the relationship of the data, we can use the summary command to explore the data further:
This will give us the results, which are given here as follows:
Let’s look at the results in closer detail:
Next, we can create a model that will use the lm function to create a linear regression model of the data. We will assign the results to a model called
linearregressionmodel, as follows:
linearregressionmodel <- lm(weight ~ height, data=women)
What does the model produce? We can use the summary command again, and this will provide some descriptive statistics about the lm model that we have generated. One of the nice, understated features of R is its ability to use variables. Here we have our variable,
linearregressionmodel – note that one word is storing a whole model!
How does this appear in the R interface? Here is an example:
What do these numbers mean? Let’s take a closer look at some of the key numbers.
Residual standard error
In the output, residual standard error is cost, which is 1.525.
Comparing actual values with predicted results
Now, we will look at real values of weight of 15 women first and then will look at predicted values. Actual values of weight of 15 women are as follows, using the following command:
When we execute the
women$weight command, this is the result that we obtain:
When we look at the predicted values, these are also read out in R:
How can we put these pieces of data together?
women$pred linearregressionmodel$fitted.values. This is a very simple merge.
When we look inside the women variable again, this is the result:
If you liked this article, please be sure to check out Advanced Analytics with R and Tableau which consists of more useful analytics techniques with R and Tableau. It will enable you to make quick, cogent, and data-driven decisions for your business using advanced analytical techniques such as forecasting, predictions, association rules, clustering, classification, and other advanced Tableau/R calculated field functions.