R Tutorial: Simple Linear Regression

Key Takeaways

now that we have inspected the correlations between the various variables we'll move on to predicting the future margin with help of the margin in year 1 we chose the margin in year 1 since the correlation between the two variables is the highest when I only use one independent variable for the prediction we call the model a simple linear regression in reality the ideal case of a perfect linear correlation that you can exactly predict Y with the given value of x is very unlikely most of the time the data points are scattered around in the form of a cloud for this we determine the direction of the relationship between x and y by fitting a straight line through the cloud this is what we use the least squares estimation procedure for this method helps us find the regression line and returned its coefficients the difference between our prediction a point on the line and the actual value a data point is called the prediction error or residual value that in a theory let's move on to some code we can specify the linear regression model using a formula object in the LM function from the stat package looking at the arguments notice that we are looking to predict future margin as a function of margin using sylvie data 1 we store the model as simple LM then we can use the summary function with simple LM as an argument to get an overview of the results take a look at the coefficient estimate for margin with the value of roughly 0.65 it is greater than zero which means that the higher the margin in year one the higher we expect the future margin to be also take a look at the multiple r-squared at the bottom of the output a value of roughly 0.32 means that about 30% of the variation and the future margin can be explained by the margin in year 1 but more on that later the ggplot function from the ggplot2 package gives us a nice visualization of the relationship here we produce a simple scatterplot of the observations using our sylvie data one data set the data is the first argument and we specify margin as the x-axis and future margin as the y-axis in the aes call this is the second argument to ggplot we also use geum smooth with method equals LM to fit a linear regression line through the data cloud before moving on to multiple linear regression let's take a look at the conditions that the data must satisfy for linear regression to be the best method the relationship between the independent variable and the dependent variable should be linear the independent variable should not contain any measurement errors the residuals should be uncorrelated one cause of correlation among the errors is violation of the linearity assumption the residuals should randomly vary around zero and their expectation should be equal to zero usually this assumption is not problematic as long as a constant is included in the model the variance of the prediction error has to be constant if not inferences made from the model can be misleading when doing statistical significance testing we also have to assume that the errors are normally distributed a well-established method to check the violation of these assumptions is a plot of the predicted values against the estimated residuals this is called a residual plot now let's try some examples