accuracy and interpretation of multiple linear regression

Introduction:

You have completed your linear regression fitting and prediction. But you want to know how to represent the accuracy of the linear regression. We discuss different accuracy metrics of linear regression in this post.

(1) adjusted R-square and mean-squared-error:

  In linear regression, the R-square is the measure of the accuracy of linear regression. The R-square can be from 0 to 1 i.e. it can also be interpreted as from 0 to 100%. Roughly speaking, R-square denotes the amount of variance in the data which is described by the linear regression. The more amount of variance gets described by the linear regression, the better the regression is. So, to describe how efficient a linear model-fitting has been, one can depend on how high the adjusted R-square percentage is. 

While adjusted R-square is accuracy-measure from the statistical point of view, the more application point of view is the mean-squared-error (mse). Mean-squared-error is the mean of the squares of the errors in prediction using the linear model over training data. A small mse means that when the linear regression is used, the error in prediction which happens is going to be near root of mse i.e. rmse. So, the smaller mse directly presents a smaller error in prediction.

(2) coefficients and their significance:

In linear regression, the coefficients of an independent variable denote the effect of the independent variable on the dependent variable. i.e. consider that in a regression problem, the coefficient of X is 0.2, which in simple terms denote that on 1 increment of X, Y will increase by 0.2. While this is roughly correct, assumptions of independence of the independent variables are not often strongly held in practical applications, but we will come to that later. Also, sometimes, we try to see whether coefficients are significant statistically. In that case, we run a linear regression and check the p-value of the coefficients. If p-values are higher than 0.05, then one can say that the corresponding coefficients are not significant.

So these are mainly two ways in which you can discuss the results of your linear regression. You can either talk about the overall performance of the linear regression or discuss the significance of the predictor variables one vs one. There are a few other techniques which can be a bit business-specific.

median correlation metrics:

Sometimes, you will be more concerned about maintaining the order of linear regression prediction with the original target variable. In such cases, you will have to consider the average correlation across multiple folds of data and then report that as median correlation.

bucket based rmse:

More than often in business, it is not only enough to have rmse good in overall, but you also have to check that whether your rmse holds for all the buckets of the target variable. i.e. let's say you are trying to estimate network traffic for some telecom company. Now, the value varies over time from quite low to quite high. Now, your overall estimation should have lesser rmse, as well as there should be lesser rmse for high values as well as low values. If someone doesn't properly take care of the regression model, then it may happen so that the model predicts quite inaccurate values for lower values but it is well enough for higher values. So bucket wise rmse is also important.

There can be many other use-cases for interpreting and checking the accuracy of multiple linear regression. But I have mentioned the theoretical and two of the common business-related cases I have come across. Please comment, share and subscribe to my blog to read more such content. Also to know more about linear regression, follow this linear regression blog written by me. Thanks for reading!