# Interpreting the evaluation result of multiple linear regression

I am learning the multiple linear regression model. I've built a model and using R command:

summary(model)

I got this result:

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 253.2 on 44 degrees of freedom Multiple
R-squared:  0.3336, Adjusted R-squared:  0.2579  F-statistic: 4.405 on
5 and 44 DF,  p-value: 0.002444

How can I interpret this result in order to have a decision regarding the goodness of the model? In specific, what is the 44 degrees of freedom means for this case?

Also, why do we have adjusted and multiple r squared parameters?

• you need to give input on how did you do it and what are the number of data elements to help others offer help. – Espanta Jul 13 '15 at 1:48

First, what do the 44 degrees of freedom mean?

It simply means that the model you built is constructed by using 44 independent variables. For example a model that looks like y = ax + b has 1 independent variable (i.e. a) and thus 1 degree of freedom. A model that looks like y= ax1 + b*x2 + c would have 2 independent variables (i.e. a and b) and thus 2 degrees of freedom.

Second, what is the Multiple R-squared?

Here for the purpose of interpretating it Multiple R-squared is equivalent to the (simple) R-squared you would have for a linear regression model with 1 degree of freedom. Multiple R-squared tells us the share of the observed variance that is explained by the model. For example if you have a Multiple R-squared of 0.79 it means that your model explains 79% of the observed variance in your data.

Third, what is the Adjusted R-squared and why do we need it?

There are several problems with Multiple R-squared.

Problem 1: Every time you add a predictor to a model, the R-squared increases, even if due to chance alone. It never decreases. Consequently, a model with more independent variables (more degrees of freedom) may appear to have a better fit simply because it has more independent variables.

Problem 2: If a model has too many predictors and higher order polynomials, it begins to model the random noise in the data. This condition is known as overfitting the model and misleadingly high R-squared values and a lessened ability to make predictions.

Problem 1 is caused by Problem 2. And this is where Adjusted R-squared is coming in handy. Adjusted R-squared is an attempt at fixing these problems by factoring in the number of independent variables. The adjusted R-squared tells you the percentage of variation explained by only the independent variables that actually affect the dependent variable.

where:

• n is the number of data points you have,
• and k is the number of independent variables used to explain their distribution, excluding the constant

If you add more and more useless variables to a model, adjusted r-squared will decrease. If you add more useful variables, adjusted r-squared will increase. Adjusted R-squared will always be less than or equal to R-squared . You only need R-squared when working with samples. In other words, R-squared isn’t necessary when you have data from an entire population.

Here is an interesting series of articles which will help you understand how to use R-squared to interpret the results of your model even better.

The details are just a help(summary.lm) in R. The meaning of the codes and the terms are as follows:

Considering R = Correlation Coefficient

1. The residual standard error is nothing more than the positive square root of the mean square error.
2. Regression degrees of freedom = number of independent variables (factors) in $y$ = $a_1x_1 + a_2x_2 + a_3x3 + ...$ Wikipedia link for a better in-depth explanation
3. R Square = $(Multiple R)^2$ = $R^2$ = $1 - (Residual SS / Total SS) = (Regress SS / Total SS)$
4. Adjusted R Square = $1 - (Total df / Residual df)(Residual SS / Total SS)$

A wonderful post on how to understand degrees of freedom.