Excel provides both descriptive and inferential information in its output. This page focuses on the descriptive measures. A separate lesson is designed to explain the inferential measures shown on the Excel output.
Regression Statistics
|
Interpretation
|
|
| Multiple
R |
0.96 |
r= Coefficient of
Simple Correlation = the
positive square root of r-squared |
|
R Square |
0.93
= 93% |
r-square =
Coefficient of Simple Determination = percent of the variation in the y-variable that is explained by the
x-variable |
|
Adjusted R Square |
0.91 |
r-square
adjusted = version of r-square that has been adjusted for the number of predictors in the
model. r-square tends to
over estimate the strength of the association, especially when there are
more than one independent variables |
|
Standard Error |
3.78 |
standard error = square root of the sum of the square of the residuals (i.e. the actual y-values minus the predicted y-values) divided by the degrees of freedom. For simple linear regression there are n-2 degrees of freedom. |
|
Observations |
7 |
Number of paired
data items – number of observations in the sample |
*R-square reduces to
r-square for simple linear regression when there is only one independent
variable in the model.
r-square = coefficient of
simple determination
When all the
observations fall directly on the fitted response surface
If the slope = 0, then r-square = 0 because there is no linear association between the x (temperature) and y (water consumption) variables.
Some misunderstandings in the interpretation of r-sq:
A large r-square does not necessarily imply that the fitted model is a useful one. One example of this occurs if the observations are taken at a narrow interval, and the predictions are wanted outside the region of observations. If the MSE (mean square error) is too large and high precision is required, then even though r-square is large, the inferences may not be useful.
Adding more variables to a model can only increase r-square and never reduce it. This is because the SSE (sum of the squared errors) can never become larger with more independent variables, and SSTO (the total of the sum squares) is always the same for a given set of responses. Therefore we need an adjusted coefficient of multiple determination.
R-square adjusted
R- square adjusted may
actually become smaller when another independent variable is added to the
model, because the decrease in SSE may be more than offset by the loss of a degree of
freedom in the denominator, n-p.
Learn the Procedure for calculating the Regression Equation
Simple Linear Regression Menu Dictionary