Regression Tutorial
(This includes material from Microsoft Excel Help)
Getting Started
Open excel. Click on Tools. On the drop down menu click “Add-ins.” Check “Analysis
ToolPak” and click O.K. The next time you click on Tools, a “Data Analysis” option should
appear. Click it. Select the appropriate tool and click on it. A window will open asking for
information on data ranges, and other parameters.
Regression
The Regression analysis tool performs linear regression analysis by using the "least squares" method to fit a
line through a set of observations. You can analyze how a single dependent variable is affected by the
values of one or more independent variables.
For example, you can analyze how an athlete's performance in a one mile run is affected by such factors as
age, height, and weight. The dependent variable, in this case is the amount of time it takes an individual to
run a mile. The independent variables are the variables that influence an athlete’s speed in this example are
age, height and weight. You can apportion shares in the performance measure to each of these three
factors, based on a set of performance data, and then use the results to predict the performance of a new,
untested athlete.
Two types of information is available from regression analysis. One type of information is designed to
determine how strongly the independent variables are related to performance. The other type of
information involves predicting the performance of an athlete based on their age, height and weight --without actually measuring how long it takes them to run a mile.
Mechanics
Create an excel table with the dependent variable, time, T-seconds is in the left most column. In the next
several columns list the independent variables, age, height and weight.
1
2
3
4
5
6
7
8
9
10
11
A
T-seconds
115
100
95
85
80
68
75
100
110
110
B
Age
60
45
45
35
30
17
24
50
55
55
C
Height
70
68
72
71
74
76
72
68
70
59
D
Weight
200
190
180
175
175
165
175
190
195
210
After you create a table like the one above, click Tools then click Data Analysis. Scroll down to
Regression, and click OK. A box will pop up in which you must specify certain information. The Y
variable is the dependent variable. Click the Labels box, so that the output is labeled with what ever is in
the first cell in the column. Here, for Y variables we put in A1:A11. For the X range we input B1:D11.
This will allow us to test the relationship of all three independent variables: age, height and weight at the
same time. Name the output sheet and click OK. The following table will appear.
SUMMARY OUTPUT
Regression Statistics
Multiple R
1.00
R Square
0.99
Adjusted R Square
0.99
Standard Error
Observations
1.51
10.00
ANOVA
df
SS
MS
Regression
3.00
2325.98
775.33
Residual
6.00
13.62
2.27
Total
9.00
2339.60
Coefficients
Intercept
Standard Error
t Stat
F
Significance F
341.50
P-value
0.00
Lower 95%
Upper 95%
-51.14
42.20
-1.21
0.27
-154.40
52.12
Age
0.84
0.10
8.33
0.00
0.59
1.08
Height
0.46
0.26
1.73
0.13
-0.19
1.10
Weight
0.42
0.16
2.71
0.04
0.04
0.80
t-statistic
The t-statistic is a measure of how strongly a particular independent variable explains variations in the
dependent variable. The larger the t-statistic the better the independent variable’s explanatory power. Next
to each t-stat is a P-value. The P-value is used to interpret the t-stat. In short, the P-value is the probability
that the independent variable in question has nothing to do with the dependent variable. Generally, we look
for a P-value of less than .05, which means there is a 5% chance that the dependent variable is unrelated to
the dependent variable. If the P-value is higher .10, a strong argument can be made for eliminating this
particular independent variable from a model because it “isn’t statistically significant.”
In the above model we can see that Age and Weight are statistically significant, but Weight isn’t
statistically significant.
F-statistic
F is similar to the t-stat, but F looks at the quality of the entire model, meaning with all independent
variables included. The larger the F the better. By eliminating independent variables with a low t-stat, you
can improve the quality of the overall model, as seen by an increasing F.
The term “Significance F” serves roughly the same function as the P-value. If it is less than .10 then we
say the model is statistically significant.
R Square and Adjusted R Square
R Square is another measure of the explanatory power of the model. In theory, R square compares the
amount of the error explained by the model as compared to the amount of error explained by averages. The
higher the R-Square the better. An R-Square above .5 is generally considered quite good, but may not be
attainable for every model. Statisticians have noticed that as more data points are analyzed, R Square
values rise, even though no more of the model error is explained. Adjusted R Square is a modified version
of R Square, and has the same meaning, but includes computations that prevent a high volume of data
points from artificially driving up the measure of explanatory power. An Adjust R Square above .20 is
generally considered quite good. Some published research has an Adjusted R-Square as low as .07.
Using the Model to Predict Information
Assume, for the moment, that all variables in our example: Age, Height and Weight are significant. We
would be able to predict the performance of an athlete we did not test if we knew their Age, Height and
Weight. The model uses the Coefficients from the Summary Output above.
Time-Seconds
= .84 x Age + .46 x Height + .42 x Weight -51.1
So if we had an untested athlete 40 years old, 5’10 inches tall or 60 inches, with a weight of 190 pounds we
could predict their time to run the mile as follows:
= .84 x 40 + .46 x 60 + .42 x 190 -51.1
= 33.6 + 27.6 + 79.8 – 51.1
= 98.9 seconds