Download Chapter 14, part C

Chapter 14, part C
Goodness of Fit.
III. Coefficient of Determination
We developed an equation, but we don’t really know
how well it fits the data.
The coefficient of variation gives us a measure of the
goodness of fit for an estimated regression
equation.
How closely an estimate, ŷi comes to the actual
value, yi is called a residual.
A. Total Sum of Squares (SST)
If you had to estimate repair cost but had no
knowledge of the car’s age, what would be your
best guess? Probably the mean repair cost.
If we subtract each yi from the mean, we calculate
the error involved in using the mean to estimate
cost. I hope our regression equation does a better
job of estimating repair cost than just using the
mean!
The calculation of SST
n
SST   ( yi  yi ) 2
i 1
For the 4th observation, this difference is 300-276=24.
Do this for each observation, square it and sum them and
you calculate SST=62,870.
B. Sum of Squares due to Error
(SSE)
Every ith observation has a residual. The process of
Least Squares minimizes the sum of the squared
residuals.
n
2
SSE   ( yi  yˆ i )
i 1
Some observations will be overestimated, some
underestimated. A predicted yi that is $20 too high
is just as large of a “miss” as $20 too low. So
squaring each residual gives equal weight to
positive and negative residuals of equal
magnitude.
Take the 4th
observation. The
estimated repair cost
for a 4-year old car is
$351.50, but the actual
data for y4=$300.
Repair Cost and Age
y = 75.5x + 49.5
$500
Cost
$400
=51.50
$300
y =276
$200
$100
$0
1
2
3
Age (years)
4
5
6
So the residual is
51.50. Square this for
every observation and
sum them and you get
SSE=5867.50.
You can also see the
variation around the
mean, 276.
C. Sum of Squares due to
Regression (SSR)
So SSE measures how closely observations are
clustered around the regression line, SST measures
how closely they are clustered around the mean.
What’s left over is called SSR.
n
SST = SSE + SSR, where SSR   ( yˆ i  yi ) 2 =57,002.5
i 1
Since our regression model is designed to minimize
SSE, I would hope that SSR would make up the
bulk of the total variation in y.
D. Coefficient of Determination
(R2)
All of the variation in y is represented by SST, and
since least squares is designed to minimize SSE,
then a very good model is one that explains most
of the variation in y and would thus have a very
small SSE.
Equivalently, you could think of a good model as
having a large SSR, relative to SSE. If so,
SSR/SST is very close to being equal to 1.
This ratio, of SSR to SST is called R2, the coefficient
of determination.
SSR
0 R 
1
SST
2
A terrible model has a very large SSE, and
a very small SSR, so R2 is very close to
zero. An excellent model has a R2 very
close to 1.
Interpretation of R2
• In the repair cost example, R2=.9067. This means
that 90.67% of the total sum of squares can be
explained by using the estimated regression
equation between age and repair cost.
Excel Output
I’ve highlighted the relevant information in the table
of regression output. Can you pick out the
important information that we have been
discussing?
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.95219352
R Square
0.906672499
Adjusted R Square
0.875563332
Standard Error
44.2248045
Observations
5
ANOVA
df
Regression
Residual
Total
Intercept
Age
1
3
4
SS
MS
F
57002.5 57002.5 29.14487
5867.5 1955.833
62870
Coefficients
Standard Error
t Stat
P-value
49.5
46.38336627 1.067193 0.364141
75.5
13.98511113 5.398599 0.012457