Page 106 - Applied Statistics with R

106 CHAPTER 7. SIMPLE LINEAR REGRESSION

SSE / (n - 2)

## [1] 236.5317

2
We can use R to verify that this matches our previous calculation of .

s2_e == SSE / (n - 2)

## [1] TRUE

These three measures also do not have an important practical interpretation in-
dividually. But together, they’re about to reveal a new statistic to help measure
the strength of a SLR model.

7.3.1 Coefficient of Determination

2
The coefficient of determination, , is defined as

SSReg ∑ ( ̂ − ̄) 2

2
= = =1

SST ∑ ( − ̄) 2

=1
SST − SSE SSE
= = 1 −
SST SST
2 2
∑ ( − ̂ ) ∑

= 1 − =1 = 1 − =1

∑ ( − ̄) 2 ∑ ( − ̄) 2
=1 =1
The coefficient of determination is interpreted as the proportion of observed
variation in that can be explained by the simple linear regression model.
R2 = SSReg / SST
R2
## [1] 0.6510794
2
For the cars example, we calculate = 0.65. We then say that 65% of the
observed variability in stopping distance is explained by the linear relationship
with speed.
The following plots visually demonstrate the three “sums of squares” for a sim-
2
ulated dataset which has = 0.92 which is a somewhat high value. Notice
in the final plot, that the orange arrows account for a larger proportion of the
total arrow.