Computing R
2
/The evils of R
2
—
Page 1
Computing R
2
/Evils of R
2
C
OMPUTING
R
2
.
Here are some of the many formulas for R
2
: Our knowledge of path analysis now
makes it possible to prove many of these formulas. See the optional appendix if you are
interested.
NOTE: I use the notation b’
k
for the standardized coefficients and b
k
for the non
standardized, aka metric coefficients. p
k
is another notation often used for the standardized (aka
path) coefficients.
R
2
= SSR/SST
Explained sum of squares over total sum of
squares, i.e. the ratio of the explained
variability to the total variability.
R
F
K
N
K
F
K
2
1
*
(
)
(
*
)
This can be useful if F, N, and K are known
R
b r
k
yk
k
K
2
1
Also,
R
bb r
i
j ij
j
K
i
K
2
1
1
These formulas uses the standardized
coefficients.and the zeroorder correlations
betwee
n y and the x’s. The
se (esoteric)
formulas can be useful when doing path
analysis.
Two IV case only:
R
b
b
bb r
2
1
2
2
2
1 2 12
2
This is a special case of the last formula.
One IV case only:
R
b
2
2
Remember that, in standardized form,
correlations and covariances are the same.
One IV case only:
)
(
)
(
)
(
)
(
)
ˆ
(
2
2
2
u
V
X
V
X
V
Y
V
Y
V
R
We’ll use this formula in the rest of this
handout. Technically I should use rho
2
(ρ
2
)
here or else put hats over the other parameters
(to distinguish between population parameters
and sample estimates) but since that seems to
confuse some people I won’t!
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Computing R
2
/The evils of R
2
—
Page 2
E
VILS OF
R
2
(In General)
While R
2
has its uses, it is far more important to have a correctly
specified model than it is to have a large R
2
. For example, consider the following model:
u
X2
X1
X4
w
X3
v
You can increase R
2
by regressing a variable on variables that are causally subsequent to it,
e.g. regress X1 on X2 and X3. This would be foolish, since X2 and X3 are consequence of
X1, not causes of it.
Remember, the data can’t tell you what the proper causal ordering is, you
have to decide that for yourself.
You should regress X3 on X1.
If you instead regress X3 on X2, you’ll get an R
2
, but it will be
meaningless (unless, perhaps, you are intentionally using X2 as a proxy for the unmeasured
X1.) Such questionable modeling occurs when, say, one attitude is regressed on another,
when in reality both attitudes are functions of something else.
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 Spring '11
 RichardWilliams
 Regression Analysis

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