12
Take a look at the pickup data, with
price
grouped by
make
.
> boxplot(data$price ~ data$make, ylab="price")
Dodge
Ford
GMC
5000
10000
15000
20000
price
Doesn’t look like there is much going on.
13
R can fit the ANOVA model with the
lm
function:
> model < lm(data$price ~ data$make)
> coef(model)
(Intercept) data$makeFord
data$makeGMC
6554.200
2313.717
1442.008
> mean(data$price[data$make=="Dodge"])
[1] 6554.2
> mean(data$price[data$make=="Ford"])
[1] 8867.917
> mean(data$price[data$make=="GMC"])
[1] 7996.208
14
And the
anova
function is useful to make a pretty table:
> anova(model)
Analysis of Variance Table
Response: data$price
Df
Sum Sq
Mean Sq F value Pr(>F)
data$make
2
29571553 14785776
0.4628 0.6326
Residuals 43 1373653582 31945432
> 29571553/(1373653582+29571553)
[1] 0.02107399
This last number is
SSR
/
(
SSE
+
SSR
) =
SSR
/
SST
:
I
Brand explains only 2% of our observed variability!
15
What else is going on in the ANOVA table?
I
Mean Square values are
sums of squares
(
SSR
and
SSE
)
divided by the
degrees of freedom
(
R

1 and
n

R
).
I
If
β
1
=
· · ·
=
β
R

1
= 0
, then
MSR
/
MSE
is an “
F
”
random variable, so the topright value is
P
(
F
>
MSR
/
MSE
).
I
This is the probability of observing a larger
MSR
/
MSE
,
if
the groupings do not matter.
I
We have
P
(
F
>
MSR
/
MSE
) =
.
63, which does
not
indicate strong evidence against
β
1
. . .
=
β
R

1
= 0.
Some of this should seem familiar; we’ll see more detail later.
16
Learning Check
1.
ThinkPairShare
: In your own words, what is the
difference between a conditional and a marginal
distribution? Come up with your own example.
17
Correlation and covariance
Cov
(
X
,
Y
) =
E
[
(
X

E
[
X
])
(
Y

E
[
Y
])
]
0.0
0.2
0.4
0.6
0.8
1.0
1
0
1
2
3
4
x
y
E[Y]
E[X]
X
and
Y
vary with each other around their means.
18
Correlation is the standardized covariance:
corr
(
X
,
Y
) =
cov
(
X
,
Y
)
p
var
(
X
)
var
(
Y
)
=
cov
(
X
,
Y
)
sd
(
X
)
sd
(
Y
)
The correlation is scale invariant and the units of
measurement don’t matter:
I
It is always true that

1
≤
corr
(
X
,
Y
)
≤
1.
Correlation gives
I
the direction (

or
+
)
I
and strength (0
→
1)
of the
linear
relationship between
X
and
Y
.
19
Sample correlation and std. deviation
Recall:
I
Sample Covariance is
s
xy
=
∑
n
i
=1
(
X
i

¯
X
)(
Y
i

¯
Y
)
n

1
.
(in units
X
times units
Y
)
I
Sample Standard Deviation is
s
x
=
s
∑
n
i
=1
(
X
i

¯
X
)
2
n

1
.
(in units
X
)
I
Sample Correlation is
r
xy
=
s
xy
s
x
s
y
=
1
n

1
n
X
i
=1
(
X
i

¯
X
)
s
x
(
Y
i

¯
Y
)
s
y
.
(correlation is scale free!)
20
3
2
1
0
1
2
3
3
2
1
0
1
2
3
corr = 1
3
2
1
0
1
2
3
3
2
1
0
1
2
3
corr = .5
3
2
1
0
1
2
3
3
2
1
0
1
2
3
corr = .8
3
2
1
0
1
2
3
3
2
1
0
1
2
3
corr = .8
21
Correlation only measures
linear
relationships:
I
corr(
X
,
Y
) = 0 does not mean the variables are unrelated!
3
2
1
0
1
2
8
6
4
2
0
corr = 0.01
0
5
10
15
20
0
5
10
15
20
corr = 0.72
Also be careful with influential observations.
22
Correlation and regression
“Imagine” that
Y
=
b
0
+
b
1
X
+
e
:
cov
(
X
,
Y
)
=
cov
(
X
,
b
0
+
b
1
X
+
e
)
=
cov
(
X
,
b
1
X
)
=
b
1
var
(
X
)
Thus
corr
(
X
,
Y
) =
b
1
σ
x
σ
y
⇔
b
1
=
r
xy
s
y
s
x
.
That is,
b
1
is correlation times units
Y
per units
X
.
23
We used the definition of covariance to suggest what the
slope,
b
1
should be. What about the intercept,
b
0
?
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 Spring '14