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Example
Growth of Hudson River striped bass; years 1965–1976.
yt = incremental growth rate in year t
x1t = population density in year t
x2t = rise in temperature in year t
x3t = a measure of river flow in year t
Fit the model
where ǫt ∼ iid Normal(0, σ 2)
yt = β0 + β1x1t + β2x2t + β3x3t + ǫt
Plot y vs each x
20
20
20
growth
25
growth
25
growth
25
15
15
15
10
10
10
0
5
10
15
density
20
25
30
0.15
0.20
0.25
0.30
0.35
temp.rise
0.40
0.45
0.50
−2
0
2
flow
4
Results
Est
SE
t-value
P value
Intercept
17.00
3.22
5.3
0.001
density
–0.37
0.11
–3.2
0.015
temp.rise
25.36
9.01
2.8
0.026
0.45
0.47
1.0
0.362
flow
Diagnostics
Assumptions
Diagnostics
ǫ’s independent
Plot residuals vs time
ǫ’s normally distributed
QQ plot of residuals
ǫ’s have constant SD
Plot residuals vs fitted values
y’s linear in each of the x’s
Plot residuals vs each x
No other x’s belong in the model
Plot residuals vs other x’s
residuals vs time
4
Residual
2
0
−2
−4
−6
2
4
6
8
10
Time index
residual(t) vs residual(t+1)
4
residual(t+1)
2
0
−2
−4
corr = −0.09
−6
−6
−4
−2
0
residual(t)
2
4
Permutation test
P−value = 78%
−1.0
−0.5
0.0
0.5
1.0
correlation
QQ plot of residuals
4
Sample Quantiles
2
0
−2
−4
−6
−1.5
−1.0
−0.5
0.0
Theoretical Quantiles
0.5
1.0
1.5
Residuals vs fitted values
4
2
Residuals
0
−2
−4
−6
10
15
20
25
Fitted values
4
4
2
2
2
0
0
0
Residual
4
Residual
Residual
Residuals vs x’s
−2
−2
−2
−4
−4
−4
−6
−6
0
5
10
15
Density
20
25
30
−6
0.15
0.20
0.25
0.30
0.35
0.40
Rise in temperature
0.45
0.50
−2
0
2
River flow
4
A fake example
20
y
15
10
5
0
−5
15
20
25
30
x1
Intercept
x1
Est
-4.29
0.58
SE
4.97
0.25
t-val
-0.86
2.36
P-val
0.40
0.03
Residuals vs another x
10
Residuals
5
0
−5
−10
35
40
45
50
x2
55
Regress y on both x’s
Intercept
x1
x2
Est
18.02
1.21
-0.77
SE
8.50
0.29
0.26
t-val
2.12
4.13
-3.00
P-val
0.0491
0.0007
0.0080
One last example
Sediment ingestion by the mud snail, Hyrobia minuta.
y = Amount ingested
x = Time allowed to eat
800
y
600
400
200
10
20
30
40
50
x
60
70
80
A model
Let’s consider the model
where ǫt ∼ iid Normal(0, σ 2)
yi = β0 + β1xi + β2x2i + β3x3i + ǫi
800
y
600
400
200
10
20
30
40
50
60
70
80
x
Estimated coefficients
Intercept
time
timeˆ2
timeˆ3
Est
-339
75.7
-1.55
0.010
SE
127
15.4
0.48
0.004
t-val
-2.66
4.91
-3.22
2.52
P-val
0.019
<0.001
0.006
0.024
Diagnostic plots
QQ plot of residuals
Residuals vs. time
200
200
150
150
150
100
100
100
50
50
50
0
Residuals
200
Residuals
Residuals
Residuals vs. Fitted
0
0
−50
−50
−50
−100
−100
−100
−150
−150
−150
200
400
600
Fitted values
800
−2
−1
0
Normal quantiles
1
2
10
20
30
40
x1
50
60
70
80