Freund & Wilson (1997) : Prediction of weight of wood from trees (Table 8.24)
Observation |
Dbh |
Weight |
Dbh*Dbh |
Wt*Wt |
Dbh*Wt |
Predicted |
Residual |
1 |
5.7 |
174 |
32.49 |
30276 |
991.8 |
288.42 |
-114.42 |
2 |
8.1 |
745 |
65.61 |
555025 |
6034.5 |
716.97 |
28.03 |
3 |
8.3 |
814 |
68.89 |
662596 |
6756.2 |
752.68 |
61.32 |
4 |
7.0 |
408 |
49.00 |
166464 |
2856.0 |
520.55 |
-112.55 |
5 |
6.2 |
226 |
38.44 |
51076 |
1401.2 |
377.7 |
-151.7 |
6 |
11.4 |
1675 |
129.96 |
2805625 |
19095.0 |
1306.23 |
368.77 |
7 |
11.6 |
1491 |
134.56 |
2223081 |
17295.6 |
1341.94 |
149.06 |
8 |
4.5 |
121 |
20.25 |
14641 |
544.5 |
74.14 |
46.86 |
9 |
3.5 |
58 |
12.25 |
3364 |
203.0 |
-104.42 |
162.42 |
10 |
6.2 |
278 |
38.44 |
77284 |
1723.6 |
377.7 |
-99.7 |
11 |
5.7 |
220 |
32.49 |
48400 |
1254.0 |
288.42 |
-68.42 |
12 |
6.0 |
342 |
36.00 |
116964 |
2052.0 |
341.99 |
0.01 |
13 |
5.6 |
209 |
31.36 |
43681 |
1170.4 |
270.56 |
-61.56 |
14 |
4.0 |
84 |
16.00 |
7056 |
336.0 |
-15.14 |
99.14 |
15 |
6.7 |
313 |
44.89 |
97969 |
2097.1 |
466.98 |
-153.98 |
16 |
4.0 |
60 |
16.00 |
3600 |
240.0 |
-15.14 |
75.14 |
17 |
12.1 |
1692 |
146.41 |
2862864 |
20473.2 |
1431.22 |
260.78 |
18 |
4.5 |
74 |
20.25 |
5476 |
333.0 |
74.14 |
-0.14 |
19 |
8.6 |
515 |
73.96 |
265225 |
4429.0 |
806.25 |
-291.25 |
20 |
9.3 |
766 |
86.49 |
586756 |
7123.8 |
931.25 |
-165.25 |
21 |
6.5 |
345 |
42.25 |
119025 |
2242.5 |
431.27 |
-86.27 |
22 |
5.6 |
210 |
31.36 |
44100 |
1176.0 |
270.56 |
-60.56 |
23 |
4.3 |
100 |
18.49 |
10000 |
430.0 |
38.43 |
61.57 |
24 |
4.5 |
122 |
20.25 |
14884 |
549.0 |
74.14 |
47.86 |
25 |
7.7 |
539 |
59.29 |
290521 |
4150.3 |
645.54 |
-106.54 |
26 |
8.8 |
815 |
77.44 |
664225 |
7172.0 |
841.96 |
-26.96 |
27 |
5.0 |
194 |
25.00 |
37636 |
970.0 |
163.42 |
30.58 |
28 |
5.4 |
280 |
29.16 |
78400 |
1512.0 |
234.85 |
45.15 |
29 |
6.0 |
296 |
36.00 |
87616 |
1776.0 |
341.99 |
-45.99 |
30 |
7.4 |
462 |
54.76 |
213444 |
3418.8 |
591.98 |
-129.98 |
31 |
5.6 |
200 |
31.36 |
40000 |
1120.0 |
270.56 |
-70.56 |
32 |
5.5 |
229 |
30.25 |
52441 |
1259.5 |
252.7 |
-23.7 |
33 |
4.3 |
125 |
18.49 |
15625 |
537.5 |
38.43 |
86.57 |
34 |
4.2 |
84 |
17.64 |
7056 |
352.8 |
20.57 |
63.43 |
35 |
3.7 |
70 |
13.69 |
4900 |
259.0 |
-68.71 |
138.71 |
36 |
6.1 |
224 |
37.21 |
50176 |
1366.4 |
359.84 |
-135.84 |
37 |
3.9 |
99 |
15.21 |
9801 |
386.1 |
-33 |
132 |
38 |
5.2 |
200 |
27.04 |
40000 |
1040.0 |
199.14 |
0.86 |
39 |
5.6 |
214 |
31.36 |
45796 |
1198.4 |
270.56 |
-56.56 |
40 |
7.8 |
712 |
60.84 |
506944 |
5553.6 |
663.4 |
48.6 |
41 |
6.1 |
297 |
37.21 |
88209 |
1811.7 |
359.84 |
-62.84 |
42 |
6.1 |
238 |
37.21 |
56644 |
1451.8 |
359.84 |
-121.84 |
43 |
4.0 |
89 |
16.00 |
7921 |
356.0 |
-15.14 |
104.14 |
44 |
4.0 |
76 |
16.00 |
5776 |
304.0 |
-15.14 |
91.14 |
45 |
8.0 |
614 |
64.00 |
376996 |
4912.0 |
699.11 |
-85.11 |
46 |
5.2 |
194 |
27.04 |
37636 |
1008.8 |
199.14 |
-5.14 |
47 |
3.7 |
66 |
13.69 |
4356 |
244.2 |
-68.71 |
134.71 |
Sum |
289.2 |
17359 |
1981.98 |
13537551 |
142968.3 |
Sum = |
0 |
Mean |
6.15 |
369.34 |
42.17 |
288033 |
3041.9 |
SS = |
670190.7322 |
n |
47 |
47 |
47 |
47 |
47 |
|
|
Intermediate Calculations
Sum
X
=
289.2
Sum Y =
17359
Sum XY
=
142968.3
n =
47
Correction factors and Corrected values (Sums of squares and
cross-products)
CF for X
= Cxx =
1779.502979
Corrected SS X = Sxx
= 202.4770213
CF for Y
= Cyy =
6411380.447
Corrected SS Y = Syy
= 7126170.553
CF for XY
= Cxy =
106813.2511
Corrected SS XY = Sxy =
36155.04894
Model Parameter Estimates
ANOVA Table
SSRegression
36155.048942 / 202.4770213 = 6455979.821
Source
df
SS
MS
F
Regression
1
6455979.821
6455979.821
433.4871821
Error
45
670190.7322 14893.12738
Standard error of b1 :
where t(0.05/2,
45 df) = 2.014103 ;
P(178.5637 - 2.0141*8.5764 £ b1 £ 178.5637 + 2.0141*8.5764 ) = 0.95
<>P( 161.289956 £ b1 £ 195.8375 ) = 0.95
<>
Testing b1 against a specified value : H0: b1 = 200 versus H1: b1 ¹ 200
Note that t2 = F = 6.247251 ; This test would be done in SAS as an F statement
The variance of a linear combination is given by the sum of the variances plus twice the covariances.
e.g. for A = aX + bY + cZ
then Var(A) = a2s2X + b2s2Y + c2s2Z + 2(absXY + acsXZ + bcsYZ)
where the covariances are equal to zero if the variables are independent
For the linear combination
Standard error of the regression line (
The calculation above DOES NOT assume that the covariances of the regression
coefficients are independent. However, for the variance of individual
points the linear combination is
Standard error of b0 is the same as the standard error of the
regression line where Xi = 0
SQRT(14893.12738(0.021276596
+ (0 - 37.8617655) / 202.4770213 = 55.69366336
Confidence interval on b0
where b0 = -729.3963003 and t(0.05/2, 45 df)
= 2.014103
P(-729.3963
- 2.0141*55.6937 < b0 < -729.3963
+ 2.0141*55.6937)=0.95
P( -841.5690916 < b0 < -617.223509 ) = 0.95
Estimate and standard error of an individual observation (e.g. the weight
of wood for a ten-inch-diameter tree)
Y =-729.3963003 + 178.5637141* X = -729.3963003 + 178.5637141 * 10 =1056.240841
se(bx=10)
=14893.1274*(1 + 0.02128 + (10 - 14.79794) / 202.4770) = 127.6654
P(1056.2408-2.0141*127.6654
< µx=10 <
1056.2408+2.0141*127.6654)=0.95
P(
799.1094964 < µx=10 < 1313.372185 ) = 0.95
Calculate the coefficient of Determination and correlation
R2
= 0.905953594
or
90.59535936 %
r = 0.951815945