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 X² =         1981.98                       Sum Y² =         13537551       
            Mean X=         6.153191489               Mean Y=         369.3404255
            Sum XY =        142968.3                     n =       47       
 
Correction factors and Corrected values (Sums of squares and cross-products)
            CF for X =       C_xx =    1779.502979               Corrected SS X =        S_xx =    202.4770213
            CF for Y =       C_yy =    6411380.447               Corrected SS Y =        S_yy =    7126170.553
            CF for XY =    C_xy =    106813.2511               Corrected SS XY =     S_xy =    36155.04894
 
Model Parameter Estimates
            Slope = b₁ =     36155.04894 / 202.4770213   =178.5637141
            Intercept = b₀ =            369.3404255 - 178.5637141 * 6.153191489 = -729.3963003
            Regression Line            Y_i = b₀ + b₁ * X_i  + e_i
                                                Y_i = -729.3963003 + 178.5637141 * X_i  + e_i 
 
ANOVA Table
            SSTotal =7126170.553
            SSRegression   36155.04894² / 202.4770213 =  6455979.821
            SSError = 7126170.553 - 6455979.821 = 670190.7322
 
            Source              df                           SS                        MS                               F
            Regression          1           6455979.821         6455979.821            433.4871821
            Error                 45          670190.7322         14893.12738
            Total                 46          7126170.553
 

Standard error of b₁ :             where t_{(0.05/2, 45 df)} = 2.014103 ;     = 8.576401034

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Testing b₁ against a specified value :  H₀: b₁ = 200 versus H₁: b₁ ¹ 200

            = (178.5637141 - 200) / 8.576401034 = -2.49945

            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) = a²s²_X  +  b²s²_Y  +  c²s²_Z  +  2(abs_XY  +  acs_XZ  +  bcs_YZ)

            where the covariances are equal to zero if the variables are independent

For the linear combination , the standard error of   is as follows. 

            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 .  For this linear combination the terms for the predicted value and residuals are assumed independent (i.e.  is independent of e_i). 
           

Standard error of b₀ is the same as the standard error of the regression line where X_i = 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(b_x=10) =14893.1274*(1 + 0.02128 + (10 - 14.79794) / 202.4770) = 127.6654

            P(1056.2408-2.0141*127.6654 <  µ_x=10 < 1056.2408+2.0141*127.6654)=0.95

            P( 799.1094964   <  µ_x=10 <   1313.372185 ) = 0.95

 

Calculate the coefficient of Determination and correlation

            R² =     0.905953594               or  90.59535936 %

            r =        0.951815945