Plot of height by diameter



The SAS System

The REG Procedure
Model: MODEL1
Dependent Variable: height

Number of Observations Read 36
Number of Observations Used 36

Analysis of Variance
Source DF Sum of
Squares
Mean
Square
F Value Pr > F
Model 1 183.24469 183.24469 65.10 <.0001
Error 34 95.70281 2.81479    
Corrected Total 35 278.94750      

Root MSE 1.67773 R-Square 0.6569
Dependent Mean 17.90833 Adj R-Sq 0.6468
Coeff Var 9.36845    

Parameter Estimates
Variable DF Parameter
Estimate
Standard
Error
t Value Pr > |t|
Intercept 1 9.14684 1.12131 8.16 <.0001
diameter 1 0.48147 0.05967 8.07 <.0001



The SAS System

The REG Procedure
Model: MODEL1
Dependent Variable: height

Durbin-Watson D 1.709
Number of Observations 36
1st Order Autocorrelation 0.128



The SAS System

The REG Procedure
Model: MODEL1
Dependent Variable: height

Output Statistics
Obs Dependent
Variable
Predicted
Value
Std Error
Mean Predict
95% CL Mean 95% CL Predict Residual
1 20.0000 18.2467 0.2827 17.6721 18.8213 14.7891 21.7043 1.7533
2 16.8000 16.6097 0.3226 15.9540 17.2654 13.1377 20.0817 0.1903
3 20.2000 18.4874 0.2887 17.9008 19.0741 15.0278 21.9471 1.7126
4 20.0000 18.7763 0.2996 18.1675 19.3852 15.3128 22.2398 1.2237
5 20.2000 23.4948 0.7467 21.9773 25.0123 19.7628 27.2268 -3.2948
6 18.0000 18.6800 0.2955 18.0794 19.2806 15.2180 22.1421 -0.6800
7 17.8000 18.9208 0.3065 18.2979 19.5436 15.4548 22.3868 -1.1208
8 19.2000 18.7763 0.2996 18.1675 19.3852 15.3128 22.2398 0.4237
9 22.3000 19.7393 0.3601 19.0074 20.4711 16.2521 23.2265 2.5607
10 18.9000 20.5096 0.4268 19.6423 21.3769 16.9915 24.0278 -1.6096
11 13.3000 16.2727 0.3454 15.5708 16.9745 12.7916 19.7537 -2.9727
12 20.6000 20.0763 0.3878 19.2882 20.8644 16.5768 23.5758 0.5237
13 19.0000 18.0541 0.2802 17.4847 18.6236 14.5973 21.5109 0.9459
14 19.2000 19.4985 0.3421 18.8033 20.1938 16.0188 22.9783 -0.2985
15 16.1000 16.2727 0.3454 15.5708 16.9745 12.7916 19.7537 -0.1727
16 19.9000 17.6689 0.2812 17.0975 18.2404 14.2118 21.1261 2.2311
17 20.4000 19.2578 0.3258 18.5956 19.9200 15.7845 22.7311 1.1422
18 17.6000 16.8023 0.3114 16.1694 17.4352 13.3345 20.2701 0.7977
19 21.4000 22.3874 0.6216 21.1242 23.6506 18.7513 26.0234 -0.9874
20 19.2000 18.9208 0.3065 18.2979 19.5436 15.4548 22.3868 0.2792
21 19.8000 20.1726 0.3962 19.3675 20.9777 16.6693 23.6759 -0.3726
22 18.5000 15.9356 0.3714 15.1808 16.6905 12.4435 19.4278 2.5644
23 12.1000 14.0097 0.5583 12.8752 15.1443 10.4164 17.6031 -1.9097
24 8.0000 11.9394 0.7909 10.3322 13.5466 8.1700 15.7088 -3.9394
25 17.4000 19.1134 0.3170 18.4691 19.7576 15.6435 22.5833 -1.7134
26 18.4000 17.7171 0.2806 17.1468 18.2874 14.2602 21.1740 0.6829
27 17.3000 14.6356 0.4927 13.6344 15.6368 11.0821 18.1892 2.6644
28 16.6000 16.0801 0.3599 15.3486 16.8115 12.5929 19.5672 0.5199
29 12.9000 15.5986 0.4002 14.7853 16.4118 12.0934 19.1038 -2.6986
30 17.5000 17.7171 0.2806 17.1468 18.2874 14.2602 21.1740 -0.2171
31 19.4000 19.1134 0.3170 18.4691 19.7576 15.6435 22.5833 0.2866
32 15.5000 15.5504 0.4045 14.7285 16.3724 12.0432 19.0577 -0.0504
33 19.2000 20.1726 0.3962 19.3675 20.9777 16.6693 23.6759 -0.9726
34 18.8000 17.1393 0.2954 16.5389 17.7397 13.6773 20.6013 1.6607
35 16.9000 16.6097 0.3226 15.9540 17.2654 13.1377 20.0817 0.2903
36 16.3000 15.7430 0.3876 14.9554 16.5307 12.2437 19.2424 0.5570

Sum of Residuals 0
Sum of Squared Residuals 95.70281
Predicted Residual SS (PRESS) 119.76223

Panel of fit diagnostics for height.

Scatter plot of residuals by diameter for height.

Scatterplot of height by diameter overlaid with the fit line, a 95% confidence band and lower and upper 95% prediction limits.



The SAS System

The UNIVARIATE Procedure
Variable: resid (Residual)

Moments
N 36 Sum Weights 36
Mean 0 Sum Observations 0
Std Deviation 1.65359181 Variance 2.73436588
Skewness -0.5246475 Kurtosis -0.0325076
Uncorrected SS 95.7028059 Corrected SS 95.7028059
Coeff Variation . Std Error Mean 0.27559864

Basic Statistical Measures
Location Variability
Mean 0.000000 Std Deviation 1.65359
Median 0.282938 Variance 2.73437
Mode . Range 6.60374
    Interquartile Range 2.02403

Tests for Location: Mu0=0
Test Statistic p Value
Student's t t 0 Pr > |t| 1.0000
Sign M 2 Pr >= |M| 0.6177
Signed Rank S 23 Pr >= |S| 0.7233

Tests for Normality
Test Statistic p Value
Shapiro-Wilk W 0.964272 Pr < W 0.2901
Kolmogorov-Smirnov D 0.105307 Pr > D >0.1500
Cramer-von Mises W-Sq 0.063514 Pr > W-Sq >0.2500
Anderson-Darling A-Sq 0.3939 Pr > A-Sq >0.2500

Quantiles (Definition 5)
Quantile Estimate
100% Max 2.664354
99% 2.664354
95% 2.564374
90% 2.231066
75% Q3 1.044044
50% Median 0.282938
25% Q1 -0.979991
10% -2.698594
5% -3.294772
1% -3.939390
0% Min -3.939390

Extreme Observations
Lowest Highest
Value Obs Value Obs
-3.93939 24 1.75330 1
-3.29477 5 2.23107 16
-2.97266 11 2.56073 9
-2.69859 29 2.56437 22
-1.90973 23 2.66435 27

Plots for resid



The SAS System

The CAPABILITY Procedure
Variable: resid (Residual)

Moments
N 36 Sum Weights 36
Mean 0 Sum Observations 0
Std Deviation 1.65359181 Variance 2.73436588
Skewness -0.5246475 Kurtosis -0.0325076
Uncorrected SS 95.7028059 Corrected SS 95.7028059
Coeff Variation . Std Error Mean 0.27559864

Basic Statistical Measures
Location Variability
Mean 0.000000 Std Deviation 1.65359
Median 0.282938 Variance 2.73437
Mode . Range 6.60374
    Interquartile Range 2.02403

Tests for Location: Mu0=0
Test Statistic p Value
Student's t t 0 Pr > |t| 1.0000
Sign M 2 Pr >= |M| 0.6177
Signed Rank S 23 Pr >= |S| 0.7233

Tests for Normality
Test Statistic p Value
Shapiro-Wilk W 0.964272 Pr < W 0.2901
Kolmogorov-Smirnov D 0.105307 Pr > D >0.1500
Cramer-von Mises W-Sq 0.063514 Pr > W-Sq >0.2500
Anderson-Darling A-Sq 0.393900 Pr > A-Sq >0.2500

Quantiles (Definition 5)
Quantile Estimate
100% Max 2.664354282
99% 2.664354282
95% 2.564373750
90% 2.231066374
75% Q3 1.044044118
50% Median 0.282938415
25% Q1 -0.979990660
10% -2.698594261
5% -3.294772307
1% -3.939389800
0% Min -3.939389800

Extreme Observations
Lowest Highest
Value Obs Value Obs
-3.93938980 24 1.75329725 1
-3.29477231 5 2.23106637 16
-2.97265824 11 2.56072701 9
-2.69859426 29 2.56437375 22
-1.90972917 23 2.66435428 27



The SAS System

Histogram for resid



The SAS System

Q-Q plot for resid



The SAS System

Cumulative Distribution Plot for resid



Plot of resid by pred



Plot of sqres by pred



Plot of resid by residr



The SAS System

The CORR Procedure

2 Variables: resid residr

Simple Statistics
Variable N Mean Std Dev Sum Minimum Maximum Label
resid 36 0 1.65359 0 -3.93939 2.66435 Residual
residr 36 0 0.97625 0 -2.11438 2.11438 Rank for Variable resid

Pearson Correlation Coefficients, N = 36
Prob > |r| under H0: Rho=0
  resid residr
resid
Residual
1.00000
 
0.98458
<.0001
residr
Rank for Variable resid
0.98458
<.0001
1.00000
 



The SAS System

The MODEL Procedure

Model Summary
Model Variables 1
Parameters 2
Equations 1
Number of Statements 1

Model Variables height
Parameters b0 b1
Equations height

The Equation to Estimate is
height = F(b0(1), b1(diameter))

NOTE: At OLS Iteration 1 CONVERGE=0.001 Criteria Met.



The SAS System

The MODEL Procedure
OLS Estimation Summary

Data Set Options
DATA= TREE

Minimization Summary
Parameters Estimated 2
Method Gauss
Iterations 1

Final Convergence Criteria
R 0
PPC 0
RPC(b0) 90561.77
Object 0.991905
Trace(S) 2.814788
Objective Value 2.658411

Observations
Processed
Read 36
Solved 36



The SAS System

The MODEL Procedure

Nonlinear OLS Summary of Residual Errors 
Equation DF Model DF Error SSE MSE Root MSE R-Square Adj R-Sq
height 2 34 95.7028 2.8148 1.6777 0.6569 0.6468

Nonlinear OLS Parameter Estimates
Parameter Estimate Approx Std Err t Value Approx
Pr > |t|
b0 9.146839 1.1213 8.16 <.0001
b1 0.481474 0.0597 8.07 <.0001

Number of Observations Statistics for System
Used 36 Objective 2.6584
Missing 0 Objective*N 95.7028

Heteroscedasticity Test
Equation Test Statistic DF Pr > ChiSq Variables
height White's Test 18.83 2 <.0001 Cross of all vars
  Breusch-Pagan 2.83 1 0.0925 1, diameter



The SAS System

The MODEL Procedure

Panel 1

Panel 2



The SAS System

The GLM Procedure

Class Level Information
Class Levels Values
grp 2 1 2

Number of Observations Read 36
Number of Observations Used 36



The SAS System

The GLM Procedure
 
Dependent Variable: resid Residual

Source DF Sum of Squares Mean Square F Value Pr > F
Model 1 1.25123772 1.25123772 0.45 0.5067
Error 34 94.45156821 2.77798730    
Corrected Total 35 95.70280593      

R-Square Coeff Var Root MSE resid Mean
0.013074 8.44457E17 1.666730 1.9737E-16

Source DF Type I SS Mean Square F Value Pr > F
grp 1 1.25123772 1.25123772 0.45 0.5067

Source DF Type III SS Mean Square F Value Pr > F
grp 1 1.25123772 1.25123772 0.45 0.5067

Fit Plot for Residual by grp



The SAS System

The GLM Procedure

Brown and Forsythe's Test for Homogeneity of resid Variance
ANOVA of Absolute Deviations from Group Medians
Source DF Sum of Squares Mean Square F Value Pr > F
grp 1 0.000098 0.000098 0.00 0.9929
Error 34 41.8332 1.2304    



The SAS System

The GLM Procedure

Distribution of resid by grp

Level of
grp
N resid
Mean Std Dev
1 18 0.18643123 1.64245053
2 18 -0.18643123 1.69065989



The SAS System

The RSREG Procedure

Coding Coefficients for the Independent
Variables
Factor Subtracted off Divided by
diameter 17.800000 12.000000

Response Surface for Variable height
Response Mean 17.908333
Root MSE 1.381773
R-Square 0.7741
Coefficient of Variation 7.7158

Regression DF Type I Sum of Squares R-Square F Value Pr > F
Linear 1 183.244694 0.6569 95.97 <.0001
Quadratic 1 32.695972 0.1172 17.12 0.0002
Crossproduct 0 0 0.0000 . .
Total Model 2 215.940666 0.7741 56.55 <.0001

Residual DF Sum of Squares Mean Square F Value Pr > F
Lack of Fit 26 55.196834 2.122955 1.90 0.1929
Pure Error 7 7.810000 1.115714    
Total Error 33 63.006834 1.909298    

Parameter DF Estimate Standard Error t Value Pr > |t| Parameter Estimate
from Coded Data
Intercept 1 0.860896 2.205022 0.39 0.6987 18.320080
diameter 1 1.469592 0.243786 6.03 <.0001 5.905366
diameter*diameter 1 -0.027457 0.006635 -4.14 0.0002 -3.953845

Factor DF Sum of Squares Mean Square F Value Pr > F
diameter 2 215.940666 107.970333 56.55 <.0001



The SAS System

The RSREG Procedure
Canonical Analysis of Response Surface Based on Coded Data

Factor Critical Value
Coded Uncoded
diameter 0.746788 26.761454
Predicted value at stationary point:
20.525108

Eigenvalues Eigenvectors
diameter
-3.953845 1.000000
Stationary point is a maximum.



The SAS System

The TRANSREG Procedure

Box Cox Plot for height


Dependent Variable BoxCox(height)

Number of Observations Read 36
Number of Observations Used 36

Model Statement Specification Details
Type DF Variable Description Value
Dep 1 BoxCox(height) Lambda Used 1
      Lambda 1.85
      Log Likelihood -10.8496
      Conv. Lambda 1
      Conv. Lambda LL -11.6412
      CI Limit -12.7704
      Alpha 0.05
      Options Convenient Lambda Used
Ind 1 Qpoint.diameter DF 1
Ind 1 Qpoint.diameter_2 DF 1


The TRANSREG Procedure Hypothesis Tests for BoxCox(height)

Univariate ANOVA Table Based on the Usual Degrees of Freedom
Source DF Sum of Squares Mean Square F Value Liberal p
Model 2 215.9407 107.9703 56.55 >= <.0001
Error 33 63.0068 1.9093    
Corrected Total 35 278.9475      
The above statistics are not adjusted for the fact that the dependent variable was transformed and so are generally liberal.

Root MSE 1.38177 R-Square 0.7741
Dependent Mean 16.90833 Adj R-Sq 0.7604
Coeff Var 8.17214 Lambda 1.0000

Univariate Regression Table Based on the Usual Degrees of Freedom
Variable DF Coefficient Type II
Sum of
Squares
Mean Square F Value Liberal p
Intercept 1 -0.1391042 0.0076 0.0076 0.00 >= 0.9501
Qpoint.diameter 1 1.4695922 69.3827 69.3827 36.34 >= <.0001
Qpoint.diameter_2 1 -0.0274573 32.6960 32.6960 17.12 >= 0.0002

The above statistics are not adjusted for the fact that the dependent variable was transformed and so are generally liberal.




The SAS System

The TRANSREG Procedure

Transformation Plot for height

Observed By Predicted for height