SAS Output

Chapter 6 : Hubble example

Raw data listing

Obs	VELOCITY	DISTANCE	anotherx
1	-220	0.27500	-220
2	-185	0.27500	-185
3	-130	0.21400	-130
4	-70	0.26300	-70
5	-30	0.90000	-30
6	150	0.90000	150
7	170	0.03200	170
8	200	0.45000	200
9	200	0.63000	200
10	270	0.50000	270
11	290	0.03400	290
12	290	0.50000	290
13	300	0.80000	300
14	450	1.10000	450
15	500	0.90000	500
16	500	1.10000	500
17	500	1.40000	500
18	500	2.00000	500
19	650	0.90000	650
20	800	2.00000	800
21	850	2.00000	850
22	920	1.00000	920
23	960	1.70000	960
24	1090	2.00000	1090
25	1200	.	.

Chapter 6 : Hubble example

Raw data listing

                        Plot of DISTANCE*VELOCITY.  Legend: A = 1 obs, B = 2 obs, etc.                         
                                                                                                               
    2.00 +                                                        A                 A  A             A         
         |                                                                                                     
         |                                                                                                     
         |                                                                                                     
         |                                                                                                     
    1.75 +                                                                                                     
         |                                                                                    A                
         |                                                                                                     
         |                                                                                                     
         |                                                                                                     
    1.50 +                                                                                                     
         |                                                                                                     
DISTANCE |                                                        A                                            
         |                                                                                                     
         |                                                                                                     
    1.25 +                                                                                                     
         |                                                                                                     
         |                                                                                                     
         |                                                     A  A                                            
         |                                                                                                     
    1.00 +                                                                                 A                   
         |                                                                                                     
         |                        A          A                    A        A                                   
         |                                                                                                     
         |                                            A                                                        
    0.75 +                                                                                                     
         |                                                                                                     
         |                                      A                                                              
         |                                                                                                     
         |                                                                                                     
    0.50 +                                          AA                                                         
         |                                      A                                                              
         |                                                                                                     
         |                                                                                                     
         |             A A                                                                                     
    0.25 +                      A                                                                              
         |                  A                                                                                  
         |                                                                                                     
         |                                                                                                     
         |                                    A      A                                                         
    0.00 +                                                                                                     
         ---+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+--
          -400        -200          0          200         400         600         800        1000        1200 
                                                                                                               
                                                        VELOCITY                                               
                                                                                                               
NOTE: 1 obs had missing values.

Chapter 6 : Hubble example

Regression without labels and with output statement

The REG Procedure

Model: MODEL1

Dependent Variable: DISTANCE

Number of Observations Read	25
Number of Observations Used	24
Number of Observations with Missing Values	1

Analysis of Variance
Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	5.97547	5.97547	36.44	<.0001
Error	22	3.60782	0.16399
Corrected Total	23	9.58329

Root MSE	0.40496	R-Square	0.6235
Dependent Mean	0.91137	Adj R-Sq	0.6064
Coeff Var	44.43383

Parameter Estimates
Variable	DF	Parameter Estimate	Standard Error	t Value	Pr > \|t\|	95% Confidence Limits
Intercept	1	0.39910	0.11847	3.37	0.0028	0.15341	0.64479
VELOCITY	1	0.00137	0.00022744	6.04	<.0001	0.00090125	0.00184

Chapter 6 : Hubble example

Regression without labels and with output statement

The REG Procedure

Model: MODEL1

Dependent Variable: DISTANCE

Output Statistics
Obs	VELOCITY	Dependent Variable	Predicted Value	Std Error Mean Predict	95% CL Mean		95% CL Predict		Residual
1	-220	0.2750	0.0971	0.1582	-0.2311	0.4252	-0.8046	0.9987	0.1779
2	-185	0.2750	0.1451	0.1515	-0.1691	0.4593	-0.7516	1.0418	0.1299
3	-130	0.2140	0.2206	0.1412	-0.0721	0.5134	-0.6688	1.1100	-0.006617
4	-70	0.2630	0.3030	0.1303	0.0327	0.5733	-0.5793	1.1853	-0.0400
5	-30	0.9000	0.3579	0.1234	0.1019	0.6139	-0.5201	1.2359	0.5421
6	150	0.9000	0.6050	0.0970	0.4039	0.8062	-0.2585	1.4686	0.2950
7	170	0.0320	0.6325	0.0947	0.4361	0.8289	-0.2300	1.4950	-0.6005
8	200	0.4500	0.6737	0.0916	0.4838	0.8636	-0.1873	1.5347	-0.2237
9	200	0.6300	0.6737	0.0916	0.4838	0.8636	-0.1873	1.5347	-0.0437
10	270	0.5000	0.7698	0.0859	0.5916	0.9480	-0.0887	1.6283	-0.2698
11	290	0.0340	0.7972	0.0848	0.6214	0.9731	-0.0608	1.6553	-0.7632
12	290	0.5000	0.7972	0.0848	0.6214	0.9731	-0.0608	1.6553	-0.2972
13	300	0.8000	0.8110	0.0843	0.6361	0.9858	-0.0469	1.6688	-0.0110
14	450	1.1000	1.0169	0.0845	0.8417	1.1921	0.1590	1.8748	0.0831
15	500	0.9000	1.0856	0.0876	0.9040	1.2671	0.2263	1.9448	-0.1856
16	500	1.1000	1.0856	0.0876	0.9040	1.2671	0.2263	1.9448	0.0144
17	500	1.4000	1.0856	0.0876	0.9040	1.2671	0.2263	1.9448	0.3144
18	500	2.0000	1.0856	0.0876	0.9040	1.2671	0.2263	1.9448	0.9144
19	650	0.9000	1.2915	0.1039	1.0760	1.5070	0.4245	2.1586	-0.3915
20	800	2.0000	1.4974	0.1275	1.2330	1.7619	0.6170	2.3779	0.5026
21	850	2.0000	1.5661	0.1364	1.2833	1.8489	0.6799	2.4523	0.4339
22	920	1.0000	1.6622	0.1493	1.3525	1.9719	0.7671	2.5573	-0.6622
23	960	1.7000	1.7171	0.1570	1.3915	2.0427	0.8164	2.6179	-0.0171
24	1090	2.0000	1.8956	0.1828	1.5165	2.2747	0.9742	2.8170	0.1044
25	1200	.	2.0466	0.2054	1.6206	2.4727	1.1049	2.9883	.

Sum of Residuals	0
Sum of Squared Residuals	3.60782
Predicted Residual SS (PRESS)	4.20513

Chapter 6 : Hubble example

Regression without labels and with output statement

The REG Procedure

Model: MODEL1

Test 1 Results for Dependent Variable DISTANCE
Source	DF	Mean Square	F Value	Pr > F
Numerator	1	0.05118	0.31	0.5820
Denominator	22	0.16399

Chapter 6 : Hubble example

Regression without labels and with output statement

The UNIVARIATE Procedure

Variable: resid

Moments
N	24	Sum Weights	24
Mean	0	Sum Observations	0
Std Deviation	0.39605753	Variance	0.15686157
Skewness	0.09440336	Kurtosis	0.24944333
Uncorrected SS	3.60781603	Corrected SS	3.60781603
Coeff Variation	.	Std Error Mean	0.0808449

Basic Statistical Measures
Location		Variability
Mean	0.00000	Std Deviation	0.39606
Median	-0.00880	Variance	0.15686
Mode	.	Range	1.67768
		Interquartile Range	0.48319

Tests for Location: Mu0=0
Test	Statistic		p Value
Student's t	t	0	Pr > \|t\|	1.0000
Sign	M	-1	Pr >= \|M\|	0.8388
Signed Rank	S	-2	Pr >= \|S\|	0.9559

Tests for Normality
Test	Statistic		p Value
Shapiro-Wilk	W	0.981984	Pr < W	0.9294
Kolmogorov-Smirnov	D	0.122752	Pr > D	>0.1500
Cramer-von Mises	W-Sq	0.03757	Pr > W-Sq	>0.2500
Anderson-Darling	A-Sq	0.220155	Pr > A-Sq	>0.2500

Quantiles (Definition 5)
Quantile	Estimate
100% Max	0.91443373
99%	0.91443373
95%	0.54208985
90%	0.50255290
75% Q3	0.23645454
50% Median	-0.00879778
25% Q1	-0.24673821
10%	-0.60049735
5%	-0.66219944
1%	-0.76324968
0% Min	-0.76324968

Extreme Observations
Lowest		Highest
Value	Obs	Value	Obs
-0.763250	11	0.314434	17
-0.662199	22	0.433906	21
-0.600497	7	0.502553	20
-0.391507	19	0.542090	5
-0.297250	12	0.914434	18

Missing Values
Missing Value	Count	Percent Of
Missing Value	Count	All Obs	Missing Obs
.	1	4.00	100.00

   Stem Leaf                     #  Boxplot                        Normal Probability Plot                                          
      8 1                        1     |         0.9+                                            * +++++                            
      6                                |            |                                         +++++                                 
      4 304                      3     |            |                                   *+*++*                                      
      2 91                       2  +-----+         |                               +*+*+                                           
      0 18038                    5  |  +  |      0.1+                         ++*****                                               
     -0 944211                   6  *-----*         |                    ***** *                                                    
     -2 9072                     4  +-----+         |               **+**                                                           
     -4                                |            |          +++++                                                                
     -6 660                      3     |        -0.7+     +*+++*  *                                                                 
        ----+----+----+----+                         +----+----+----+----+----+----+----+----+----+----+                            
    Multiply Stem.Leaf by 10**-1                         -2        -1         0        +1        +2

Chapter 6 : Hubble example

Regression without labels and with output statement

                          Plot of resid*VELOCITY.  Legend: A = 1 obs, B = 2 obs, etc.                          
                                                                                                               
 resid |                                                                                                       
       |                                                                                                       
  1.00 +                                                                                                       
       |                                                                                                       
       |                                                        A                                              
       |                                                                                                       
       |                                                                                                       
  0.75 +                                                                                                       
       |                                                                                                       
       |                                                                                                       
       |                                                                                                       
       |                        A                                                                              
  0.50 +                                                                          A                            
       |                                                                             A                         
       |                                                                                                       
       |                                                                                                       
       |                                   A                    A                                              
  0.25 +                                                                                                       
       |             A                                                                                         
       |               A                                                                                       
       |                                                     A                                     A           
       |                                                                                                       
  0.00 +------------------A-------------------------A-----------A---------------------------A----------------  
       |                      A               A                                                                
       |                                                                                                       
       |                                                                                                       
       |                                      A                 A                                              
 -0.25 +                                          A                                                            
       |                                           A                                                           
       |                                                                                                       
       |                                                                 A                                     
       |                                                                                                       
 -0.50 +                                                                                                       
       |                                                                                                       
       |                                    A                                                                  
       |                                                                                 A                     
       |                                                                                                       
 -0.75 +                                           A                                                           
       |                                                                                                       
       ---+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+--  
        -400        -200          0          200         400         600         800        1000        1200   
                                                                                                               
                                                      VELOCITY                                                 
                                                                                                               
NOTE: 1 obs had missing values.

Chapter 6 : Hubble example

Test of Lack of Fit

The GLM Procedure

Class Level Information
Class	Levels	Values
anotherx	19	-220 -185 -130 -70 -30 150 170 200 270 290 300 450 500 650 800 850 920 960 1090

Number of Observations Read	25
Number of Observations Used	24

Chapter 6 : Hubble example

Test of Lack of Fit

The GLM Procedure

Dependent Variable: DISTANCE

Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	18	8.76851166	0.48713954	2.99	0.1146
Error	5	0.81477801	0.16295560
Corrected Total	23	9.58328967

R-Square	Coeff Var	Root MSE	DISTANCE Mean
0.914979	44.29325	0.403678	0.911375

Source	DF	Type I SS	Mean Square	F Value	Pr > F
VELOCITY	1	5.97547364	5.97547364	36.67	0.0018
anotherx	17	2.79303802	0.16429635	1.01	0.5484

Source	DF	Type III SS	Mean Square	F Value	Pr > F
VELOCITY	0	0.00000000	.	.	.
anotherx	17	2.79303802	0.16429635	1.01	0.5484

Chapter 6 : Hubble example

Regression forced through the origin

The REG Procedure

Model: MODEL1

Dependent Variable: DISTANCE

Number of Observations Read	25
Number of Observations Used	24
Number of Observations with Missing Values	1

No intercept in model. R-Square is redefined.

Analysis of Variance
Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	24.04889	24.04889	101.14	<.0001
Error	23	5.46890	0.23778
Uncorrected Total	24	29.51780

Root MSE	0.48763	R-Square	0.8147
Dependent Mean	0.91137	Adj R-Sq	0.8067
Coeff Var	53.50434

Parameter Estimates
Variable	DF	Parameter Estimate	Standard Error	t Value	Pr > \|t\|	95% Confidence Limits
VELOCITY	1	0.00192	0.00019109	10.06	<.0001	0.00153	0.00232

Chapter 6 : Hubble example

Box-Cox transformation with PROC TRANSREG

The TRANSREG Procedure

Transformation Information for BoxCox(DISTANCE)
Lambda		R-Square	Log Like
-3.00		0.02	-142.489
-2.75		0.02	-126.999
-2.50		0.02	-111.702
-2.25		0.02	-96.634
-2.00		0.02	-81.840
-1.75		0.02	-67.378
-1.50		0.03	-53.325
-1.25		0.04	-39.778
-1.00		0.05	-26.873
-0.75		0.09	-14.787
-0.50		0.15	-3.762
-0.25		0.24	5.880
0.00		0.36	13.735
0.25		0.48	19.364
0.50		0.56	22.467	*
0.75		0.61	23.099	<
1.00	+	0.62	21.695	*
1.25		0.62	18.836
1.50		0.61	15.022
1.75		0.59	10.597
2.00		0.57	5.776
2.25		0.55	0.688
2.50		0.53	-4.589
2.75		0.51	-10.009
3.00		0.49	-15.541
< - Best Lambda * - Confidence Interval + - Convenient Lambda

Chapter 6 : Hubble example

Box-Cox transformation with PROC TRANSREG

The TRANSREG Procedure

TRANSREG Univariate Algorithm Iteration History for BoxCox(DISTANCE)
Iteration Number	Average Change	Maximum Change	R-Square	Criterion Change	Note
1	0.22699	2.41957	0.50355
2	0.00000	0.00000	0.67509	0.17154	Converged

Algorithm converged.

Chapter 6 : Hubble example

Box-Cox transformation with PROC TRANSREG

The TRANSREG Procedure

Transformation Information for BoxCox(DISTANCE)
Lambda		R-Square	Log Like
0.250		0.48	19.36384
0.500		0.56	22.46662	*
0.600		0.58	22.99847	*
0.650		0.59	23.12123	*
0.675		0.60	23.14838	*
0.676		0.60	23.14900	*
0.677		0.60	23.14958	*
0.678		0.60	23.15014	*
0.679		0.60	23.15065	*
0.680		0.60	23.15113	*
0.681		0.60	23.15158	*
0.682		0.60	23.15199	*
0.683		0.60	23.15236	*
0.684		0.60	23.15270	*
0.685		0.60	23.15300	*
0.686		0.60	23.15327	*
0.687		0.60	23.15351	*
0.688		0.60	23.15371	*
0.689		0.60	23.15387	*
0.690		0.60	23.15400	*
0.691		0.60	23.15409	*
0.692		0.60	23.15415	*
0.693		0.60	23.15418	<
0.694		0.60	23.15417	*
0.695		0.60	23.15412	*
0.696		0.60	23.15404	*
0.697		0.60	23.15393	*
0.698		0.60	23.15378	*
0.699		0.60	23.15359	*
0.700		0.60	23.15338	*
0.750		0.61	23.09910	*
0.800		0.61	22.96291	*
0.900		0.62	22.46393	*
1.000	+	0.62	21.69528	*
1.250		0.62	18.83617
< - Best Lambda * - Confidence Interval + - Convenient Lambda

Chapter 6 : Hubble example

Box-Cox transformation with PROC TRANSREG

The TRANSREG Procedure

TRANSREG Univariate Algorithm Iteration History for BoxCox(DISTANCE)
Iteration Number	Average Change	Maximum Change	R-Square	Criterion Change	Note
1	0.22988	2.43057	0.49746
2	0.00000	0.00000	0.66899	0.17154	Converged

Algorithm converged.

Simple Linear Regression Example

Hubbles nebula distance data

Listing of results from the regression output statement

Obs	VELOCITY	DISTANCE	anotherx	yhat	stdp	lclm	uclm	lcli	ucli	resid	stdi
1	-220	0.27500	-220	0.09705	0.15821	-0.23106	0.42517	-0.80460	0.99871	0.17795	0.43477
2	-185	0.27500	-185	0.14511	0.15148	-0.16905	0.45926	-0.75156	1.04177	0.12989	0.43236
3	-130	0.21400	-130	0.22062	0.14117	-0.07214	0.51338	-0.66878	1.11001	-0.00662	0.42886
4	-70	0.26300	-70	0.30299	0.13035	0.03267	0.57332	-0.57928	1.18526	-0.03999	0.42542
5	-30	0.90000	-30	0.35791	0.12345	0.10189	0.61393	-0.52008	1.23590	0.54209	0.42336
6	150	0.90000	150	0.60504	0.09700	0.40388	0.80620	-0.25855	1.46863	0.29496	0.41641
7	170	0.03200	170	0.63250	0.09470	0.43611	0.82889	-0.22999	1.49499	-0.60050	0.41588
8	200	0.45000	200	0.67369	0.09156	0.48380	0.86357	-0.18735	1.53472	-0.22369	0.41518
9	200	0.63000	200	0.67369	0.09156	0.48380	0.86357	-0.18735	1.53472	-0.04369	0.41518
10	270	0.50000	270	0.76979	0.08593	0.59159	0.94799	-0.08874	1.62832	-0.26979	0.41397
11	290	0.03400	290	0.79725	0.08480	0.62139	0.97311	-0.06080	1.65530	-0.76325	0.41374
12	290	0.50000	290	0.79725	0.08480	0.62139	0.97311	-0.06080	1.65530	-0.29725	0.41374
13	300	0.80000	300	0.81098	0.08432	0.63611	0.98584	-0.04687	1.66882	-0.01098	0.41364
14	450	1.10000	450	1.01692	0.08449	0.84170	1.19214	0.15900	1.87484	0.08308	0.41368
15	500	0.90000	500	1.08557	0.08755	0.90399	1.26714	0.22633	1.94480	-0.18557	0.41432
16	500	1.10000	500	1.08557	0.08755	0.90399	1.26714	0.22633	1.94480	0.01443	0.41432
17	500	1.40000	500	1.08557	0.08755	0.90399	1.26714	0.22633	1.94480	0.31443	0.41432
18	500	2.00000	500	1.08557	0.08755	0.90399	1.26714	0.22633	1.94480	0.91443	0.41432
19	650	0.90000	650	1.29151	0.10392	1.07600	1.50702	0.42446	2.15855	-0.39151	0.41808
20	800	2.00000	800	1.49745	0.12751	1.23300	1.76189	0.61696	2.37793	0.50255	0.42456
21	850	2.00000	850	1.56609	0.13637	1.28328	1.84891	0.67992	2.45227	0.43391	0.42730
22	920	1.00000	920	1.66220	0.14935	1.35247	1.97192	0.76707	2.55732	-0.66220	0.43162
23	960	1.70000	960	1.71712	0.15700	1.39151	2.04272	0.81637	2.61786	-0.01712	0.43433
24	1090	2.00000	1090	1.89560	0.18281	1.51648	2.27471	0.97416	2.81704	0.10440	0.44431
25	1200	.	.	2.04662	0.20543	1.62058	2.47266	1.10490	2.98834	.	0.45409