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Multiple Linear Regression Analysis

EXAMPLES

**********************************
*  Example  1  originates from:  *
*  DE JONGE [4], pp. 472 & 479.  *
**********************************






"Model"   y  =  c * Log (x)  +  a  +  b * x;





"Input"   5 * ([x], 10 * [y]);





"Options" Transformed data matrix, Correlation matrix,
          Residual analysis, Process submodels (1, 2);
Transformed data matrix
=======================

obs.no.      c           a           b           dep.var.     repeats

     1          1.398       1.000      25.000       0.790      10.000
     2          1.699       1.000      50.000       0.984      10.000
     3          1.903       1.000      80.000       1.058      10.000
     4          2.114       1.000     130.000       1.163      10.000
     5          2.255       1.000     180.000       1.209      10.000
Control information
===================

transformed variable
denoted by parameter          mean           standard deviation                minimum                  maximum

 c                          1.873843                 0.306746                 1.397940                 2.255273
 a                          1.000000                 0.000000                 1.000000                 1.000000
 b                         93.000000                56.387870                25.000000               180.000000
 dep.var.                   1.040800                 0.163655                 0.670000                 1.330000

Number of observations :    5





Correlation matrix of the variables
===================================

             c           a           b           dep.var.

 c           1.000000
 a               *       1.000000
 b           0.962417        *       1.000000
 dep.var.    0.907742        *       0.849838    1.000000





Multiple correlation coefficient     0.911959    (adjusted   0.908023)
================================





Proportion of variation explained    0.831669    (adjusted   0.824506)
=================================





Standard deviation of the error term              0.068558
====================================
Regression parameters
=====================
                                                                                                    right  tail
parameter                 estimate           standard deviation              F - ratio              probability

 c                      0.6499168512             0.1175695440                30.558070                 0.000001

 a                     -0.0899819314             0.1641470240                 0.300500                 0.586163

 b                     -0.0009361326             0.0006395700                 2.142390                 0.149935






Correlation matrix of the estimates
===================================

             c           a           b

 c           1.000000
 a          -0.993392    1.000000
 b          -0.962417    0.929333    1.000000





Analysis of variance
====================

source of                                                                                           right  tail
variation          df          sum of squares           mean square             F - ratio           probability

---------------------------------------------------------------------------------------------------------------

total              50               55.475600

---------------------------------------------------------------------------------------------------------------

mean                1               54.163232             54.163232          11523.444701              0.000000
regression          2                1.091456              0.545728            116.105776              0.000000
residual           47                0.220912              0.004700

---------------------------------------------------------------------------------------------------------------

 lack of fit        2                0.005012              0.002506              0.522336              0.596686
 pure error        45                0.215900              0.004798

---------------------------------------------------------------------------------------------------------------


regression null hypothesis :  c = b = 0
Residual analysis
=================
                                                                                              standardized         studentized
obs.no.        observation        fitted value    standard deviation          residual            residual            residual

     1            0.790000            0.795160            0.020789           -0.005160           -0.118992           -0.078976
     2            0.984000            0.967401            0.013590            0.016599            0.382824            0.247021
     3            1.058000            1.071978            0.015165           -0.013978           -0.322363           -0.209059
     4            1.163000            1.162208            0.012847            0.000792            0.018260            0.011757
     5            1.209000            1.207254            0.019954            0.001746            0.040272            0.026622

                                                       sum of residuals :     0.000000

                  Upper bound for the right tail probability of the largest absolute studentized residual (no. 2) :   1.000000
Control information  -  submodel  1
===================

transformed variable
denoted by parameter          mean           standard deviation                minimum                  maximum

 b           omitted

 c                          1.873843                 0.306746                 1.397940                 2.255273
 a                          1.000000                 0.000000                 1.000000                 1.000000
 dep.var.                   1.040800                 0.163655                 0.670000                 1.330000

Number of observations :    5





Multiple correlation coefficient     0.907742    (adjusted   0.905720)
================================





Proportion of variation explained    0.823996    (adjusted   0.820329)
=================================





Standard deviation of the error term              0.069370
====================================
Regression parameters
=====================
                                                                                                    right  tail
parameter                 estimate           standard deviation              F - ratio              probability

 c                      0.4842988398             0.0323066319               224.720913                 0.000000

 a                      0.1332999205             0.0613273100                 4.724458                 0.034701






Correlation matrix of the estimates
===================================

             c           a

 c           1.000000
 a          -0.987122    1.000000





Analysis of variance
====================

source of                                                                                           right  tail
variation          df          sum of squares           mean square             F - ratio           probability

---------------------------------------------------------------------------------------------------------------

total              50               55.475600

---------------------------------------------------------------------------------------------------------------

mean                1               54.163232             54.163232          11255.564569              0.000000
regression          1                1.081386              1.081386            224.720852              0.000000
residual           48                0.230982              0.004812

---------------------------------------------------------------------------------------------------------------

 lack of fit        3                0.015082              0.005027              1.047838              0.380681
 pure error        45                0.215900              0.004798

---------------------------------------------------------------------------------------------------------------

 reduction          1                0.010070              0.010070              2.142390              0.149935

---------------------------------------------------------------------------------------------------------------


regression null hypothesis :  c = 0  (in the reduced model)

 reduction null hypothesis :  b = 0  (in the original model)
Residual analysis
=================
                                                                                              standardized         studentized
obs.no.        observation        fitted value    standard deviation          residual            residual            residual

     1            0.790000            0.810321            0.018238           -0.020321           -0.378175           -0.303615
     2            0.984000            0.956109            0.011321            0.027891            0.519060            0.407526
     3            1.058000            1.054964            0.009856            0.003036            0.056497            0.044211
     4            1.163000            1.157080            0.012506            0.005920            0.110169            0.086758
     5            1.209000            1.225526            0.015751           -0.016526           -0.307551           -0.244617

                                                       sum of residuals :    -0.000000

                  Upper bound for the right tail probability of the largest absolute studentized residual (no. 2) :   1.000000
Control information  -  submodel  2
===================

transformed variable
denoted by parameter          mean           standard deviation                minimum                  maximum

 a           omitted
 b           omitted

 c                          1.873843                 0.306746                 1.397940                 2.255273
 dep.var.                   1.040800                 0.163655                 0.670000                 1.330000

Number of observations :    5





There is no constant independent variable in the transformed (sub)model    (message)





Multiple correlation coefficient     0.997711    (adjusted   0.997664)
================================





Proportion of variation explained    0.995427    (adjusted   0.995333)
=================================





Standard deviation of the error term              0.071958
====================================
Regression parameters
=====================
                                                                                                    right  tail
parameter                 estimate           standard deviation              F - ratio              probability

 c                      0.5536156656             0.0053607978             10664.926934                 0.000000






Correlation matrix of the estimates
===================================

             c

 c           1.000000





Analysis of variance
====================

source of                                                                                           right  tail
variation          df          sum of squares           mean square             F - ratio           probability

---------------------------------------------------------------------------------------------------------------

total              50               55.475600

---------------------------------------------------------------------------------------------------------------

regression          1               55.221883             55.221883          10664.926934              0.000000
residual           49                0.253717              0.005178

---------------------------------------------------------------------------------------------------------------

 lack of fit        4                0.037817              0.009454              1.970525              0.115263
 pure error        45                0.215900              0.004798

---------------------------------------------------------------------------------------------------------------

 reduction          2                0.032804              0.016402              3.489644              0.038633

---------------------------------------------------------------------------------------------------------------


regression null hypothesis :  c = 0  (in the reduced model)

 reduction null hypothesis :  a = b = 0  (in the original model)
Residual analysis
=================
                                                                                              standardized         studentized
obs.no.        observation        fitted value    standard deviation          residual            residual            residual

     1            0.790000            0.773921            0.007494            0.016079            0.249818            0.224666
     2            0.984000            0.940576            0.009108            0.043424            0.674690            0.608353
     3            1.058000            1.053580            0.010202            0.004420            0.068669            0.062046
     4            1.163000            1.170312            0.011332           -0.007312           -0.113612           -0.102902
     5            1.209000            1.248554            0.012090           -0.039554           -0.614569           -0.557614

                                                       sum of residuals :     0.170553

                  Upper bound for the right tail probability of the largest absolute studentized residual (no. 2) :   1.000000





End of job :  1
**********************************
*  Example  2  originates from:  *
*  SEARLE [11], pp. 121-123      *
**********************************






"Model 1"  y  =  a3 * x3  +  a2 * x2  +  a1 * x1;





"Input"  5 * [y, x1, x2, x3];





"Options"  Save original model, Process submodels (1);
Control information
===================

transformed variable
denoted by parameter          mean           standard deviation                minimum                  maximum

 a3                         3.400000                 1.949359                 1.000000                 6.000000
 a2                         1.000000                 2.549510                -3.000000                 4.000000
 a1                         1.000000                 1.224745                -1.000000                 2.000000
 dep.var.                   9.000000                 2.236068                 6.000000                12.000000

Number of observations :    5





There is no constant independent variable in the transformed (sub)model    (message)





Multiple correlation coefficient     0.936662    (adjusted   0.832670)
================================





Proportion of variation explained    0.877336    (adjusted   0.693339)
=================================





Standard deviation of the error term              5.105507
====================================
Regression parameters
=====================
                                                                                                    right  tail
parameter                 estimate           standard deviation              F - ratio              probability

 a3                     2.5446171560             0.9982125895                 6.498286                 0.125553

 a2                     0.2665515256             1.0423373169                 0.065395                 0.822061

 a1                    -1.3851468048             2.3646149361                 0.343140                 0.617320






Analysis of variance
====================

source of                                                                                           right  tail
variation          df          sum of squares           mean square             F - ratio           probability

---------------------------------------------------------------------------------------------------------------

total               5              425.000000

---------------------------------------------------------------------------------------------------------------

regression          3              372.867588            124.289196              4.768212              0.178233
residual            2               52.132412             26.066206

---------------------------------------------------------------------------------------------------------------


regression null hypothesis :  a3 = a2 = a1 = 0
Control information  -  submodel  1
===================

transformed variable
denoted by parameter          mean           standard deviation                minimum                  maximum

 a1          omitted

 a3                         3.400000                 1.949359                 1.000000                 6.000000
 a2                         1.000000                 2.549510                -3.000000                 4.000000
 dep.var.                   9.000000                 2.236068                 6.000000                12.000000

Number of observations :    5





There is no constant independent variable in the transformed (sub)model    (message)





Multiple correlation coefficient     0.925359    (adjusted   0.872057)
================================





Proportion of variation explained    0.856290    (adjusted   0.760483)
=================================





Standard deviation of the error term              4.512086
====================================
Regression parameters
=====================
                                                                                                    right  tail
parameter                 estimate           standard deviation              F - ratio              probability

 a3                     2.1052066559             0.5820385900                13.082354                 0.036325

 a2                     0.4159957059             0.8931663715                 0.216927                 0.673122






Analysis of variance
====================

source of                                                                                           right  tail
variation          df          sum of squares           mean square             F - ratio           probability

---------------------------------------------------------------------------------------------------------------

total               5              425.000000

---------------------------------------------------------------------------------------------------------------

regression          2              363.923242            181.961621              8.937686              0.054479
residual            3               61.076758             20.358919

---------------------------------------------------------------------------------------------------------------

 reduction          1                8.944346              8.944346              0.343140              0.617320

---------------------------------------------------------------------------------------------------------------


regression null hypothesis :  a3 = a2 = 0  (in the reduced model)

 reduction null hypothesis :  a1 = 0  (in the original model)





End of job :  2




"Model 2"  y - 4 * x1  =  b2 * (x1 + x2) + b3 * x3;   (eqn. 118, p. 121)





"Input"  5 * [y, x1, x2, x3];





"Options"  Test reduced model, Transformed data matrix;
Transformed data matrix
=======================

obs.no.      b2          b3          dep.var.

     1          3.000       4.000       0.000
     2          1.000       1.000      14.000
     3         -2.000       4.000       5.000
     4          3.000       2.000      -2.000
     5          5.000       6.000       8.000
Control information
===================

transformed variable
denoted by parameter          mean           standard deviation                minimum                  maximum

 b2                         2.000000                 2.645751                -2.000000                 5.000000
 b3                         3.400000                 1.949359                 1.000000                 6.000000
 dep.var.                   5.000000                 6.403124                -2.000000                14.000000

Number of observations :    5





There is no constant independent variable in the transformed (sub)model    (message)





Proportion of variation explained    0.293057    (adjusted  -0.178239)
=================================





Standard deviation of the error term              8.252406
====================================
Regression parameters
=====================
                                                                                                    right  tail
parameter                 estimate           standard deviation              F - ratio              probability

 b2                    -0.2325836533             1.6513864104                 0.019836                 0.896920

 b3                     1.1991223258             1.3390842284                 0.801883                 0.436516






Analysis of variance
====================

source of                                                                                           right  tail
variation          df          sum of squares           mean square             F - ratio           probability

---------------------------------------------------------------------------------------------------------------

total               5              289.000000

---------------------------------------------------------------------------------------------------------------

regression          2               84.693363             42.346681              0.621811              0.594397
residual            3              204.306637             68.102212

---------------------------------------------------------------------------------------------------------------

 reduction          1              152.174225            152.174225              5.837989              0.136963

---------------------------------------------------------------------------------------------------------------


regression null hypothesis :  b2 = b3 = 0





End of job :  3
****************************************
*  Example  3  originates from:        *
*  AFIFI & AZEN [1], pp. 88 & 93-100.  *
****************************************






"Model"  y  =  alfa0  +  alfa1 * x;





"Input"  5 * ([x], n, n * [y]);





"Option" Transformed data matrix, Print input data;






"Data"

       1.000       4.000       1.100       0.700       1.800       0.400       3.000       5.000       3.000       1.400       4.900
       4.400       4.500       5.000       3.000       7.300       8.200       6.200      10.000       4.000      12.000      13.100
      12.600      13.200      15.000       4.000      18.700      19.700      17.400      17.100
Transformed data matrix
=======================

obs.no.      alfa0       alfa1       dep.var.     repeats

     1          1.000       1.000       1.000       4.000
     2          1.000       3.000       3.640       5.000
     3          1.000       5.000       7.233       3.000
     4          1.000      10.000      12.725       4.000
     5          1.000      15.000      18.225       4.000
Control information
===================

transformed variable
denoted by parameter          mean           standard deviation                minimum                  maximum

 alfa0                      1.000000                 0.000000                 1.000000                 1.000000
 alfa1                      6.700000                 5.262579                 1.000000                15.000000
 dep.var.                   8.385000                 6.545571                 0.400000                19.700000

Number of observations :    5





Multiple correlation coefficient     0.987051    (adjusted   0.986326)
================================





Proportion of variation explained    0.974269    (adjusted   0.972840)
=================================





Standard deviation of the error term              1.078736
====================================
Regression parameters
=====================
                                                                                                    right  tail
parameter                 estimate           standard deviation              F - ratio              probability

 alfa0                  0.1594830832             0.3968072487                 0.161536                 0.692478

 alfa1                  1.2276890919             0.0470261690               681.549798                 0.000000






Analysis of variance
====================

source of                                                                                           right  tail
variation          df          sum of squares           mean square             F - ratio           probability

---------------------------------------------------------------------------------------------------------------

total              20             2220.210000

---------------------------------------------------------------------------------------------------------------

mean                1             1406.164500           1406.164500           1208.387113              0.000000
regression          1              793.099430            793.099430            681.549798              0.000000
residual           18               20.946070              1.163671

---------------------------------------------------------------------------------------------------------------

 lack of fit        3                4.252403              1.417468              1.273658              0.319196
 pure error        15               16.693667              1.112911

---------------------------------------------------------------------------------------------------------------


regression null hypothesis :  alfa1 = 0





End of job :  4
***  Marten van Gelderen; Mathematisch Centrum  ***