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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  ***

```