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decuslib10-08
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43,50504/mulexa.pri
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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 ***