Regression.pdf
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Sep 28, 2023
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About This Presentation
Biostatics
Slide Content
Faculty of Pharmacy
Regression
92
Regression
92
Faculty of Pharmacy
What is Regression?
•Regressionisthemeasureoftheaverage
relationshipbetweentwoormorevariables.
•RegressionAnalysismeasuresthenatureand
extentoftherelationshipbetweentwoormore
variables,thusenablesustomakepredictions.
93
What is Regression?
•Regressionisthemeasureoftheaverage
relationshipbetweentwoormorevariables.
•RegressionAnalysismeasuresthenatureand
extentoftherelationshipbetweentwoormore
variables,thusenablesustomakepredictions.
93
Faculty of Pharmacy
Application of Regression
•Degree&Natureofrelationship
•Estimationofrelationship
•Prediction
•UsefulinEconomic&BusinessResearch
94
Application of Regression
•Degree&Natureofrelationship
•Estimationofrelationship
•Prediction
•UsefulinEconomic&BusinessResearch
94
Faculty of Pharmacy
DIFFERENCE BETWEEN CORRELATION
& REGRESSION
•Degree&NatureofRelationship
•Correlationisameasureofdegreeofrelationship
betweenX&Y
•Regressionstudiesthenatureofrelationshipbetweenthe
variablessothatonemaybeabletopredictthevalueof
onevariableonthebasisofanother.
•Cause&EffectRelationship
•Correlationdoesnotalwaysassumecauseandeffect
relationshipbetweentwovariables.
•Regressionclearlyexpressesthecauseandeffect
relationshipbetweentwovariables.Theindependent
variableisthecauseanddependentvariableiseffect.
95
DIFFERENCE BETWEEN CORRELATION
& REGRESSION
•Degree&NatureofRelationship
•Correlationisameasureofdegreeofrelationship
betweenX&Y
•Regressionstudiesthenatureofrelationshipbetweenthe
variablessothatonemaybeabletopredictthevalueof
onevariableonthebasisofanother.
•Cause&EffectRelationship
•Correlationdoesnotalwaysassumecauseandeffect
relationshipbetweentwovariables.
•Regressionclearlyexpressesthecauseandeffect
relationshipbetweentwovariables.Theindependent
variableisthecauseanddependentvariableiseffect.
95
Faculty of Pharmacy
DIFFERENCE BETWEEN CORRELATION
& REGRESSION
•Prediction
•Correlationdoesn’thelpinmakingpredictions
•Regressionenableustomakepredictionsusingregression
line
•Symmetric
•Correlationcoefficientsaresymmetricali.e.rxy=ryx.
•Regressioncoefficientsarenotsymmetricali.e.bxy≠byx.
•Origin&Scale
•Correlationisindependentofthechangeoforiginand
scale
•Regressioncoefficientisindependentofchangeoforigin
butnotofscale
96
DIFFERENCE BETWEEN CORRELATION
& REGRESSION
•Prediction
•Correlationdoesn’thelpinmakingpredictions
•Regressionenableustomakepredictionsusingregression
line
•Symmetric
•Correlationcoefficientsaresymmetricali.e.rxy=ryx.
•Regressioncoefficientsarenotsymmetricali.e.bxy≠byx.
•Origin&Scale
•Correlationisindependentofthechangeoforiginand
scale
•Regressioncoefficientisindependentofchangeoforigin
butnotofscale
96
Faculty of Pharmacy
Types of Regression Analysis
•Simple&MultipleRegression
•Linear&NonLinearRegression
•Partial&TotalRegression
97
Types of Regression Analysis
•Simple&MultipleRegression
•Linear&NonLinearRegression
•Partial&TotalRegression
97
Faculty of Pharmacy
Simple Linear Regression
Simple Linear
Regression
Regression
Lines
Regression
Equations
Regression
Coefficient
98
Simple Linear Regression
Simple Linear
Regression
Regression
Lines
Regression
Equations
Regression
Coefficient
98
Faculty of Pharmacy
Regression Lines
99
•Theregressionlineshowstheaveragerelationshipbetween
twovariables.ItisalsocalledLineofBestFit.
•IftwovariablesX&Yaregiven,thentherearetworegression
lines:
•RegressionLineofXonY
•RegressionLineofYonX
•NatureofRegressionLines
•Ifr=±1,thenthetworegressionlinesarecoincident.
•Ifr=0,thenthetworegressionlinesintersecteachotherat
90°.
•Thenearertheregressionlinesaretoeachother,thegreater
willbethedegreeofcorrelation.
•Ifregressionlinesrisefromlefttorightupward,then
correlationispositive.
Regression Lines
99
•Theregressionlineshowstheaveragerelationshipbetween
twovariables.ItisalsocalledLineofBestFit.
•IftwovariablesX&Yaregiven,thentherearetworegression
lines:
•RegressionLineofXonY
•RegressionLineofYonX
•NatureofRegressionLines
•Ifr=±1,thenthetworegressionlinesarecoincident.
•Ifr=0,thenthetworegressionlinesintersecteachotherat
90°.
•Thenearertheregressionlinesaretoeachother,thegreater
willbethedegreeofcorrelation.
•Ifregressionlinesrisefromlefttorightupward,then
correlationispositive.
Faculty of Pharmacy
Regression Equations
10
0
•RegressionEquationsarethealgebraicformulationofregressionlines.
•Therearetworegressionequations:
•RegressionEquationofYonX
Y = a + bX
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
????????????−�????????????=????????????.
????????????
????????????
????????????
????????????
(????????????−�????????????)
•RegressionEquationofXonY
X = a + bY
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
????????????−�????????????=????????????.
????????????
????????????
????????????
????????????
(????????????−�????????????)
Regression Equations
10
0
•RegressionEquationsarethealgebraicformulationofregressionlines.
•Therearetworegressionequations:
•RegressionEquationofYonX
Y = a + bX
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
????????????−�????????????=????????????.
????????????
????????????
????????????
????????????
(????????????−�????????????)
•RegressionEquationofXonY
X = a + bY
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
????????????−�????????????=????????????.
????????????
????????????
????????????
????????????
(????????????−�????????????)
Faculty of Pharmacy
Regression Coefficients
10
1
•Regressioncoefficientmeasurestheaverage
changeinthevalueofonevariableforaunit
changeinthevalueofanothervariable.
•Theserepresenttheslopeofregressionline
•Therearetworegressioncoefficients:
•RegressioncoefficientofYonX:byx=????????????.σ????????????/σ????????????
•RegressioncoefficientofXonY:bxy=????????????.σ????????????/σ????????????
Regression Coefficients
10
1
•Regressioncoefficientmeasurestheaverage
changeinthevalueofonevariableforaunit
changeinthevalueofanothervariable.
•Theserepresenttheslopeofregressionline
•Therearetworegressioncoefficients:
•RegressioncoefficientofYonX:byx=????????????.σ????????????/σ????????????
•RegressioncoefficientofXonY:bxy=????????????.σ????????????/σ????????????
Faculty of Pharmacy
Properties of Regression Coefficients
10
2
•Coefficientofcorrelationisthegeometricmeanofthe
regressioncoefficients.i.e.r=????????????
????????????????????????.????????????
????????????????????????
•Boththeregressioncoefficientsmusthavethesame
algebraicsign.
•Coefficientofcorrelationmusthavethesamesignasthat
oftheregressioncoefficients.
•Boththeregressioncoefficientscannotbegreaterthan
unity.
•Arithmeticmeanoftworegressioncoefficientsisequalto
orgreaterthanthecorrelationcoefficient.
•i.e.????????????????????????????????????+????????????????????????????????????/2≥r
•Regressioncoefficientisindependentofchangeoforigin
butnotofscale.
Properties of Regression Coefficients
10
2
•Coefficientofcorrelationisthegeometricmeanofthe
regressioncoefficients.i.e.r=????????????
????????????????????????.????????????
????????????????????????
•Boththeregressioncoefficientsmusthavethesame
algebraicsign.
•Coefficientofcorrelationmusthavethesamesignasthat
oftheregressioncoefficients.
•Boththeregressioncoefficientscannotbegreaterthan
unity.
•Arithmeticmeanoftworegressioncoefficientsisequalto
orgreaterthanthecorrelationcoefficient.
•i.e.????????????????????????????????????+????????????????????????????????????/2≥r
•Regressioncoefficientisindependentofchangeoforigin
butnotofscale.
Faculty of Pharmacy
Obtaining Regression Equations
10
3
Regression
Equation
Using Normal
Equation
Using Regression
Coefficient
Using Actual
values of X
and Y
Using
deviations from
actual means
Using r, σx,
σy
Using
deviations
from
Assumed
Means
Obtaining Regression Equations
10
3
Regression
Equation
Using Normal
Equation
Using Regression
Coefficient
Using Actual
values of X
and Y
Using
deviations from
actual means
Using r, σx,
σy
Using
deviations
from
Assumed
Means
Faculty of Pharmacy
Regression Equations in Individual Series Using Normal
Equations
10
4
•ThismethodisalsocalledasLeastSquareMethod.
•Inthismethod,regressionequationscanbecalculatedby
solvingtwonormalequations:
•ForregressionequationYonX:Y=a+bX
•Σ????????????=????????????????????????+????????????Σ????????????
•Σ????????????????????????=????????????Σ????????????+????????????Σ????????????
2
•AnotherMethod:
•????????????
????????????????????????=
????????????∑????????????????????????−∑????????????∑????????????
∑????????????
2
−
∑????????????
2
•HereaistheY–intercept,indicatestheminimumvalueof
YforX=0
•bistheslopeoftheline,indicatestheabsoluteincreasein
YforaunitincreaseinX.
Regression Equations in Individual Series Using Normal
Equations
10
4
•ThismethodisalsocalledasLeastSquareMethod.
•Inthismethod,regressionequationscanbecalculatedby
solvingtwonormalequations:
•ForregressionequationYonX:Y=a+bX
•Σ????????????=????????????????????????+????????????Σ????????????
•Σ????????????????????????=????????????Σ????????????+????????????Σ????????????
2
•AnotherMethod:
•????????????
????????????????????????=
????????????∑????????????????????????−∑????????????∑????????????
∑????????????
2
−
∑????????????
2
•HereaistheY–intercept,indicatestheminimumvalueof
YforX=0
•bistheslopeoftheline,indicatestheabsoluteincreasein
YforaunitincreaseinX.
Faculty of Pharmacy
Regression Equations in Individual Series Using Normal
Equations
10
5
•ForregressionequationXonY:X=a+bY
•ΣX=????????????????????????+????????????ΣY
•Σ????????????????????????=????????????ΣY+????????????ΣY
2
•AnotherMethod:
•????????????
????????????????????????=
????????????∑????????????????????????−∑????????????∑????????????
∑????????????
2
−
∑????????????
2
•HereaistheX–intercept,indicatesthe
minimumvalueofXforY=0
•bistheslopeoftheline,indicatestheabsolute
increaseinXforaunitincreaseinY.
Regression Equations in Individual Series Using Normal
Equations
10
5
•ForregressionequationXonY:X=a+bY
•ΣX=????????????????????????+????????????ΣY
•Σ????????????????????????=????????????ΣY+????????????ΣY
2
•AnotherMethod:
•????????????
????????????????????????=
????????????∑????????????????????????−∑????????????∑????????????
∑????????????
2
−
∑????????????
2
•HereaistheX–intercept,indicatesthe
minimumvalueofXforY=0
•bistheslopeoftheline,indicatestheabsolute
increaseinXforaunitincreaseinY.
Faculty of Pharmacy
Regression Equations Using Regression Coefficients
(Using Actual Values)
10
6
•RegressionEquationofYonX
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
•RegressionEquationofXonY
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
Regression Equations Using Regression Coefficients
(Using Actual Values)
10
6
•RegressionEquationofYonX
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
•RegressionEquationofXonY
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
Faculty of Pharmacy
Regression Equations Using Regression Coefficients
(Using Deviations Actual Values)
10
7
•RegressionEquationofYonX
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
????????????
????????????????????????=
∑????????????????????????
∑????????????
2•RegressionEquationofXonY
????????????−�????????????=????????????????????????????????????????????????−�????????????
????????????
????????????????????????=
∑????????????????????????
∑????????????
2
Regression Equations Using Regression Coefficients
(Using Deviations Actual Values)
10
7
•RegressionEquationofYonX
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
????????????
????????????????????????=
∑????????????????????????
∑????????????
2•RegressionEquationofXonY
????????????−�????????????=????????????????????????????????????????????????−�????????????
????????????
????????????????????????=
∑????????????????????????
∑????????????
2
Faculty of Pharmacy
Regression Equations Using Regression Coefficients
(Using Deviations from Assumed Mean)
10
8
•RegressionEquationofYonX
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
????????????
????????????????????????=
????????????∑????????????????????????????????????????????????−∑????????????????????????∑????????????????????????
????????????∑????????????????????????
2
−
∑????????????????????????
2
•RegressionEquationofXonY
????????????−�????????????=????????????????????????????????????????????????−�????????????
????????????
????????????????????????=
????????????∑????????????????????????????????????????????????−∑????????????????????????∑????????????????????????
????????????∑????????????????????????
2
−
∑????????????????????????
2
Regression Equations Using Regression Coefficients
(Using Deviations from Assumed Mean)
10
8
•RegressionEquationofYonX
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
????????????
????????????????????????=
????????????∑????????????????????????????????????????????????−∑????????????????????????∑????????????????????????
????????????∑????????????????????????
2
−
∑????????????????????????
2
•RegressionEquationofXonY
????????????−�????????????=????????????????????????????????????????????????−�????????????
????????????
????????????????????????=
????????????∑????????????????????????????????????????????????−∑????????????????????????∑????????????????????????
????????????∑????????????????????????
2
−
∑????????????????????????
2
Faculty of Pharmacy
Regression Equations Using Regression Coefficients
(Using Standard Deviations)
10
9
•RegressionEquationofYonX
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
????????????
????????????????????????=????????????.
????????????
????????????
????????????
????????????
•RegressionEquationofXonY
????????????−�????????????=????????????????????????????????????????????????−�????????????
????????????
????????????????????????=????????????.
????????????
????????????
????????????
????????????
Regression Equations Using Regression Coefficients
(Using Standard Deviations)
10
9
•RegressionEquationofYonX
????????????−�????????????=????????????????????????????????????(????????????−�????????????)
????????????
????????????????????????=????????????.
????????????
????????????
????????????
????????????
•RegressionEquationofXonY
????????????−�????????????=????????????????????????????????????????????????−�????????????
????????????
????????????????????????=????????????.
????????????
????????????
????????????
????????????
Faculty of Pharmacy
Examples
11
0
•Example1:Fromfollowingdatacalculatethe
linesofregression.
•EstimatevalueofYwhenX=25
•EstimatevalueofXwhenY=50
X 16 20 15 20 18 25
Y 50 60 35 50 50 60
Examples
11
0
•Example1:Fromfollowingdatacalculatethe
linesofregression.
•EstimatevalueofYwhenX=25
•EstimatevalueofXwhenY=50
X 16 20 15 20 18 25
Y 50 60 35 50 50 60
Faculty of Pharmacy
Examples
11
1
X Y dx= X-A dy= Y-B X*X Y*Y X*Y
16 50 256 2500 800
20 60 400 3600 1200
15 35 225 1225 525
20 50 400 2500 1000
18 50 324 2500 900
25 60 625 3600 1500
ƩX= 114ƩY= 305 ƩX
2
=2230ƩY
2
=15925ƩXY=5925
Regression line Y on X:
????????????−�????????????=????????????
????????????????????????????????????−�????????????
????????????
????????????????????????=
????????????∑????????????????????????−∑????????????∑????????????
????????????∑????????????
2
−
∑????????????2
B
yx=
6∗5925−114∗305
6∗2230−114∗114
=0.792
Examples
11
1
X Y dx= X-A dy= Y-B X*X Y*Y X*Y
16 50 256 2500 800
20 60 400 3600 1200
15 35 225 1225 525
20 50 400 2500 1000
18 50 324 2500 900
25 60 625 3600 1500
ƩX= 114ƩY= 305 ƩX
2
=2230ƩY
2
=15925ƩXY=5925
Regression line Y on X:
????????????−�????????????=????????????
????????????????????????????????????−�????????????
????????????
????????????????????????=
????????????∑????????????????????????−∑????????????∑????????????
????????????∑????????????
2
−
∑????????????2
B
yx=
6∗5925−114∗305
6∗2230−114∗114
=0.792
Faculty of Pharmacy
Examples
11
2
X Y dx= X-A dy= Y-B X*X Y*Y X*Y
16 50 256 2500 800
20 60 400 3600 1200
15 35 225 1225 525
20 50 400 2500 1000
18 50 324 2500 900
25 60 625 3600 1500
ƩX= 114ƩY= 305 ƩX
2
=2230ƩY
2
=15925ƩXY=5925
Regression line Y on X:
????????????−�????????????=????????????
????????????????????????????????????−�????????????
????????????−50.8=0.792????????????−19
Y-50.8=0.792X -15.048 --------------------------(1)
B
yx=0.792
Y –50.8= 0.792*25-15.048 Y= 55.55
Examples
11
2
X Y dx= X-A dy= Y-B X*X Y*Y X*Y
16 50 256 2500 800
20 60 400 3600 1200
15 35 225 1225 525
20 50 400 2500 1000
18 50 324 2500 900
25 60 625 3600 1500
ƩX= 114ƩY= 305 ƩX
2
=2230ƩY
2
=15925ƩXY=5925
Regression line Y on X:
????????????−�????????????=????????????
????????????????????????????????????−�????????????
????????????−50.8=0.792????????????−19
Y-50.8=0.792X -15.048 --------------------------(1)
B
yx=0.792
Y –50.8= 0.792*25-15.048 Y= 55.55
Faculty of Pharmacy
Examples
11
3
X Y dx= X-A dy= Y-B X*X Y*Y X*Y
16 50 256 2500 800
20 60 400 3600 1200
15 35 225 1225 525
20 50 400 2500 1000
18 50 324 2500 900
25 60 625 3600 1500
ƩX= 114ƩY= 305 ƩX
2
=2230ƩY
2
=15925ƩXY=5925
Regression line X on Y:
????????????−�????????????=????????????
????????????????????????????????????−�????????????
????????????
????????????????????????=
????????????∑????????????????????????−∑????????????∑????????????
????????????∑????????????
2
−
∑????????????2
B
yx=
6∗5925−114∗305
6∗15925−305∗305
=0.308
Examples
11
3
X Y dx= X-A dy= Y-B X*X Y*Y X*Y
16 50 256 2500 800
20 60 400 3600 1200
15 35 225 1225 525
20 50 400 2500 1000
18 50 324 2500 900
25 60 625 3600 1500
ƩX= 114ƩY= 305 ƩX
2
=2230ƩY
2
=15925ƩXY=5925
Regression line X on Y:
????????????−�????????????=????????????
????????????????????????????????????−�????????????
????????????
????????????????????????=
????????????∑????????????????????????−∑????????????∑????????????
????????????∑????????????
2
−
∑????????????2
B
yx=
6∗5925−114∗305
6∗15925−305∗305
=0.308
Faculty of Pharmacy
Examples
11
4
X Y dx= X-A dy= Y-
B
X*X=dx2 Y*Y=dy2X*Y=dxdy
16 50 1 15 256 2500 800
20 60 5 25 400 3600 1200
15 35 0 0 225 1225 525
20 50 5 15 400 2500 1000
18 50 3 15 324 2500 900
25 60 10 25 625 3600 1500
ƩX= 114ƩY= 305Ʃdx= Ʃdy= ƩX
2
=2230= ƩY
2
=15925
=
ƩXY=5925=
Regression line X on Y:
????????????−�????????????=????????????
????????????????????????????????????−�????????????
X-19 = 0.308 (Y- 50.8)
X-19 =0.308Y –15.64 --------------------------(2)
B
xy=0.308
X-19 = 0.308*50- 15.64 X= 18.76
Examples
11
4
X Y dx= X-A dy= Y-
B
X*X=dx2 Y*Y=dy2X*Y=dxdy
16 50 1 15 256 2500 800
20 60 5 25 400 3600 1200
15 35 0 0 225 1225 525
20 50 5 15 400 2500 1000
18 50 3 15 324 2500 900
25 60 10 25 625 3600 1500
ƩX= 114ƩY= 305Ʃdx= Ʃdy= ƩX
2
=2230= ƩY
2
=15925
=
ƩXY=5925=
Regression line X on Y:
????????????−�????????????=????????????
????????????????????????????????????−�????????????
X-19 = 0.308 (Y- 50.8)
X-19 =0.308Y –15.64 --------------------------(2)
B
xy=0.308
X-19 = 0.308*50- 15.64 X= 18.76
Faculty of Pharmacy
Examples
11
5
•Example2:fromfollowingdatacalculatethe
linesofregression.
•Correlationcoefficientvalueis0.8=r
•EstimatevalueofYwhenX=60
•EstimatevalueofXwhenY=80
MeanS.D
X 50 5
Y 20 4
Examples
11
5
•Example2:fromfollowingdatacalculatethe
linesofregression.
•Correlationcoefficientvalueis0.8=r
•EstimatevalueofYwhenX=60
•EstimatevalueofXwhenY=80
MeanS.D
X 50 5
Y 20 4
Faculty of Pharmacy
Examples
11
6
•Wehave
•�????????????=50
•�????????????=20
•σX=5
•σY=4
•????????????
????????????????????????=????????????
????????????????????????
????????????
????????????
=0.8∗
4
5
=0.64
•????????????
????????????????????????=????????????
????????????
????????????
????????????????????????
=0.8∗
5 4
=1
Examples
11
6
•Wehave
•�????????????=50
•�????????????=20
•σX=5
•σY=4
•????????????
????????????????????????=????????????
????????????????????????
????????????
????????????
=0.8∗
4
5
=0.64
•????????????
????????????????????????=????????????
????????????
????????????
????????????????????????
=0.8∗
5 4
=1
Faculty of Pharmacy
Examples
11
7
•Wehave
•�????????????=50
•�????????????=20
•σX=5
•σY=4
•????????????
????????????????????????=????????????
????????????????????????
????????????
????????????
=0.8∗
4
5
=0.64
•????????????
????????????????????????=????????????
????????????
????????????
????????????????????????
=0.8∗
5 4
=1
Examples
11
7
•Wehave
•�????????????=50
•�????????????=20
•σX=5
•σY=4
•????????????
????????????????????????=????????????
????????????????????????
????????????
????????????
=0.8∗
4
5
=0.64
•????????????
????????????????????????=????????????
????????????
????????????
????????????????????????
=0.8∗
5 4
=1
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