epi 809/spring 2008 1 chapter 11 regression and correlation methods
TRANSCRIPT
EPI 809/Spring 2008EPI 809/Spring 2008 11
Chapter 11Chapter 11
Regression and Correlation Regression and Correlation methodsmethods
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Learning ObjectivesLearning Objectives
1.1. Describe the Linear Regression ModelDescribe the Linear Regression Model
2.2. State the Regression Modeling StepsState the Regression Modeling Steps
3.3. Explain Ordinary Least SquaresExplain Ordinary Least Squares
4.4. Compute Regression CoefficientsCompute Regression Coefficients
5.5. Understand and check model assumptionsUnderstand and check model assumptions
6.6. Predict Response VariablePredict Response Variable
7.7. Comments of SAS OutputComments of SAS Output
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Learning Objectives… Learning Objectives…
8.8. Correlation ModelsCorrelation Models
9.9. Link between a correlation model and a Link between a correlation model and a regression modelregression model
10.10. Test of coefficient of CorrelationTest of coefficient of Correlation
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ModelsModels
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What is a Model?What is a Model?
1.1. Representation of Representation of
Some PhenomenonSome Phenomenon
Non-Math/Stats ModelNon-Math/Stats Model
1.1. Representation of Representation of
Some PhenomenonSome Phenomenon
Non-Math/Stats ModelNon-Math/Stats Model
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What is a Math/Stats Model?What is a Math/Stats Model?
1.1. Often Describe Relationship between Often Describe Relationship between VariablesVariables
2.2. TypesTypes- Deterministic Models (no randomness)Deterministic Models (no randomness)
- Probabilistic Models (with randomness)Probabilistic Models (with randomness)
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Deterministic ModelsDeterministic Models
1.1. Hypothesize Exact RelationshipsHypothesize Exact Relationships2.2. Suitable When Prediction Error is NegligibleSuitable When Prediction Error is Negligible3.3. Example: Body mass index (BMI) is measure of Example: Body mass index (BMI) is measure of
body fat basedbody fat based
Metric Formula: BMI = BMI = Weight in KilogramsWeight in Kilograms (Height in Meters) (Height in Meters)22
Non-metric Formula: BMI = BMI = Weight (pounds)x703Weight (pounds)x703 (Height in inches)(Height in inches)22
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Probabilistic ModelsProbabilistic Models
1.1. Hypothesize 2 ComponentsHypothesize 2 Components• DeterministicDeterministic• Random ErrorRandom Error
2.2. Example: Systolic blood pressure of newborns Example: Systolic blood pressure of newborns Is 6 Times the Age in days + Random ErrorIs 6 Times the Age in days + Random Error
• SBPSBP = 6xage(d) = 6xage(d) + + • Random Error May Be Due to Factors Random Error May Be Due to Factors
Other Than age in days (e.g. Birthweight)Other Than age in days (e.g. Birthweight)
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Types of Types of Probabilistic ModelsProbabilistic Models
ProbabilisticModels
RegressionModels
CorrelationModels
OtherModels
ProbabilisticModels
RegressionModels
CorrelationModels
OtherModels
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Regression ModelsRegression Models
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Types of Types of Probabilistic ModelsProbabilistic Models
ProbabilisticModels
RegressionModels
CorrelationModels
OtherModels
ProbabilisticModels
RegressionModels
CorrelationModels
OtherModels
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Regression ModelsRegression Models
Relationship between one Relationship between one dependentdependent variablevariable and and explanatory variable(s)explanatory variable(s)
Use equation to set up relationshipUse equation to set up relationship• NumericalNumerical Dependent (Response) Variable Dependent (Response) Variable• 1 or More Numerical or Categorical Independent 1 or More Numerical or Categorical Independent
(Explanatory) Variables(Explanatory) Variables
Used Mainly for Prediction & EstimationUsed Mainly for Prediction & Estimation
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Regression Modeling Steps Regression Modeling Steps
1.1. Hypothesize Deterministic Hypothesize Deterministic ComponentComponent
• Estimate Unknown ParametersEstimate Unknown Parameters
2.2. Specify Probability Distribution of Specify Probability Distribution of Random Error TermRandom Error Term
• Estimate Standard Deviation of ErrorEstimate Standard Deviation of Error
3.3. Evaluate the fitted ModelEvaluate the fitted Model 4.4. Use Model for Prediction & Use Model for Prediction &
Estimation Estimation
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Model SpecificationModel Specification
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Specifying the deterministic Specifying the deterministic componentcomponent
1.1. Define the dependent variable and Define the dependent variable and independent variableindependent variable
2.2. Hypothesize Nature of RelationshipHypothesize Nature of Relationship Expected Effects (i.e., Coefficients’ Signs)Expected Effects (i.e., Coefficients’ Signs) Functional Form (Linear or Non-Linear)Functional Form (Linear or Non-Linear) InteractionsInteractions
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Model Specification Model Specification Is Based on TheoryIs Based on Theory
1.1. Theory of Field (e.g., Theory of Field (e.g., Epidemiology)Epidemiology)
2.2. Mathematical TheoryMathematical Theory 3.3. Previous ResearchPrevious Research 4.4. ‘Common Sense’‘Common Sense’
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Thinking Challenge: Thinking Challenge: Which Is More Logical?Which Is More Logical?
Years since seroconversion
CD+ counts
CD+ counts
Years since seroconversion
Years since seroconversion
Years since seroconversion
CD+ counts
CD+ counts
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OB/GYN Study OB/GYN Study
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Types of Types of Regression ModelsRegression Models
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Types of Types of Regression ModelsRegression Models
RegressionModels
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Types of Types of Regression ModelsRegression Models
RegressionModels
Simple
1 Explanatory1 ExplanatoryVariableVariable
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Types of Types of Regression ModelsRegression Models
RegressionModels
2+ Explanatory2+ ExplanatoryVariablesVariables
Simple Multiple
1 Explanatory1 ExplanatoryVariableVariable
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Types of Types of Regression ModelsRegression Models
RegressionModels
Linear
2+ Explanatory2+ ExplanatoryVariablesVariables
Simple Multiple
1 Explanatory1 ExplanatoryVariableVariable
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Types of Types of Regression ModelsRegression Models
RegressionModels
LinearNon-
Linear
2+ Explanatory2+ ExplanatoryVariablesVariables
Simple Multiple
1 Explanatory1 ExplanatoryVariableVariable
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Types of Types of Regression ModelsRegression Models
RegressionModels
LinearNon-
Linear
2+ Explanatory2+ ExplanatoryVariablesVariables
Simple Multiple
Linear
1 Explanatory1 ExplanatoryVariableVariable
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Types of Types of Regression ModelsRegression Models
RegressionModels
LinearNon-
Linear
2+ Explanatory2+ ExplanatoryVariablesVariables
Simple Multiple
Linear
1 Explanatory1 ExplanatoryVariableVariable
Non-Linear
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Linear Regression Linear Regression ModelModel
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Types of Types of Regression ModelsRegression Models
RegressionModels
LinearNon-
Linear
2+ ExplanatoryVariables
Simple
Non-Linear
Multiple
Linear
1 ExplanatoryVariable
RegressionModels
LinearNon-
Linear
2+ ExplanatoryVariables
Simple
Non-Linear
Multiple
Linear
1 ExplanatoryVariable
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Y
Y = mX + b
b = Y-intercept
X
Changein Y
Change in X
m = Slope
Linear EquationsLinear Equations
© 1984-1994 T/Maker Co.
YY XXii ii ii 00 11
Linear Regression ModelLinear Regression Model
1.1. Relationship Between Variables Is Relationship Between Variables Is a Linear Functiona Linear Function
Dependent Dependent (Response) (Response) VariableVariable(e.g., CD+ c.)(e.g., CD+ c.)
Independent Independent (Explanatory) Variable (Explanatory) Variable (e.g., Years s. serocon.)(e.g., Years s. serocon.)
Population Population SlopeSlope
Population Population Y-InterceptY-Intercept
Random Random ErrorError
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Population & Sample Population & Sample Regression ModelsRegression Models
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Population & Sample Population & Sample Regression ModelsRegression Models
PopulationPopulation
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Population & Sample Population & Sample Regression ModelsRegression Models
Unknown Relationship
PopulationPopulation
Y Xi i i 0 1
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Population & Sample Population & Sample Regression ModelsRegression Models
Unknown Relationship
PopulationPopulation Random SampleRandom Sample
Y Xi i i 0 1
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Population & Sample Population & Sample Regression ModelsRegression Models
Unknown Relationship
PopulationPopulation Random SampleRandom Sample
Y Xi i i 0 1
Y Xi i i 0 1Y Xi i i 0 1
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Y
X
Y
X
Population Linear Regression Population Linear Regression ModelModel
Y Xi i i 0 1Y Xi i i 0 1
iXYE 10 iXYE 10
ObservedObservedvaluevalue
Observed valueObserved value
ii = Random error= Random error
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Y
X
Y
X
Y Xi i i 0 1Y Xi i i 0 1
Sample Linear Regression Sample Linear Regression ModelModel
Y Xi i 0 1 Y Xi i 0 1
Unsampled Unsampled observationobservation
ii = Random = Random
errorerror
Observed valueObserved value
^̂
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Estimating Parameters:Estimating Parameters:Least Squares MethodLeast Squares Method
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0204060
0 20 40 60
X
Y
Scatter plotScatter plot
1.1. Plot of All (Plot of All (XXii, , YYii) Pairs) Pairs
2.2. Suggests How Well Model Will FitSuggests How Well Model Will Fit
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Thinking ChallengeThinking Challenge
How would you draw a line through the How would you draw a line through the points? How do you determine which line points? How do you determine which line ‘fits best’? ‘fits best’?
0204060
0 20 40 60
X
Y
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Thinking ChallengeThinking ChallengeHow would you draw a line through the How would you draw a line through the points? How do you determine which line points? How do you determine which line ‘fits best’?‘fits best’?
0204060
0 20 40 60
X
YSlope changed
Intercept unchanged
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Thinking ChallengeThinking Challenge
How would you draw a line through the How would you draw a line through the points? How do you determine which line points? How do you determine which line ‘fits best’?‘fits best’?
0204060
0 20 40 60
X
Y
Slope unchanged
Intercept changed
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Thinking ChallengeThinking ChallengeHow would you draw a line through the How would you draw a line through the points? How do you determine which line points? How do you determine which line ‘fits best’?‘fits best’?
0204060
0 20 40 60
X
YSlope changed
Intercept changed
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Least SquaresLeast Squares
1.1. ‘Best Fit’ Means Difference Between ‘Best Fit’ Means Difference Between Actual Y Values & Predicted Y Values Are Actual Y Values & Predicted Y Values Are a Minimum. a Minimum. ButBut Positive Differences Off- Positive Differences Off-Set Negative onesSet Negative ones
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Least SquaresLeast Squares
1.1. ‘Best Fit’ Means Difference Between ‘Best Fit’ Means Difference Between Actual Y Values & Predicted Y Values is a Actual Y Values & Predicted Y Values is a Minimum. Minimum. ButBut Positive Differences Off-Set Positive Differences Off-Set Negative ones. Negative ones. So square errors!So square errors!
n
ii
n
iii YY
1
2
1
2ˆˆ
n
ii
n
iii YY
1
2
1
2ˆˆ
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Least SquaresLeast Squares
1.1. ‘Best Fit’ Means Difference Between ‘Best Fit’ Means Difference Between Actual Y Values & Predicted Y Values Are Actual Y Values & Predicted Y Values Are a Minimum. a Minimum. ButBut Positive Differences Off- Positive Differences Off-Set Negative. So square errors!Set Negative. So square errors!
2.2. LS Minimizes the Sum of the LS Minimizes the Sum of the Squared Differences (errors) (SSE)Squared Differences (errors) (SSE)
n
ii
n
iii YY
1
2
1
2ˆˆ
n
ii
n
iii YY
1
2
1
2ˆˆ
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Least Squares GraphicallyLeast Squares Graphically
2
Y
X
1 3
4
^^
^̂2
Y
X
1 3
4
^^
^^
Y X2 0 1 2 2 Y X2 0 1 2 2
Y Xi i 0 1 Y Xi i 0 1
LS minimizes ii
n2
112
22
32
42
LS minimizes ii
n2
112
22
32
42
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Coefficient EquationsCoefficient Equations
Prediction equationPrediction equation
Sample slopeSample slope
Sample Y - interceptSample Y - intercept
ii xy 10ˆˆˆ
21̂
xx
yyxxSS
SS
i
ii
xx
xy
xy 10 ˆˆ
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Derivation of Parameters (1)Derivation of Parameters (1)
Least Squares (L-S): Least Squares (L-S):
Minimize squared errorMinimize squared error
xy 10 ˆˆ
220 1
0 0
0 1
0
2
i i iy x
ny n n x
220 1
1 1
n n
i i ii i
y x
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Derivation of Parameters (1)Derivation of Parameters (1)
Least Squares (L-S): Least Squares (L-S):
Minimize squared errorMinimize squared error
220 1
1 1
0 1
1 1
0
2
2
i i i
i i i
i i i
y x
x y x
x y y x x
1
1
1̂
i i i i
i i i i
xy
xx
x x x x y y
x x x x x x y y
SS
SS
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Computation TableComputation Table
Xi Yi Xi2 Yi
2 XiYi
X1 Y1 X12 Y1
2 X1Y1
X2 Y2 X22 Y2
2 X2Y2
: : : : :
Xn Yn Xn2 Yn
2 XnYn
XiYi
Xi2 Yi
2 XiYi
Xi Yi Xi2 Yi
2 XiYi
X1 Y1 X12 Y1
2 X1Y1
X2 Y2 X22 Y2
2 X2Y2
: : : : :
Xn Yn Xn2 Yn
2 XnYn
XiYi
Xi2 Yi
2 XiYi
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Interpretation of CoefficientsInterpretation of Coefficients
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Interpretation of CoefficientsInterpretation of Coefficients
1.1. Slope (Slope (11)) Estimated Estimated YY Changes by Changes by 11 for Each 1 Unit for Each 1 Unit
Increase in Increase in XX• If If 11 = 2, then = 2, then YY Is Expected to Increase by 2 for Is Expected to Increase by 2 for
Each 1 Unit Increase in Each 1 Unit Increase in XX
^̂
^̂
^̂
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Interpretation of CoefficientsInterpretation of Coefficients
1.1. Slope (Slope (11)) Estimated Estimated YY Changes by Changes by 11 for Each 1 Unit for Each 1 Unit
Increase in Increase in XX• If If 11 = 2, then = 2, then YY Is Expected to Increase by 2 for Is Expected to Increase by 2 for
Each 1 Unit Increase in Each 1 Unit Increase in XX
2.2. Y-Intercept (Y-Intercept (00)) Average Value of Average Value of YY When When XX = 0 = 0
• If If 00 = 4, then Average = 4, then Average YY Is Expected to Be Is Expected to Be
4 When 4 When XX Is 0 Is 0
^̂
^̂
^̂
^̂
^̂
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Parameter Estimation ExampleParameter Estimation Example Obstetrics:Obstetrics: What is the What is the relationshiprelationship between between
Mother’s Estriol level & Birthweight using the Mother’s Estriol level & Birthweight using the following data?following data?
EstriolEstriol BirthweightBirthweight
(mg/24h)(mg/24h) (g/1000)(g/1000)
11 1122 1133 2244 2255 44
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0
1
2
3
4
0 1 2 3 4 5 6
Scatterplot Scatterplot Birthweight vs. Estriol level Birthweight vs. Estriol level
Birthweight
Estriol level
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Parameter Estimation Solution Parameter Estimation Solution TableTable
Xi Yi Xi2 Yi
2 XiYi
1 1 1 1 1
2 1 4 1 2
3 2 9 4 6
4 2 16 4 8
5 4 25 16 20
15 10 55 26 37
Xi Yi Xi2 Yi
2 XiYi
1 1 1 1 1
2 1 4 1 2
3 2 9 4 6
4 2 16 4 8
5 4 25 16 20
15 10 55 26 37
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Parameter Estimation SolutionParameter Estimation Solution
10.0370.02ˆˆ
70.0
515
55
51015
37ˆ
10
2
1
2
12
11
11
XY
n
X
X
n
YX
YX
n
i
n
ii
i
n
ii
n
iin
iii
10.0370.02ˆˆ
70.0
515
55
51015
37ˆ
10
2
1
2
12
11
11
XY
n
X
X
n
YX
YX
n
i
n
ii
i
n
ii
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iin
iii
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Coefficient Interpretation Coefficient Interpretation SolutionSolution
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Coefficient Interpretation Coefficient Interpretation SolutionSolution
1.1. Slope (Slope (11)) Birthweight (Birthweight (YY) Is Expected to Increase by .7 ) Is Expected to Increase by .7
Units for Each 1 unit Increase in Estriol (Units for Each 1 unit Increase in Estriol (XX))
^̂
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Coefficient Interpretation Coefficient Interpretation SolutionSolution
1.1. Slope (Slope (11)) Birthweight (Birthweight (YY) Is Expected to Increase by .7 ) Is Expected to Increase by .7
Units for Each 1 unit Increase in Estriol (Units for Each 1 unit Increase in Estriol (XX))
2.2. Intercept (Intercept (00)) Average Birthweight (Average Birthweight (YY) Is -.10 Units When ) Is -.10 Units When
Estriol level (Estriol level (XX) Is 0) Is 0• Difficult to explainDifficult to explain• The birthweight should always be positiveThe birthweight should always be positive
^̂
^̂
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SAS codes for fitting a simple linear SAS codes for fitting a simple linear regressionregression
DataData BW; /*Reading data in SAS*/ BW; /*Reading data in SAS*/ input input estriol birthwestriol birthw@@;@@; cards;cards; 11 11 2 2 1 1 33 22
44 2 2 5 5 44 ; ; runrun;;
PROC REGPROC REG data=BW data=BW; /*Fitting linear regression ; /*Fitting linear regression models*/models*/
model model birthwbirthw==estriolestriol;; runrun; ;
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Parameter EstimatesParameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 -0.10000 0.63509 -0.16 0.8849
Estriol 1 0.70000 0.19149 3.66 0.0354
Parameter Estimation Parameter Estimation SAS Computer OutputSAS Computer Output
0^̂ 1
^̂
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Parameter Estimation Thinking Parameter Estimation Thinking ChallengeChallenge
You’re a Vet epidemiologist for the county You’re a Vet epidemiologist for the county cooperative. You gather the following data:cooperative. You gather the following data:
Food (lb.)Food (lb.) Milk yield (lb.)Milk yield (lb.) 4 4 3.03.0 6 6 5.55.51010 6.56.51212 9.09.0
What is the What is the relationshiprelationship between cows’ food intake and milk yield?between cows’ food intake and milk yield?
© 1984-1994 T/Maker Co.
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02468
10
0 5 10 15
02468
10
0 5 10 15
Scattergram Scattergram Milk Yield vs. Food intake*Milk Yield vs. Food intake*
M. Yield (lb.)M. Yield (lb.)M. Yield (lb.)M. Yield (lb.)
Food intake (lb.)Food intake (lb.)Food intake (lb.)Food intake (lb.)
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Parameter Estimation Solution Parameter Estimation Solution Table*Table*
Xi Yi Xi2 Yi
2 XiYi
4 3.0 16 9.00 12
6 5.5 36 30.25 33
10 6.5 100 42.25 65
12 9.0 144 81.00 108
32 24.0 296 162.50 218
Xi Yi Xi2 Yi
2 XiYi
4 3.0 16 9.00 12
6 5.5 36 30.25 33
10 6.5 100 42.25 65
12 9.0 144 81.00 108
32 24.0 296 162.50 218
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Parameter Estimation Solution*Parameter Estimation Solution*
80.0865.06ˆˆ
65.0
432
296
42432
218ˆ
10
2
1
2
12
11
11
XY
n
X
X
n
YX
YX
n
i
n
ii
i
n
ii
n
iin
iii
80.0865.06ˆˆ
65.0
432
296
42432
218ˆ
10
2
1
2
12
11
11
XY
n
X
X
n
YX
YX
n
i
n
ii
i
n
ii
n
iin
iii
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Coefficient Interpretation Coefficient Interpretation Solution*Solution*
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Coefficient Interpretation Coefficient Interpretation Solution*Solution*
1.1. Slope (Slope (11)) Milk Yield (Milk Yield (YY) Is Expected to Increase ) Is Expected to Increase
by .65 lb. for Each 1 lb. Increase in Food by .65 lb. for Each 1 lb. Increase in Food intake (intake (XX))
^̂
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Coefficient Interpretation Coefficient Interpretation Solution*Solution*
1.1. Slope (Slope (11)) Milk Milk Yield (Yield (YY) Is Expected to Increase ) Is Expected to Increase
by .65 lb. for Each 1 lb. Increase inby .65 lb. for Each 1 lb. Increase in Food Food intakeintake ( (XX))
2.2. Y-Intercept (Y-Intercept (00)) Average Milk yield (Average Milk yield (YY) Is Expected to Be 0.8 ) Is Expected to Be 0.8
lb. When Food intake (lb. When Food intake (XX) Is 0) Is 0
^̂
^̂