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Chapter 11Regression and Correlation methods
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Learning ObjectivesDescribe the Linear Regression ModelState the Regression Modeling StepsExplain Ordinary Least SquaresCompute Regression CoefficientsUnderstand and check model assumptionsPredict Response VariableComments of SAS Output
EPI 809/Spring 2008As a result of this class, you will be able to...
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Learning Objectives Correlation ModelsLink between a correlation model and a regression modelTest of coefficient of Correlation
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Models
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What is a Model?
Representation of Some Phenomenon
Non-Math/Stats Model
Representation of Some Phenomenon
Non-Math/Stats Model
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What is a Math/Stats Model?Often Describe Relationship between Variables
TypesDeterministic Models (no randomness)
Probabilistic Models (with randomness)
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Deterministic ModelsHypothesize Exact RelationshipsSuitable When Prediction Error is NegligibleExample: Body mass index (BMI) is measure of body fat based
Metric Formula: BMI = Weight in Kilograms (Height in Meters)2
Non-metric Formula: BMI = Weight (pounds)x703
(Height in inches)2
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Probabilistic ModelsHypothesize 2 ComponentsDeterministicRandom ErrorExample: Systolic blood pressure of newborns Is 6 Times the Age in days + Random ErrorSBP = 6xage(d) + Random Error May Be Due to Factors Other Than age in days (e.g. Birthweight)
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Types of Probabilistic Models
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Regression Models
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Types of Probabilistic Models
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Regression ModelsRelationship between one dependent variable and explanatory variable(s)Use equation to set up relationship
Numerical Dependent (Response) Variable1 or More Numerical or Categorical Independent (Explanatory) VariablesUsed Mainly for Prediction & Estimation
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Regression Modeling Steps 1.Hypothesize Deterministic Component
Estimate Unknown Parameters2.Specify Probability Distribution of Random Error Term
Estimate Standard Deviation of Error3.Evaluate the fitted Model4.Use Model for Prediction & Estimation
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Model Specification
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Specifying the deterministic component1.Define the dependent variable and independent variable
2.Hypothesize Nature of RelationshipExpected Effects (i.e., Coefficients Signs)Functional Form (Linear or Non-Linear)Interactions
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Model Specification Is Based on Theory1.Theory of Field (e.g., Epidemiology)2.Mathematical Theory3.Previous Research4.Common Sense
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Thinking Challenge: Which Is More Logical?
Years since seroconversionCD+ countsCD+ countsYears since seroconversionYears since seroconversionYears since seroconversionCD+ countsCD+ counts
EPI 809/Spring 200817With positive linear relationship, sales increases infinitely.Discuss concept of relevant range.
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OB/GYN Study
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Types of Regression Models
EPI 809/Spring 200818This teleology is based on the number of explanatory variables & nature of relationship between X & Y.
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Types of Regression ModelsRegressionModels
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Types of Regression ModelsRegressionModelsSimple1 ExplanatoryVariable
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Types of Regression ModelsRegressionModels2+ ExplanatoryVariablesSimpleMultiple1 ExplanatoryVariable
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Types of Regression ModelsRegressionModelsLinear2+ ExplanatoryVariablesSimpleMultiple1 ExplanatoryVariable
EPI 809/Spring 200822This teleology is based on the number of explanatory variables & nature of relationship between X & Y.
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Types of Regression ModelsRegressionModelsLinearNon-Linear2+ ExplanatoryVariablesSimpleMultiple1 ExplanatoryVariable
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Types of Regression ModelsRegressionModelsLinearNon-Linear2+ ExplanatoryVariablesSimpleMultipleLinear1 ExplanatoryVariable
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Types of Regression ModelsRegressionModelsLinearNon-Linear2+ ExplanatoryVariablesSimpleMultipleLinear1 ExplanatoryVariableNon-Linear
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Linear Regression Model
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Types of Regression Models
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Linear Equations 1984-1994 T/Maker Co.
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Linear Regression Model1.Relationship Between Variables Is a Linear Function
YXiii01Dependent (Response) Variable(e.g., CD+ c.)Independent (Explanatory) Variable (e.g., Years s. serocon.)Population SlopePopulation Y-InterceptRandom Error
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Population & Sample Regression Models
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Population & Sample Regression ModelsPopulation
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Population & Sample Regression ModelsUnknown RelationshipPopulation
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Population & Sample Regression ModelsUnknown RelationshipPopulationRandom Sample
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Population & Sample Regression ModelsUnknown RelationshipPopulationRandom Sample
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Population Linear Regression ModelObservedvalueObserved valuei = Random error
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Sample Linear Regression ModelUnsampled observationi = Random errorObserved value^
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Estimating Parameters:Least Squares Method
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Scatter plot1.Plot of All (Xi, Yi) Pairs2.Suggests How Well Model Will Fit
02040600204060XY
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Thinking ChallengeHow would you draw a line through the points? How do you determine which line fits best? 02040600204060XY
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Thinking ChallengeHow would you draw a line through the points? How do you determine which line fits best?
02040600204060XYSlope changedIntercept unchanged
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Thinking ChallengeHow would you draw a line through the points? How do you determine which line fits best?
02040600204060XYSlope unchangedIntercept changed
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Thinking ChallengeHow would you draw a line through the points? How do you determine which line fits best?
02040600204060XYSlope changedIntercept changed
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Least Squares1.Best Fit Means Difference Between Actual Y Values & Predicted Y Values Are a Minimum. But Positive Differences Off-Set Negative ones
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Least Squares1.Best Fit Means Difference Between Actual Y Values & Predicted Y Values is a Minimum. But Positive Differences Off-Set Negative ones. So square errors!
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Least Squares1.Best Fit Means Difference Between Actual Y Values & Predicted Y Values Are a Minimum. But Positive Differences Off-Set Negative. So square errors!
2.LS Minimizes the Sum of the Squared Differences (errors) (SSE)
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Least Squares Graphically
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Coefficient EquationsPrediction equation
Sample slope
Sample Y - intercept
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Derivation of Parameters (1)Least Squares (L-S):
Minimize squared error
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Derivation of Parameters (1)Least Squares (L-S):
Minimize squared error
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Computation Table
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Xi
Yi
Xi2
Yi2
XiYi
X1
Y1
X12
Y12
X1Y1
X2
Y2
X22
Y22
X2Y2
:
:
:
:
:
Xn
Yn
Xn2
Yn2
XnYn
SXi
SYi
SXi2
SYi2
SXiYi
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Interpretation of Coefficients
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Interpretation of Coefficients1.Slope (1)Estimated Y Changes by 1 for Each 1 Unit Increase in X
If 1 = 2, then Y Is Expected to Increase by 2 for Each 1 Unit Increase in X^^^
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Interpretation of Coefficients1.Slope (1)Estimated Y Changes by 1 for Each 1 Unit Increase in X
If 1 = 2, then Y Is Expected to Increase by 2 for Each 1 Unit Increase in X2.Y-Intercept (0)Average Value of Y When X = 0
If 0 = 4, then Average Y Is Expected to Be 4 When X Is 0^^^^^
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Parameter Estimation ExampleObstetrics: What is the relationship betweenMothers Estriol level & Birthweight using the following data?
Estriol Birthweight (mg/24h)(g/1000) 1121324254
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Scatterplot Birthweight vs. Estriol levelBirthweightEstriol level
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Parameter Estimation Solution Table
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Xi
Yi
Xi2
Yi2
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 Solution
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Coefficient Interpretation Solution
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Coefficient Interpretation Solution1.Slope (1)Birthweight (Y) Is Expected to Increase by .7 Units for Each 1 unit Increase in Estriol (X)
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Coefficient Interpretation Solution1.Slope (1)Birthweight (Y) Is Expected to Increase by .7 Units for Each 1 unit Increase in Estriol (X)2.Intercept (0)Average Birthweight (Y) Is -.10 Units When Estriol level (X) Is 0
Difficult to explainThe birthweight should always be positive^^
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SAS codes for fitting a simple linear regressionData BW; /*Reading data in SAS*/input estriol birthw@@;cards;11 21 32 42 54; run;
PROC REG data=BW; /*Fitting linear regression models*/model birthw=estriol;run;
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Parameter Estimation SAS Computer Output
Parameter 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 0.03540^1^
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Parameter Estimation Thinking ChallengeYoure a Vet epidemiologist for the county cooperative. You gather the following data:Food (lb.) Milk yield (lb.) 43.0 65.5106.5129.0What is the relationship between cows food intake and milk yield?
1984-1994 T/Maker Co.
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Scattergram Milk Yield vs. Food intake*M. Yield (lb.)Food intake (lb.)
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Parameter Estimation Solution Table*
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Xi
Yi
Xi2
Yi2
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*
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Coefficient Interpretation Solution*
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Coefficient Interpretation Solution*1.Slope (1)Milk Yield (Y) Is Expected to Increase by .65 lb. for Each 1 lb. Increase in Food intake (X)
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Coefficient Interpretation Solution*1.Slope (1)Milk Yield (Y) Is Expected to Increase by .65 lb. for Each 1 lb. Increase in Food intake (X)
2.Y-Intercept (0)Average Milk yield (Y) Is Expected to Be 0.8 lb. When Food intake (X) Is 0
^^
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17With positive linear relationship, sales increases infinitely.Discuss concept of relevant range.18This teleology is based on the number of explanatory variables & nature of relationship between X & Y.19This teleology is based on the number of explanatory variables & nature of relationship between X & Y.20This teleology is based on the number of explanatory variables & nature of relationship between X & Y.21This teleology is based on the number of explanatory variables & nature of relationship between X & Y.22This teleology is based on the number of explanatory variables & nature of relationship between X & Y.23This teleology is based on the number of explanatory variables & nature of relationship between X & Y.24This teleology is based on the number of explanatory variables & nature of relationship between X & Y.24This teleology is based on the number of explanatory variables & nature of relationship between X & Y.26
27This teleology is based on the number of explanatory variables & nature of relationship between X & Y.28
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