Linear Modelling: Simple Regression - GitHub Pages · Simple Regression: 21 Aims: • To investigate linear correlation between two variables in more detail • Be able to predict

Linear Modelling: Simple Regression10th of May 2018 R. Nicholls / D.-L. Couturier / M. Fernandes

Introduction:

ANOVA•  Usedfortestinghypothesesregardingdifferencesbetweengroups•  Considersthevariationwithinandbetweengroups

Regression•  Usedforrevealingandinvestigatingrelationshipsbetweeninputandoutputvariables•  Modeldata,andextrapolateasmuchinformationaspossible

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Howtomeasurethestrengthofalinearrelationshipbetweenvariables?

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Positivelycorrelated:

Negativelycorrelated:

Uncorrelated:

Correlation:

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Pearson’sproduct-momentcorrelationcoefficient:

CoefficientofVariation(R2value):

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Correlation:

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r=0.931R2=0.866

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r=-0.949R2=0.901

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Correlation:

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r=-0.060R2=0.004

r=0.106R2=0.011

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Correlation:

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data:xandyt=17.613,df=48,p-value<2.2e-16alternativehypothesis:truecorrelationisnotequalto095percentconfidenceinterval:0.88025560.9602168sampleestimates:cor0.9305923

data:xandyt=1.5609,df=48,p-value=0.1251alternativehypothesis:truecorrelationisnotequalto095percentconfidenceinterval:-0.062380660.46941403sampleestimates:cor0.2197833

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CanIsaywhethermydataarecorrelated?Isanobservedcorrelationsignificant?

Simple Regression:

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Aims:•  Toinvestigatelinearcorrelationbetweentwovariablesinmoredetail•  Beabletopredictresponsegivenaknowledgeoftheindependentvariable

PredictorvariableIndependentvariable

ResponsevariableDependentvariable

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Simple Regression:

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εi=errors,residuals

Fortheithobservation:

yi

εixi

Simple Regression:

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Sohowdowefittheregressionline?Supposeweknowparameterestimatesand

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Observations:

yi

εixi

= !(! + !!+ !|!,!)

Simple Regression:

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Observations:Fittedvalues:

yi

εi

ŷi

xi


Simple Regression:

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! = !− !Residuals:

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εi

ŷi

xi


Simple Regression:

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εi

! = !− !

ŷi

Residuals: xi! = !+ !

Sohowdowefittheregressionline?Supposeweknowparameterestimatesand ! = !− !

Simple Regression:

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εi

! = !− !

ŷi


! ~ !(!,!!)


Simple Regression:

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εi

! = !− !

ŷi


! ~ !(!,!!)

!(!|!;!,!) = 12!!!

!!(!!!)!!!!


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Simple Regression:

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Sohowdowefittheregressionline?ObtainestimatesandMaximiselikelihoodofparametersgiventhedata

yi

εi

ŷi

xi

! = !− !

!(!|!;!,!) = 12!!!

!!(!!!)!!!!

! !,! !,! = !(!!|!!;!,!)!

= 12!!!!

!!(!!!!!)!

!!!

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yi

εi

ŷi

xi

! = !− !

!(!|!;!,!) = 12!!!

!!(!!!)!!!!

! !,! !,! = !(!!|!!;!,!)!

= 12!!!!

!!(!!!!!)!

!!!

ln! !,! !,! = !!! ln 2!!! − (!!!!!)!

!!!!

= !!! log 2!!

! − !!!! (!! − !!)!

!

! !,! !,! !"#

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Simple Regression:

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Optimalparameters:minimiseresidualsumofsquaresMaximumLikelihoodandLeastSquaresestimatesareequivalent(forGaussianerrorsmodel)

yi

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xi

! = !− !


! !,! !,! !"#

Simple Regression:

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Optimalparameters:minimiseresidualsumofsquaresMaximumLikelihoodandLeastSquaresestimatesareequivalent(forGaussianerrormodel)

! = !− !


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!

Simple Regression:

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! = !− !

Sohowdowefittheregressionline?ObtainestimatesandMaximiselikelihoodofparametersgiventhedataMinimisesumofsquaredresiduals

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!

Finalanswer:

Simple Regression:

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Example:Predictingtimbervolumeoffelledblackcherrytrees

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GirthVolume

> cor(trees$Volume,trees$Girth)[1] 0.9671194

> m1 = lm(Volume~Girth,data=trees)> summary(m1)

Call:lm(formula = Volume ~ Girth, data = trees)

Residuals: Min 1Q Median 3Q Max -8.065 -3.107 0.152 3.495 9.587

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -36.9435 3.3651 -10.98 7.62e-12 ***Girth 5.0659 0.2474 20.48 < 2e-16 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.252 on 29 degrees of freedomMultiple R-squared: 0.9353, Adjusted R-squared: 0.9331 F-statistic: 419.4 on 1 and 29 DF, p-value: < 2.2e-16

Response: y=VolumePredictor: x=Girth

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GirthVolume

Simple Regression:

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Residuals

Frequency

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Simple Regression:

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! = 4.252!! = 18.1

Residuals

Frequency

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Simple Regression:

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95%within±8.5

! = 4.252!! = 18.1

Linear Regression:

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Assumptions:1.  Modelislinearinparameters.

Linear Regression:

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Linear Regression:

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Linear Regression:

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Linear Regression:

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Linear Regression:

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2.  Gaussianerrormodel.

Linear Regression:

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3.  Additiveerrormodel.

Linear Regression:

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4.  Independenceoferrors.

Noautocorrelation–whenoneobservationdependsonthelast

Linear Regression:

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4.  Independenceoferrors.

Noautocorrelation–whenoneobservationdependsonthelast

5.  Homoscedasticity.Homogeneity/stabilityofvarianceoftheresiduals

Testing Assumptions: diagnostic plots

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1.  ResidualsvsFittedValues

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Fitted values

Residuals

lm(Volume ~ Girth)

Residuals vs Fitted

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•  Shouldnotberelated•  Novisiblepattern•  Meanresidual=zero•  Constantvariance

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Theoretical Quantiles

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lm(Volume ~ Girth)

Normal Q-Q

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1.  ResidualsvsFittedValues2.  NormalQuantile-Quantileplot

•  VisualtestforNormality•  Nostrongtrends/departures

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Scale-Location31

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1.  ResidualsvsFittedValues2.  NormalQuantile-QuantilePlot3.  Scale-LocationPlot

•  Testforhomoscedasticity•  Shouldbeconstant,≈1•  Notrend

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1.  ResidualsvsFittedValues2.  NormalQuantile-QuantilePlot3.  Scale-LocationPlot4.  IndexPlotofCook’sDistance

•  Measurestheinfluenceofaparticularobservation

•  Extremex-vals:highleverage•  Mayinformoutlierrejection

Modelling Non-Linear Relationships

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Linearmodelscanbeusedtodescribenon-linearrelationships…

Applyingtransformationstoresponseand/orpredictorvariablescanbeusefulto:•  Linearisethedata,i.e.maketherelationshipbetweenvariablesmorelinear.•  Stabilisethevarianceoftheresiduals,sothatσ2doesn’tdependonthe

independentvariable.•  Normalisethedistributionoftheresiduals


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Example:Stoppingdistanceofcarsversusspeed(mph)

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speed

dist

Response: y=distancePredictor: x=speed

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speed

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lm(dist ~ speed)

Residuals vs Fitted

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R2=0.651

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Residuals

lm(sqrt(dist) ~ speed)

Residuals vs Fitted

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speed

sqrt(dist)


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R2=0.651R2=0.709

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lm(log(dist) ~ log(speed))

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log(speed)

log(dist)


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R2=0.651R2=0.709R2=0.733

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R2=0.651R2=0.709R2=0.733

Call:lm(formula = log(dist) ~ log(speed), data = cars)

Residuals: Min 1Q Median 3Q Max -1.00215 -0.24578 -0.02898 0.20717 0.88289

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.7297 0.3758 -1.941 0.0581 . log(speed) 1.6024 0.1395 11.484 2.26e-15 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.4053 on 48 degrees of freedomMultiple R-squared: 0.7331, Adjusted R-squared: 0.7276 F-statistic: 131.9 on 1 and 48 DF, p-value: 2.259e-15


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Canyouusesimpleregressiontofitthismodel?

Non-linearMultiplicativeerrormodel


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Yes,solongasErrormodelislog-Normal.

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R2=0.651R2=0.709R2=0.733

Call:lm(formula = log(dist) ~ log(speed), data = cars)

Residuals: Min 1Q Median 3Q Max -1.00215 -0.24578 -0.02898 0.20717 0.88289

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.7297 0.3758 -1.941 0.0581 . log(speed) 1.6024 0.1395 11.484 2.26e-15 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.4053 on 48 degrees of freedomMultiple R-squared: 0.7331, Adjusted R-squared: 0.7276 F-statistic: 131.9 on 1 and 48 DF, p-value: 2.259e-15

! = !!!!!!

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R functions:

plot(x,y)cor(x,y)cor.test(x,y)

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data:xandyt=17.613,df=48,p-value<2.2e-16alternativehypothesis:truecorrelationisnotequalto095percentconfidenceinterval:0.88025560.9602168sampleestimates:cor0.9305923

Simple Regression in R:

Correlation Coefficients:

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R functions:

plot(x,y)m1 = lm(y~x)abline(m1)

summary(m1)





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R functions:


summary(m1)

r1 = residuals(r1)hist(r1)

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R functions:


summary(m1)

r1 = residuals(r1)hist(r1)

plot(m1)

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Residuals

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Scale-Location31

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Documents

Linear Modelling: Simple Regression - GitHub Pages · Simple Regression: 21 Aims: • To investigate linear correlation between two variables in more detail • Be able to predict