adecuacion del modelo en reg

7/30/2019 Adecuacion Del Modelo en Reg

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Chapter 4 Model Adequacy Checking

Ray-Bing Chen

Institute of StatisticsNational University of Kaohsiung


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4.1 Introduction

The major assumptions:

1. The relationship between y and xs is linear.

2. The error term has zero mean.

3. The error term has constant variance, 2

4. The errors are uncorrelated

5. The errors are normally distributed. 4. and 5. imply the errors are independent, and 5.

for the hypothesis testing and C.I.


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Gross violations of the assumptions may yield an

unstable model with opposite conclusions.

The standard summary statistics: t-; F- statisticsand R2 can not detect the departures from the

underlying assumptions.

Based on the study of the model residuals.


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4.2 Residual Analysis

4.2.1 Definition of Residuals

Residual:

The deviation between the data and the fit

A measure of the variability in the response

variable not explained by the regression model.

The realized or observed values of the modelerrors.

Plot residuals

niyyeiii

,,1,


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Properties:


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4.2.2 Methods for Scaling Residuals

Scaling Residuals is helpful in detecting the

outliers or extreme values.

Standardized Residuals:


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Studentized Residuals:

The residual vector:

e=(I-H)y,

where H=X(XX)-1X is the hat matrix.

Since H is symmetric and idempotent, (I-H) is

also symmetric and idempotent.Then


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Hence

Var(ei) = 2(1-hii)

Cov(ei, ej) = -2hijStudentized residuals:


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ri > di

Constant variance, Var(ri) = 1

ri and di may be little difference and oftenconvey equivalent information.

PRESS Residuals:The prediction error (PRESS residuals)

is the fitted value of the ith response basedon all observations except the ith one.

From Appendix C7,

)()(

iii yye

)(

i

y

ii

ii

h

ee

1

)(


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The variance of the ith PRESS residual is

A standardized PRESS residual:

A studentized PRESS residual:

iiii

ii

hh

eVareVar

1

)

1

()(2

)(

)1()( 2

)(

)(

ii

i

i

i

h

e

eVar

e

)1()( Re)(

)(

iis

i

i

i

hMS

e

eVar

e


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R-Student:

Estimate variance based on a data set with the

ith observation removed.From Appendix C.8,

R-student

If the ith observation is influential, then candiffer significantly from MSRes , and thus R-

student statistic will be more sensitive to this

point.

1

)1/()( Re2)(

pn

heMSpnS iiisi

)1(2)( iii

ii

hS

et

2

)(iS


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Example 4.1 The Delivery Time Data

See Table 4.1


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4.2.3 Residual Plot

Graphical analysis is a very effective way to

investigate the adequacy of the fit of a regressionmodel and to check the underlying assumption.

Normal Probability Plot:If the errors come from a distribution with

thicker or heavier tails than the normal, LS fit

may be sensitive to a small subset of the data.

Heavy-tailed error distributions often generateoutliers that pull LS fit too much in their

direction.


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Normal probability plot: a simple way to check

the normal assumption.

Ranked residuals: e[1]< < e[n]Plot e[i] against Pi = (i-1/2)/n

Sometimes plot e[i] against -1[ (i-1/2)/n]

Plot nearly a straight line for large sample n >32 if e[i] normal

Small sample (n


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Fitting the parameters tends to destroy the

evidence of nonnormality in the residuals, and

we cannot always rely on the normalprobability to detect departures from normality.

Defect: Occurrence of one or two large

residuals. Sometimes this is an indication that

the corresponding observations are outliers.


Fig 4.2 (a) The original LS residuals

Fig 4.2 (b) The R-student residuals

There may be one or more outliers in the data.


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Plot of Residuals against the Fitted Values:

From Fig 4.3:

1. Fig 4.3a: Satisfactory

2. Fig 4.3b: Variance is an increase function of y

3. Fig 4.3c: Often occurs when y is a proportionbetween 0 and 1.

4. Fig 4.3d: Indicate nonlinearity.

For 2. and 3., use suitable transformations toeither the regressor or the response variable or

use the method of weighted LS.

For 4., except the above two methods, the

other regressors are needed in the model.

correlatedareandanded,uncorrelatareand iiii yeye


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Fig 4.4 (a)

Fig 4.4 (b)

Both plots do not exhibit any strong unusual

pattern.

Plot of Residuals against the Regressor:

These plots often exhibit patterns such as those

in Fig 4.3.

In the simple linear regressor case, it is not

necessary to plot residuals v.s. both fitted

values and the regressors.

iyv.s.ie

iyv.s.it


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Fig 4.5(a): Plot R-student v.s. case

Fig 4.5(b): Plot R-student v.s. distance

Plot of Residuals in Time Sequence:

The time sequence plot of residuals mayindicate that the errors at one time period are

correlated with those at other time periods.

Autocorrelation: The correlation between

model errors at different time periods.

Fig 4.6(a) positive autocorrelation

Fig 4.6(b) negative autocorrelation


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4.2.4 Partial Regression and Partial Residual Plots

The plots in Section 4.2.3 may not completely

show the correct or complete marginal effect of aregressor given the other regressors in the model.

A partial regression plot(added variable plot or

adjusted variable plot) is a variation of the plot

of residuals v.s. the predictor.

This plot can be used to provide information about

the marginal usefulness of a variable that is not

currently in the model. This plot consider the marginal role of xj given

other regressors that are already in the model.


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Consider y = 0 + 1 x1 + 2 x2 + . We concern

the relationship y and x1

First regress y on x2, and obtain the fitted valuesand residuals:

Then regress x1 on x2, and calculate the residuals:nixyyxye

xxy

iii

ii

,,2,1),(

)|(

)(

22

2102

nixxxxxe

xxx

iii

ii

,,2,1),(

)|(

)(

21121

21021


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Plot the y residuals ei(y|x2) against the x1 residuals

ei(x1|x2).

If the regressor x1 enters the model linearly, thenthe partial regression plot should show a linear

relationship.

If the partial regression plot shows a curvilinear

band, then the higher-order terms in x1 or a

transformation may be helpful.

When x1 is a candidate variable begin considered

for inclusion in the model, a horizontal bandindicates that there is no additional useful

information in x1 for predicting y.


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Fig 4.7 (a) for x1

Fig 4.7 (b) for x2The linear relationship between cases and

distance is clearly evident in both of these plots.

Obs. 9 falls somewhat off the straight line thatapparently well describes the rest of the data.


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Some Comments on Partial Regression Plots:

Partial regression plots need to be used with

caution as they only suggest possible relationship

between the regressor and the response. These

plots may not give information about the proper

form of the relationship if several variablesalready in the model are incorrectly specified. It

will usually be necessary to investigate several

alternate forms for the relationship between the

regressor and y or several transformations.Residual plots for these subsequent models should

be examined to identify the best relationship or

transformation.


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Partial regression plots will not, in general, detect

interaction effects among the regressors.

The presence of strong multicollinearity can causepartial regression plots to give incorrect

information about the relationship between the

response and the regressor variables.

It is fairly easy to give a general development of

the partial regression plotting concept that shows

clearly why the slope of the plot should be the

regression coefficient for the variable of interest.


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Partial regression plot: e[y|X(j)] v.s. e[xj|X(j)]

y = X + = X(j) + j xj +

(I - H(j)) y = (I - H(j)) X(j) + j (IH(j)) xj +(I_H(j))

Then (IH(j)) y = j (IH(j)) xj + (I_H(j))

That is e[y|X(j)] = j e[xj|X(j)] + *

This suggests that a partial regression plot shouldhave slope, j.

Partial Residual plot:

The partial residual for regressor xj

ej are the residuals of full model.

ijjiji xexye )|(*


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4.2.5 Other Residual Plotting and Analysis Methods

It may be very useful to construct a scatterplot of

regressors xi v.s. xj. This plot may be useful instudying the relationship between regressor

variables and the disposition of the data in the x-

space.

Fig 4.9 is a scatterrplot of x1 v.s. x2 for the

delivery time data from Example 3.1

The problem situation often suggests other types

of residual plots. Fig 4.10: plot the residuals by sites.


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4.3 The Press Statistic

The PRESS residuals:

The PRESS statistic:

PRESS is generally regarded as a measure of how

well a regression model will perform in predictingnew data.

Small PRESS

)()(

iii yye

2

11

2)(

1)(

n

i ii

in

i

iih

eyyPRESS


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Example 4.6 Delivery Time Data


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An R2- like statistic for prediction (based on PRESS):

In Example 3.1,

We expect this model to explain about 92.09% of

variability in predicting new observations.

Use PRESS to compare models: A model with

small PRESS is preferable to one with large

PRESS.

Tediction SS

PRESSR 12

Pr

9209.012Pr T

ediction

SS

PRESSR


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4.4 Detection and Treatment of Outliers

An outlier is an extreme observation with larger

residual (in absolute value) than others, say 3 or 4

standard deviations from the mean. Outliers are data points that are not typical of the

rest of the data.

Identifying outliers: Residual plots against fitted

values, normal probability plot, examining scaledresiduals (studentized and R-student residuals)


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Bad values?? (A result of unusual but explainable

event) Discard bad values!!

An unusual but perfectly plausible observations Outliers may control many key model properties

and may also point out inadequancies in the model.

Various statistical tests have been proposed for

detecting and rejecting outliers:

Identifying the outliers based on the maximum

normed residual

n

i

ii ee1

2/


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Outliers: Keep or drop??

Effect of outliers on regression may be checked

by dropping these points and refitting theregression equation.

t-, F-statistics, R2 and residual mean square

may be very sensitive to the outliers.

Situation in which a relatively small % of the

data has a significant impact on the model may

not be acceptable to the user of the regression

equation. Generally we are happier about assuming that a

regression equation is valid if it is not overly

sensitive to a few observations.


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Example 4.7 The Rocket Propellant data

Fig 4-11 and Fig 4-12

From Fig 4-11, observation 5 and 6 should be theoutliers.


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Deleting observation 5 and 6 has almost no effect

on the estimate of the regression coefficients.

A dramatic reduction in MSRes.

A one-third reduction in the standard error of

estimate of1.

Observation 5 and 6 are not overly influential. No particular reason for unusually low propellant

shear strengths obtained for observation 5 and 6.

Should not discard these two points.


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4.5 Lack of Fit the Regression Model

4.5.1 A Formal Test for Lack of Fit

Assume normality, independence, and constant

variance. Only the simple linear relationship is in doubt.

See Fig 4.13

Requirement: have replicate observations on y forat least one level of x.

True replication: Run n separate experiments at x.


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These replicated observations are used to obtain a

model-independent estimate of 2.

There are ni observations on the response at the ithlevel of the regressor xi, i=1,2,,m.

Let yij be the jth observation on the response at xi,

i=1,2,,m and j =1, , ni.

Hence there are n = (n1+ + nm) total

observations.

Partition SSRes into two components:

SSRes = SSPE + SSLOF

where

m

i

iii

m

i

n

j

m

i

n

j

iijiij yynyyyyi i

1

2

1 1 1 1

22 )()()(


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The pure error sum of squares (model-independentmeasure of pure error)

The degree of freedom for pure error:

The sum of squares due to lack of fit with degreeof freedom m-2

If the fitted values are closed to the correspondingaverage responses, then the regression functionshould be linear.

m

i

n

j

iijPE

i

yySS1 1

2

)(

m

iiiiLOF yynSS 1

2

)

(

m

i

i mnn

1

)1(


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The test statistic for lack of fit:

If we conclude that the regression function is not

linear, then the tentative model must be abandonedand attempts made to find a more appropriate

equation.

mnm

PE

LOF

PE

LOF FMS

MS

mnSS

mSSF

,20 ~

)/(

)2/(


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Even though F-ratio for lack of fit is not

significant, and the hypothesis of significance of

regression is rejected, this still does not guarantee

that the model will be satisfactory as a prediction

equation.

The variation of the predicted values is large

relative to the random error. The work of Box and Wetz (1973) suggests that

the observed F ratio must be at least four or five

times the critical value from the F table if the

regression model is to be useful as a predictor.


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A simple measure of potential prediction

performance: Compare the range of the fitted

values, , to their average standard error.

The average variance of the fitted values:

Satisfactory predictor: the range of the fitted

values ( ) is large relative to their

average estimated standard error , where

is a model-independent estimate of the errorvariance.

minmax

yy

minmax yy

np / 2

2


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Example 4.8 Testing for Lack of Fit (Data in Fig

4.13)


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i i f f i hb


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4.5.2 Estimation of Pure Error from Near-Neighbors

In above subsection,

SSRes = SSPE + SSLOFSSPE is computed using responses at repeat

observations at the same level of x. This is a

model-independent estimate of

2

. Repeat observations on y at the same xsome of

rows of X are the same!

Daniel and Wood (1980) and Joglekar et al. (1989)

have investigated methods for obtaining a model-independent estimate of error when there are no

exact repeat points.


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Near-neighbors: sets of observations that have

been taken with nearly identical levels of x1, ,

xk. Then the observations yi from such near-neighbors can be considered as repeat points and

used to obtain an estimate of pure error.

The weighted sum of squared distance (WSSD):

Near-neighbors: small value of . That is pointsare relatively close together in x-space. If is

large, the points are widely separated in x-space.

k

j s

jiijj

iiMS

xxD

1

2

Re

'2

'

)(

2'iiD

2

'iiD


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The estimate is based on

For samples of size 2,

Algorithm:

Arrange the data points xi1, , xikin order of

increasing the predictor,

Compute the values . Finally there are 4n-10

values.

Arrange the 4n-10 values in ascending order.

Let Eu be the range of the residuals at thesepoints.

'iii eeE

EE 886.0)128.1(

1

iy

2

'iiD


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Then the estimate of the standard deviation of

pure error is


Table 4.3

Use 15 smallest values of

In this case,

If there is no appreciable lack of fit, we would

expect

Here

Some lack of fit!!!

m

u

uEm 1

886.0

2

'iiD

969.1

sMSRe

259.3Re sMS

adecuacion del modelo en reg

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