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Residual AnalysisChapter 8



Model Assumptions Independence (response variables yi are

independent)- this is a design issue

Normality (response variables are normallydistributed)

Homoscedasticity (the response variables

have the same variance)



Best way to chec assumptions! chec the

assumptions on the random errors "hey are independent

"hey are normally distributed

"hey have a constant variance σ2 for all settings of the

independent variables (Homoscedasticity)

"hey have a zero mean.

I# these assumptions are satis#ied$ we may use the normal density

as the woring appro%imation #or the random component& 'o$the residuals are distributed as!

i N(*$+,)



lotting Residuals "o chec #or homoscedasticity (constant variance)!

roduce a scatterplot o# the standardi.ed residuals against

the #itted values&

roduce a scatterplot o# the standardi.ed residuals against

each o# the independent variables&

I# assumptions are satis#ied$ residuals should vary

randomly around .ero and the spread o# the residualsshould be about the same throughout the plot (no

systematic patterns&)



Homoscedasticity is probably violated i#/

"he residuals seem to increase or decrease inaverage magnitude with the #itted values$ it is

an indication that the variance o# the residualsis not constant&

"he points in the plot lie on a curve around

.ero$ rather than #luctuating randomly& A #ew points in the plot lie a long way #rom

the rest o# the points&



Heteroscedasticity Not #atal to an analysis0 the analysis is

weaened$ not invalidated& 1etected with scatterplots and recti#ied through

trans#ormation&

http!22www&p#c&c#s&nrcan&gc&ca2pro#iles2wulder2mvstats2trans#orm3e&html

http!22www&ru#&rice&edu2lane2stat3sim2trans#ormations2inde%&html



4autions with trans#ormations 1i##iculty o# interpretation o# trans#ormed

variables&

"he scale o# the data in#luences the utility o#trans#ormations& I# a scale is arbitrary$ a trans#ormation can be more

e##ective

I# a scale is meaning#ul$ the di##iculty o# interpretationincreases



Normality "he random errors can be regarded as a random

sample #rom a N(*$+,) distribution$ so we can chec

this assumption by checing whether the residuals

might have come #rom a normal distribution&

5e should loo at the standardi.ed residuals

6ptions #or looing at distribution!

Histogram$ 'tem and lea# plot$ Normal plot o# residuals

http!22statmaster&sdu&d2courses2st7772module*82inde%&html



Histogram or 'tem and 9ea# lot 1oes distribution o# residuals appro%imate a

normal distribution:

Regression is robust with respect to

nonnormal errors (in#erences typically still

valid unless the errors come #rom a highlysewed distribution)



Normal lot o# Residuals A normal probability plot is #ound by plotting the

residuals o# the observed sample against the

corresponding residuals o# a standard normal

distribution N (*$7) I# the plot shows a straight line$ it is reasonable to

assume that the observed sample comes #rom a normal

distribution&

I# the points deviate a lot #rom a straight line$ there is

evidence against the assumption that the random errors

are an independent sample #rom a normal distribution& http!22www&symar&com2resources2tools2normal3test3plot&asp



!;ariance o# the random error ε I# this value e<uals zero

all random errors e<ual *

rediction e<uation ( ŷ) will be e<ual to mean value E(y) I# this value is large

9arge (absolute values) o# ε

9arger deviations between ŷ and the mean value E(y).

"he larger the value o# $ the greater the error in estimating

the model parameters and the error in predicting a value o#

y #or speci#ic values o# x&

,σ

,σ



;ariance is estimated with (M'=)

"he units o# the estimated variance are squared units o# the dependent variable y.

>or a more meaning#ul measure o# variability$we use s or Root M'=.

"he interval ?, s provides a rough estimationwith which the model will predict #uturevalues o# y #or given values o# %&

, s



5hy do residual analysis: >ollowing any modeling procedure$ it is a

good idea to assess the validity o# your model&

Residuals and diagnostic statistics allow youto identi#y patterns that are either poorly #it bythe model$ have a strong in#luence upon theestimated parameters$ or which have a high

leverage& It is help#ul to interpret thesediagnostics @ointly to understand any potential problems with the model&



6utliers 5hat!

An observation with a residual that is larger than s or astandardi.ed residual larger than (absolute value)

5hy! A data entry or recording error$ 'ewness o# the

distribution$ 4hance$ nassignable causes

"hen what:

=liminate: 4orrect: Analy.e them: How much in#luence do they have: How do I now: Minitab 'torage! 1iagnostics (9everages$ 4ooCs

1istance$ 1>>I"')



9everages- p& 8* Identi#ies observations with unusual or outlying x-

values&

9everages #all between * and 7& A value greater than,(p2n) is large enough to suggest you should

e%amine the corresponding observation& p! number o# predictors (including constant)

n! number o# observations Minitab identi#ies observations with leverage over

(p2n) with an D in the table o# unusual observations



4ooCs 1istance (1)- p& 8*E An overall measure o# the combined impact o# each

observation on the #itted values& 4alculated using

leverage values and standardi.ed residuals&

4onsiders whether an observation is unusual with

respect to both x- and y-values&

6bservations with large 1 values may be outliers&

4ompare 1 to >-distribution with (p$ n-p) degrees o##reedom& 1etermine corresponding percentile& 9ess than ,*F- little in#luence

Greater than E*F- ma@or in#luence



1>>I"'- p& 8* "he di##erence between the predicted value when all

observations are included and when the ith observation isdeleted&

4ombines leverage and 'tudenti.ed residual (deleted tresiduals) into one overall measure o# how unusual anobservation is

Represent roughly the number o# standard deviations that the#itted value changes when each case is removed #rom the dataset& 6bservations with 1>>I"' values greater than , times

the s<uare root o# (p2n) are considered large and should bee%amined& p is number o# predictors (including the constant) n is the number o# observations



'ummary o# what to loo #or 'tart with the plot and brush outliers

9oo #or values that stand out in diagnostic measures

Rules o# thumb 9everages (HI)! values greater than ,(p2n)

4ooCs 1istance! values greater than E*F o# comparable >

(p$ n-p) distribution

1>>I"'! values greater than n

p

,



9etCs do an e%ample

Returning to the Naval Base data set



>inally$ are the residuals correlated:

1urbin-5atson d statistic (p& 87E)

Range! * d 8

ncorrelated! d is close to , ositively correlated! d is closer to .ero

Negatively correlated! d is closer to 8&

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