Linear Lack of Fit (LOF) Test

Linear Lack of Fit (LOF) Test

An F test for checking whether a linear regression function is inadequate in

describing the trend in the data

Where does this topic fit in?

• Model formulation

• Model estimation

• Model evaluation

• Model use

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Latitude

Mo

rta

lity

S = 19.1150 R-Sq = 68.0 % R-Sq(adj) = 67.3 %Mortality = 389.189 - 5.97764 Latitude

Regression Plot

Example 1

Do the data suggest that a linear function is inadequate in describing the relationship between skin cancer mortality and latitude?

Example 2

Do the data suggest that a linear function is inadequate in describing the relationship between the length and weight of an alligator?

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Length

We

ight

S = 54.0115 R-Sq = 83.6 % R-Sq(adj) = 82.9 %Weight = -393.264 + 5.90235 Length

Regression Plot

Example 3

Do the data suggest that a linear function is inadequate in describing the relationship between iron content and weight loss due to corrosion?

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Iron content

We

ight

loss

S = 3.05778 R-Sq = 97.0 % R-Sq(adj) = 96.7 %wgtloss = 129.787 - 24.0199 iron

Regression Plot

Some notation

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Size of minimum deposit

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xy 49.07.50ˆ

2811 y

4212 y

351 y

12462 y

1146 y

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2.87ˆˆ 1211 yy 1.148ˆˆ 6261 yy

Decomposing the error

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i jijij yy

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i jiji yy

11482

i jiij yy


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The basic idea

• Break down the residual error (“error sum of squares – SSE) into two components:– a component that is due to lack of model fit

(“lack of fit sum of squares” – SSLF)– a component that is due to pure random error

(“pure error sum of squares” – SSPE)

• If the lack of fit sum of squares is a large component of the residual error, it suggests that a linear function is inadequate.

A geometric decomposition

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iijijiijij yyyyyy ˆˆ

ijy

ijij xbby 10ˆ

iy

The decomposition holds for the sum of the squared deviations, too:

c

i

n

jiij

c

i

n

jiji

c

i

n

jijij

iii

yyyyyy1 1

2

1 1

2

1 1

2 ˆˆ

Error sum of squares (SSE)

Lack of fit sum of squares (SSLF)

Pure error sum of squares (SSPE)

SSPESSLFSSE

Breakdown of degrees of freedom

cncn 22

Degrees of freedom associated with SSE

Degrees of freedom associated with SSLF

Degrees of freedom associated with SSPE

Definitions of Mean Squares

And, the pure error mean square (MSPE) is defined as:

The lack of fit mean square (MSLF) is defined as:

cn

SSPE

cn

yyMSPE iij

2

22

ˆ 2

c

SSLF

c

yyMSLF iji

Expected Mean Squares

2

)(2

102

c

XnMSLFE iii

2)( MSPEE

• If μi = β0+β1Xi, we’d expect the ratio MSLF/MSPE to be …

• If μi ≠ β0+β1Xi, we’d expect the ratio MSLF/MSPE to be …

• Use ratio, MSLF/MSPE, to reject whether or not μi = β0+β1Xi.

Expanded Analysis of Variance Table

Source DF SS MS F

Regression 1

Residual error

n-2

Lack of fit c-2

Pure error n-c

Total n-1

c

i

n

jijij

i

yySSE1 1

2ˆ

c

i

n

jij

i

yySSTO1 1

2

c

i

n

jij

i

yySSR1 1

2ˆ1

SSRMSR

2n

SSEMSE

MSE

MSRF

c

i

n

jiji

i

yySSLF1 1

2ˆ

c

i

n

jiij

i

yySSPE1 1

2

2c

SSLFMSLF

cn

SSPEMSPE

MSPE

MSLFF

The formal lack of fit F-test

Null hypothesis H0: μi = β0+β1XiAlternative hypothesis HA: μi ≠ β0+β1Xi

Test statisticMSPE

MSLFF *

P-value = What is the probability that we’d get an F* statistic as large as we did, if the null hypothesis is true?

The P-value is determined by comparing F* to an F distribution with c-2 numerator degree of freedom and n-c denominator degrees of freedom.

LOF Test in Minitab

• Stat >> Regression >> Regression …

• Specify predictor and response.

• Under Options…– under Lack of Fit Tests, select the box labeled

Pure error.

• Select OK. Select OK.


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Is there lack of linear fit?

Analysis of Variance

Source DF SS MS F PRegression 1 5141 5141 3.14 0.110Residual Error 9 14742 1638 Lack of Fit 4 13594 3398 14.80 0.006 Pure Error 5 1148 230Total 10 19883

1 rows with no replicates


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i jijij yy

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Is there lack of linear fit?


Source DF SS MS F PRegression 1 5448.9 5448.9 1087.06 0.000Residual Error 9 45.1 5.0 Lack of Fit 4 6.6 1.7 0.21 0.919 Pure Error 5 38.5 7.7Total 10 5494.0


Example 1

Do the data suggest that a linear function is not adequate in describing the relationship between skin cancer mortality and latitude?

504030

200

150

100

Latitude

Mo

rta

lity

S = 19.1150 R-Sq = 68.0 % R-Sq(adj) = 67.3 %Mortality = 389.189 - 5.97764 Latitude

Regression Plot

Example 1: Mortality and Latitude




Example 2

Do the data suggest that a linear function is not adequate in describing the relationship between the length and weight of an alligator?

150140130120110100 90 80 70 60

700

600

500

400

300

200

100

0

Length

We

ight

S = 54.0115 R-Sq = 83.6 % R-Sq(adj) = 82.9 %Weight = -393.264 + 5.90235 Length

Regression Plot

Example 2: Alligator length and weight




Example 3

Do the data suggest that a linear function is not adequate in describing the relationship between iron content and weight loss due to corrosion?

210

130

120

110

100

90

80

Iron content

We

ight

loss

S = 3.05778 R-Sq = 97.0 % R-Sq(adj) = 96.7 %wgtloss = 129.787 - 24.0199 iron

Regression Plot

Example 3: Iron and corrosion


Source DF SS MS F PRegression 1 3293.8 3293.8 352.27 0.000Residual Error 11 102.9 9.4 Lack of Fit 5 91.1 18.2 9.28 0.009 Pure Error 6 11.8 2.0Total 12 3396.6


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mileage

gro

ove

Example 4

Do the data suggest that a linear function is not adequate in describing the relationship between mileage and groove depth?

Example 4: Tread wear


Source DF SS MS F PRegression 1 50887 50887 140.71 0.000Residual Error 7 2532 362Total 8 53419

No replicates. Cannot do pure error test.

When is it okay to perform the LOF Test?

• When the “INE” part of the “LINE” assumptions are met.

• The LOF test requires repeat observations, called replicates, for at least one of the values of the predictor X.

Documents

Linear Lack of Fit (LOF) Test