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Lack of Fit (LOF) Test. A formal F test for checking whether a specific type of regression function adequately fits the data. Example 1. Do the data suggest that a linear function is adequate in describing the relationship between skin cancer mortality and latitude?. Example 2.

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lack of fit lof test

Lack of Fit (LOF) Test

A formal F test for checking whether a specific type of regression function adequately fits the data

slide2

Example 1

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

slide3

Example 2

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

slide4

Example 3

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

lack of fit test for a linear function the basic idea
Lack of fit test for a linear function … the basic idea
  • Use general linear test approach.
  • Full model is most general model with no restrictions on the means μj at each Xj level.
  • Reduced model assumes that the μj are a linear function of the Xj, i.e., μj = β0+ β1Xj.
  • Determine SSE(F), SSE(R), and F statistic.
  • If the P-value is small, reject the reduced model (H0: No lack of fit (linear)) in favor of the full model (HA: Lack of fit (not linear)).
assumptions and requirements
Assumptions and requirements
  • The Y observations for a given X level are independent.
  • The Y observations for a given X level are normally distributed.
  • The distribution of Y for each level of X has the same variance.
  • LOF test requires repeat observations, called replications (or replicates), for at least one of the X values.
notation
Notation

iron wgtloss

0.01 127.6

0.01 130.1

0.01 128.0

0.48 124.0

0.48 122.0

0.71 110.8

0.71 113.1

0.95 103.9

1.19 101.5

1.44 92.3

1.44 91.4

1.96 83.7

1.96 86.2

  • c different levels of X (c=7 with X1=0.01, X2=0.48, …, X7=1.96)
  • nj = number of replicates for jth level of X (Xj) (n1=3, n2=2, …, n7=2) for a total of n = n1 + … + nc observations.
  • Yij = observed value of the responsevariable for the ith replicate of Xj (Y11=127.6, Y21=130.1, …, Y27=86.2)
the full model
The Full Model

Assume nothing about (or “put no structure on”) the means of the responses, μj, at the jth level of X:

Make usual assumptions about error terms (εij): normal, mean 0, constant variance σ2.

Least squares estimates of μj are sample means of responses at Xj level.

“Pure error sum of squares”

the reduced model
The Reduced Model

Assume the means of the responses, μj, are linearly related to the jth level of X (same model as before, just modified subscripts):

Make usual assumptions about error terms (εij): normal, mean 0, constant variance σ2.

Least squares estimates of μj are as usual.

“Error sum of squares”

error sum of squares decomposition
Error sum of squares decomposition

error deviation

pure error deviation

lack of fit deviation

the decision intuitively
The Decision (Intuitively)
  • If the largest portion of the error sum of squares is due to lack of fit, the F test should be large.
  • A large F* statistic leads to a small P-value (determined by F(c-2, n-2) distribution).
  • If P-value is small, reject null and conclude significant lack of (linear) fit.
lof test in minitab
LOF Test in Minitab
  • Stat >> Regression >> Regression …
  • Specify predictor and response.
  • Under Options…, under Lack of Fit Tests, select box labeled “Pure error.”
  • Select OK. Select OK. ANOVA table appears in session window.
slide15

Example 1

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

example 1 mortality and latitude
Example 1: Mortality and Latitude

Analysis of Variance

Source DF SS MS F P

Regression 1 36464 36464 99.80 0.000

Residual Error 47 17173 365

Lack of Fit 30 12863 429 1.69 0.128

Pure Error 17 4310 254

Total 48 53637

19 rows with no replicates

slide17

Example 2

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

example 2 alligator length and weight
Example 2: Alligator length and weight

Analysis of Variance

Source DF SS MS F P

Regression 1 342350 342350 117.35 0.000

Residual Error 23 67096 2917

Lack of Fit 17 66567 3916 44.36 0.000

Pure Error 6 530 88

Total 24 409446

14 rows with no replicates

slide19

Example 3

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

example 3 iron and corrosion
Example 3: Iron and corrosion

Analysis of Variance

Source DF SS MS F P

Regression 1 3293.8 3293.8 352.27 0.000

Residual Error 11 102.9 9.4

Lack of Fit 5 91.1 18.2 9.28 0.009

Pure Error 6 11.8 2.0

Total 12 3396.6

2 rows with no replicates

closing comment 1
Closing comment #1
  • The t-test or F=MSR/MSE test only tests whether there is a linear relation between the predictor and response (β1≠0) or not (β1=0).
  • Failing to reject the null does not imply that there is no relation between the predictor and response.
example closing comment 123
Example: Closing comment #1

The regression equation is

Y* = 14.1 - 0.100 X

Predictor Coef SE Coef T P

Constant 14.118 2.598 5.44 0.000

X -0.0998 0.6942 -0.14 0.887

S = 13.25 R-Sq = 0.1% R-Sq(adj) = 0.0%

Analysis of Variance

Source DF SS MS F P

Regression 1 3.6 3.6 0.02 0.887

Residual Error 24 4210.4 175.4

Lack of Fit 11 4188.3 380.8 223.87 0.000

Pure Error 13 22.1 1.7

Total 25 4214.0

closing comments 2 3
Closing comments #2, #3
  • We used general linear test approach to test appropriateness of a linear function. It can just as easily be used to test for appropriateness of other functions (quadratic, cubic).
  • The alternative HA: Lack of fit (not linear) includes all possible regression functions other than a linear one. Use residuals to help identify what type of function is appropriate.