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COMPLEMENTARY LOG LOG MODEL

Under the assumption of binary response, there are two alternatives to logit model: probit

model and complementary-log-log model.

They all follow the same form( ) ( )xx += (1)

for a continuous cdf .

Complementary log-log model says1

log{-log[1- ( )]} Tp n p

x X

= . The expression on

the left-hand side is called Complementary Log-Log transformation. Like the logit and

the probit transformation, the complementary log-log transformation takes a response

restricted to the (0,1) interval and converts it into something in ( , ) + interval. Here,

we need mentioned that the log of 1- (x) is always a negative number. This is changed

to a positive number before taking the log a second time. We can also write the model

down like form (1), as 1)]( ) 1 exp[ exp( Tp n px X = .

Both logit and probit links have the same property, which is link[(x)]=-link[1-(x)] .

This means that the response curve for (x) has a symmetric appearance about the point

(x) =0.5 and so (x) has the same rate for approaching 0 as well as for approaching 1.

When the data given is not symmetric in the [0,1] interval and increase slowly at small to

moderate value but increases sharply near 1. The logit and probit models areinappropriate. However, in this situation, the complementary log-log model might give a

satisfied answer.

PROBIT CLOGLOG AND LOGIT MODELS COMPARISON

0

0. 2

0. 4

0. 6

0. 8

1

1. 2

- 12 - 11 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8 9 10 11 12

Explanatory var i abl e

PI(X)

CLOGLOG

PROBI T

LOGI T

Unlike logit and probit the complementary log-log model is asymmetrical, it is frequentlyused when the probability of an event is very small or very large. Under the assumption

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that the general features are not lost, let us only consider a simple complementary log-log

model with one predictor (x)=1-exp[-exp( + x)] in the report left. The response has

an S-shaped curve, it approach 0 fairly slowly but approaching 1 quite sharply, when

>0.

compl ement ary l og l og model

0

0.2

0.4

0.6

0.8

1

1.2

- 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8

Explanatory var i abl e

PI(x)

bet a>0

bet a

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For 2 1 1x x = , the complement probability at 2x equals the complement probability at

1x raised to the power exp( ) .

Here, we will give out another related model (x)=exp[-exp( + x)] , it is called log-

log model. For it, (x) approaches 0 sharply but approaches 1 slowly. As x increases, the

curve is monotone decreasing when >0, and monotone increasing when 0

bet a

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F(t)=P(T )t . For fixed dosage x, the probability a randomly selected subject dies is

(x)=P(Y=1| X=x)=P(T x)=F(x)

That is the appropriate binary model is the one having the shape of the cdf F of the

tolerance distribution. Let denote the standard cdf for the family to which F belongs.

A common standardization uses the mean and standard deviation of T, so that

(x)=F(x)= [( ) / ]x Then, the model has form (x)= ( + x) .Let us use the beetle data as an example.

Beetles Killed after Exposure to Carbon Disulfide

Fitted Values

Log DoseNumber of

BeetlesNumber

KilledComp. Log-

LogProbit Logit

1.691 59 6 5.7 3.4 35

1.724 60 13 11.3 10.7 9.8

1.755 62 18 20.9 23.4 22.4

1.784 56 28 30.3 33.8 33.9

1.811 63 5 47.7 49.6 50.0

1.837 59 53 54.2 53.4 53.3

1.861 62 61 61.1 59.7 59.2

1.884 60 60 59.9 59.2 58.8

In the table, we find that the underlying cdf of number killed increases moderately beforex=1.811,then there is a big jump on the number of beetles killed.

SAS result:

For logistic proc:the logit model for beetles data 31

21:41 Sunday, March 14, 2004

The LOGISTIC Procedure

Model Fit Statistics

InterceptIntercept and

Criterion Only Covariates

AIC 647.441 376.354SC 651.617 384.706-2 Log L 645.441 372.354

Testing Global Null Hypothesis: BETA=0

Test Chi-Square DF Pr > ChiSq

Likeli hood Ratio 273.0869 1

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Intercep t 1 -60.7339 5.1814 137.3964

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logdos e 1 19.7408 1.4880 175.9925 ChiSq

Intercept 1 -34.9561 2.6413 -40.1330 -29.7793 175.15

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Model Information

Data Set WORK.BEETLES1Distribution BinomialLink Function LogitResponse Variable (Events) nkilledResponse Variable (Trials) nbeetlesObservations Used 8Number Of Events 291Number Of Trials 481

Criteria For Assessing Goodness Of Fit

Criterion DF Value Value/DF

Deviance 6 11.1156 1.8526Scaled Deviance 6 11.1156 1.8526Pearson Chi-Square 6 9.9067 1.6511Scaled Pearson X2 6 9.9067 1.6511Log Likelihood -186.1771

Algorithm converged.Analysis Of Parameter Estimates

Standard Wald 95% Confidence Chi-Parameter DF Estimate Error Limits Square Pr > ChiSq

Intercept 1 -60.7401 5.1819 -70.8964 -50.5838 137.40

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of survival is^

1 (x)=exp[-exp(-39.5224+22.0148 1.7)] = 0.884445 where at dosage

=1.8 it is 0.3296029, and at dosage =1.9,it is 4.39966e-05, the probability of survival at

dosage+0.1 equals the probability at dosage raised to the power

Exp(22.0148 0.1)=9.03838 . For instance, 9.038380.3296029 (0.884445)

(9.03838

0.3296027=(0.884445) )

Underlying the LOGISTIC proc:

Model intercept logdose Standard ErrorAIC with Intercept

and Covariates

logit -60.7339 34.2824 5.1814/2.9129 376.354

probit -34.9557 19.7408 2.6490/1.4880 375.226

Complementarylog-log

-39.5224 22.0148 3.2356/1.7970 368.753

Underlying the GENMOD proc:

Model intercept logdoseStandard

Error2Deviance-G DF

logit -60.7401 34.2859 5.1819 /2.9132 11.1156 6

probit -34.9561 19.7410 2.6413 /1.4853 9.9870 6

Complementary

log-log-39.5223 22.0148 3.2229/1.7899 3.5143 6

From the table we can find that under logistic proc, complementary log-log has the

smallest AIC =368.753, under Genmod proc, it is still complementary log-log that has the

smallest 2G 3.5143= .The last few things I need mentioned are there are

The reason is that GENMOD uses the Newton-Raphson algorithm to get the ML

estimates, and LOGISTIC uses iteratively reweighed least squares(also called Fisherscoring ), these two algorithms are equivalent for logit models but diverge for any other

model.(That is because logit is the unique canonical link function).

(With the coefficients, we can see with the one unit change on x, the logit will

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?????, the compare between coefficients and standard error)

Finally, we need mentioned that complementary log-log model is not only used for binaryrespond but also can be used for ordinal responses with cumulative link, the form is

j+log{ log[1 ( | x)]}= xTP Y j

and the ordinal model using this link is sometimes called a Proportional hazards model

for survival data to handle grouped survival times.

model using cumulative link

Bookmark:

Proof for symmetric property of logit distribution and probit model:

logit[(x)]=log[ (x)/(1- (x))]

=-log[(1- (x))/ (x)]

=-logit[1-(x)]

1

1

1 1

(x) + x

(x) ( + x)

(x) (x)

[ ]

[1 ]

[ ] [1 ]

=

=

=

log{ log[ (x)]}=log{-log(1-exp[-exp( + x)])}

The model says that

for large p, log-log close to logit

/*************storage plant***********************/

F(x)=exp{-exp[-(x-a)/b]}

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2 1 2 1

2

2 1

1

2 1

2 1

exp[ ( )]

( )

( )

log[ log(1 ( ))] log[ log(1 ( ))]

log[1 ( )]exp[ ]

log[1 ( )]

1 ( ) [1 ( )]

link[(x)]=-link[1-(x)](x)=exp[-exp( + x)]

log[-log(1- (x))]= + x

F(x)=exp{-exp[-(x-a)

x x

x x

xx

x

x x

xx

x

=

=

=

2 1

/b]}

1

(x)=exp[-exp( + x)]

F(t)=P(T )

(x)=P(Y=1| X=x)=P(T x)=F(x)

(x)=F(x)= [( ) / ]

(x)=1-exp[-exp( + x)]

(x)=1-exp[-exp( + x)]

link[(x)]=-link[1-(x)]

logit[(x)]=log[ (x)/(1- (x))]

x x

t

x

=

^

1

1

1)]

( , )

=-log[(1- (x))/ (x)]

=-logit[1-(x)]

log[-log( (x))]=

( ) 1 exp[ exp(

1 (x)=exp[-exp(-39.5224+22.0148 1.7)]

exp( )

(x) + x

(x) ( + x)

x

p [0.1 0.9]

[ ]

[1 ]

T

Tp n px X

+

=

=

=

1 1

9.03838

9.03838

2

(x) (x)log{ log[ (x)]}=log{-log(1-exp[-exp( + x)])}

log[-log(1- (x))]=-39.5224+22.0148 logdose

Exp(22.0148 0.1)=9.03838

0.3296029 (0.884445)

0.3296027=(0.884445)

Deviance-G 3.51

[ ] [1 ]

=

=

j

+

43

log{ log[1 ( | x)]}= xTP Y j

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2 1

2 1

2

1

2 1

2 1

2 1

exp[ ( )]

( )

( )

log[ log(1 ( ))] log[ log(1 ( ))]

log[1 ( )]exp[ ]

log[1 ( )]

1 ( ) [1 ( )]

link[(x)]=-link[1-(x)](x)=exp[-exp( + x)]

log[-log( (x))]= + x

F(x)=exp{-exp[-(x-a)/b

x x

x x

xx

x

x x

xx

x

=

=

=

2 1

]}

1

(x)=exp[-exp( + x)]

F(t)=P(T )

(x)=P(Y=1| X=x)=P(T x)=F(x)

(x)=F(x)= [( ) / ]

(x)=1-exp[-exp( + x)]

link[(x)]=-link[1-(x)]

logit[(x)]=log[ (x)/(1- (x))]

=-log[(1- (x))/ (x)

x x

t

x

=

1)]

( , )

]

=-logit[1-(x)]

log[-log( (x))]=

( ) 1 exp[ exp(

x

p [0.1 0.9]

T

Tp n px X

+

=

/*************************************/

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APPENDIX I

For logistic proc/ *t he begi n of sas codet he code i s used f or t wo goal , so ther e i s some r esul tt hat got f r om i t and use i t back t o t he code agai n f r o pl ot t i ng*/ data beet l es1;i nput l ogdose nbeet l es nki l l ed;nsur vi ve=nbeet l es- nki l l ed;/ *f or sampl e pr opor t i on*/

r pr ob1=nki l l ed/ nbeet l es;/ *f or l ogi t model */ yl ogi t =exp( - 60.7339+34.2824*l ogdose) / (1+exp( - 60.7339+34.2824*l ogdose) ) ;/ *f or compl ement ar y l og l og*/ ycl l = 1- exp( - exp( - 39.5224+22.0148*l ogdose) ) ;dat al i nes;

1. 691 59 61. 724 60 131. 755 62 181. 784 56 281. 811 63 521. 837 59 531. 861 62 61

1. 884 60 60;/ ***usi ng l ogi st i c pr oc f or t est compar i ng t he t hr ee ki nds of model */ proc logistic data= beet l es1; / * for l ogi t * / model nki l l ed/ nbeet l es = l ogdose ;t i t l e t he l ogi t model f or beet l es dat a ;

proc logistic dat a= beet l es1; / *f or cl ogl og*/ model nki l l ed/ nbeet l es = l ogdose / l i nk=cl ogl og OUTROC=cl l pl ot d;t i t l e t he compl ement ar y l og l og model f or beet l es dat a ;

proc logistic dat a= beet l es1; / *f or probi t */ model nki l l ed/ nbeet l es = l ogdose / l i nk=pr obi t ;t i t l e t he pr obi t model f or beet l es dat a ;

/ *pl ot a cur ve*/

symbol 1 col or =r ed val ue=st ar i nt erpol =NONE hei ght =1 wi dt h=1;symbol 2 col or =gr een val ue=pl us i nt er pol =spl i ne hei ght =1 wi dt h=1;symbol 3 col or =bl ue val ue=DI AMOND i nterpol =spl i ne hei ght =1 wi dt h=1;proc gplot data=beet l es1;pl ot r pr ob1*l ogdose yl ogi t *l ogdose ycl l *l ogdose/haxi s=1.65 t o 1.90 by.05 over l ay l egend=l egend2;t i t l e ' sampl e pr oport i on, cl l and l ogi t model compar i son' ;

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run;/ ***t he end of usi ng l ogi st i c f or t est compar i ng t he three ki nd ofmodel */ / *t he end of sas code*/

/*result begin*//*for logistic proc*/

the logit model for beetles data 3121:41 Sunday, March 14, 2004


Model Information

Data Set WORK.BEETLES1Response Variable (Events) nkilledResponse Variable (Trials) nbeetlesNumber of Observations 8Model binary logitOptimization Technique Fisher's scoring

Response Profile

Ordered Binary TotalValue Outcome Frequency

1 Event 2912 Nonevent 190

Model Convergence Status

Convergence criterion (GCONV=1E-8) satisfied.




AIC 647.441 376.354

SC 651.617 384.706-2 Log L 645.441 372.354



Likelihood Ratio 273.0869 1

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Association of Predicted Probabilities and Observed Responses

Percent Concordant 87.0 Somers' D 0.802Percent Discordant 6.8 Gamma 0.856Percent Tied 6.3 Tau-a 0.384Pairs 55290 c 0.901

-------------------------------------------------------------------------------------------------------------------the complementary log log model for beetles data 33

21:41 Sunday, March 14, 2004


Model Information

Data Set WORK.BEETLES1Response Variable (Events) nkilledResponse Variable (Trials) nbeetlesNumber of Observations 8Model binary cloglogOptimization Technique Fisher's scoring

Response Profile








AIC 647.441 368.753SC 651.617 377.105-2 Log L 645.441 364.753




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Model Information

Data Set WORK.BEETLES1Response Variable (Events) nkilledResponse Variable (Trials) nbeetlesNumber of Observations 8Model binary probitOptimization Technique Fisher's scoring

Response Profile








AIC 647.441 375.226SC 651.617 383.577-2 Log L 645.441 371.226




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1. 691 59 61. 724 60 131. 755 62 181. 784 56 281. 811 63 521. 837 59 531. 861 62 611. 884 60 60;

/ *t he end of sas code*/ proc genmod dat a= beet l es1;model nki l l ed/ nbeet l es= l ogdose / di st =bi n l i nk=pr obi t ;

proc genmod data= beet l es1;model nki l l ed/ nbeet l es= l ogdose / di st =bi n l i nk=cl ogl og;

proc genmod data= beet l es1;model nki l l ed/ nbeet l es= l ogdose / di st =bi n l i nk=l ogi t ;qui t ;

run;/ ** *t he end of usi ng genmode f or t est compar i ng t he thr ee ki nd of

model */

***************************************************************************************************************************The SAS System 23:18 Monday, March 15, 2004 4

The GENMOD Procedure

Model Information

Data Set WORK.BEETLES1Distribution BinomialLink Function ProbitResponse Variable (Events) nkilledResponse Variable (Trials) nbeetlesObservations Used 8Number Of Events 291

Number Of Trials 481






Intercept 1 -34.9561 2.6413 -40.1330 -29.7793 175.15

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Response Variable (Trials) nbeetlesObservations Used 8Number Of Events 291Number Of Trials 481






Intercept 1 -39.5223 3.2229 -45.8391 -33.2055 150.38

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clog log

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