Introduction to Probability and Statistics for Engineers and Scientists

412 Chapter 9: Regression

LetYibe the result (either 1 if a success, or 0 if a failure) of the experiment when performed
at levelxi. Then, using the Bernoulli density function (that is, the binomial density for
a single trial), gives

P{Yi=yi}=[p(xi)]yi[ 1 −p(xi)]^1 −yi=

(

ea+bx

1 +ea+bx

)yi(

1

1 +ea+bx

) 1 −yi

, yi=0, 1

Thus, the probability that the experiment at levelxi results in outcomeyi, for all
i=1,...,k,is

P{Yi=yi,i=1,...,k}=

∏

i

(

ea+bxi

1 +ea+bxi

)yi(

1

1 +ea+bxi

) 1 −yi

=

∏

i

(

ea+bxi

)yi

1 +ea+bxi

Taking logarithms gives that

log

(

P{Yi=yi,i=1,...,k}

)

=

∑k

i= 1

yi(a+bxi)−

∑k

i= 1

log

(

1 +ea+bxi

)

The maximum likelihood estimates can now be obtained by numerically finding the values
ofaandbthat maximize the preceding likelihood. However, because the likelihood
is nonlinear this requires an iterative approach; consequently, one typically resorts to
specialized software to obtain the estimates.
Whereas the logistic regression model is the most frequently used model when the
response data are binary, other models are often employed. For instance in situations
where it is reasonable to suppose thatp(x), the probability of a positive response when the
input level isx, is an increasing function ofx, it is often supposed thatp(x) has the form of
a specified probability distribution function. Indeed, whenb>0, the logistic regression
model is of this form becausep(x) is equal to the distribution function of a logistic random
variable (Section 5.9) with parametersμ=−a/b,ν=1/b. Another model of this type is
theprobitmodel, which supposes that for some constants,α,β> 0

p(x)= (α+βx)=

1

√

2 π

∫α+βx

−∞

e−y

(^2) /2
dy
In other wordsp(x) is equal to the probability that a standard normal random variable
is less thanα+βx.
EXAMPLE 9.11a A common assumption for whether an animal becomes sick when
exposed to a chemical at dosage levelxis to assume a threshold model, which supposes