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mast30027 Binomial Regression
创建日期：2024-05-31 16:29:51
所属分类：数学mathematics
标签： mast30027
Binomial Regression

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Binomial Regression 

Reminder: questions from Challenger disaster
Forecast probability of an O-ring being damaged when the launch
temperature is 29 oF .
How good is our forecast? Can we provide a confidence interval?
Is temperature useful to predict the O-ring failing?
Binomial Regression II 2 / 17
Is temperature useful to predict the O-ring failing?
Yi , the number of damaged O-rings on the i-th launch, has distribution
Yi ∼ bin(6, pi )
where
log pi/(1− pi ) = ηi = β0 + β1ti .
Test for association between the number of damaged O-rings and
temperature:
H0 : β1 = 0 Ha : β1 6= 0
Wald Test
Likelihood Ratio Test
Binomial Regression II 3 / 17
Wald Test
Reminder: asymptotic normality of MLE
θˆi ∼ asy . N(θ∗i , [I(θˆ)−1]i ,i ).
Test for association between the number of damaged O-rings and
temperature
H0 : β1 = 0 Ha : β1 6= 0
Wald test statistic:
z∗ =
βˆ1
se(βˆ1)
∼ asy . N(0, 1) under H0.
Challenger disaster
See R script and result in “Wald Test” of Challenger.pdf
|z∗| = 4.07 > 1.96 (critical value N(0, 1) at α = 0.05) ⇒ reject H0.
p-value = 0.0000476.
Binomial Regression II 4 / 17
Likelihood Ratio Test (LRT)
Test for association between the number of damaged O-rings and
temperature
H0 : β1 = 0 Ha : β1 6= 0
Full model (F):
ηi = β0 + β1ti
Maximum log likelihood: logL(βˆF )
βˆ
F
: MLE of the parameters in the full model.
Reduced model (R):
ηi = β0
Maximum log likelihood: logL(βˆR)
βˆ
R
: MLE of the parameters in the reduced model.
Compare two models.
Likelihood ratio test statistic:
LR∗ = −2
[
logL(βˆR)− logL(βˆF )
]
∼ asy . χ21 under H0.
LR∗ > critical value from χ21 at α ⇒ reject H0.
Binomial Regression II 5 / 17
Wald test vs Likelihood Ratio Test (LRT)
Wald test and LRT are asymptotically equivalent.
[z∗]2 =
[
βˆ1
se(βˆ1)
]2
∼ asy . χ21
Precisely speaking, two tests are asymptotically equivalent in the
sense that under H0 they reach the same decision with probability
approaching 1 as n goes to infinite.
However, the chi-squared approximation to the log likelihood ratio is
generally better than the normal approximation to the MLE.
Binomial Regression II 6 / 17
Likelihood Ratio Test (LRT): Challenger disaster
See R script and result in “Likelihood Ratio test” and “Wald Test vs
Likelihood Ratio test” of Challenger.pdf
LR∗ = 21.98 > 3.84 (critical value from χ21 at α = 0.05) ⇒ reject H0.
p-value = 0.0000027.
Binomial Regression II 7 / 17
Likelihood Ratio Test (LRT) for model selection
In general, likelihood ratio test is used to select between two nested
models (one model can be obtained by constraining parameters of another
model).
Full model (F):
Maximum log likelihood: logL(θˆF )
Reduced model (R):
Maximum log likelihood: logL(θˆR)
Let k indicate the difference in the number of parameters between two
models.

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