Detailed
Outline
I. Overview(page 132)
Focus:
Testing hypotheses
Computing confidence intervals
II. Information for making statistical inferences
(pages 132–133)
A. Maximized likelihood value:Lð^uÞ.
B. Estimated variance–covariance matrix:V^ðu^Þ
contains variances of estimated coefficients on
the diagonal and covariances between
coefficients off the diagonal.
C. Variable listing: contains each variable followed
by ML estimate, standard error, and other
information.
III. Models for inference-making(pages 133–134)
A. Model 1: logitP(X)¼aþb 1 X 1 þb 2 X 2 ;
Model 2: logitP(X)¼aþb 1 X 1 þb 2 X 2 þb 3 X 3 ;
Model 3: logitP(X)¼aþb 1 X 1 þb 2 X 2 þb 3 X 3
þb 4 X 1 X 3 þb 5 X 2 X 3.
B. L^ 1 ,L^ 2 ,L^ 3 are maximized likelihoods (L^) for models
1–3, respectively.
C. L^is similar toRsquare:L^ 1 L^ 2 L^ 3.
D. 2lnL^ 3 2lnL^ 2 2lnL^ 1 ,
where2lnL^is called the log likelihood statistic.
IV. The likelihood ratio (LR) test(pages 134–138)
A. LR statistic compares two models: full (larger)
model vs. reduced (smaller) model.
B. H 0 : some parameters in full model are equal to 0.
C. df¼number of parameters in full model set
equal to 0 to obtain reduced model.
D. Model 1 vs. Model 2: LR¼2lnL^ 1 (2lnL^ 2 ),
whereH 0 :b 3 ¼0. This LR has approximately a
chi-square distribution with one df under the null
hypothesis.
E. 2 lnL^ 1 2 lnL^ 2¼ 2 lnL^ 1 =L^ 2;
whereL^ 1 =L^ 2 is a ratio of likelihoods.
F. How the LR test works: LR works like a chi-square
statistic. For highly significant variables, LR is
large and positive; for nonsignificant variables,
LR is close to 0.
G. Model 2 vs. Model 3: LR¼2lnL^ 2 (2lnL^ 3 ),
whereH 0 :b 4 ¼b 5 ¼0. This LR has
approximately a chi-square distribution with 2 df
under the null hypothesis.
H. Computer prints2lnL^separately for each
model, so LR test requires only subtraction.154 5. Statistical Inferences Using Maximum Likelihood Techniques