Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

EXAMPLE (continued) Model 3: logit P(X)¼aþbEþgV þdEV Largest possible model containing binaryEand binaryV Note:E^2 ¼EandV^2 ¼V F ...

Model A: X¼(X 1 , X 2 , X 3 ) fully parameterized but Model B:X¼(X 1 ,X 2 ,X 3 ,X 4 ,X 5 ,X 6 ) “better fit” than Model A EXAMPL ...

Model 3.Fully parameterized model + #of expected cases ¼#of observed cases for each covariate pattern Covariate patternXg:ngsubj ...

Classical GOF approach: Saturated model gives perfect fit for individual subjects Why? Yi¼0 or 1 only possible outcomes for subj ...

L^SS max¼ Y^40 i¼ 1 YiYið 1 YiÞ^1 Yi For any binary logistic model: Yi¼ 1 :YiYið 1 YiÞ^1 Yi¼ 11 ð 1  1 Þ^1 ^1 ¼ 1 Yi¼ 0 :Y ...

Properties of deviance similar to properties ofw^2 statistic Common to test for GOF by comparing deviance withw^2 nk 1 value: ...

Alternative (Events–Trials) Devi- ance formula: for the logistic model: Gcovariate patterns Xg¼ðXg 1 ;Xg 2 ;...;XgpÞ;g¼ 1 ; 2 ;. ...

EXAMPLE “group-saturated”: # of parameters = # of groups sample size = 4 Model 3: logit P(X)¼aþbEþgV þdEV vs. Model 4 (perfectly ...

EXAMPLE Covariate pattern Obs. risk Pred. risk X 1 :E¼1, V¼ 1 ^p 1 ¼ 0 : 6 P^ðX 1 Þ¼ 0 : 6 X 2 :E¼0, V¼ 1 P^ 2 ¼ 0 : 4 P^ðX 2 Þ¼ ...

Deviance not always appropriate for logistic regression GOF Alternative approach: Hosmer–Lemeshow WhenG<<n, we can assume ...

IV. The Hosmer– Lemeshow (HL) Statistic  Alternative to questionable use of deviance  Available in most computer packages HL w ...

Step 3: FormQpercentile group- ings. Typically,Q¼10, i.e., deciles of risk e.g.,n¼ 200 Decile No. of subjects 1  20 2  20 ... ...

EXAMPLE e.g.,q¼3,n 3 ¼4,^PðXiÞ: 0.30, 0.35, 0.40, 0.45 Ec 3 ¼~ n 3 i¼ 1 ^PðXi 3 Þ¼ 0 : 30 þ 0 : 35 þ 0 : 40 þ 0 : 45 ¼ 1 : 50 an ...

EXAMPLE (continued) Datalines in events trials format (n 5 609) Group (g) dg ng CAT AGEG ECG 1 17 274 0 0 0 2 15 122 0 1 0 3 7 5 ...

EXAMPLE Edited Output (Model EC2): (Variables – CAT, AGE, ECG, AGE×ECG, CAT×AGE, AGE×ECG, and CAT×AGE×ECG) 0.0000 4 1.0000 0.000 ...

Pred¼obs for each g ðphat¼pÞ + d^g¼dgcases EXAMPLE Fully parameterized model (w/o conts. Xs):  Above results support its use as ...

EXAMPLE G¼ 599 n(¼609):  Deviance nor Pearson not approximatelyw^2  HL statistic approximatelyw^2 (can use HL for GOF test) ...

EXAMPLE Edited Output (Model EC4): (Variables – CAT, AGE, ECG, SMK, CHL, HPT CAT×CHL, and CAT× HPT) 7.9914 8 0.4343 347.2295 590 ...

VI. SUMMARY 3 Chapter 9: Assessing Goodness of Fit for Logistic Regression Saturated model:  Contains as many parameters (pþ1) ...

Chapter 10: Assessing Discrimi- natory Performance of a Binary Logistic Model We suggest that you review the material cov- ered ...