Applied Logistic Regression, Second Edition by Hosmer and Lemeshow Chapter 1: Introduction to the logistic regression model

page 3 Table 1.1 Age and coronary heart disease (chd) status of 100 subjects.

data chdage1;
  set 'd:hosmerdatachdage';
  agrp = .;
  if age le 29 then agrp = 1;
  if age ge 30 and age < 35 then agrp = 2;
  if age ge 35 and age < 40 then agrp = 3;
  if age ge 40 and age < 45 then agrp = 4;
  if age ge 45 and age < 50 then agrp = 5;
  if age ge 50 and age < 55 then agrp = 6;
  if age ge 55 and age < 60 then agrp = 7;
  if age ge 60 then agrp = 8;
run;
proc print data=chdage1;
  var id age agrp chd;
run;

Obs ID AGE agrp CHD
1 1 20 1 0
2 2 23 1 0
3 3 24 1 0
4 4 25 1 0
5 5 25 1 1
6 6 26 1 0
7 7 26 1 0
8 8 28 1 0
9 9 28 1 0
10 10 29 1 0
11 11 30 2 0
12 12 30 2 0
13 13 30 2 0
14 14 30 2 0
15 15 30 2 0
16 16 30 2 1
17 17 32 2 0
18 18 32 2 0
19 19 33 2 0
20 20 33 2 0
21 21 34 2 0
22 22 34 2 0
23 23 34 2 1
24 24 34 2 0
25 25 34 2 0
26 26 35 3 0
27 27 35 3 0
28 28 36 3 0
29 29 36 3 1
30 30 36 3 0
31 31 37 3 0
32 32 37 3 1
33 33 37 3 0
34 34 38 3 0
35 35 38 3 0
36 36 39 3 0
37 37 39 3 1
38 38 40 4 0
39 39 40 4 1
40 40 41 4 0
41 41 41 4 0
42 42 42 4 0
43 43 42 4 0
44 44 42 4 0
45 45 42 4 1
46 46 43 4 0
47 47 43 4 0
48 48 43 4 1
49 49 44 4 0
50 50 44 4 0
51 51 44 4 1
52 52 44 4 1

Obs ID AGE agrp CHD
53 53 45 5 0
54 54 45 5 1
55 55 46 5 0
56 56 46 5 1
57 57 47 5 0
58 58 47 5 0
59 59 47 5 1
60 60 48 5 0
61 61 48 5 1
62 62 48 5 1
63 63 49 5 0
64 64 49 5 0
65 65 49 5 1
66 66 50 6 0
67 67 50 6 1
68 68 51 6 0
69 69 52 6 0
70 70 52 6 1
71 71 53 6 1
72 72 53 6 1
73 73 54 6 1
74 74 55 7 0
75 75 55 7 1
76 76 55 7 1
77 77 56 7 1
78 78 56 7 1
79 79 56 7 1
80 80 57 7 0
81 81 57 7 0
82 82 57 7 1
83 83 57 7 1
84 84 57 7 1
85 85 57 7 1
86 86 58 7 0
87 87 58 7 1
88 88 58 7 1
89 89 59 7 1
90 90 59 7 1
91 91 60 8 0
92 92 60 8 1
93 93 61 8 1
94 94 62 8 1
95 95 62 8 1
96 96 63 8 1
97 97 64 8 0
98 98 64 8 1
99 99 65 8 1
100 100 69 8 1

page 4 Figure 1.1 Scatterplot of chd by age for 100 subjects.

filename outgraph 'd:https://stats.idre.ucla.edu/wp-content/uploads/2016/02/hlch1sas1.gif';
goptions gsfname=outgraph dev=gif373;
symbol1 value=circle;
proc gplot data=chdage1;
  plot chd*age;
run;
quit;

page 4 Table 1.2 Frequency table of age group by chd.

NOTE: The row percent gives the data listed in the ‘mean’ column. Because SAS gives a
percent instead of a proportion (as shown in the text) the decimal is moved over two places.

proc freq data=chdage1;
  tables agrp*chd / out=chdage2 outpct;
run;

page 5 Figure 1.2 Plot of the percentage of subjects with CHD in each age group.

filename outgraph 'd:https://stats.idre.ucla.edu/wp-content/uploads/2016/02/hlch1sas2.gif';
goptions gsfname=outgraph dev=gif373;
symbol1 color=black i=none value=circle height=1;
axis1 order=(0 to 100 by 10);
axis2 order=(1 to 8 by 1);
proc gplot data=chdage2;
  where chd = 1;
  plot pct_row*agrp / haxis=axis2 vaxis=axis1;
run;
quit;

page 10 Table 1.3 Results of fitting the logistic regression model to the data in Table 1.1.

NOTE: To get the Wald tests shown in the text, take the square root of the chi-squares given
in the SAS output.

NOTE: We have bolded the relevant output.

proc logistic data=chdage1 covout outest=chdage3 descending;
  model chd = age;
run;
quit;

The LOGISTIC Procedure

Model Information
Data Set WORK.CHDAGE1
Response Variable CHD
Number of Response Levels 2
Number of Observations 100
Link Function Logit
Optimization Technique Fisher’s scoring
Response Profile

Ordered Total
Value CHD Frequency
1 1 43
2 0 57
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics

Intercept
Intercept and
Criterion Only Covariates
AIC 138.663 111.353
SC 141.268 116.563
-2 Log L 136.663 107.353
Testing Global Null Hypothesis: BETA=0

Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 29.3099 1 <.0001
Score 26.3989 1 <.0001
Wald 21.2541 1 <.0001
Analysis of Maximum Likelihood Estimates

Standard
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 -5.3095 1.1337 21.9350 <.0001
AGE 1 0.1109 0.0241 21.2541 <.0001

The LOGISTIC Procedure

Odds Ratio Estimates

Point 95% Wald
Effect Estimate Confidence Limits
AGE 1.117 1.066 1.171
Association of Predicted Probabilities and Observed Responses
Percent Concordant 79.0 Somers’ D 0.600
Percent Discordant 19.0 Gamma 0.612
Percent Tied 2.0 Tau-a 0.297
Pairs 2451 c 0.800

page 20 Table 1.4 Estimated covariance matrix of the estimated coefficients in Table 1.3.

proc print data=chdage3;
run;

Obs _LINK_ _TYPE_ _STATUS_ _NAME_ Intercept AGE _LNLIKE_

1 LOGIT PARMS 0 Converged CHD -5.30945 0.11092 -53.6765
2 LOGIT COV 0 Converged Intercept 1.28517 -0.02668 -53.6765
3 LOGIT COV 0 Converged AGE -0.02668 0.00058 -53.6765