Applied Logistic Regression, Second Edition by Hosmer and Lemeshow Chapter 2: Multiple logistic regression

page 32 Table 2.1 An example of the coding of the design variables for race, coded at three levels.

data lowbwt1;
  set 'd:hosmerdatalowbwt';
  if race = 1 then do; race2 = 0; race3 = 0; end;
  if race = 2 then do; race2 = 1; race3 = 0; end;
  if race = 3 then do; race2 = 0; race3 = 1; end;
run;
proc print data=lowbwt1 (obs=3);
var race race2 race3;
run;

Obs    RACE    race2    race3

  1      2       1        0
  2      3       0        1
  3      1       0        0

page 36 Table 2.2 Estimated coefficients for a multiple logistic regression model using the variables
age, weight at last menstrual period (lwt), race and number of first trimester physician
visits from the low birth weight study.

NOTE: We have bolded the relevant output.

proc logistic data=lowbwt1 descending;
model low = age lwt race2 race3 ftv;
run;
quit;

The LOGISTIC Procedure

Model Information
Data Set WORK.LOWBWT1
Response Variable LOW < 2500g
Number of Response Levels 2
Number of Observations 189
Link Function Logit
Optimization Technique Fisher's scoring

Response Profile

Ordered Total
Value LOW Frequency
1 1 59
2 0 130

Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.

Model Fit Statistics

Intercept
Intercept and
Criterion Only Covariates
AIC 236.672 234.573
SC 239.914 254.023
-2 Log L 234.672 222.573

Testing Global Null Hypothesis: BETA=0

Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 12.0991 5 0.0335
Score 11.3876 5 0.0442
Wald 10.6964 5 0.0577

The LOGISTIC Procedure

Analysis of Maximum Likelihood Estimates

Standard
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 1.2953 1.0714 1.4616 0.2267
AGE 1 -0.0238 0.0337 0.4988 0.4800
LWT 1 -0.0142 0.00654 4.7428 0.0294
race2 1 1.0039 0.4979 4.0660 0.0438
race3 1 0.4331 0.3622 1.4296 0.2318
FTV 1 -0.0493 0.1672 0.0869 0.7681

Odds Ratio Estimates

Point 95% Wald
Effect Estimate Confidence Limits
AGE 0.976 0.914 1.043
LWT 0.986 0.973 0.999
race2 2.729 1.029 7.240
race3 1.542 0.758 3.136
FTV 0.952 0.686 1.321

Association of Predicted Probabilities and Observed Responses
Percent Concordant 65.1 Somers' D 0.308
Percent Discordant 34.3 Gamma 0.310
Percent Tied 0.6 Tau-a 0.133
Pairs 7670 c 0.654

page 38 Table 2.3 Estimated coefficients for a multiple logistic regression model using the variables
lwt and race from the low birth weight study.

proc logistic data=lowbwt1 descending covout outest=lowbwt2;
model low = lwt race2 race3;
run;
quit;

The LOGISTIC Procedure

Model Information
Data Set WORK.LOWBWT1
Response Variable LOW < 2500g
Number of Response Levels 2
Number of Observations 189
Link Function Logit
Optimization Technique Fisher's scoring

Response Profile

Ordered Total
Value LOW Frequency
1 1 59
2 0 130

Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.

Model Fit Statistics

Intercept
Intercept and
Criterion Only Covariates
AIC 236.672 231.259
SC 239.914 244.226
-2 Log L 234.672 223.259

Testing Global Null Hypothesis: BETA=0

Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 11.4129 3 0.0097
Score 10.7572 3 0.0131
Wald 10.1316 3 0.0175

The LOGISTIC Procedure

Analysis of Maximum Likelihood Estimates

Standard
Parameter DF Estimate Error Chi-Square Pr > ChiSq
Intercept 1 0.8057 0.8452 0.9088 0.3404
LWT 1 -0.0152 0.00644 5.5886 0.0181
race2 1 1.0811 0.4881 4.9065 0.0268
race3 1 0.4806 0.3567 1.8156 0.1778

Odds Ratio Estimates

Point 95% Wald
Effect Estimate Confidence Limits
LWT 0.985 0.973 0.997
race2 2.948 1.133 7.672
race3 1.617 0.804 3.253

Association of Predicted Probabilities and Observed Responses
Percent Concordant 64.1 Somers' D 0.293
Percent Discordant 34.8 Gamma 0.296
Percent Tied 1.1 Tau-a 0.127
Pairs 7670 c 0.647

page 42 Table 2.4 Estimated covariance matrix of the estimated coefficients in Table 2.3.

proc print data=lowbwt2;

  where _type_='COV';

  var _name_ intercept lwt race2 race3;

run;
Obs    _NAME_       Intercept            LWT      race2       race3
 2     Intercept      0.71430    -.005213648     0.02260    -0.10350

 3     LWT           -0.00521    0.000041465    -0.00065     0.00036

 4     race2          0.02260    -.000647028     0.23819     0.05320

 5     race3         -0.10350    0.000355854     0.05320     0.12722