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| The PHREG Procedure |
Example 64.5 Conditional Logistic Regression for m:n Matching
Conditional logistic regression is used to investigate the relationship between an outcome and a set of prognostic factors in matched case-control studies. The outcome is whether the subject is a case or a control. If there is only one case and one control, the matching is 1:1. The m:n matching refers to the situation in which there is a varying number of cases and controls in the matched sets. You can perform conditional logistic regression with the PHREG procedure by using the discrete logistic model and forming a stratum for each matched set. In addition, you need to create dummy survival times so that all the cases in a matched set have the same event time value, and the corresponding controls are censored at later times.
Consider the following set of low infant birth-weight data extracted from Appendix 1 of Hosmer and Lemeshow (1989). These data represent 189 women, of whom 59 had low-birth-weight babies and 130 had normal-weight babies. Under investigation are the following risk factors: weight in pounds at the last menstrual period (LWT), presence of hypertension (HT), smoking status during pregnancy (Smoke), and presence of uterine irritability (UI). For HT, Smoke, and UI, a value of 1 indicates a "yes" and a value of 0 indicates a "no." The woman’s age (Age) is used as the matching variable. The SAS data set LBW contains a subset of the data corresponding to women between the ages of 16 and 32.
data LBW;
input id Age Low LWT Smoke HT UI @@;
Time=2-Low;
datalines;
25 16 1 130 0 0 0 143 16 0 110 0 0 0
166 16 0 112 0 0 0 167 16 0 135 1 0 0
189 16 0 135 1 0 0 206 16 0 170 0 0 0
216 16 0 95 0 0 0 37 17 1 130 1 0 1
45 17 1 110 1 0 0 68 17 1 120 1 0 0
71 17 1 120 0 0 0 83 17 1 142 0 1 0
93 17 0 103 0 0 0 113 17 0 122 1 0 0
116 17 0 113 0 0 0 117 17 0 113 0 0 0
147 17 0 119 0 0 0 148 17 0 119 0 0 0
180 17 0 120 1 0 0 49 18 1 148 0 0 0
50 18 1 110 1 0 0 89 18 0 107 1 0 1
100 18 0 100 1 0 0 101 18 0 100 1 0 0
132 18 0 90 1 0 1 133 18 0 90 1 0 1
168 18 0 229 0 0 0 205 18 0 120 1 0 0
208 18 0 120 0 0 0 23 19 1 91 1 0 1
33 19 1 102 0 0 0 34 19 1 112 1 0 1
85 19 0 182 0 0 1 96 19 0 95 0 0 0
97 19 0 150 0 0 0 124 19 0 138 1 0 0
129 19 0 189 0 0 0 135 19 0 132 0 0 0
142 19 0 115 0 0 0 181 19 0 105 0 0 0
187 19 0 235 1 1 0 192 19 0 147 1 0 0
193 19 0 147 1 0 0 197 19 0 184 1 1 0
224 19 0 120 1 0 0 27 20 1 150 1 0 0
31 20 1 125 0 0 1 40 20 1 120 1 0 0
44 20 1 80 1 0 1 47 20 1 109 0 0 0
51 20 1 121 1 0 1 60 20 1 122 1 0 0
76 20 1 105 0 0 0 87 20 0 105 1 0 0
104 20 0 120 0 0 1 146 20 0 103 0 0 0
155 20 0 169 0 0 1 160 20 0 141 0 0 1
172 20 0 121 1 0 0 177 20 0 127 0 0 0
201 20 0 120 0 0 0 211 20 0 170 1 0 0
217 20 0 158 0 0 0 20 21 1 165 1 1 0
28 21 1 200 0 0 1 30 21 1 103 0 0 0
52 21 1 100 0 0 0 84 21 1 130 1 1 0
88 21 0 108 1 0 1 91 21 0 124 0 0 0
128 21 0 185 1 0 0 131 21 0 160 0 0 0
144 21 0 110 1 0 1 186 21 0 134 0 0 0
219 21 0 115 0 0 0 42 22 1 130 1 0 1
67 22 1 130 1 0 0 92 22 0 118 0 0 0
98 22 0 95 0 1 0 137 22 0 85 1 0 0
138 22 0 120 0 1 0 140 22 0 130 1 0 0
161 22 0 158 0 0 0 162 22 0 112 1 0 0
174 22 0 131 0 0 0 184 22 0 125 0 0 0
204 22 0 169 0 0 0 220 22 0 129 0 0 0
17 23 1 97 0 0 1 59 23 1 187 1 0 0
63 23 1 120 0 0 0 69 23 1 110 1 0 0
82 23 1 94 1 0 0 130 23 0 130 0 0 0
139 23 0 128 0 0 0 149 23 0 119 0 0 0
164 23 0 115 1 0 0 173 23 0 190 0 0 0
179 23 0 123 0 0 0 182 23 0 130 0 0 0
200 23 0 110 0 0 0 18 24 1 128 0 0 0
19 24 1 132 0 1 0 29 24 1 155 1 0 0
36 24 1 138 0 0 0 61 24 1 105 1 0 0
118 24 0 90 1 0 0 136 24 0 115 0 0 0
150 24 0 110 0 0 0 156 24 0 115 0 0 0
185 24 0 133 0 0 0 196 24 0 110 0 0 0
199 24 0 110 0 0 0 225 24 0 116 0 0 0
13 25 1 105 0 1 0 15 25 1 85 0 0 1
24 25 1 115 0 0 0 26 25 1 92 1 0 0
32 25 1 89 0 0 0 46 25 1 105 0 0 0
103 25 0 118 1 0 0 111 25 0 120 0 0 1
120 25 0 155 0 0 0 121 25 0 125 0 0 0
169 25 0 140 0 0 0 188 25 0 95 1 0 1
202 25 0 241 0 1 0 215 25 0 120 0 0 0
221 25 0 130 0 0 0 35 26 1 117 1 0 0
54 26 1 96 0 0 0 75 26 1 154 0 1 0
77 26 1 190 1 0 0 95 26 0 113 1 0 0
115 26 0 168 1 0 0 154 26 0 133 1 0 0
218 26 0 160 0 0 0 16 27 1 150 0 0 0
43 27 1 130 0 0 1 125 27 0 124 1 0 0
4 28 1 120 1 0 1 79 28 1 95 1 0 0
105 28 0 120 1 0 0 109 28 0 120 0 0 0
112 28 0 167 0 0 0 151 28 0 140 0 0 0
159 28 0 250 1 0 0 212 28 0 134 0 0 0
214 28 0 130 0 0 0 10 29 1 130 0 0 1
94 29 0 123 1 0 0 114 29 0 150 0 0 0
123 29 0 140 1 0 0 190 29 0 135 0 0 0
191 29 0 154 0 0 0 209 29 0 130 1 0 0
65 30 1 142 1 0 0 99 30 0 107 0 0 1
141 30 0 95 1 0 0 145 30 0 153 0 0 0
176 30 0 110 0 0 0 195 30 0 137 0 0 0
203 30 0 112 0 0 0 56 31 1 102 1 0 0
107 31 0 100 0 0 1 126 31 0 215 1 0 0
163 31 0 150 1 0 0 222 31 0 120 0 0 0
22 32 1 105 1 0 0 106 32 0 121 0 0 0
134 32 0 132 0 0 0 170 32 0 134 1 0 0
175 32 0 170 0 0 0 207 32 0 186 0 0 0
;
The variable Low is used to determine whether the subject is a case (Low=1, low-birth-weight baby) or a control (Low=0, normal-weight baby). The dummy time variable Time takes the value 1 for cases and 2 for controls.
The following statements produce a conditional logistic regression analysis of the data. The variable Time is the response, and Low is the censoring variable. Note that the data set is created so that all the cases have the same event time and the controls have later censored times. The matching variable Age is used in the STRATA statement so that each unique age value defines a stratum. The variables LWT, Smoke, HT, and UI are specified as explanatory variables. The TIES=DISCRETE option requests the discrete logistic model.
proc phreg data=LBW; model Time*Low(0)= LWT Smoke HT UI / ties=discrete; strata Age; run;
The procedure displays a summary of the number of event and censored observations for each stratum. These are the number of cases and controls for each matched set shown in Output 64.5.1.
| Model Information | |
|---|---|
| Data Set | WORK.LBW |
| Dependent Variable | Time |
| Censoring Variable | Low |
| Censoring Value(s) | 0 |
| Ties Handling | DISCRETE |
| Summary of the Number of Event and Censored Values | |||||
|---|---|---|---|---|---|
| Stratum | Age | Total | Event | Censored | Percent Censored |
| 1 | 16 | 7 | 1 | 6 | 85.71 |
| 2 | 17 | 12 | 5 | 7 | 58.33 |
| 3 | 18 | 10 | 2 | 8 | 80.00 |
| 4 | 19 | 16 | 3 | 13 | 81.25 |
| 5 | 20 | 18 | 8 | 10 | 55.56 |
| 6 | 21 | 12 | 5 | 7 | 58.33 |
| 7 | 22 | 13 | 2 | 11 | 84.62 |
| 8 | 23 | 13 | 5 | 8 | 61.54 |
| 9 | 24 | 13 | 5 | 8 | 61.54 |
| 10 | 25 | 15 | 6 | 9 | 60.00 |
| 11 | 26 | 8 | 4 | 4 | 50.00 |
| 12 | 27 | 3 | 2 | 1 | 33.33 |
| 13 | 28 | 9 | 2 | 7 | 77.78 |
| 14 | 29 | 7 | 1 | 6 | 85.71 |
| 15 | 30 | 7 | 1 | 6 | 85.71 |
| 16 | 31 | 5 | 1 | 4 | 80.00 |
| 17 | 32 | 6 | 1 | 5 | 83.33 |
| Total | 174 | 54 | 120 | 68.97 | |
Results of the conditional logistic regression analysis are shown in Output 64.5.2. Based on the Wald test for individual variables, the variables LWT, Smoke, and HT are statistically significant while UI is marginal.
The hazard ratios, computed by exponentiating the parameter estimates, are useful in interpreting the results of the analysis. If the hazard ratio of a prognostic factor is larger than 1, an increment in the factor increases the hazard rate. If the hazard ratio is less than 1, an increment in the factor decreases the hazard rate. Results indicate that women were more likely to have low-birth-weight babies if they were underweight in the last menstrual cycle, were hypertensive, smoked during pregnancy, or suffered uterine irritability.
For matched case-control studies with one case per matched set (1:n matching), the likelihood function for the conditional logistic regression reduces to that of the Cox model for the continuous time scale. For this situation, you can use the default TIES=BRESLOW.
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