# Modeling Time in the Discrete Time Analysis using Left-truncated Data

I am doing multiple-spell, discrete-time analysis to examine factors shaping time to exiting a renter spell, with time-varying covariates at monthly interval. My data has left-truncated spells, meaning some people have already started their renter spell before the start of two-year observation period.

I have followed Stephen Jenkins' advice (http://citeseerx.ist.psu.edu/viewdoc...=rep1&type=pdf) on setting up left-truncated data for discrete time analysis, which makes observations in the left-truncated spell start to contribute to person-observation from the time when they enter the study.

My problem starts from here. I have now left with observations with elapsed months since risk onset much longer (25 percentile = 9 months, 50 percentile = 30 months, 75 percentile = 80 months, 99 percentile = 657 months) than two-year study period.

I am not very comfortable modeling time that goes way over study time period, but at the same time, I would like to keep left-truncated spells in my analysis. I would like to have your thoughts or direction on how to model time when left-truncated data makes elapsed time since risk onset far stretched beyond the length of observation period.

Here's event occurrence for each month since risk onset:

tab timesince monthlymove

|      monthlymove
timesince |         0          1 |     Total
-----------+----------------------+----------
1 |     3,915         35 |     3,950
2 |     3,864         39 |     3,903
3 |     3,745         64 |     3,809
4 |     3,525         31 |     3,556
5 |     3,236         41 |     3,277
6 |     3,247         48 |     3,295
7 |     3,040         68 |     3,108
8 |     2,996         74 |     3,070
9 |     2,867         44 |     2,911
10 |     2,673         74 |     2,747
11 |     2,556         45 |     2,601
12 |     2,435         59 |     2,494
13 |     1,952         87 |     2,039
14 |     1,806         43 |     1,849
15 |     1,682         48 |     1,730
16 |     1,573         45 |     1,618
17 |     1,441         28 |     1,469
18 |     1,401         33 |     1,434
19 |     1,343         26 |     1,369
20 |     1,333         10 |     1,343
21 |     1,299         19 |     1,318
22 |     1,270         21 |     1,291
23 |     1,246         16 |     1,262
24 |     1,225         19 |     1,244
25 |     1,074         29 |     1,103
26 |       987         25 |     1,012
27 |       933         12 |       945
28 |       905          9 |       914
29 |       884         10 |       894
30 |       837          6 |       843
31 |       817          8 |       825
32 |       828          6 |       834
33 |       800         12 |       812
34 |       785          5 |       790
35 |       776         10 |       786
36 |       760          7 |       767
37 |       736         17 |       753
38 |       729          4 |       733
39 |       712          8 |       720
40 |       695          9 |       704
41 |       676          8 |       684
42 |       639          5 |       644
43 |       616          6 |       622
44 |       608          3 |       611
45 |       600          4 |       604
46 |       566          6 |       572
47 |       566          5 |       571
48 |       539          3 |       542
49 |       526         10 |       536
50 |       514          2 |       516
51 |       516          5 |       521
52 |       500         10 |       510
53 |       488          7 |       495
54 |       487         10 |       497
55 |       488          1 |       489
56 |       489          7 |       496
57 |       493          2 |       495
58 |       469         13 |       482
59 |       452          7 |       459
60 |       438         11 |       449
61 |       416          2 |       418
62 |       397          5 |       402
63 |       387          6 |       393
64 |       381          5 |       386
65 |       358          8 |       366
66 |       361          2 |       363
67 |       351         10 |       361
68 |       336          3 |       339
69 |       324          6 |       330
70 |       326          1 |       327
71 |       315          2 |       317
72 |       309          6 |       315
73 |       301          5 |       306
74 |       294          2 |       296
75 |       285          1 |       286
76 |       277          5 |       282
77 |       283          3 |       286
78 |       273          1 |       274
79 |       273          2 |       275
80 |       265          5 |       270
81 |       252          1 |       253
82 |       247          2 |       249
83 |       242          1 |       243
84 |       245          0 |       245
85 |       243          1 |       244
86 |       239          2 |       241
87 |       237          0 |       237
88 |       227          3 |       230
89 |       225          2 |       227
90 |       217          5 |       222
91 |       207          3 |       210
92 |       205          1 |       206
93 |       206          0 |       206
94 |       200          0 |       200
95 |       193          0 |       193
96 |       183          2 |       185
97 |       181          5 |       186
98 |       185          0 |       185
99 |       183          0 |       183
100 |       180          2 |       182
101 |       175          1 |       176
102 |       167          2 |       169
103 |       160          2 |       162
104 |       181          1 |       182
105 |       185          2 |       187
106 |       184          2 |       186
107 |       182          0 |       182
108 |       177          4 |       181
109 |       169          2 |       171
110 |       165          1 |       166
111 |       159          2 |       161
112 |       148          2 |       150
113 |       146          1 |       147
114 |       148          2 |       150
115 |       144          2 |       146
116 |       143          1 |       144
117 |       144          2 |       146
118 |       149          1 |       150
119 |       152          1 |       153
120 |       153          2 |       155
121 |       155          1 |       156
122 |       152          1 |       153
123 |       150          1 |       151
124 |       148          1 |       149
125 |       144          0 |       144
126 |       137          1 |       138
127 |       135          1 |       136
128 |       130          0 |       130
129 |       128          0 |       128
130 |       124          1 |       125
131 |       116          0 |       116
132 |       111          0 |       111
133 |       114          1 |       115
134 |       106          1 |       107
135 |       107          2 |       109
136 |       103          0 |       103
137 |       103          1 |       104
138 |       103          1 |       104
139 |       103          1 |       104
140 |       102          1 |       103
141 |        91          1 |        92
142 |        94          1 |        95
143 |        97          0 |        97
144 |        96          0 |        96
145 |        91          0 |        91
146 |        98          1 |        99
147 |        99          0 |        99
148 |       104          1 |       105
149 |       107          0 |       107
150 |       106          1 |       107
151 |       110          0 |       110
152 |       126          0 |       126
153 |       126          2 |       128
154 |       126          2 |       128
155 |       127          2 |       129
156 |       127          1 |       128
157 |       131          0 |       131
158 |       130          1 |       131
159 |       129          1 |       130
160 |       123          0 |       123
161 |       120          2 |       122
162 |       122          1 |       123
163 |       116          0 |       116
164 |       123          0 |       123
165 |       126          0 |       126
166 |       116          1 |       117
167 |       116          2 |       118
168 |       115          0 |       115
169 |       110          4 |       114
170 |       102          1 |       103
171 |       106          0 |       106
172 |       103          1 |       104
173 |        97          0 |        97
174 |        89          1 |        90
175 |        83          0 |        83
176 |        70          2 |        72
177 |        69          0 |        69
178 |        67          1 |        68
179 |        64          0 |        64
180 |        64          0 |        64
181 |        64          0 |        64
182 |        62          0 |        62
183 |        57          0 |        57
184 |        56          0 |        56
185 |        52          1 |        53
186 |        53          0 |        53
187 |        54          0 |        54
188 |        46          1 |        47
189 |        45          0 |        45
190 |        47          1 |        48
191 |        44          0 |        44
192 |        47          0 |        47
193 |        46          0 |        46
194 |        45          0 |        45
195 |        44          0 |        44
196 |        41          0 |        41
197 |        43          0 |        43
198 |        48          0 |        48
199 |        47          0 |        47
200 |        53          1 |        54
201 |        53          1 |        54
202 |        52          2 |        54
203 |        53          0 |        53
204 |        52          1 |        53
205 |        52          1 |        53
206 |        53          2 |        55
207 |        55          0 |        55
208 |        50          4 |        54
209 |        49          0 |        49
210 |        50          0 |        50
211 |        46          1 |        47
212 |        60          1 |        61
213 |        58          1 |        59
214 |        55          0 |        55
215 |        56          0 |        56
216 |        54          0 |        54
217 |        55          1 |        56
218 |        51          1 |        52
219 |        48          2 |        50
220 |        50          1 |        51
221 |        52          0 |        52
222 |        50          1 |        51
223 |        54          0 |        54
224 |        59          0 |        59
225 |        58          0 |        58
226 |        60          1 |        61
227 |        64          1 |        65
228 |        65          0 |        65
229 |        70          0 |        70
230 |        68          0 |        68
231 |        69          0 |        69
232 |        68          1 |        69
233 |        67          1 |        68
234 |        62          2 |        64
235 |        63          0 |        63
236 |        55          0 |        55
237 |        52          0 |        52
238 |        54          0 |        54
239 |        50          0 |        50
240 |        48          0 |        48
241 |        53          0 |        53
242 |        53          1 |        54
243 |        51          2 |        53
244 |        49          0 |        49
245 |        45          3 |        48
246 |        44          0 |        44
247 |        40          0 |        40
248 |        46          0 |        46
249 |        46          0 |        46
250 |        45          1 |        46
251 |        41          1 |        42
252 |        41          0 |        41
253 |        37          0 |        37
254 |        37          0 |        37
255 |        35          1 |        36
256 |        34          1 |        35
257 |        31          2 |        33
258 |        34          0 |        34
259 |        34          0 |        34
260 |        32          0 |        32
261 |        32          0 |        32
262 |        28          0 |        28
263 |        27          1 |        28
264 |        26          0 |        26
265 |        21          0 |        21
266 |        20          0 |        20
267 |        23          0 |        23
268 |        23          0 |        23
269 |        24          0 |        24
270 |        24          0 |        24
271 |        27          0 |        27
272 |        28          0 |        28
273 |        28          0 |        28
274 |        28          0 |        28
275 |        30          1 |        31
276 |        28          1 |        29
277 |        29          0 |        29
278 |        27          0 |        27
279 |        28          0 |        28
280 |        28          0 |        28
281 |        28          1 |        29
282 |        28          0 |        28
283 |        24          0 |        24
284 |        24          0 |        24
285 |        24          1 |        25
286 |        25          0 |        25
287 |        25          0 |        25
288 |        26          0 |        26
289 |        27          1 |        28
290 |        27          0 |        27
291 |        25          0 |        25
292 |        27          0 |        27
293 |        26          0 |        26
294 |        27          0 |        27
295 |        28          0 |        28
296 |        29          0 |        29
297 |        30          0 |        30
298 |        30          0 |        30
299 |        28          0 |        28
300 |        26          0 |        26
301 |        26          0 |        26
302 |        25          0 |        25
303 |        25          0 |        25
304 |        27          0 |        27
305 |        25          1 |        26
306 |        24          1 |        25
307 |        25          0 |        25
308 |        23          0 |        23
309 |        23          0 |        23
310 |        23          0 |        23
311 |        23          0 |        23
312 |        22          0 |        22
313 |        18          2 |        20
314 |        19          0 |        19
315 |        19          0 |        19
316 |        17          0 |        17
317 |        19          0 |        19
318 |        15          0 |        15
319 |        13          0 |        13
320 |        10          0 |        10
321 |         9          0 |         9
322 |        11          1 |        12
323 |        12          0 |        12
324 |        12          0 |        12
325 |        11          1 |        12
326 |        12          0 |        12
327 |        15          0 |        15
328 |        16          1 |        17
329 |        18          0 |        18
330 |        19          0 |        19
331 |        20          0 |        20
332 |        26          1 |        27
333 |        26          1 |        27
334 |        28          0 |        28
335 |        32          0 |        32
336 |        32          0 |        32
337 |        35          0 |        35
338 |        33          1 |        34
339 |        33          0 |        33
340 |        35          0 |        35
341 |        35          0 |        35
342 |        35          0 |        35
343 |        35          0 |        35
344 |        33          2 |        35
345 |        35          0 |        35
346 |        31          2 |        33
347 |        27          2 |        29
348 |        29          0 |        29
349 |        26          1 |        27
350 |        26          0 |        26
351 |        26          0 |        26
352 |        23          1 |        24
353 |        23          0 |        23
354 |        22          1 |        23
355 |        21          1 |        22
356 |        18          0 |        18
357 |        16          0 |        16
358 |        14          0 |        14
359 |        14          0 |        14
360 |        13          0 |        13
361 |        14          0 |        14
362 |        16          0 |        16
363 |        15          0 |        15
364 |        13          0 |        13
365 |        13          0 |        13
366 |        14          0 |        14
367 |        16          0 |        16
368 |        15          0 |        15
369 |        14          1 |        15
370 |        14          0 |        14
371 |        15          0 |        15
372 |        15          0 |        15
373 |        15          0 |        15
374 |        15          0 |        15
375 |        15          0 |        15
376 |        16          0 |        16
377 |        15          0 |        15
378 |        14          0 |        14
379 |        16          0 |        16
380 |        16          0 |        16
381 |        17          0 |        17
382 |        18          0 |        18
383 |        18          0 |        18
384 |        17          1 |        18
385 |        16          1 |        17
386 |        17          0 |        17
387 |        17          0 |        17
388 |        16          0 |        16
389 |        20          0 |        20
390 |        19          0 |        19
391 |        16          0 |        16
392 |        17          0 |        17
393 |        17          0 |        17
394 |        18          1 |        19
395 |        18          0 |        18
396 |        19          0 |        19
397 |        19          1 |        20
398 |        20          0 |        20
399 |        21          0 |        21
400 |        23          0 |        23
...........................................................
600 |         3          0 |         3
601 |         3          0 |         3
602 |         3          0 |         3
603 |         3          0 |         3
604 |         3          0 |         3
605 |         4          0 |         4
606 |         3          0 |         3
607 |         3          0 |         3
608 |         2          1 |         3
609 |         1          1 |         2
610 |         2          0 |         2
611 |         2          0 |         2
612 |         2          0 |         2
613 |         2          0 |         2
614 |         2          0 |         2
615 |         2          0 |         2
616 |         2          0 |         2
617 |         2          0 |         2
618 |         2          0 |         2
619 |         2          0 |         2
620 |         1          0 |         1
621 |         1          0 |         1
622 |         2          0 |         2
623 |         2          0 |         2
624 |         2          0 |         2
625 |         2          0 |         2
626 |         2          0 |         2
627 |         2          0 |         2
628 |         2          0 |         2
629 |         2          0 |         2
630 |         3          0 |         3
631 |         5          0 |         5
632 |         6          0 |         6
633 |         7          0 |         7
634 |         6          0 |         6
635 |         6          0 |         6
636 |         6          0 |         6
637 |         6          0 |         6
638 |         6          0 |         6
639 |         6          0 |         6
640 |         6          0 |         6
641 |         5          1 |         6
642 |         5          0 |         5
643 |         5          0 |         5
644 |         6          1 |         7
645 |         6          0 |         6
646 |         5          0 |         5
647 |         4          0 |         4
648 |         4          0 |         4
649 |         4          0 |         4
650 |         5          0 |         5
651 |         5          0 |         5
652 |         6          0 |         6
653 |         6          0 |         6
654 |         6          0 |         6
655 |         5          0 |         5
656 |         3          0 |         3
657 |         4          0 |         4
658 |         5          0 |         5
659 |         5          0 |         5
660 |         5          0 |         5
661 |         5          0 |         5
662 |         8          0 |         8
663 |         9          0 |         9
664 |        13          0 |        13
665 |        14          1 |        15
666 |        16          0 |        16
667 |        19          0 |        19
668 |        31          0 |        31
669 |        36          0 |        36
670 |        37          0 |        37
671 |        39          1 |        40
672 |        44          0 |        44
673 |        47          0 |        47
674 |        45          0 |        45
675 |        45          0 |        45
676 |        44          0 |        44
677 |        43          0 |        43
678 |        42          0 |        42
679 |        42          0 |        42
680 |        40          0 |        40
681 |        37          0 |        37
682 |        36          0 |        36
683 |        36          0 |        36
684 |        36          0 |        36
685 |        35          0 |        35
686 |        33          0 |        33
687 |        32          0 |        32
688 |        28          0 |        28
689 |        27          0 |        27
690 |        23          2 |        25
691 |        20          0 |        20
692 |        11          0 |        11
693 |         9          0 |         9
694 |         8          0 |         8
695 |         5          0 |         5
696 |         2          0 |         2
-----------+----------------------+----------
Total |   110,911      1,608 |   112,519