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  • 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:

    Code:
     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 
    .....................................................
    
    
           500 |         8          0 |         8 
           501 |         8          0 |         8 
           502 |         8          0 |         8 
           503 |         8          0 |         8 
           504 |         8          0 |         8 
           505 |         8          1 |         9 
           506 |         8          0 |         8 
           507 |        10          0 |        10 
           508 |         9          0 |         9 
           509 |         9          0 |         9 
           510 |         9          0 |         9 
           511 |        11          0 |        11 
           512 |        15          0 |        15 
           513 |        17          0 |        17 
           514 |        16          0 |        16 
           515 |        16          0 |        16 
           516 |        15          0 |        15 
           517 |        15          0 |        15 
           518 |        15          0 |        15 
           519 |        14          0 |        14 
           520 |        14          0 |        14 
           521 |        14          0 |        14 
           522 |        15          0 |        15 
           523 |        14          0 |        14 
           524 |        12          0 |        12 
           525 |        12          0 |        12 
           526 |        14          0 |        14 
           527 |        14          1 |        15 
           528 |        14          0 |        14 
           529 |        15          0 |        15 
           530 |        16          0 |        16 
           531 |        15          0 |        15 
           532 |        13          2 |        15 
           533 |        13          0 |        13 
           534 |        12          0 |        12 
           535 |        11          0 |        11 
           536 |        10          0 |        10 
           537 |        10          0 |        10 
           538 |        10          0 |        10 
           539 |        12          0 |        12 
           540 |        12          0 |        12 
           541 |        12          0 |        12 
           542 |        13          0 |        13 
           543 |        13          0 |        13 
           544 |        12          0 |        12 
           545 |        13          0 |        13 
           546 |        14          0 |        14 
           547 |        14          0 |        14 
           548 |        14          0 |        14 
           549 |        14          0 |        14 
           550 |        12          0 |        12 
           551 |        11          0 |        11 
           552 |        10          1 |        11 
           553 |         9          0 |         9 
           554 |         9          0 |         9 
           555 |         9          0 |         9 
           556 |        10          0 |        10 
           557 |         9          0 |         9 
           558 |         9          0 |         9 
           559 |         9          0 |         9 
           560 |         9          0 |         9 
           561 |         8          0 |         8 
           562 |         8          0 |         8 
           563 |         8          0 |         8 
           564 |         8          1 |         9 
           565 |         8          0 |         8 
           566 |         7          0 |         7 
           567 |         6          0 |         6 
           568 |         7          0 |         7 
           569 |         8          0 |         8 
           570 |         6          0 |         6 
           571 |         7          0 |         7 
           572 |         7          0 |         7 
           573 |         8          0 |         8 
           574 |         8          0 |         8 
           575 |         8          0 |         8 
           576 |         8          0 |         8 
           577 |         9          0 |         9 
           578 |         9          0 |         9 
           579 |         9          0 |         9 
           580 |         8          0 |         8 
           581 |         8          0 |         8 
           582 |         9          0 |         9 
           583 |        10          0 |        10 
           584 |        11          0 |        11 
           585 |         9          0 |         9 
           586 |         9          0 |         9 
           587 |        10          0 |        10 
           588 |        10          0 |        10 
           589 |         9          1 |        10 
           590 |         9          0 |         9 
           591 |        10          0 |        10 
           592 |        10          0 |        10 
           593 |         7          2 |         9 
           594 |         6          0 |         6 
           595 |         5          0 |         5 
           596 |         5          0 |         5 
           597 |         5          0 |         5 
           598 |         5          0 |         5 
           599 |         4          0 |         4 
           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 
Working...
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