6) Predict a … xڝXYs�6~�_�Gv���u�*��ɤ���qOR��>�ݲ[^v�T�����>��A��G T$��}�wя��e$3�d����T\Q,E�M�/�d?�b�%��f����U���}�}��Ѱ�OW����$�:�b%y!����_?�Z�~�"����8�tI�ן?\��@��k� % 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 /Type/Font endobj /Widths[719.7 539.7 689.9 950 592.7 439.2 751.4 1138.9 1138.9 1138.9 1138.9 339.3 277.8 500] 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 endobj /Type/Font The cumulative hazard has a less clear understanding than the survival functions, but the hazard functions are based on more advanced survival analysis techniques. In the latter case, the relia… 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 By Property 1 of Survival Analysis Basic Concepts, the baseline cumulative hazard function is. /LastChar 196 18 0 obj 33 0 obj An example will help fix ideas. The hazard function describes the ‘intensity of death’ at the time tgiven that the individual has already survived past time t. There is another quantity that is also common in survival analysis, the cumulative hazard function. endobj 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 In the first year, that’s 15/500. Cumulative hazard function: H(t) def= Z t … 877 0 0 815.5 677.6 646.8 646.8 970.2 970.2 323.4 354.2 569.4 569.4 569.4 569.4 569.4 Changing hazards Sometimes the hazard function will not be constant, which will result in the gradient/slope of the cumulative hazard function changing over time. /BaseFont/KSDXMI+CMR7 /FirstChar 33 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 << /FirstChar 33 The survival function is then a by product. Plot estimated survival curves, and for parametric survival models, plothazard functions. >> Our first year hazard, the probability of finishing within one year of advancement, is.03. endobj 600.2 600.2 507.9 569.4 1138.9 569.4 569.4 569.4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Type/Font /Filter /FlateDecode 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 306.7 766.7 511.1 511.1 766.7 743.3 703.9 715.6 755 678.3 652.8 773.6 743.3 385.6 Value. 542.4 542.4 456.8 513.9 1027.8 513.9 513.9 513.9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 endobj 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 Notice that the predicted hazard (i.e., h(t)), or the rate of suffering the event of interest in the next instant, is the product of the baseline hazard (h 0 (t)) and the exponential function of the linear combination of the predictors. 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 Example: The simplest possible survival distribution is obtained by assuming a constant risk over time, so the hazard is λ (t) = λ 892.9 892.9 723.1 328.7 617.6 328.7 591.7 328.7 328.7 575.2 657.4 525.9 657.4 543 /FirstChar 33 << 511.1 511.1 511.1 831.3 460 536.7 715.6 715.6 511.1 882.8 985 766.7 255.6 511.1] Estimate cumulative hazard and fit Weibull cumulative hazard functions. 21 0 obj << Example: The simplest possible survival distribution is obtained by assuming a constant risk … 285.5 799.4 485.3 485.3 799.4 770.7 727.9 742.3 785 699.4 670.8 806.5 770.7 371 528.1 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 742.3 799.4 0 0 742.3 599.5 571 571 856.5 856.5 285.5 314 513.9 513.9 513.9 513.9 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 >> I fit to that data a Kaplan Meier model and a Cox proportional hazards model—and I plot the associated survival curves. /FirstChar 33 /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 endobj 8888 University Drive Burnaby, B.C. 530.4 539.2 431.6 675.4 571.4 826.4 647.8 579.4 545.8 398.6 442 730.1 585.3 339.3 >> 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 endobj In the Cox-model the maximum-likelihood estimate of the cumulated hazard function is a step function..." But without an estimate of the baseline hazard (which cox is not concerned with), how contrive the cumulative hazard for a set of covariates? 761.6 272 489.6] Terms and conditions © Simon Fraser University >> In , the cause-specific hazard function λ k (t) on the right-hand side makes the probability density function for cause-specific events of type k improper whenever λ k < ∑ k λ k.Therefore, the cumulative incidence function in may also be improper. 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 12 0 obj /Name/F11 >> /Name/F8 /FontDescriptor 32 0 R /FirstChar 33 /Widths[360.2 617.6 986.1 591.7 986.1 920.4 328.7 460.2 460.2 591.7 920.4 328.7 394.4 /LastChar 196 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 �yNf\t�0�uj*e�l���}\v}e[��4ոw�]��j���������/kK��W�`v��Ej�3~g%�q�Wk�I�H�|%5Wzj����0�v;.�YA 323.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 323.4 323.4 460 511.1 306.7 306.7 460 255.6 817.8 562.2 511.1 511.1 460 421.7 408.9 332.2 536.7 >> /BaseFont/HPIIHH+CMSY10 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 610.8 925.8 710.8 1121.6 924.4 888.9 808 888.9 886.7 657.4 823.1 908.6 892.9 1221.6 >> 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 706.4 938.5 877 781.8 754 843.3 815.5 877 815.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 However, these values do not correspond to probabilities and might be greater than 1. xڵWK��6��W�VX�$E�@.i���E\��(-�k��R��_�e�[��`���!9�o�Ro���߉,�%*��vI��,�Q�3&�$�V����/��7I�c���z�9��h�db�y���dL /Type/Font /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 /Subtype/Type1 The hazard function at any time tj is the number of deaths at that time divided by the number of subjects at risk, i.e. /FontDescriptor 20 0 R 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 Cumulative Hazard Function The formula for the cumulative hazard function of the Weibull distribution is \( H(x) = x^{\gamma} \hspace{.3in} x \ge 0; \gamma > 0 \) The following is the plot of the Weibull cumulative hazard function with the same values of γ as the pdf plots above. 388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 If T1 and T2 are two independent survival times with hazard functions h1(t) and h2(t), respectively, then T = min(T1,T2) has a hazard function hT (t) = h1(t)+ h2(t). 788.9 924.4 854.6 920.4 854.6 920.4 0 0 854.6 690.3 657.4 657.4 986.1 986.1 328.7 /LastChar 196 %PDF-1.2 27 0 obj Melchers, 1999) Load and organize sample data. It is used in survival theory, reliability engineering and life insurance to estimate the cumulative number of expected events. The cumulative hazard function is H(t) = Z t 0 306.7 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 306.7 306.7 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 /LastChar 196 << 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 The hazard function is related to the probability density function, f(t), cumulative distribution function, F(t), and survivor function, S(t), as follows: Relationship between Survival and hazard functions: t S t t S t f t S t t S t t S t. ∂ ∂ =− ∂ =− ∂ = ∂ ∂ log ( ) ( ) ( ) ( ) ( ) ( ) log ( ) … In principle the hazard function or hazard rate may be interpreted as the frequency of failure per unit of time. /BaseFont/CKCRPC+CMMI10 If dj > 1, we can assume that at exactly at time tj only one subject dies, in which case, an alternative value is We assume that the hazard function is constant in the interval [tj, tj+1), which produces a step function. >> 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 756 339.3] stream /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 /Subtype/Type1 460 664.4 463.9 485.6 408.9 511.1 1022.2 511.1 511.1 511.1 0 0 0 0 0 0 0 0 0 0 0 /BaseFont/UCURDE+CMR12 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 770.7 628.1 285.5 513.9 285.5 513.9 285.5 285.5 513.9 571 456.8 571 457.2 314 513.9 /FirstChar 33 The cumulative hazard plot consists of a plot of the cumulative hazard \(H(t_i)\) versus the time \(t_i\) of the \(i\)-th failure. 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 /LastChar 196 /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 339.3 892.9 585.3 892.9 585.3 610.1 859.1 863.2 819.4 934.1 838.7 724.5 889.4 935.6 Plotting cumulative hazard function using the Nelson Aalen estimator for a time-varing exposure Posted 01-22-2019 09:38 PM (898 views) Hi, I am trying to create a plot of the cumulative hazard of an outcome over time for a time-varying exposure using the Nelson-Aalen estimator in SAS. The cumulative hazard has less obvious understanding than the survival functions, but the hazard functions is the basis of more advanced techniques in survival analysis. Canada V5A 1S6. Additional properties of hazard functions If H(t) is the cumulative hazard function of T, then H(T) ˘ EXP (1), the unit exponential distribution. It's like summing up probabilities, but since Δ t is very small, these probabilities are also small numbers (e.g. /Widths[323.4 569.4 938.5 569.4 938.5 877 323.4 446.4 446.4 569.4 877 323.4 384.9 /FontDescriptor 26 0 R (12) and (13), we get the unconditional bivariate survival functions at time t1j > 0 and t2j > 0 as, (23) S(t1j, t2j) = [1 + θηj{α1 ln (1 + λ1tγ11j) + α2 ln (1 + λ2tγ22j)}] − 1 θ Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. 692.5 323.4 569.4 323.4 569.4 323.4 323.4 569.4 631 507.9 631 507.9 354.2 569.4 631 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] %PDF-1.5 /FontDescriptor 38 0 R The Nelson–Aalen estimator is a non-parametric estimator of the cumulative hazard rate function in case of censored data or incomplete data. 24 0 obj >> As I said, not that realistic, but this could be just as well applied to machine failures, etc. Estimate and plot cumulative distribution function for each gender. �TP��p�G�$a�a���=}W� Here we can see that the cumulative hazard function is a straight line, a consequence of the fact that the hazard function is constant. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 627.2 817.8 766.7 692.2 664.4 743.3 715.6 /Subtype/Type1 /Subtype/Type1 This might be a bit confusing, so to make the statement a bit simpler (yet not that realistic) you can think of the cumulative hazard function as the expected number of deaths of an individual up to time t, if the individual could to be resurrected after each death without resetting the time. 15 finished out of the 500 who were eligible. Definition of Survival and hazard functions: ( ) Pr | } ( ) ( ) lim ( ) Pr{ } 1 ( ) 0S t f t u t T t u T t t S t T t F t. u. λ. 361.6 591.7 591.7 591.7 591.7 591.7 892.9 525.9 616.8 854.6 920.4 591.7 1071 1202.5 /Name/F9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 693.8 954.4 868.9 36 0 obj /Type/Font Plot survivor functions. 525 768.9 627.2 896.7 743.3 766.7 678.3 766.7 729.4 562.2 715.6 743.3 743.3 998.9 By Property 2, it follows that. << /BaseFont/PEMUMN+CMR9 Curves are automaticallylabeled at the points of maximum separation (using the labcurvefunction), and there are many other options for labeling that can bespecified with the label.curvesparameter. /Widths[285.5 513.9 856.5 513.9 856.5 799.4 285.5 399.7 399.7 513.9 799.4 285.5 342.6 That is, the hazard function is a conditional den-sity, given that the event in question has not yet occurred prior to time t. Note that for continuous T, h(t) = d dt ln[1 F(t)] = d dt lnS(t). This MATLAB function returns a probability density estimate, f, for the sample data in the vector or two-column matrix x. Bdz�Iz{�! /FirstChar 33 /FontDescriptor 17 0 R hazard rate of dying may be around 0.004 at ages around 30). d dtln(S(t)) The hazard function is also known as the failure rate or hazard rate. The cumulative hazard function (CHF), is the total number of failures or deaths over an interval of time. 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 /Type/Font As with probability plots, the plotting positions are calculated independently of the model and a … Step 4. Thus, the predictors have a multiplicative or proportional effect on the predicted hazard. /Type/Font 1074.4 936.9 671.5 778.4 462.3 462.3 462.3 1138.9 1138.9 478.2 619.7 502.4 510.5 endobj This is the approach taken when using the non-parametric Nelson-Aalen estimator of survival.First the cumulative hazard is estimated and then the survival. << '-ro�TA�� Hazard and Survivor Functions for Different Groups; On this page; Step 1. 588.6 544.1 422.8 668.8 677.6 694.6 572.8 519.8 668 592.7 662 526.8 632.9 686.9 713.8 /LastChar 196 /Type/Font 460.2 657.4 624.5 854.6 624.5 624.5 525.9 591.7 1183.3 591.7 591.7 591.7 0 0 0 0 sts graph and sts graph, cumhaz are probably most successful at this. endobj Rodrigo says: September 17, 2020 at 7:43 pm Hello Charles, Would it be possible to add an example for this? /Name/F4 /LastChar 196 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … >> /Type/Font 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 endobj 339.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 339.3 /FontDescriptor 29 0 R Property 3: 6 Responses to Estimating the Baseline Hazard Function. This MATLAB function returns the empirical cumulative distribution function (cdf), f, evaluated at the points in x, using the data in the vector y. The sum of estimates is … Substituting cumulative hazard function for the generalized log-logistic type II and the generalized Weibull baseline distribution in Eqs. For example, differentplotting symbols can be placed at constant x-increments and a legendlinking the symbols with … /FontDescriptor 8 0 R /Subtype/Type1 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 766.7 715.6 766.7 0 0 715.6 613.3 562.2 587.8 881.7 894.4 306.7 332.2 511.1 511.1 /LastChar 196 743.3 743.3 613.3 306.7 514.4 306.7 511.1 306.7 306.7 511.1 460 460 511.1 460 306.7 570 517 571.4 437.2 540.3 595.8 625.7 651.4 277.8] 0 0 0 0 0 0 0 615.3 833.3 762.8 694.4 742.4 831.3 779.9 583.3 666.7 612.2 0 0 772.4 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 For example, survivor functions can be plotted using. /Length 2053 << /Filter[/FlateDecode] /Name/F7 /LastChar 196 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 642.3 856.5 799.4 713.6 685.2 770.7 742.3 799.4 /FirstChar 33 /Subtype/Type1 3 0 obj Fit Weibull survivor functions. /Type/Font /Subtype/Type1 Recall that we are estimating cumulative hazard functions, \(H(t)\). /FirstChar 33 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 Let F (t) be the distribution function of the time-to-failure of a random variable T, and let f (t) be its probability density function. /Subtype/Type1 799.2 642.3 942 770.7 799.4 699.4 799.4 756.5 571 742.3 770.7 770.7 1056.2 770.7 �x�+&���]\�D�E��� Z2�+� ���O\(�-ߢ��O���+qxD��(傥o٬>~�Q��g:Sѽ_�D��,+r���Wo=���P�sͲ���`���w�Z N���=��C�%P� ��-���u��Y�A ��ڕ���2� �{�2��S��̮>B�ꍇ�c~Y��Ks<>��4�+N�~�0�����>.\B)�i�uz[�6���_���1DC���hQoڪkHLk���6�ÜN�΂���C'rIH����!�ޛ� t�k�|�Lo���~o �z*�n[��%l:t��f���=y�t�$�|�2�E ����Ҁk-�w>��������{S��u���d$�,Oө�N'��s��A�9u��$�]D�P2WT Ky6-A"ʤ���$r������$�P:� Step 5. 4sts— Generate, graph, list, and test the survivor and cumulative hazard functions Comparing survivor or cumulative hazard functions sts allows you to compare survivor or cumulative hazard functions. Step 3. /Name/F10 That is the number who finished (the event occurred)/the number who were eligible to finish (the number at risk). �P�Fd��BGY0!r��a��_�i�#m��vC_�ơ�ZwC���W�W4~�.T�f e0��A$ /FirstChar 33 /Widths[306.7 514.4 817.8 769.1 817.8 766.7 306.7 408.9 408.9 511.1 766.7 306.7 357.8 T = (− ln (U) b e − X β) 1 a, where U ∼ U (0, 1), a is the Weibull shape parameter and b is the Weibull scale parameter. /Name/F1 465 322.5 384 636.5 500 277.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 That is, the survival function is the probability that the time of death is later than some specified time t. The survival function is also called the survivor function or survivorship function in problems of biological survival, and the reliability function in mechanical survival problems. /FontDescriptor 35 0 R h ( t) = lim Δ t → 0 P ( t < T ≤ t + Δ t | T > t) Δ t. Cumulative hazard is integrating (instantaneous) hazard rate over ages/time. There is an option to print the number of subjectsat risk at the start of each time interval. where S(t) = Pr(T > t) and Λ k (t) = ∫ 0 t λ k (u)du is the cumulative hazard function for the kth cause-specific event. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 710.8 986.1 920.4 827.2 9 0 obj Step 2. 843.3 507.9 569.4 815.5 877 569.4 1013.9 1136.9 877 323.4 569.4] << 41 0 obj /Subtype/Type1 571 285.5 314 542.4 285.5 856.5 571 513.9 571 542.4 402 405.4 399.7 571 542.4 742.3 /FontDescriptor 14 0 R 594.7 542 557.1 557.3 668.8 404.2 472.7 607.3 361.3 1013.7 706.2 563.9 588.9 523.6 >> The hazard function always takes a positive value. n��I4��#M����ߤS*��s�)m!�&�CeX�:��F%�b e]O��LsB&- $��qY2^Y(@{t�G�{ImT�rhT~?t��. [��FH�U���vB�H�w�`�߶��r�=,���o:vז-Z2V�>s�2��3��%���G�8t$�����uw�V[O�������k��*���'��/�O���.�W���.rP�ۺ�R��s��MF�@$�X�|�g9���a�q� AR1�ؕ���n�u%;bP a�C�< �}b�+�u�™fs8��w ��&8l�g�x�;2����4sF ���� �È�3j$��(���wD � �x��-��(����Q�By�ۺlH�] ��J��Z�k. �������ёF���ݎU�rX��`y��] ! /LastChar 196 Given the hazard, we can always integrate to obtain the cumulative hazard and then exponentiate to obtain the survival function using Equation 7.4. /Name/F6 /BaseFont/LXJWHL+CMBX12 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 /BaseFont/JVGETH+CMTI10 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 513.9 770.7 456.8 513.9 742.3 799.4 513.9 927.8 1042 799.4 285.5 513.9] The cumulative hazard function should be in the focus during the modeling process. /Name/F5 /FontDescriptor 11 0 R /BaseFont/JYBATY+CMEX10 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 An example will help x ideas. 361.6 591.7 657.4 328.7 361.6 624.5 328.7 986.1 657.4 591.7 657.4 624.5 488.1 466.8 323.4 877 538.7 538.7 877 843.3 798.6 815.5 860.1 767.9 737.1 883.9 843.3 412.7 583.3 285.5 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 285.5 285.5 639.7 565.6 517.7 444.4 405.9 437.5 496.5 469.4 353.9 576.2 583.3 602.5 494 437.5 ��B�0V�v,��f���$�r�wNwG����رj�>�Kbl�f�r6��|�YI��� For each of the hazard functions, I use F(t), the cumulative density function to get a sample of time-to-event data from the distribution defined by that hazard function. Hazard function: h(t) def= lim h#0 P[t T