olution of Assignmnt #2 Instructor: Alirza imchi Qustion #: For simplicity, assum that th distribution function of T is continuous. Th distribution function of R is: F R ( r = P( R r = P( log ( T r = P(log ( T r = P( ( T inc (T is a continuous dcrasing function, w hav P( ( T = P[ T ( ]. nc, F R ( r r r r r ( ( = r r = P( ( T r = P[ T ( ] = P[ T ( ] = Th probability dnsity function of R is distribution with paramtr λ = f R F R r ( r = ' ( r =. Thrfor, R has an xponntial, and this complts th proof. Qustion #2: (a Probability plots for xponntial, Wibull, lognormal, and log-logistic distribution ar givn in th nxt pag. It is clar that xponntial and Wibull distributions ar inadquat. Although th probability plots for lognormal and log-logistic distribution ar vry similar, but thy ar not idntical. (Compar Y-axs to s th diffrncs. It sms that lognormal distribution fit th data bttr. (b Cox-nll rsidual plots for xponntial, Wibull, lognormal, and log-logistic distribution ar givn in th nxt pags. It is clar that xponntial and Wibull distributions ar inadquat and gnralizd gamma distribution is th bst modl that fit th data. You might choos lognormal or log-logistic distribution, although th graphs is no vry last two obsrvations. r (c Th following tabl givs th log-liklihoods for xponntial, Wibull, lognormal, and gnralizd gamma distribution. Distribution Exponntial Wibull Lognormal Gnralizd Gamma Log-liklihood -4.8597-4.857-4.6769-4.37
Titl: Probability plots of urvival Tim
Titl: Cox_nll Rsidual plots of urvival Tim
(c -cont. For tsting : Exponntial distribution is an adquat modl for th data vs. : Wibull distribution is a bttr modl than xponntial distribution. th tst is T = -2[-4.8597 (-4.857] =.52. Th tst statistic has a chi-squar distribution with dgrs of frdom df = 2 - =. o P-valu = P(T >.52 =.942536. Thrfor, w fail to rjct and hnc Exponntial distribution is an adquat modl. Th following tabl givs th valu of tst statistic and P-valu for ach tsting othr distributions. Tsting Tst tatistic Df P-valu Exponntial vs. Gnralizd Gamma 3.98 2.22463 Wibull vs. Gnralizd Gamma 3.93.786297 Lognormal vs. Gnralizd Gamma.7324.39264 inc all P-valus ar gratr than.5, w fail to rjct th null hypothss. Thrfor, with liklihood ratio tst, w choos xponntial distribution. (d Th log-liklihood, BIC, and AIC for xponntial, Wibull, lognormal, log-logistic, and gnralizd gamma distribution ar givn in th following tabl. Distribution Log-Liklihhod (LL BIC = LL (p/2log(n AIC = LL 2p Exponntial -4.8597-43.5767-4.8597 Wibull -4.857-45.29-45.857 Lognormal -4.6769-44.9-44.6769 Log-logistic -4.4858-44.998-45.4858 Gnralizd Gamma -4.37-45.457-46.37
inc xponntial distribution has largst valu of BIC and AIC, w choos xponntial distribution as th modl that fit th data bst. ( Th following tabl givs slctd distribution for parts (a to (d. Part (a (b (c (d lctd distribution Lognormal Gnralizd Gamma Exponntial Exponntial It is clar that th rsults ar not consistnt. For xampl, th probability plots and Cox-nll rsidual plots shows that xponntial and Wibtull distribution ar not adquat whil w choos xponntial by liklihood ratio tst, BIC, and AIC. For this problm, th sampl siz is 3 with cnsord obsrvations. inc liklihood ratio tst, BIC, and AIC ar basd on larg sampl proprtis, w should rly mor on graphical mthods. o, lognormal distribution should b a good modl for data (In part (b, w might choos lognormal instad of gnralizd gamma distribution.. Th following tabl givs th last-squar stimats of intrcpt and slop of rgrssion lin for data (plot in part (a. (tudnts may find th stimats approximatly by fitting lins by hand. Distribution Intrcpt lop Exponntial.8429.7 Wibull -3.79484.973 Lognormal -2.4898.7922 Log-logistic -4.2765.23348 Exponntial Distribution: inc log ( t = λt, th stimat of λ is th slop of th rgrssion lin. Thrfor, th stimat of λ is.7 (Not that th intrcpt for xponntial distribution should b aro.. Th maximum liklihood stimat of λ is xp(-intrcpt = xp(-3.9692 =.889. Wibull Distribution: inc log( log ( t = logλ + logt, th stimat of is th slop of th rgrssion lin. Thrfor, th stimat of is.973. In addition, w hav.973 log( λ = 3.79484 λ =.2. Th maximum liklihood stimat of is /scal=/.9869 =.33. Th maximum liklihood stimat of λ is xp(-intrcpt = xp(-3.9677 =.892.
µ Lognormal Distribution: incφ log ( ( t = t, th stimat of σ is th invrs of th σ slop of th rgrssion lin. Thrfor, th stimat of σ is /.7992=.389. In addition, w µ µ hav = = 2.4898 µ = 3. 45747. Th maximum liklihood of σ is scal=.229 σ.389 and th maximum liklihood stimat of µ is intrcpt = 3.4868. ( t Log-logistic Distribution: inc log( = logα + logt, th stimat of is th slop of ( t th rgrssion lin. Thrfor, th stimat of is.23348. In addition, w hav log( α = 4.2765 α =.397. Th maximum liklihood stimat of is /scal = /.7538 =.3266. Th maximum liklihood stimat of α is Intrcpt 3.487 xp = xp( =.99 cal.7538 Th stimats of paramtrs by graphical mthod ar vry clos to maximum liklihood stimats of paramtrs. Not that if you find th last-squar rgrssion lin by using PROC REG for data (graphs in part (a, you can find confidnc intrvals for paramtrs of distributions using stimats and standard rror of stimats in A output. Qustion #3: (a For Wibull distribution, tsting : ( t = ( t vrsus : ( t ( t is I II I II quivalnt to tsting : λ = λ, I II = vrsus I II : λ I λ or. Th log-- II I II liklihood for (all data is -24.3. Th log-liklihoods for data for Typ I and Typ II ar - 2.748 and -.454, rspctivly. Thrfor, th tst statistic for tsting is T [ 24.3 ( 2.748.454 ] 2. 4876 = 2 = Th tst statistic has an asymptotic chi-squar distribution with dgrs of frdom of 4-2=2. Thrfor, P-valu = P(T > 2.4876 >.5 and w fail to rjct. (Not that liklihood ratio tst is not vry rliabl for this problm sinc w hav only obsrvations in ach group. W should us non-paramtric mthod.
(b Not that = = =. 358 and = = =. 8734. Ths valus I cal.743 II cal.5338 ar stimat of Wibull hap in A rsults. nc, by using A rsults, w hav. E.( =.3477 and. E.( =. 54. Thrfor, th tst statistic for tsting I II : vrsus I II : is I II I II.358.8734 T = = ( 2 Var + Var(.3477 +. 54 I II = 2 =.8534 P-valu = 2P(T >.8534 =.3934, using normal distribution. Thrfor, w fail to rjct. II : λ I inc =, tsting : ( t = ( t vrsus : ( t < ( t is quivalnt to I I II I II tsting = λ vrsus II : λ > I λ. Not that II W can find standard rrors of λ and I λ II 3.2398 = 3.725 λ =.397 and = =. 2466 I II asir to writ th null hypothsis in trms of λ. by using maximum liklihood proprtis, but it is θ I = and λ I θ II =. Not that th stimat λ II ar givn in A output. (Th stimat is calld Wibull and standard rror of θ and I θ II cal. Tsting : λ = I λ vrsus II : λ > I λ II : θ = I θ vrsus II : θ < I θ II 25.5284 4.55 = θ I θ II T = ( 2 Var θ + Var( θ 6.4489 + 7. 3676 I II is quivalnt to tsting. Using A rsults, th valu of th tst statistic is: 2 =.5342 P-valu =P(T <-.5342 =.625, using normal distribution. Thrfor, w fail to rjct. Thr is not nough vidnc to conclud that Typ II insulation is suprior.
Qustion #4: (a Th maximum liklihood stimators of paramtrs for ach group ar: 2.49 λ = =.829 = = = 2. 588 I I cal.3864.9256 λ = =.458 = = = 2. 322 II II cal.43 2.275 λ = =.33 = = = 3. 328 III III cal.392 2.45 λ = =.897 = = =. 939 IV IV cal.557 2.7943 λ = =.62 = = = 3. 6523 V V cal.2738 By dfinition = ( is th pth quantil (or pth prcntil of F( t = ( t. Th tnth prcntil, t p F p t, is sud as a rating lif. For Wibull distribution, w hav. ( =. ( =.9 xp[. ] =.9. F t ( ( = log. 9. t. λt λt log( λt = log( log.9 log( log( log.9 log( log.9. log( t = λ + = β + σ. Not that t is not a linar function of. β andσ. W can find standard rror of th stimat of t..th formula to find standard rror of th stimat of t is mor complicatd than th. formula that w discussd in th class sinc t is a function of two paramtrs instad of on. paramtrs. A find th standard rror oft by using that formula. I will show you how to. find th standard rror using A at th nd of this part. On th othr hand, in th following, I will find a confidnc intrval for t by a diffrnt mthod. Not that th confidnc intrvals. by th following mthod is diffrnt from th confidnc intrval obtaind basd on A rsults dirctly. t is a linar function of β andσ, it is asir to find a confidnc intrval for inc log(. log( t first and thn find a confidnc intrval. fort..
Group I: An stimat of t is. t = xp[ + log( log.9] = xp[2.49 + (.3864 log( log.9]. β σ t = xp(.6245 = 5.554. Not that an stimat of log(. t is.6245. Put a = log( log(.9 = 2.254. Thn th standard rror of th stimat of var[ β β 2 + σ log( log.9] = + var( σ + 2aCov(, = var( σ a β log( t is..642 + ( 2.254 2 (.62 + 2( 2.254(.3955 =.84835 nc a 95% confidnc intrval for log( t is. Estimat ± Z.E.(Estimat =.6245 ±.96 (.84835 (-.423, 3.2832. A 95%. 25.423 3.2832 confidnc intrval for t is (, = (.9586,26.66. imilarly, w can find. confidnc intrvals for othr groups. Group II: An stimat of t is. = xp[.9256 + (.43 log( log.9] = xp(.9557 = 2.65. An stimat of log(. t is..9557. Thn th standard rror of th stimat of log( t is. t.272 + ( 2.254 2 (.9489 + 2( 2.254(.4866 =.37 nc a 95% confidnc intrval for log( t is:..9557 ±.96 (.37 (.3643,.547. A 95% confidnc intrval for t..3643.547 is (, = (.4395,4.6973.
Group III: An stimat of t is. t = xp[2.275 + (.392 log( log.9] = xp(.5522 = 4.728. An stimat of log(. t is..5522. Thn th standard rror of th stimat of log( t is..278 + ( 2.254 2 (.6825 + 2( 2.254(.2723 =.243 nc a 95% confidnc intrval for log( t is..5522 ±.96 (.243 (.78, 2.246. A 95% confidnc intrval for t..798 2.246 is (, = (2.944,,7.573. Group IV: An stimat of t is. = xp[2.45 + (.557 log( log.9] = xp(.25 = 3.493. An stimat of log(. t is..25. Thn th standard rror of th stimat of log( t is. t.334 2 + ( 2.254 (.23 + 2( 2.254(.678. nc a 95% confidnc intrval for log( t is. =.34926.25 ±.96 (.34926 (.5654,.9345. A 95% confidnc intrval for t..5654.9345 is (, = (.762,6.926. Group V: An stimat of t is. = xp[2.7943 + (.2738 log( log.9] = xp(2.78 = 8.8299. An stimat of log(. t is. 2.78. Thn th standard rror of th stimat of log( t is. t.8298 + ( 2.254 2 (.4928 + 2( 2.254(.984 =.2539 nc a 95% confidnc intrval for log( t is.
2.78 ±.96 (.2539 (.7755, 2.587. A 95% confidnc intrval for t..7755 2.587 is (, = (5.932,3.264. Not that prcntils and thir standard rrors can b obtaind in A asily and you do not hav to do this long procss. You should writ Proc lifrg data=b; By Typ; Modl Tim*tatus(=/covb d=wibull; Output out=a p=t quantil=. std=s; Run; quit; Proc print data=a; BY Typ; Var Tim tatus _Prob_ t s; Run; quit; Th P = t in th output statmnt rqusts prcntils. By dfaults, PROC LIFEREG calculats th 5 th prcntil, which is th mdian. I rqustd tnth prcntil with th QUANTILE kyword, as dscribd in th PROC LIFEREG documntation. Th word t in th OUTPUT statmnt is just th variabl nam that I chos for tnth prcntil. Th TD kyword rqusts th standard rror of th prcntil. In th OUTPUT, _PROB_ is th rqustd quantil (.. Th stimat of tnth prcntil and its standard dviation, and a 95% confidnc intrval for tnth prcntil for ach group, using A, ar givn in th following tabl. Group Estimat tandard Error Confidnc Intrval I 5.55.49955 5.55 ±.96(.49955 II 2.6.7846 2.6 ±.96(.7846 III 4.72.385 4.72 ±.96(.385 IV 3.49.297 3.49 ±.96(.297 V 8.829.8328 8.829 ±.96(.8328 Ths confidnc intrvals ar diffrnt from th confidnc intrvals obtaind by using log transformation. (b Th plot of survivor functions of failur tim for fiv diffrnt compounds is givn in th nxt pag. If ( t, i =, K,5, is survivor function of compound I, thn i ( t < ( t ( t ( t 2 < 3 < 5
Th survivor function of compound 4 is gratr than survivor function of compound 2. urvivor function of compound 4 is lss than survivor function of compound 3 most of th tim. Th p- valus for long-rank tst and Wilcoxon tst for tsting quality of survivor functions ar.5 and.3, rspctivly. Thrfor, w strongly rjct th quality of survivor functions. Titl: Th Plot of urvivor Functions of Failur Tim for Fiv Diffrnt Compounds For Wibull distribution, tsting : ( t = ( t, i j, i =, K,5, j =,,5 vrsus K i j : ( t ( t, for som i i j is quivalnt to tsting j : λ = i λ, j = for all i j, i =, K,5, j =, K,5 for vrsus λ i λ j i : or for som i j. Th log--liklihood (all data is -39.8296. Th log-liklihoods for data for Typs I, II, III, IV, and typ V ar -6.767, -6.4549, -4.5925, -7.8387 and -3.27, rspctivly. Thrfor, th tst statistic for tsting is T = 2 [ 39.8296 (-6.767-6.4549-4.5925-7.8387-3.27 ] = 22. 396 j i j
Th tst statistic has an asymptotic chi-squar distribution with dgrs of frdom of -2=8. P-valu = P(T > 22.396 =.44<.5 and w rjct strongly. Thrfor, thr is strong vidnc to conclud that thr is a diffrnc in th survivor functions of th failur tim of fiv compounds. (c W should chck that lognormal distribution is an adquat modl for data for ach group. W can compar lognormal with gamma distribution for ach compound. This nds tsting fiv null hypothss. If th failur tim for diffrnt compound is indpndnt from th failur tims of othr groups and th distribution of all data is lognormal, thn distribution of data in ach group is also lognormal. (This proprty may not b tru for othr distributions. inc indpndncy assumption sms a rasonabl assumption, w only nd to compar lognormal with gamma distribution for all data. Th log-liklihood for, using all data, for lognormal and gamma distribution ar -37.92776 and - 37.88796, rspctivly. Thrfor, for r tsting : Lognormal distribution is an adquat modl for th data vs. distribution. : Gnralizd gamma distribution is a bttr modl than xponntial th tst is T = -2[-37.92776 (-37.8879] =.796. Th tst statistic has a chi-squar distribution with dgrs of frdom df = 3-2 =. o P-valu = P(T >.796 =.7778. Thrfor, w fail to rjct and hnc lognormal distribution is an adquat modl. Not that this dos not man th lognormal is a good modl. It only compar lognormal with gnralizd gamma distribution and it only mans that, compar to gnralizd gamma, distribution, w can us th simplr modl, log-normal distribution. Th probability plots for lognormal distribution (for all data and for Wibull distribution (for ach group ar givn in th nxt pag. It is clar that lognormal is fit th data vry wll and, in fact; it fits bttr than Wibull distribution. Thrfor, w should xpct that th confidnc intrvals basd lognormal distribution ar narrowr that th confidnc intrvals obtaind from Wibull distribution. inc Log T has normal distribution with man µ and standard dviationσ, th th quantil is log( t obtain by solving µ. = Φ (. =.5398278, whr Φ is th standard normal σ distribution function. Lt a =.5398278.Thn log( t = µ + aσ. A givs th stimats of. µ andσ. W can us th sam mthod in part (a to find a confidnc intrval for log( t and. thn find a confidnc intrval fort. inc this is similar to what w did in part (a, I lav it. to you to complt it and, in th following; I only giv you A rsults. Not that th stimats from abov mthod ar th sam as th A rsults, but th standard rrors and confidnc intrvals ar diffrnt.
Th stimat of tnth prcntil and its standard dviation, and a 95% confidnc intrval for tnth prcntil for ach group, using A for lognormal distribution, ar givn in th following tabl. As w xpct, th confidnc intrvals basd on lognormal distribution ar narrowr. Group Estimat tandard Error Confidnc Intrval I 4.75996.8927 4.75996±.96(.8927 II 3.36783.56237 3.36783 ±.96(.56237 III 4.782.82937 4.782 ±.96(.82937 IV 5.26383.976 5.26383 ±.96(.976 V 8.8268.33558 8.8268 ±.96(.33558 Titl: Probability plots of FailurTim
Qustion #5: Tratmnt R: Th log-liklihood for xponntial, Wibull, lognormal, log-logistic, and gamma distributions ar -68.284, -67.9342, -67.843, -67.968, and -67.968, rspctivly. For tsting : Exponntial distribution is an adquat modl for th data vs. : Wibull distribution is a bttr modl than xponntial distribution. th tst is T = -2[-68.284 (-67.9432] =.374. Th tst statistic has a chi-squar distribution with dgrs of frdom df = 2 - =. o P-valu = P(T >.374 =.5427863. Thrfor, w fail to rjct and hnc Exponntial distribution is an adquat modl. Th following tabl givs th valu of tst statistic and P-valu for ach tsting othr distributions. Tsting Tst tatistic Df P-valu Exponntial vs. Gnralizd Gamma.5852 2.74632 Wibull vs. Gnralizd Gamma.248.6433 Lognormal vs. Gnralizd Gamma.44.94483 Log-logistic vs. Gnralizd Gamma.252.6567 inc all P-valus ar gratr than.5, w fail to rjct th null hypothss. Thrfor, with liklihood ratio tst, w choos xponntial distribution for group R. Tratmnt RC: Th log-liklihood for xponntial, Wibull, lognormal, and log-logistic distributions ar -82.864, -73.2674, -74.2923, and -73.6248, rspctivly For tsting : Exponntial distribution is an adquat modl for th data vs. : Wibull distribution is a bttr modl than xponntial distribution. th tst is T = -2[-82.864 (-73.2674] = 7.838. Th tst statistic has a chi-squar distribution with dgrs of frdom df = 2 - =. o P-valu = P(T > 7.838 =.2. Thrfor, w strongly rjct and hnc Exponntial distribution is not an adquat modl. If w compar othr distributions, w find that lognormal distribution is an adquat modl for data. inc th lognormal was not rjctd for th tratmnt R, w choos log-normal for both groups. I chos lognormal distribution sinc it is vry simpl. (Log T has a normal distribution. tudnts might choos Wibull distribution or log-logistic distribution. For Lognormal distribution, tsting : ( t = ( t vrsus : ( t ( t is R RC R RC quivalnt to tsting : µ = µ, R RC σ = R σ vrsus RC : µ µ or R RC σ R σ. Th log-- RC liklihood for (all data is -48.7927. Th log-liklihoods for data for Tratmnts R and RC ar -67.843 and -74.2923, rspctivly. Thrfor, th tst statistic for tsting is
T [ 48.7927 ( 67.843 74.2923 ] 3. 348 = 2 = Th tst statistic has an asymptotic chi-squar distribution with dgrs of frdom of 4-2=2. Thrfor, P-valu = P(T > 3.348 =.28<.5 and w strongly rjct. Thrfor, thr is strong vidnc to conclud that two survival functions ar not qual. Th stimatd paramtrs ar: µ = Intrcpt = 3.752, = Intrcpt =. 278 R µ, σ = cal =. 278, =. 6392 R σ RC RC Not that PROC LIFETET cannot handl intrval cnsoring. Thrfor, w cannot us nonparamtric mthods, probability plots, and Cox-nll rsidual plots. Qustion #6: (a Th plot of survivor function is givn in th following. Th Kaplan-Mir stimat of survivor function can b obtaind by A. I do not copy th stimat to sav pags.
f i ( = ' ' (b Lt b = ( p, λ,,, λ and ( t, b = (, whr 2 2 i t ( t = xp[ i λt ], i, 2. Thn, log-liklihood function is: LogL( b = n i= Log( f ( t, b = i n i= log [ p ( t + ( p ( t ] Now, w nd to find vctorb that maximiz log-liklihood function. This can b don by Nwton-Raphson s mthod. Lt U (b b th vctor of first drivativs of log L(b with rspct to b, and lt I (b b th matrix of scond drivativs of log L with rspct to b. Th Nwton- Raphson algorithm is thn b b j I ( b b f = U (, whr + j j I is th invrs of I. (tudnts j should gt full crdit if thy calculat U(b and I(b. W can also us optimization softwar to maximiz log L(b. Th maximum liklihood stimats of paramtrs ar: λ, =. 66 =. 36 λ, =. 4 2 2 p =.37, =. 48 Th Cox-nll rsidual plot of data is givn in th following. It shows that th mixd modl fit th data wll. f 2