V4 1 SAS 2000.11.18
4
( ) (Survival Time)
1 (Event)
Start of Study Start of Observation Died Died Died Lost End Time Censor Died Died Censor Died Time
Start of Study End Start of Observation Censor Died Died Censor Died Time
()
Survival Time Analysis Proportion Survival Function 5000 10
,Weibull
(Survival Function) S(t) t (Proportion) F(t) S(t)=1-F(t) f(t) f(t) t
(Survival Function) S(t) 1 S(t) 1-S(t)=F(t) t
(hazard function) (rate) h(t) f(t)/s(t) 40 40 40 1
S(t) ( ) ds t dt S(t)
( ) = Pr ( ) ( ) = 1 St ( ) St obt t Ft ( ), ( ) = ( ) f t Ft f udu ( ) ht t 0 ( ) ( + ) ( ) ( < + ) Pr ob t T t tt t = lim t 0 t ( ) ( ) ( ) ( log St) St St t ds t 1 d = lim = = t 0 t St dt St dt
ht { Ht } ( ) ( ) ( ) ( logst ( )) Ht = hudu= log St St = exp d = t ( ) ( ) ( ) 0 dt
( ) ( ) ( ) ( ) ha t a ht ha ( t) = a ht ( ), = = a ht ht t t ( ) = exp ( ) = exp ( ) Sa t a h u du a h u du 0 0 t = exp h u du = 0 ( ) S( t) a a a a
3
t
Weibull Weibull ( ) f t S t β 1 β t t = exp α α α t = 1 F t = exp α ( ) ( ) β β h t ( ) f ( t) β t = = S( t) α α β 1
Weibull Weibull3 0 Weibull 1 1 1
Weibull f
Weibull 63.2 JMP(Beta) JMP(Alpha)
1 Weibull.JMP Weibull 1 2 3 3
time 10000 2.5 Weibull time f(t),f(t),s(t) h1(t) 2.5 h2(t) 1.0 h3(t) 0.5 40
1.1 1.Overlay Plot 2.
1.2 3.f(t) 1.Time 4.Y 2.X 5.OK
1.3 1.Connect Thru Missing 2.
1 Weibull.JMP h(t) 1 1 1
Calculator Cols Column Info. Current Properties Formula Edit Formula, 2 Col Name
LnNormal.JMP,, f t ( ) 1 1 logt = exp σt 2π 2 σ µ 2
( x µ ) 2 1 f( x) = exp, x 2 < <+ σ 2π 2σ ( t µ ) 2 1 x 2 σ 2π 2σ F( x) = exp dt, < x<+
( ) ( ) f t = λexp λt t 0, λ> 0 ( ) ( λ ) F t f(t) = 1 exp t t 0, λ > 0 t
( ln t µ ) 2 1 1 f() t = exp t 0 2 σ 2π t > 2σ ( ) F t ( ln x µ ) 2 1 t 1 = exp dxt 0 2 > 0 σ 2π x 2σ f(t)
49
V4, weibull,3 weibull,, Weibull
Weibull S t t = 1 F t = exp α ( ) ( ) ( ) log S t t = α β β t α log S( t) = log{ log ( )} S t β log t = α log logs t = β logα + β logt { ( )} β
2 Weibull.JMP 2 Weibull Column 1 Calculator,-lnS(t) Analyze Fit Y X,X time,y -lns(t)
2 Weibull.JMP Weibull time S(time)
Wiring JMP Time Analyze Distribution Fit DistributionWeibull
Weibull Weibull
Part II Survival 57
1 2 3 4
Survival Survival Distribution Kaplan-Meier Parametric Regression Proportional Hazards Cox Recurrence
Weibull 64
(censor) (censor) 1. 2.
Censor Censor 0 1 2 0
3 Wiring.JMP, Survival Disuribution Survival Weibull Weibull Weibull
(Al ). Time( ) Censor( ) Distribution
3.1 1.Survival Distribution 2.Time 3.Y 4.OK
3.2 Weibull Weibull Plot Weibull Fit
3.3 Weibull
3.4 1.Survival Distribution 3.Y 2.Time 4.Censor 6.OK 5.Censor
3.5 1.Plot Options Show Confid Interval 2. Kaplan-Meier, 95
3.6 Weibull 1.Weibull Plot,Weibull Fit 2.Weibull 3.Save Estimates
3.7
3 Wiring.JMP Extreme-value Parameter Estimates Weibull Parameter Estimates Weibull 95 Delta Weibull Beta=1/Delta Lambda Weibull 63.2 Alpha=e Lambda Weibull Exponential Plot LogNormal Plot
Kaplan Meier n ( ) { t i} i= 1 t i d i t i n i,, ( ) ˆ d d d d S t = 1 1 1 = 1 n n i L n1 n2 nn ti < t ni ( ) 1 2
Kaplan Meier ˆ d d d d S t = 1 1 1 = 1 n n i L n1 n2 nn ti < t ni ( ) 1 2, n ˆ d i logs( t) = log 1 i= 1 n log ˆ d V S = V log 1 i= 1 n n i i i
Kaplan-Meier 1 V log Sˆ V Sˆ ˆ 2 S, V log Sˆ = V Sˆ = Sˆ 2 n d i di V log 1 = ni ni ni di d n i i= 1 ni ni i di n n d ( ) i= 1 i i i ( d ) ( )
Kaplan-Meier Sˆ = n 1 n i= 1 1 n d i ( ) 1 / 1 V Sˆ = Sˆ Sˆ n
t i ht ( ) i Dt = t, t ( ) i i t, n i + Dt 0 H t ( ) i i tk = k= 1 t 1 Ht ( i) = k= 1 i
39 79 90 115 66 96 ) Excel
=count( ) If F(t) A$1-C3+1 E3/D3 If
MON j = MON + j 1 n+ 1 MON n+ 1 ( i 1) MON(Mean Order Number) F(t) j 1
87
1 Kaplan-Meier Wilcoxon
Wilcoxon
j ( j 1,2,, k) = L t j t j 1 d 1 11 n11 d11 n d 11 1j n1j d1j n1j 1 d1k n1k d1k n1k L L 2 d 2 d 21 n21 d21 n 21 2 j n2 j d2 j n2 j 2 d2 k n2k d2k n2k d n d n d n d n d n d n j 1 1 1 1 j j j j k k k k m = d n / n, m = d n / n 1j j 1j j 2 j j 2 j j
( ) ( 1) n n n-d n n d n d v d n n n n 1j 1j j j 1j 2j j j j 1 j = j 1 = 2 j j n-1 j j j ------------------- ( ) ( 1) n n n-d n n d n d v d n n n n 2 j = j 2 j 2j j j 1j 2j j j j 1 = 2 j j n-1 j j j ------------------- n= d, p= n / n j 1 j j,
δ = d m, δ = d m 1j 1j 1j 2 j 2 j 2 j δ1j = ( d1j m1j), δ2j = ( d2j m2j)
n 1 n 1j n n 1j j δ1j = d1j dj = d1j = n j n j n n j n n v = 1j 2 j 1 2 n j 1 j 1 j
n δ1j = wj( d m 1j 1j) = wj d 1j n ( ) ( nj 1) n1jn2jd j nj dj v1 j = wj, wj = 1 n δ ( ) 1 1 j = w d m = w d 1j 1 j j j j j ( ) ( nj 1) n n d n d v w w n n 1j 2 j j j j 1 j = j, j = j j n 1 j j 1 j n j
δ n 1 n n = n = w d = ( nd n ) = S 1j 1 j 2 n 1 j 1j j j 1j n j ( ) ( nj 1) n n d n d 1j 2 j j j j 1j = wj = n1jn2 j n j j 1j 2j 1 j
4 Rats.JMP, Survival Time Modeling 2 Survival
2 day Group ) Censor )
4.1 1.Survival Distribution 2.days 3.Y 4.Censor 6.Group 8.OK 5.Censor 7.Grouping
5 4.2 2
4.3 ( ) Plot Options Show Conbined
4 Rats.JMP
days. 2Survival Survival=(At Risk - N Failed)/At Risk 3Failure Survival+Failure=1 SurvStdErr(Survival Standard Error)
N Failed 1 N Censored 1 At Risk Quantiles Mean StdDev Standard Deviation
PWB JMP Weibull
PWB 65 75 85
111
(, )
1
,
( ),m=1( ),
A
( ) ( ) No. time x; V d1: d2 1 140 2 170 i n 230 390 35 1 0 50 0 1 M M M M M 40 0 1 M M M M M 50 0 1
5 Unit.JMP Survival Time Modeling Survival Omit
Unit.JMP
2 time ) failure ) 2 Condenser Relay
5.1 1.Survival Distribution 2.time 4.OK 3.Y 5.Competing Causes 6.Falure
5.2 Weibull Plot Weibull
1.Omit Causes 5.3 Weibull Weibull 2.Condenser 3.OK
5.4
5 Unit.JMP Competing Causes condenser Omit condenser Failure Grouping
Part III 127
1
S ( ) a S ( t) o t = a S t = as t a ( ) ( ) o h ( ) a h ( t) o t = a S t = S t a ( ) ( ) a o
Weibull ( ) ( ) a Sa t = S0 t lns t = lns t lns t = ln S t a ( ) ( ) ( ) ( ) 0 a 0 { ( )} ( ) a { a } ( ) 0 0 { } ln lns t = ln lns t = ln aln S t a β t = lna+ ln = ( βlnα + lna) + β ln t α ln a,, ( ) a
Weibull, { S ( t) } S ( t) ln ln ln ln a { } β t t = lna+ ln ln = ln a α α Weibull Weibull 0 β
Ht ( ) = βt Ht ( ) = βln ( t/ α) α Ht ( ) e βt β Ht = t/ α = ( ) ( ) β
h ( ) ( ) ( ) x t = h0 t exp bx h 0 (t) x 0 h 0 (t) b hi( t) h0 ( t) exp( bxi) = = exp{ b( x )} i xj i j h ( t) h t exp bx j 0 ( ) ( ) j
PWB JMP Survival Distribution Y Time1 Grouping /kt K-M Log(Time) Log(-logS(t)) Group 1/kT
Weibull log Ht () = 9.326269+ 2.0330log( Time) log Ht () = 11.76942+ 2.0537log( Time) log Ht () = 12.98731+ 2.0451log( Time)
1/kT=32.** ( ) 33.** 32.** = 2.443 34.** 32.** = 3.661 1 1.92 kt COX
COX -2.1750,COX
L,,,
, 2 2
Arrhenius Life= exp b + b 1/ kt Eˆ = b, k = 8.62 10 { ( )} 5 0 1 a 1 Life= exp { ( ) } ˆ b0 + b1 ln 2 T b1 = 1/ θ Eyring Life/ T = exp{ b + b ( 1/ kt) } Eˆ a = b (n) Life= exp{ b ( )} ˆ 0 + b1 ln T n= b1 0 1 1
t Log-Normal Distribution Acceleated x
R= R0 exp B T T 0 α = 1 R log y = b + b x R dr dt 1 1 0 1 T x = 1/ ( 273+ t) T = B T logyˆ = 3.455+ 7.614log x R 1 yˆ R = 0.0316exp 2026.1 xt 2 T
PWB JMP Weibull
PWB 65 75 85
PWB,?
PWB?,
PWB!!!
PWB
( ),
6 Creep.JMP 1 ( ), Survival Time Modeling (Weibull )
. temp ( ) 1/ time Censor( ) Life= exp(b 0 +b 1 /(273+temp)) exp(b 0 +b 1 /T)
T ( C + log L) = Q L = exp( Q/T - C) T L
Larson-Miller Larson-Miller
8.Run Mode 6.1 1.Proportional Hazards 2.time 3.Time to Even 5.Censor 4.Censor 6.1/T 7.Add
6.2 65 10 75 Risk = h h 75 65 ( t) ( t) ( ) { } = exp 18303.289 0.002874 0.002959 = 4.74 4.74
8.Weibull 9.Run Model 5.Censor 6.3 1.Parametric Regression 2.time 3.Time to Event 4.Censor 6.1/T 7.Add
6.4 Weibull =1/ 95 1 (1 ), 1 ˆ α = exp 23.42711+ 10266.8942 T 1.844
6.5 1. Columns 2 2., 3.Formula, 1 αˆ = exp 23.42711+ 10266.8942 T ln( time) ln ( ) res = exp 4. Survival Distribution,Y,Censor Censor,Weibull 5. 11 Weibull
6.5 Weibull
6 Creep.JMP Parameter Estimates 1/ 95. RiskRatio (exp(-16603.743)). Baseline Survival at time. weibull Parameter Estimates 1/
{ 5 T } α ˆ = exp 1.0266 10 / 23.42711 ˆ exp 1.0266 10 A F 1 1 T1 T2 5 = ˆ h ( t) exp 1.8303 10 1 1 1 5 HR = = h ( ) 2 t T 1 T 2
167
Survival fit model
h i 1 0 exp( 0 1 0 0 β1 Ti Ea 1 = h 0( ti ) exp( β1 / Ti ) = h0 ( ti ) exp( ) k T ( t) = h ( t) β + β ) = h ( t) exp( β ) exp( / T ) i E = b k a 1 i AF = 1 exp β1 Ti 1 T j = E exp k a 1 Ti 1 T j
( 1/ T ) ( ) i + β2 Si β RHi zi = β 0 + β1 ln + 3 ( ) ( ) ( ) ( ) β t h t exp exp / T S ( RH ) 2 = β β β h i 0 j 0 1 i i exp 3 i AF S β 2 i = exp β S 1 exp j Ti T 3 j 1 1 { β ( S S )} i j
7 Reliable.JMP ( ) Survival Weibull
.. Glue( )---- Temp( ) 1/(k ) RH( ) Day Censor( )
7.1.Day 1.Parametric Regression 3.Time to Event 8.Weibull 9.Run Model 5.Censor 7.Add.Censor 6.glue,1/kT,RH
7.2 Weibull
7 Reliable.JMP 1/ =1.8596. 1 exp(0.2575423)=1.294 1.3. exp(-4.8603917+0.28443285/kt-0.0330083rh) 0.28443285(eV) 95 0.1033674 0.4646598(eV)
LnReg.JMP,,,,,, 1. 2.1/ T,, 3., 4.
1 1 L exp β 0 exp β 1 exp 2 log10 T β T σ = ln L = C + 1 log T + 1 β β σ T + 1 2 10 ε N(0, 2 ) 3962.6295 1 σ T T ˆ 4 = 1.0345 10 exp exp 1063.091 log10 L
L T E a n = expa exp T I kt B ( ) ( ) E a n L= expa exp T I kt B ( ) ( ) temp 1/ k T, T ln T I ln I B k k
1/(k B T) ln( ) ln( T) m 7 0.1578 ηˆ = 3.0578 10 exp T kt B 1.7651 1.8129 I ( ) ( )
1/(k B T) ln( ) ln( T) m 4 0.1920 ηˆ = 2.8338 10 T exp T I kt B ( ) ( ) 1.7650 1.8126
184
( ) Prentice1974.JMP
VA Lung Cancer.JMP. COX,