JMP V4 による生存時間分析
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- かずただ うみのなか
- 5 years ago
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1 V4 1 SAS
2
3
4 4
5 ( ) (Survival Time)
6 1 (Event)
7
8 Start of Study Start of Observation Died Died Died Lost End Time Censor Died Died Censor Died Time
9 Start of Study End Start of Observation Censor Died Died Censor Died Time
10
11
12
13
14
15
16 ()
17 Survival Time Analysis Proportion Survival Function
18 ,Weibull
19 (Survival Function) S(t) t (Proportion) F(t) S(t)=1-F(t) f(t) f(t) t
20 (Survival Function) S(t) 1 S(t) 1-S(t)=F(t) t
21 (hazard function) (rate) h(t) f(t)/s(t)
22 S(t) ( ) ds t dt S(t)
23 ( ) = 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
24 ht { Ht } ( ) ( ) ( ) ( logst ( )) Ht = hudu= log St St = exp d = t ( ) ( ) ( ) 0 dt
25
26 ( ) ( ) ( ) ( ) 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
27 3
28
29 t
30
31
32 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
33 Weibull Weibull3 0 Weibull 1 1 1
34 Weibull f
35 Weibull 63.2 JMP(Beta) JMP(Alpha)
36 1 Weibull.JMP Weibull
37 time Weibull time f(t),f(t),s(t) h1(t) 2.5 h2(t) 1.0 h3(t)
38 1.1 1.Overlay Plot 2.
39 1.2 3.f(t) 1.Time 4.Y 2.X 5.OK
40 1.3 1.Connect Thru Missing 2.
41
42 1 Weibull.JMP h(t) 1 1 1
43 Calculator Cols Column Info. Current Properties Formula Edit Formula, 2 Col Name
44 LnNormal.JMP,, f t ( ) 1 1 logt = exp σt 2π 2 σ µ 2
45 ( 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<+
46 ( ) ( ) f t = λexp λt t 0, λ> 0 ( ) ( λ ) F t f(t) = 1 exp t t 0, λ > 0 t
47 ( ln t µ ) 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)
48
49 49
50
51 V4, weibull,3 weibull,, Weibull
52 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
53 2 Weibull.JMP 2 Weibull Column 1 Calculator,-lnS(t) Analyze Fit Y X,X time,y -lns(t)
54 2 Weibull.JMP Weibull time S(time)
55 Wiring JMP Time Analyze Distribution Fit DistributionWeibull
56 Weibull Weibull
57 Part II Survival 57
58
59
60
61
62
63 Survival Survival Distribution Kaplan-Meier Parametric Regression Proportional Hazards Cox Recurrence
64 Weibull 64
65 (censor) (censor) 1. 2.
66 Censor Censor
67 3 Wiring.JMP, Survival Disuribution Survival Weibull Weibull Weibull
68 (Al ). Time( ) Censor( ) Distribution
69 3.1 1.Survival Distribution 2.Time 3.Y 4.OK
70 3.2 Weibull Weibull Plot Weibull Fit
71 3.3 Weibull
72 3.4 1.Survival Distribution 3.Y 2.Time 4.Censor 6.OK 5.Censor
73 3.5 1.Plot Options Show Confid Interval 2. Kaplan-Meier, 95
74 3.6 Weibull 1.Weibull Plot,Weibull Fit 2.Weibull 3.Save Estimates
75 3.7
76 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
77 Kaplan Meier n ( ) { t i} i= 1 t i d i t i n i,, ( ) ˆ d d d d S t = = 1 n n i L n1 n2 nn ti < t ni ( ) 1 2
78 Kaplan Meier ˆ d d d d S t = = 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
79 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 ) ( )
80 Kaplan-Meier Sˆ = n 1 n i= 1 1 n d i ( ) 1 / 1 V Sˆ = Sˆ Sˆ n
81 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
82 ) Excel
83 =count( ) If F(t) A$1-C3+1 E3/D3 If
84 MON j = MON + j 1 n+ 1 MON n+ 1 ( i 1) MON(Mean Order Number) F(t) j 1
85
86
87 87
88 1 Kaplan-Meier Wilcoxon
89 Wilcoxon
90
91 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 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
92 ( ) ( 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,
93 δ = d m, δ = d m 1j 1j 1j 2 j 2 j 2 j δ1j = ( d1j m1j), δ2j = ( d2j m2j)
94 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
95 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
96 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
97
98 4 Rats.JMP, Survival Time Modeling 2 Survival
99 2 day Group ) Censor )
100 4.1 1.Survival Distribution 2.days 3.Y 4.Censor 6.Group 8.OK 5.Censor 7.Grouping
101
102 4.3 ( ) Plot Options Show Conbined
103 4 Rats.JMP
104 days. 2Survival Survival=(At Risk - N Failed)/At Risk 3Failure Survival+Failure=1 SurvStdErr(Survival Standard Error)
105 N Failed 1 N Censored 1 At Risk Quantiles Mean StdDev Standard Deviation
106 PWB JMP Weibull
107 PWB
108
109
110
111 111
112 (, )
113
114 1
115 ,
116 ( ),m=1( ),
117 A
118 ( ) ( ) No. time x; V d1: d i n M M M M M M M M M M
119 5 Unit.JMP Survival Time Modeling Survival Omit
120 Unit.JMP
121 2 time ) failure ) 2 Condenser Relay
122 5.1 1.Survival Distribution 2.time 4.OK 3.Y 5.Competing Causes 6.Falure
123 5.2 Weibull Plot Weibull
124 1.Omit Causes 5.3 Weibull Weibull 2.Condenser 3.OK
125 5.4
126 5 Unit.JMP Competing Causes condenser Omit condenser Failure Grouping
127 Part III 127
128 1
129 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
130 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
131 Weibull, { S ( t) } S ( t) ln ln ln ln a { } β t t = lna+ ln ln = ln a α α Weibull Weibull 0 β
132 Ht ( ) = βt Ht ( ) = βln ( t/ α) α Ht ( ) e βt β Ht = t/ α = ( ) ( ) β
133 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
134 PWB JMP Survival Distribution Y Time1 Grouping /kt K-M Log(Time) Log(-logS(t)) Group 1/kT
135 Weibull log Ht () = log( Time) log Ht () = log( Time) log Ht () = log( Time)
136 1/kT=32.** ( ) 33.** 32.** = ** 32.** = kt COX
137 COX ,COX
138 L,,,
139 , 2 2
140 Arrhenius Life= exp b + b 1/ kt Eˆ = b, k = { ( )} 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= b
141 t Log-Normal Distribution Acceleated x
142
143 R= R0 exp B T T 0 α = 1 R log y = b + b x R dr dt T x = 1/ ( 273+ t) T = B T logyˆ = log x R 1 yˆ R = exp xt 2 T
144
145
146 PWB JMP Weibull
147 PWB
148 PWB,?
149 PWB?,
150 PWB!!!
151 PWB
152 ( ),
153 6 Creep.JMP 1 ( ), Survival Time Modeling (Weibull )
154 . temp ( ) 1/ time Censor( ) Life= exp(b 0 +b 1 /(273+temp)) exp(b 0 +b 1 /T)
155
156 T ( C + log L) = Q L = exp( Q/T - C) T L
157 Larson-Miller Larson-Miller
158
159 8.Run Mode Proportional Hazards 2.time 3.Time to Even 5.Censor 4.Censor 6.1/T 7.Add
160 Risk = h h ( t) ( t) ( ) { } = exp =
161 8.Weibull 9.Run Model 5.Censor Parametric Regression 2.time 3.Time to Event 4.Censor 6.1/T 7.Add
162 6.4 Weibull =1/ 95 1 (1 ), 1 ˆ α = exp T 1.844
163 Columns 2 2., 3.Formula, 1 αˆ = exp T ln( time) ln ( ) res = exp 4. Survival Distribution,Y,Censor Censor,Weibull Weibull
164 6.5 Weibull
165 6 Creep.JMP Parameter Estimates 1/ 95. RiskRatio (exp( )). Baseline Survival at time. weibull Parameter Estimates 1/
166 { 5 T } α ˆ = exp / ˆ exp A F 1 1 T1 T2 5 = ˆ h ( t) exp HR = = h ( ) 2 t T 1 T 2
167 167
168 Survival fit model
169 h i 1 0 exp( 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
170 ( 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
171 7 Reliable.JMP ( ) Survival Weibull
172 .. Glue( )---- Temp( ) 1/(k ) RH( ) Day Censor( )
173 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
174 7.2 Weibull
175 7 Reliable.JMP 1/ = exp( )= exp( /kt rh) (eV) (eV)
176 LnReg.JMP,,,,,, / T,, 3., 4.
177 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 ε N(0, 2 ) σ T T ˆ 4 = exp exp log10 L
178
179
180
181 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
182 1/(k B T) ln( ) ln( T) m ηˆ = exp T kt B I ( ) ( )
183 1/(k B T) ln( ) ln( T) m ηˆ = T exp T I kt B ( ) ( )
184 184
185 ( ) Prentice1974.JMP
186 VA Lung Cancer.JMP. COX,
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1 25/5/3-6/3 1 1 1.1.................................. 1 1.2.................................. 4 2 5 2.1.............................. 5 2.2.............................. 6 3 Black Scholes 7 3.1 BS............................
More information2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n
. X {x, x 2, x 3,... x n } X X {, 2, 3, 4, 5, 6} X x i P i. 0 P i 2. n P i = 3. P (i ω) = i ω P i P 3 {x, x 2, x 3,... x n } ω P i = 6 X f(x) f(x) X n n f(x i )P i n x n i P i X n 2 G(k) e ikx = (ik) n
More information(1) 1 y = 2 = = b (2) 2 y = 2 = 2 = 2 + h B h h h< h 2 h
6 6.1 6.1.1 O y A y y = f() y = f() b f(b) B y f(b) f() = b f(b) f() f() = = b A f() b AB O b 6.1 2 y = 2 = 1 = 1 + h (1 + h) 2 1 2 (1 + h) 1 2h + h2 = h h(2 + h) = h = 2 + h y (1 + h) 2 1 2 O y = 2 1
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