A fun¸c˜ao de sobrevivˆencia da vari´avel W ficar´a dessa forma: SW(w) = Z ∞ w (s1 s2) s1evs1 B(s1, s2)[1 + (ss12)e v](s1+s2)dv
A.6 Demonstra¸c˜ao para a equa¸c˜ao 3.4 35 SW(w) = Z ∞ w (s1 s2)e v 1 + (s1 s2)e v s1 1 1 + (s1 s2)e v s2 B(s1, s2) −1 dv Fazendo u = 1 1 +s1 s2 ev = s2 s2+ s1ev e du = − 1 1 +s1 s2 ev s1 s2 ev 1 +s1 s2 ev dv, temos que: SW(w) = − Z 0 s2(s2+s1ew)−1 us2−1 (1 − u)s1−1 B(s1, s2) −1 du SW(w) = Z s2(s2+s1ew)−1 0 us2−1 (1 − u)s1−1 B(s1, s2) −1 du SW(w) = Ik(s2, s1) sendo k=s2(s2+ s1ew)−1.
Apˆendice B
Comandos no R
Nessa se¸c˜ao mostramos a rotina utilizada para a obten¸c˜ao das estimativas tanto da simula¸c˜ao da Se¸c˜ao 3, quanto da Se¸c˜ao 4. Para isso, ´e preciso que baixe o pacote gfcure e execute no software R 32-bits. Para mais informa¸c˜oes de onde est´a dispon´ıvel o pacote, consulte Peng (1999).
require(flexsurv) # Chamar o pacote flexsurv, j´a com o survival incluso. attach("SUA_BIBLIOTECA_DO_R \\ gfcure\\.RData")
load.gfcure("SUA_BIBLIOTECA_DO_R \\ gfcure") #### Rotina para a simula¸c~ao na Se¸c~ao 3.2.1
### Fun¸c~ao a partir da gera¸c~ao de n´umeros aleat´orios da F ## a = 20 (mu = 3), b = 5, s1 = 10 e s2 = 7
rm(list = ls())
D = function(n, a, b, s1, s2, p, tau){ N <- rbinom(n, 1, 1-p)
C <- runif(n, 0, tau)
T <- vector(); y <- vector(); d <- vector(); cv <- vector() T[N==1] <- a*rf(sum(N),2*s1,2*s2)b b T[N==0] <- C[N==0] y <- apply(cbind(T,C), 1, min) d <- ifelse(T < C, 1, 0) cv <- ifelse(y = C, 1, 0) return(list(y = y, d = d, pc1=sum(cv)/sum(1-d),pc2=mean(1-d))) } 36
37 #### n = 50
### Censura = 30 ## % de cura = 10
d <- D(50, 20, 5, 10, 7, 0.1, 250);d$pc1;d$pc2 # Verificar a censura. n=50 # Escolha do tamanho da amostra
mod=gfcure(Surv(d$y, d$d)∼1, cureform=∼1, dist="gf", sait = 0, temp = 10, ntemp = 100);mod
(AIC=-2*mod$log+2*5) (BIC=-2*mod$log+5*log(n)) ## % de cura = 30
d <- D(50, 20, 5, 10, 7, 0.3, 450);d$pc1;d$pc2 n=50
mod=gfcure(Surv(d$y, d$d)∼1, cureform=∼1, dist="gf", sait = 0, temp = 10, ntemp = 100);mod
(AIC=-2*mod$log+2*5) (BIC=-2*mod$log+5*log(n)) ## % de cura = 50
d <- D(50, 20, 5, 10, 7, 0.5, 500000);d$pc1;d$pc2 n=50
mod=gfcure(Surv(d$y, d$d)∼1, cureform=∼1, dist="gf", sait = 0, temp = 0, ntemp = 200);mod
(AIC=-2*mod$log+2*5) (BIC=-2*mod$log+5*log(n)) ### Censura = 50 ## % de cura = 10 d <- D(50, 20, 5, 10, 7, 0.1, 100);d$pc1;d$pc2 n=50
mod=gfcure(Surv(d$y, d$d)∼1, cureform=∼1, dist="gf", sait = 0, temp = 0, ntemp = 0);mod
38 (AIC=-2*mod$log+2*5) (BIC=-2*mod$log+5*log(n)) ## % de cura = 30 d <- D(50, 20, 5, 10, 7, 0.3, 500);d$pc1;d$pc2 n=50
mod=gfcure(Surv(d$y, d$d)∼1, cureform=∼1, dist="gf", sait = 0, temp = 0, ntemp = 0);mod
(AIC=-2*mod$log+2*5) (BIC=-2*mod$log+5*log(n)) ## % de cura = 50
d <- D(50, 20, 5, 10, 7, 0.5, 500000);d$pc1;d$pc2 n=50
mod=gfcure(Surv(d$y, d$d)∼1, cureform=∼1, dist="gf", sait = 0, temp = 0, ntemp = 0);mod
(AIC=-2*mod$log+2*5) (BIC=-2*mod$log+5*log(n)) #### Rotina para a Se¸c~ao 4.1 ### Dados sobre c^ancer de ov´ario str(ovarian)
t=seq(0:1200)
### Ajuste usando o Kaplan-Meier
ekm=survfit(Surv(futime, fustat)∼1, conf.type="none", data=ovarian) plot(ekm, main="Estimador de Kaplan-Meier",ylab="S(t)", xlab="Tempos (em dias)")
##### Modelos a serem considerados no estudo #### Gama Generalizada
### Ajuste usando o gfcure
mod1=gfcure(Surv(futime, fustat)∼1, cureform=∼1, data=ovarian, dist ="egg",sait = 0, temp = 0, ntemp = 0)
39 mod1
s=mod1$coef[1]
sigma=exp(mod1$coef[2]) mu=mod1$coef[3]
y1=(1-pegg(t, s, sigma, mu))*(1-mod1$cure)+mod1$cure #### Weibull
### Ajuste usando o gfcure
mod2=gfcure(Surv(futime, fustat)∼1, cureform=∼1, data=ovarian) mod2
a=1/exp(mod2$coef[1]);a b=exp(mod2$coef[2]);b
y2=(exp(-(t/b)b a)*(1-mod2$cure))+mod2$cure #### log-log´ıstica
### Ajuste usando o gfcure
mod3=gfcure(Surv(futime, fustat)∼1, cureform=∼1, data=ovarian, dist="loglogistic") mod3 a=1/exp(mod3$coef[1]);a b=exp(mod3$coef[2]);b y3=(1/(1+(t/b)b a))*(1-mod3$cure)+mod3$cure #### F Generalizada
### Ajuste usando o gfcure
mod4=gfcure(Surv(futime, fustat)∼1, cureform=∼1, data=ovarian, dist="gf", sait = 0, temp = 0, ntemp = 0)
mod4
#### log-normal
### Ajuste usando o gfcure
mod5=gfcure(Surv(futime, fustat)∼1, cureform=∼1, data=ovarian, dist="lognormal")
mod5
mu=mod5$coef[2]
40 w = (log(t)-mu)/sigma
y5=pnorm(w, 0, 1, lower.tail=F)*(1-mod5$cure)+mod5$cure ### Gr´afico da figura 4.1
plot(ekm, main="Compara¸c~ao curvas de sobreviv^encia", ylab="S(t)", xlab="Tempos (em dias)")
lines(t,y1, col=2, lty=1) lines(t,y2, col=3, lty=2) lines(t,y3, col=4, lty=3) lines(t,y5, col=6, lty=4)
legend(700,0.9,col=c(1,2,3,4,6),lty=c(1,1,2,3,4), c("Kaplan-Meier", "gama generalizada","Weibull","log-log´ıstica","lognormal"),lwd=1, bty="n")
#### Crit´erios de Informa¸c~ao da tabela 4.1 ### AIC
(AIC1=-2*mod1$log+2*4) # Gama generalizada (AIC2=-2*mod2$log+2*3) # Weibull
(AIC3=-2*mod3$log+2*3) # log-log´ıstica (AIC4=-2*mod4$log+2*5) # F generalizada (AIC5=-2*mod5$log+2*3) # log-normal ### BIC
n=100
(BIC1=-2*mod1$log+4*log(n)) # Gama generalizada (BIC2=-2*mod2$log+3*log(n)) # Weibull
(BIC3=-2*mod3$log+3*log(n)) # log-log´ıstica (BIC4=-2*mod4$log+5*log(n)) # F generalizada (BIC5=-2*mod5$log+3*log(n)) # log-normal ####### Rotina para a se¸c~ao 4.2
##### Inserir os dados str(colon)
t=seq(0:3500)
41 ekm=survfit(Surv(time, status)∼1, conf.type="none", data=colon)
#### Gama generalizada ### Ajuste usando o gfcure
mod1=gfcure(Surv(time, status)∼1, cureform=∼1, dist="egg", data=colon, sait = 0, temp = 10, ntemp = 10)
mod1
s=mod1$coef[1]
sigma=exp(mod1$coef[2]) mu=mod1$coef[3]
y1=(1-pegg(t, s, sigma, mu))*(1-mod1$cure)+mod1$cure ##### Usando a distribui¸c~ao Weibull
### Ajuste usando o gfcure
mod2=gfcure(Surv(time, status)∼1, cureform=∼1, data=colon) mod2
a=1/exp(mod2$coef[1]);a b=exp(mod2$coef[2]);b
y2=(exp(-(t/b)b a)*(1-mod2$cure))+mod2$cure #### log-log´ıstica
### Ajuste usando o gfcure
mod3=gfcure(Surv(time, status)∼1, cureform=∼1, dist="loglogistic", data=colon) mod3 a=1/exp(mod3$coef[1]);a b=exp(mod3$coef[2]);b y3=(1/(1+(t/b)b a))*(1-mod3$cure)+mod3$cure #### F Generalizada
### Ajuste usando o gfcure
mod4=gfcure(Surv(time, status)∼1, cureform=∼1, dist="gf", data=colon, sait = 10, temp = 10, ntemp = 10)
mod4
42 ### Ajuste usando o gfcure
mod5=gfcure(Surv(time, status)∼1, cureform=∼1, dist="lognormal", data =colon) mod5 mu=mod5$coef[2] sigma=exp(mod5$coef[1]) w = (log(t)-mu)/sigma y5=pnorm(w, 0, 1, lower.tail=F)*(1-mod5$cure)+mod5$cure ### Gr´afico para a figura 4.3
plot(ekm, main="Compara¸c~ao curvas de sobreviv^encia", ylab="S(t)", xlab="Tempos (em dias)")
lines(t,y1, col=2, lty=1) lines(t,y2, col=3, lty=2) lines(t,y3, col=4, lty=3) lines(t,y5, col=6, lty=4)
legend(1500,0.9,col=c(1,2,3,4,6),lty=c(1,1,2,3,4), c("Kaplan-Meier", "gama generalizada","Weibull","log-log´ıstica","lognormal"),lwd=1,bty="n") #### Crit´erios de Informa¸c~ao da tabela 4.3
### AIC
(AIC1=-2*mod1$log+2*4) # Gama generalizada (AIC2=-2*mod2$log+2*3) # Weibull
(AIC3=-2*mod3$log+2*3) # log-log´ıstica (AIC4=-2*mod4$log+2*5) # F generalizada (AIC5=-2*mod5$log+2*3) # log-normal ### BIC
n=100
(BIC1=-2*mod1$log+4*log(n)) # Gama generalizada (BIC2=-2*mod2$log+3*log(n)) # Weibull
(BIC3=-2*mod3$log+3*log(n)) # log-log´ıstica (BIC4=-2*mod4$log+5*log(n)) # F generalizada (BIC5=-2*mod5$log+3*log(n)) # log-normal
Referˆencias
Abramowitz, M. and Stegun, I.A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York. Akaike, H. (1974). A New Look at the Statistical Model Identification. IEEE Tran-
sactions on Automatic Control, Vol. 19, No. 6, pp. 716-723.
Berkson, J. and Gage, R.P. (1952). The likelihood ratio, Wald, and Lagrange mul- tiplier tests: an expository note. Journal of American Statistical Association, Vol. 47, No. 259, pp. 501-515.
Brown, B.W., Levy, L.B., Lovato, J., Russell, K. and Spears, F.M. (1992). Algorithm 762: LLDRLF, Log-likelihood and Some Derivatives for log-F Models. ACM Transactions on Mathematical Software, Vol. 22, No. 3, pp. 372-382.
Carneiro, H.P. de A. (2012). Testes de Hip´oteses em Modelos de Sobrevivˆencia com Fra¸c˜ao de Cura. Disserta¸c˜ao de Mestrado - PPGMAE - UFRN.
Cox, C (2008). A generalized F distribution: An umbrella for parametric survival analysis. Statistics in medicine, Vol. 27, pp. 4301-4312.
Colon, A.S.C., Taylor, J.M.G., Sargent, D.J. and Yothers, G. (2011). Using cure models and multiple imputation to utilize recurrence as an auxiliary variable for overall survival. Clinical trials, Vol. 8, pp. 581-590.
Colosimo, E.A. e Giolo, S.R. (2006). An´alise de Sobrevivˆencia Aplicada. Edgard Bl¨ucher, S˜ao Paulo, SP.
Edmunson, J.H., Fleming, T.R., Decker, D.G., Malkasian, G.D., Jefferies, J.A., Webb, M.J., and Kvols, L.K. (1979). Different Chemotherapeutic Sensitivities and Host Factors Affecting Prognosis in Advanced Ovarian Carcinoma vs. Mi- nimal Residual Disease. Cancer Treatment Reports, Vol. 63, pp. 241-247. Fonseca, R.S. (2011). Modelos de Sobrevivˆencia com Fra¸c˜ao de Cura e Omiss˜ao nas
Vari´aveis. Disserta¸c˜ao de Mestrado - PPGMAE - UFRN.
44 Hogg, S.A. and Ciampi, A. (1985). GFREG: A computer program for maximum likelihood regression using the generalized F distribution. Computer Methods and Programs in Biomedicine, Vol. 20, pp. 201-215.
Hubben, G.A.A., Bishai, D., Pechlivanoglou, P., Cattelan, A.M., Grisetti, R., Fac- chin, C., Compostella, F.A., Bos, J.M., Postma, M.J. and Tramarin, A. (2008). The societal burden of HIV/AIDS in Northern Italy: An analysis of costs and quality of life. AIDS Care, Vol. 20, No. 4, pp. 449-455.
Ibrahim, J.G., Chen, M.H. and Sinha, B. (1992). Bayesian Survival Analysis. Sprin- ger, New York.
Jackson, C. (2013). Flexible parametric survival models. URL: http://cran.r- project.org/web/packages/flexsurv/index.html. Acesso: 09 de julho de 2014. Johnson, N.L., Kotz, S. and Balakrishnan, N. (1992). Continuous Univariate Dis-
tributions. Wiley, Vol.2, New York.
Kalbfleisch, J.D. and Prentice, R.L. (2002). The Statistical Analysis of Failure Time Data. Wiley, New York.
Kaplam, E.L. and Meier, P. (1958). Nonparametric Estimation from Incomplete Observation. Journal of the American Statistical Association, Vol.58, No. 282, pp. 457-481.
Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data. Wiley, New York.
Le, Q.H., Thomas, X., Ecochard, R., Iwaz, J., Lhe´eritier, V., Michallet, M., Fiere, D. (2007). Proportion of long-term event-free survivors and lifetime of adult patients not cured after a standard acute lymphoblastic leukemia therapeutic program. American Cancer Society, Vol. 109, No. 10, pp. 2058-2067.
Maller, R.A., Zhou, X. (1974). Survival analysis with long-term survivors. Wiley, New York.
Martinez, E.Z., Louren¸con, A.F., Mazucheli, J. e Ferreira, O. (2007). Modelos de Sobrevivˆencia de Longa Dura¸c˜ao Aplicados ao Estudo do Comportamento de Retorno do Doador de Sangue Volunt´ario. Revista de Matem´atica e Estat´ıstica. Vol 25, No. 1, pp. 137-154.
Moetel, C.G., Fleming, T.R., Macdonald, J.S., Hallen, D.G., Laurie, J.A., Good- man, P.J., Ungerleider, J.S., Emerson, W.A., Tormey, D.C., Glick, J.H., Veeder, M.H. and Mailliard, J.A. (1990). Levamisole and Flurouracil for Adjuvant The-
45 rapy of Resected Colon Carcinoma. The New England Journal of Medicine, Vol.8, pp. 322-352.
Mood, A.M., Graybill, F.A., Boes, D.C. (1974). Introduction To The Theory of Statistics. McGraw - Hill International Editions, 3th Edition.
Peng, Y. (1999) GFCURE - An S-PLUS Package for Parametric
Analysis of Survival Data with Possible Cured Fraction. URL: http://www.math.mun.ca/∼ypeng/research/gfcure/. Acesso: 09 de junho de 2014.
Peng, Y. and Carrier, K.C. (2002). An Empirical Comparison of Parametric and Semiparametric Cure Models. Biometrical Journal, Vol. 44, No. 8, pp. 1002-1014. Peng, Y. and Dear, K.B.G. and Denham, J.W. and others (1998). A generalized F mixture model for cure rate estimation. Statistics in medicine, Vol. 17, No. 8, pp. 813-830.
Prentice, R.L. (1975). Discrimination Among Some Parametric Models. Statistics in medicine, Vol. 62, No. 3, pp. 607-614.
R Development Core Team (2013). R: A Language and Environment of Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL: http://www.r-project.org/.
Rodrigues, J., Cancho, V.G., de Castro, M. and Louzada-Neto, F. (2009). On the unification of long-term survival models. Statistics and Probability Letters Vol. 79, pp. 753-759.
Schwartz, G. (1978). Estimating the Dimension of a Model.The Annals of Statistics Vol. 6, No. 2, pp. 461-464.
Silva, R. P. da (2013). Modelo Gama Generalizado com Longa Dura¸c˜ao: Teoria e Pr´atica. Monografia de Gradua¸c˜ao (Forma¸c˜ao em Estat´ıstica) - Curso de Esta- t´ıstica, Departamento de Estat´ıstica, Universidade Federal do Rio Grande do Norte, Natal-RN. 50 f.
Yakovlev, A.Y., Asselain, B., Bardou, V.J., Fourquet, A., Hoang, T., Rochefediere, A. and Tsodikov, A.D. (1993). A simple stochastic model of tumor recurrence and its application to data on premenopausal breast cancer. Biometric et analyse de donnes spatio-temporelles, Vol. 12, pp. 66-82.