Survival models and Cox-regression

Transcription

1 models Cox-regression Steno Diabetes Center Copenhagen, Gentofte, Denmark & Department of Biostatistics, University of Copenhagen IDEG 2017 training day, Abu Dhabi, 11 December Rates From /home/bendix/sdc/conf/ideg2017/teach/slides/slides.tex. 1/ 94

2 Rates Senior Statistician, Steno Diabetes Center models Cox-regression IDEG 2017 training day, Abu Dhabi, 11 December surv-rate

3 data Persons enter study at some date. Persons exit at a later date, eir dead or alive. Observation: Actual time span to death ( event ) or Some time alive ( at least this long ) Rates models Rates (surv-rate) 2/ 94

4 Examples of time-to-event measurements Time from diagnosis of cancer to death. Time from romisation to death in a cancer clinical trial Time from HIV infection to AIDS. Time from marriage to 1st child birth. Time from marriage to divorce. Time to re-offending after being released from jail Rates models Rates (surv-rate) 3/ 94

5 Each line a person Each blob a death Study ended at 31 Dec Calendar time Rates models Rates (surv-rate) 4/ 94

6 Ordered by date of entry models Most likely order in your database. Rates Calendar time Rates (surv-rate) 5/ 94

7 Timescale changed to Time since diagnosis. Rates models Time since diagnosis Rates (surv-rate) 6/ 94

8 Patients ordered by survival time. models Rates Time since diagnosis Rates (surv-rate) 7/ 94

9 times grouped into bs of survival. models Rates Year of follow up Rates (surv-rate) 8/ 94

10 Patients ordered by survival status within each b. models Rates Year of follow up Rates (surv-rate) 9/ 94

11 after Cervix cancer Stage I Stage II Year N D L N D L Estimated risk in year 1 for Stage I women is 5/107.5 = Estimated 1 year survival is = Rates models Rates Life-table (surv-rate) estimator. 10/ 94

12 function Persons enter at time 0: Date of birth, date of romization, date of diagnosis. How long do y survive? time T a stochastic variable. Distribution is characterized by survival function: S(t) = P {survival at least till t} = P {T > t} = 1 P {T t} = 1 F (t) F (t) is cumulative risk of death before time t. Rates models Rates (surv-rate) 11/ 94

13 Intensity / rate / hazard same same intensity or hazard function Probability of event in interval, reltive to interval length: λ(t) = P {event in (t, t + h] alive at t} /h Characterizes distribution of survival times as does f (density) or F (cumulative distibution). oretical counterpart of a(n empirical) rate. Rates models Rates (surv-rate) 12/ 94

14 Rate survival ( t ) S(t) = exp λ(s) ds 0 λ(t) = S (t) S(t) is a cumulative measure, rate is an instantaneous measure. Note: A cumulative measure requires an origin!... it is always survival since some timepoint. Rates models Rates (surv-rate) 13/ 94

15 Observed survival rate studies: Observation of (right censored) survival time: X = min(t, Z ), δ = 1{X = T } Rates sometimes conditional on T > t 0 (left truncation, delayed entry). Epidemiological studies: Observation of (components of) a rate: D/Y D: no. events, Y no of person-years, in a prespecified time-frame. models Rates (surv-rate) 14/ 94

16 Empirical for individuals At individual level we introduce empirical rate: (d, y), number of events (d {0, 1}) during y risk time. A person contributes several observations of (d, y), with associated covariate values. Empirical are responses in survival analysis. timescale t is a covariate varies within each individual: t: age, time since diagnosis, calendar time. Don t confuse with y difference between two points on any timescale we may choose. Rates models Rates (surv-rate) 15/ 94

17 Empirical by calendar time. models Rates Calendar time Rates (surv-rate) 16/ 94

18 Empirical by time since diagnosis. Rates models Time since diagnosis Rates (surv-rate) 17/ 94

19 Statistical inference: Likelihood Two things needed: Data what did we actually observe Follow-up for each person: Entry time, exit time, exit status, covariates Model how was data generated Rates as a function of time: Probability machinery that generated data Likelihood is probability of observing data, assuming model is correct. Maximum likelihood estimation is choosing parameters of model that makes likelihood maximal. Rates models Rates (surv-rate) 18/ 94

20 Likelihood from one person likelihood from several empirical from one individual is a product of conditional probabilities: P {event at t 4 t 0 } = P {survive (t 0, t 1 ) alive at t 0 } P {survive (t 1, t 2 ) alive at t 1 } P {survive (t 2, t 3 ) alive at t 2 } P {event at t 4 alive at t 3 } Log-likelihood from one individual is a sum of terms. Each term refers to one empirical rate (d, y) y = t i t i 1 mostly d = 0. t i is timescale (covariate). Rates models Rates (surv-rate) 19/ 94

21 Poisson likelihood log-likelihood contributions from follow-up of one individual: d t log ( λ(t) ) λ(t)y t, t = t 1,..., t n is also log-likelihood from several independent Poisson observations with mean λ(t)y t, i.e. log-mean log ( λ(t) ) + log(y t ) Analysis of, (λ) can be based on a Poisson model with log-link applied to empirical where: d is response variable. log(λ) is modelled by covariates log(y) is offset variable. Rates models Rates (surv-rate) 20/ 94

22 Likelihood for follow-up of many persons Adding empirical over follow-up of persons: D = d Y = y Dlog(λ) λy Rates Persons are assumed independent Contribution from same person are conditionally independent, hence give separate contributions to log-likelihood. refore equivalent to likelihood for independent Poisson variates No need to correct for dependent observations; likelihood is a product. models Rates (surv-rate) 21/ 94

23 Likelihood Probability of data parameter: Assuming rate (intensity) is constant, λ, probability of observing 7 deaths in course of 500 person-years: P {D = 7, Y = 500 λ} = λ D e λy K = λ 7 e λ500 K = L(λ data) Best guess of λ is where this function is as large as possible. Confidence interval is where it is not too far from maximum Rates models Rates (surv-rate) 22/ 94

24 Likelihood function Likelihood 8e 17 6e 17 4e 17 2e 17 0e Rates models Rate parameter, λ Rates (surv-rate) 23/ 94

25 Likelihood function Log likelihood ratio Rates models Rate parameter, λ (per 1000) Rates (surv-rate) 23/ 94

26 Example using R Poisson likelihood, for one rate, based on 17 events in PY: library( Epi ) D <- 17 ; Y < m1 <- glm( D ~ 1, offset=log(y/1000), family=poisson) ci.exp( m1 ) exp(est.) 2.5% 97.5% (Intercept) Poisson likelihood, two, or one rate RR: D <- c(17,28) ; Y <- c(843.7,632.3) ; gg <- factor(0:1) m2 <- glm( D ~ gg, offset=log(y/1000), family=poisson) ci.exp( m2 ) exp(est.) 2.5% 97.5% (Intercept) gg Rates models Rates (surv-rate) 24/ 94

27 Example using R Poisson likelihood, two, or one rate RR: D <- c(17,28) ; Y <- c(843.7,632.3) ; gg <- factor(0:1) m2 <- glm( D ~ gg, offset=log(y/1000), family=poisson) ci.exp( m2 ) exp(est.) 2.5% 97.5% (Intercept) gg m3 <- glm( D ~ gg - 1, offset=log(y/1000), family=poisson) ci.exp( m3 ) exp(est.) 2.5% 97.5% gg gg Rates models Rates (surv-rate) 25/ 94

28 Senior Statistician, Steno Diabetes Center models Cox-regression IDEG 2017 training day, Abu Dhabi, 11 December ltab

29 analysis Response variable: Time to event, T Censoring time, Z We observe (min(t, Z ), δ = 1{T < Z }). This gives time a special status, mixes response variable (risk)time with covariate time(scale). Originates from clinical trials where everyone enters at time 0, refore Y = T 0 = T Rates models (ltab) 26/ 94

30 life table method simplest analysis is by life-table method : interval alive dead cens. i n i d i l i p i /(77 2/2)= /(70 4/2)= /(59 1/2)= p i = P {death in interval i} = d i /(n i l i /2) S(t) = (1 p 1 ) (1 p t ) Rates models (ltab) 27/ 94

31 Population life table, DK Men Women a S(a) λ(a) E[l res(a)] S(a) λ(a) E[l res(a)] Rates models (ltab) 28/ 94

32 Danish life tables Mortality per 100,000 person years log 2 [mortality per 10 5 (40 85 years)] Men: age Women: age Rates models (ltab) 29/ 94 Age

33 Swedish life tables Mortality per 100,000 person years log 2 [mortality per 10 5 (40 85 years)] Men: age Women: age Rates models (ltab) 30/ 94 Age

34 Observations for lifetable Age Life table is based on person-years deaths accumulated in a short period. Age-specific cross-sectional! function: S(t) = e t 0 λ(a) da = e t 0 λ(a) assumes stability of to be interpretable for actual persons. Rates models (ltab) 31/ 94

35 Observations for lifetable 65 This is a Lexis diagram. Age 60 Rates models (ltab) 32/ 94

36 Observations for lifetable 65 This is a Lexis diagram. Age 60 Rates models (ltab) 33/ 94

37 Life table approach population experience: D: Deaths (events). Y : Person-years (risk time). classical lifetable analysis compiles se for prespecified intervals of age, computes age-specific mortality. Data are collected crossectionally, but interpreted longitudinally. are basic building bocks used for construction of: RRs cumulative measures (survival risk) Rates models (ltab) 34/ 94

38 Senior Statistician, Steno Diabetes Center models Cox-regression IDEG 2017 training day, Abu Dhabi, 11 December km-na

39 Method most common method of estimating survival function. A non-parametric method. Divides time into small intervals where intervals are defined by unique times of failure (death). Based on conditional probabilities as we are interested in probability a subject surviving next time interval given that y have survived so far. Rates models (km-na) 35/ 94

40 Kaplan method illustrated ( = failure = censored): models Cumulative 1.0 survival probability N = Steps caused by multiplying by (1 1/49) (1 1/46) respectively Late entry can also be dealt with Time Rates (km-na) 36/ 94

41 Using R: Surv() library( survival ) data( lung ) head( lung, 3 ) inst time status age sex ph.ecog ph.karno pat.karno meal.cal wt.loss NA NA 15 with( lung, Surv( time, status==2 ) )[1:10] [1] ( s.km <- survfit( Surv( time, status==2 ) ~ 1, data=lung ) ) Call: survfit(formula = Surv(time, status == 2) ~ 1, data = lung) n events median 0.95LCL 0.95UCL plot( s.km ) abline( v=310, h=0.5, col="red" ) Rates models (km-na) 37/ 94

42 Rates models (km-na) 38/ 94

43 Rates models (km-na) 39/ 94

44 Senior Statistician, Steno Diabetes Center models Cox-regression IDEG 2017 training day, Abu Dhabi, 11 December cox

45 proportional hazards model λ(t, x) = λ 0 (t) exp(x β) partial log-likelihood for regression parameters (βs): l(β) = ( e x death ) β log i R t e x iβ death times This is David Cox s invention. Extremely efficient from a computational point of view. baseline hazard λ 0 (t) is bypassed (profiled out). Rates models (cox) 40/ 94

46 Proportional Hazards model baseline hazard rate, λ 0 (t), is hazard rate when all covariates are 0. form of above equation means that covariates act multiplicatively on baseline hazard rate. Time is a covariate (albeit modeled special). baseline hazard is a function of time thus varies with time. No assumption about shape of underlying hazard function. but you will never see shape... Rates models (cox) 41/ 94

47 Interpreting Regression Coefficients If x j is binary exp(β j ) is estimated hazard ratio for subjects corresponding to x j = 1 compared to those where x j = 0. If x j is exp(β j ) is estimated increase/decrease in hazard rate for a unit change in x j. With more than one covariate interpretation is similar, i.e. exp(β j ) is hazard ratio for subjects who only differ with respect to covariate x j. Rates models (cox) 42/ 94

48 Fitting a Cox- model in R library( survival ) data(bladder) bladder <- subset( bladder, enum<2 ) head( bladder) id rx number size stop event enum Rates models (cox) 43/ 94

49 Fitting a in R c0 <- coxph( Surv(stop,event) ~ number + size, data=bladder ) c0 Call: coxph(formula = Surv(stop, event) ~ number + size, data = bladder) coef exp(coef) se(coef) z p number size Likelihood ratio test=7.04 on 2 df, p= n= 85, number of events= 47 Rates models (cox) 44/ 94

50 Plotting base survival in R plot( survfit(c0) ) lines( survfit(c0), conf.int=f, lwd=3 ) plot.coxph plots survival curve for a person with an average covariate value which is not average survival for population considered... not necessarily meaningful Rates models (cox) 45/ 94

51 Rates models (cox) 46/ 94

52 Plotting base survival in R You can plot survival curve for specific values of covariates, using newdata= argument: plot( survfit(c0) ) lines( survfit(c0), conf.int=f, lwd=3 ) lines( survfit(c0, newdata=data.frame(number=1,size=1)), lwd=2, col="limegreen" ) text( par("usr")[2]*0.98, 1.00, "number=1,size=1", col="limegreen", font=2, adj=1 ) Rates models (cox) 47/ 94

53 1.0 number=1,size=1 number=4,size=1 number=1,size= Rates models (cox) 48/ 94

54 Senior Statistician, Steno Diabetes Center models Cox-regression IDEG 2017 training day, Abu Dhabi, 11 December KMCox

55 A look at Cox model λ(t, x) = λ 0 (t) exp(x β) A model for rate as a function of t x. covariate t has a special status: Computationally, because all individuals contribute to (some of) range of t.... scale along which time is split ( risk sets) Conceptually t is just a covariate that varies within individual. Cox s approach profiles λ 0 (t) out from model Rates models (KMCox) 49/ 94

56 Cox-likelihood as profile likelihood One parameter per death time to describe effect of time (i.e. chosen timescale). log ( λ(t, x i ) ) = log ( λ 0 (t) ) + β 1 x 1i + + β p x pi = α t + η i Rates Profile likelihood: Derive estimates of αt as function of data βs assuming constant rate between death times Insert in likelihood, now only a function of data βs Turns out to be Cox s partial likelihood models (KMCox) 50/ 94

57 Cox-likelihood: mechanics of computing likelihood is computed by summing over risk-sets at each event time t: l(η) = ( e η death ) log t i R t e η i this is essentially splitting follow-up time at event- ( censoring) times... repeatedly in every cycle of iteration... simplified by not keeping track of risk time... but only works along one time scale Rates models (KMCox) 51/ 94

58 log ( λ(t, x i ) ) = log ( λ 0 (t) ) + β 1 x 1i + + β p x pi = α t + η i Suppose time scale has been divided into small intervals with at most one death in each: Empirical : (d it, y it ) each t has at most one d it = 0. Assume w.l.o.g. ys in empirical all are 1. Log-likelihood contributions that contain information on a specific time-scale parameter α t will be from: (only) empirical rate (1, 1) with death at time t. all or empirical (0, 1) from those at risk at time t. Rates models (KMCox) 52/ 94

59 Splitting dataset a priori Poisson approach needs a dataset of empirical (d, y) with suitably small values of y. each individual contributes many empirical (one per risk-set contribution in ling) From each empirical rate we get: Poisson-response d Risk time y log(y) as offset Covariate value for timescale (time since entry, current age, current date,... ) or covariates Rates models (KMCox) 53/ 94

60 Example: Mayo Clinic lung cancer after lung cancer Covariates: Age at diagnosis Sex Time since diagnosis Cox model Split data: Poisson model, time as factor Poisson model, time as spline Rates models (KMCox) 54/ 94

61 Mayo Clinic lung cancer 60 year old woman Rates Days since diagnosis models (KMCox) 55/ 94

62 Example: Mayo Clinic lung cancer I > library( survival ) > library( Epi ) > Lung <- Lexis( exit = list( tfe=time ), + exit.status = factor(status,labels=c("alive","dead")), + data = lung ) NOTE: entry.status has been set to "Alive" for all. NOTE: entry is assumed to be 0 on tfe timescale. Rates models (KMCox) 56/ 94

63 Example: Mayo Clinic lung cancer II > ml.cox <- coxph( Surv( tfe, tfe+lex.dur, lex.xst=="dead" ) ~ + age + factor( sex ), + method="breslow", eps=10^-8, iter.max=25, data=lung ) > Lung.s <- splitlexis( Lung, + breaks=c(0,sort(unique(lung$time))), + time.scale="tfe" ) > Lung.S <- splitlexis( Lung, Rates + breaks=c(0,sort(unique(lung$time[lung$lex.xst=="dead"]))), + time.scale="tfe" ) > summary( Lung.s ) Transitions: To From Alive Dead Records: Events: Risk time: Persons: Alive > summary( Lung.S ) models (KMCox) 57/ 94

64 Example: Mayo Clinic lung cancer III Transitions: To From Alive Dead Records: Events: Risk time: Persons: Alive > subset( Lung.s, lex.id==96 )[,1:11] lex.id tfe lex.dur lex.cst lex.xst inst time status age sex ph.ecog Alive Alive Alive Alive Alive Alive Alive Alive Alive Alive Alive Alive Alive Dead > nlevels( factor( Lung.s$tfe ) ) [1] 186 Rates models (KMCox) 58/ 94

65 Example: Mayo Clinic lung cancer IV > system.time( + mls.pois.fc <- glm( lex.xst=="dead" ~ factor( tfe ) + + age + factor( sex ), + offset = log(lex.dur), + family=poisson, data=lung.s, eps=10^-8, maxit=25 ) + ) user system elapsed > length( coef(mls.pois.fc) ) [1] 188 > system.time( + mls.pois.fc <- glm( lex.xst=="dead" ~ factor( tfe ) + + age + factor( sex ), + offset = log(lex.dur), + family=poisson, data=lung.s, eps=10^-8, maxit=25 ) + ) Rates models (KMCox) 59/ 94

66 Example: Mayo Clinic lung cancer V user system elapsed > length( coef(mls.pois.fc) ) [1] 142 > t.kn <- c(0,25,100,500,1000) > dim( Ns(Lung.s$tfe,knots=t.kn) ) [1] > system.time( + mls.pois.sp <- glm( lex.xst=="dead" ~ Ns( tfe, knots=t.kn ) + + age + factor( sex ), + offset = log(lex.dur), + family=poisson, data=lung.s, eps=10^-8, maxit=25 ) + ) Rates models (KMCox) 60/ 94

67 Example: Mayo Clinic lung cancer VI user system elapsed models > ests <- Rates + rbind( ci.exp(ml.cox), + ci.exp(mls.pois.fc,subset=c("age","sex")), + ci.exp(mls.pois.fc,subset=c("age","sex")), + ci.exp(mls.pois.sp,subset=c("age","sex")) ) > cmp <- cbind( ests[c(1,3,5,7),], + ests[c(1,3,5,7)+1,] ) > rownames( cmp ) <- c("cox","poisson-factor","poisson-factor (D)","Poisson-spline") > colnames( cmp )[c(1,4)] <- c("age","sex") > round( cmp, 7 ) (KMCox) 61/ 94

68 Example: Mayo Clinic lung cancer VII age 2.5% 97.5% sex 2.5% 97.5% Cox Poisson-factor Poisson-factor (D) Poisson-spline Rates models (KMCox) 62/ 94

69 models Rates Mortality rate per year Days since diagnosis Days since diagnosis (KMCox) 63/ 94

70 models Rates Mortality rate per year Days since diagnosis Days since diagnosis (KMCox) 63/ 94

71 Deriving survival function > mls.pois.sp <- glm( lex.xst=="dead" ~ Ns( tfe, knots=t.kn ) + + age + factor( sex ), + offset = log(lex.dur), + family=poisson, data=lung.s, eps=10^-8, maxit=25 ) > CM <- cbind( 1, Ns( seq(10,1000,10)-5, knots=t.kn ), 60, 1 ) > lambda <- ci.exp( mls.pois.sp, ctr.mat=cm ) > Lambda <- ci.cum( mls.pois.sp, ctr.mat=cm, intl=10 )[,-4] > survp <- exp(-rbind(0,lambda)) Code output for entire example avaiable in Rates models (KMCox) 64/ 94

72 What really is Taking life-table approach ad absurdum by: dividing time very finely modeling one covariate, time-scale, with one parameter per distinct value. model for time scale is really with exchangeable time-intervals. difficult to access baseline hazard (which looks terrible) uninitiated tempted to show survival curves where irrelevant Rates models (KMCox) 65/ 94

73 Models of this world Replace α t s by a parametric function f (t) with a limited number of parameters, for example: Piecewise constant Splines (linear, quadratic or cubic) Fractional polynomials two latter brings model into this world : smoothly varying parametric closed form representation of baseline hazard finite no. of parameters Makes it really easy to use directly in calculations of expected residual life time state occupancy probabilities in multistate models... Rates models (KMCox) 66/ 94

74 Senior Statistician, Steno Diabetes Center models Cox-regression IDEG 2017 training day, Abu Dhabi, 11 December crv-mod

75 Testis cancer Testis cancer in Denmark: > options( show.signif.stars=false ) > library( Epi ) > data( testisdk ) > str( testisdk ) 'data.frame': 4860 obs. of 4 variables: $ A: num $ P: num $ D: num $ Y: num > head( testisdk ) A P D Y Rates models (crv-mod) 67/ 94

76 Cases, PY > stat.table( list(a=floor(a/10)*10, + P=floor(P/10)*10), + list( D=sum(D), + Y=sum(Y/1000), + rate=ratio(d,y,10^5) ), + margins=true, data=testisdk ) P A Total Rates models (crv-mod) / 94

77 Linear effects in glm How do depend on age? > ml <- glm( D ~ A, offset=log(y), family=poisson, data=testisdk ) > round( ci.lin( ml ), 4 ) Estimate StdErr z P 2.5% 97.5% (Intercept) A > round( ci.exp( ml ), 4 ) exp(est.) 2.5% 97.5% (Intercept) A Linear increase of log- by age Rates models (crv-mod) 69/ 94

78 Linear effects in glm > nd <- data.frame( A=15:60, Y=10^5 ) > pr <- ci.pred( ml, newdata=nd ) > head( pr ) Estimate 2.5% 97.5% > matplot( nd$a, pr, + type="l", lty=1, lwd=c(3,1,1), col="black", log="y" ) Rates models (crv-mod) 70/ 94

79 Linear effects in glm > round( ci.lin( ml ), 4 ) Estimate StdErr z P 2.5% 97.5% (Intercept) A > Cl <- cbind( 1, nd$a ) > head( Cl ) [,1] [,2] [1,] 1 15 [2,] 1 16 [3,] 1 17 [4,] 1 18 [5,] 1 19 [6,] 1 20 > matplot( nd$a, ci.exp( ml, ctr.mat=cl ), + type="l", lty=1, lwd=c(3,1,1), col="black", log="y" ) Rates models (crv-mod) 71/ 94

80 Linear effects in glm pr nd$a > matplot( nd$a, pr, + type="l", lty=1, lwd=c(3,1,1), col="black", log="y" ) Rates models (crv-mod) 72/ 94

81 Linear effects in glm ci.exp(ml, ctr.mat = Cl) * 10^ nd$a > matplot( nd$a, ci.exp( ml, ctr.mat=cl )*10^5, + type="l", lty=1, lwd=c(3,1,1), col="black", log="y" ) Rates models (crv-mod) 73/ 94

82 Quadratic effects in glm How do depend on age? > mq <- glm( D ~ A + I(A^2), + offset=log(y), family=poisson, data=testisdk ) > round( ci.lin( mq ), 4 ) Estimate StdErr z P 2.5% 97.5% (Intercept) A I(A^2) > round( ci.exp( mq ), 4 ) exp(est.) 2.5% 97.5% (Intercept) A I(A^2) Rates models (crv-mod) 74/ 94

83 Quadratic effect in glm > round( ci.lin( mq ), 4 ) Estimate StdErr z P 2.5% 97.5% (Intercept) A I(A^2) > Cq <- cbind( 1, 15:60, (15:60)^2 ) > head( Cq, 4 ) [,1] [,2] [,3] [1,] [2,] [3,] [4,] > matplot( nd$a, ci.exp( mq, ctr.mat=cq )*10^5, + type="l", lty=1, lwd=c(3,1,1), col="black", log="y" ) Rates models (crv-mod) 75/ 94

84 Quadratic effect in glm ci.exp(mq, ctr.mat = Cq) * 10^ nd$a > matplot( nd$a, ci.exp( mq, ctr.mat=cq )*10^5, + type="l", lty=1, lwd=c(3,1,1), col="black", log="y" ) > matlines( nd$a, ci.exp( ml, ctr.mat=cl )*10^5, + type="l", lty=1, lwd=c(3,1,1), col="blue" ) Rates models (crv-mod) 76/ 94

85 Spline effects in glm > library( splines ) > ms <- glm( D ~ Ns(A,knots=seq(15,65,10)), + offset=log(y), family=poisson, data=testisdk ) > round( ci.exp( ms ), 3 ) exp(est.) 2.5% 97.5% (Intercept) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) > aa <- 15:65 > As <- Ns( aa, knots=seq(15,65,10) ) > head( As ) [1,] [2,] Rates models [3,] [4,] time scales (crv-mod) / 94

86 Spline effects in glm Testis cancer incidence rate per 100,000 PY Rates models > matplot( aa, ci.exp( ms, Age ctr.mat=cbind(1,as) )*10^5, + log="y", xlab="age", ylab="testis cancer incidence rate per 100,000timePY", scales + type="l", lty=1, lwd=c(3,1,1), col="black", ylim=c(2,20) ) > matlines( nd$a, ci.exp( mq, ctr.mat=cq )*10^5, + type="l", lty=1, lwd=c(3,1,1), col="blue" ) (crv-mod) 78/ 94

87 Adding a linear period effect > msp <- glm( D ~ Ns(A,knots=seq(15,65,10)) + P, + offset=log(y), family=poisson, data=testisdk ) > round( ci.lin( msp ), 3 ) Estimate StdErr z P 2.5% 97.5% (Intercept) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) P > Ca <- cbind( 1, Ns( aa, knots=seq(15,65,10) ), 1970 ) > head( Ca ) [1,] [2,] [3,] Rates models [4,] [5,] time 1scales (crv-mod) / 94

88 Adding a linear period effect Testis cancer incidence rate per 100,000 PY in Age > matplot( aa, ci.exp( msp, ctr.mat=ca )*10^5, + log="y", xlab="age", + ylab="testis cancer incidence rate per 100,000 PY in 1970", + type="l", lty=1, lwd=c(3,1,1), col="black", ylim=c(2,20) ) Rates models (crv-mod) 80/ 94

89 Adding a linear period effect Testis cancer incidence rate per 100,000 PY in Age > matplot( aa, ci.exp( msp, ctr.mat=ca )*10^5, + log="y", xlab="age", + ylab="testis cancer incidence rate per 100,000 PY in 1970", + type="l", lty=1, lwd=c(3,1,1), col="black", ylim=c(2,20) ) Rates models > matlines( nd$a, ci.pred( ms, newdata=nd ), + type="l", (crv-mod) lty=1, lwd=c(3,1,1), col="blue" ) 81/ 94

90 period effect > round( ci.lin( msp ), 3 ) Estimate StdErr z P 2.5% 97.5% (Intercept) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) P > pp <- seq(1945,1995,0.2) > Cp <- cbind( pp ) > head( Cp ) pp [1,] [2,] [3,] [4,] Rates models [5,] [6,] time scales (crv-mod) 82/ 94

91 Period effect Testis cancer incidence RR > matplot( pp, ci.exp( msp, Date subset="p", ctr.mat=cp ), + log="y", ylim=c(0.5,2), xlab="date", + ylab="testis cancer incidence RR", + type="l", lty=1, lwd=c(3,1,1), col="black" ) > abline( h=1, v=1970 ) Rates models (crv-mod) 83/ 94

92 A quadratic period effect > mspq <- glm( D ~ Ns(A,knots=seq(15,65,10)) + P + I(P^2), + offset=log(y), family=poisson, data=testisdk ) > round( ci.exp( mspq ), 3 ) exp(est.) 2.5% 97.5% (Intercept) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) P I(P^2) > Cq <- cbind( pp-1970, pp^2-1970^2 ) > head( Cq ) [,1] [,2] [1,] [2,] Rates models [3,] [4,] time scales (crv-mod) 84/ 94

93 A quadratic period effect Testis cancer incidence RR Date > matplot( pp, ci.exp( mspq, subset="p", ctr.mat=cq ), + log="y", ylim=c(0.5,2), xlab="date", + ylab="testis cancer incidence RR", + type="l", lty=1, lwd=c(3,1,1), col="black" ) > abline( h=1, v=1970 ) Rates models (crv-mod) 85/ 94

94 A spline period effect Because we have age-effect with rate dimension, period effect is a RR > msps <- glm( D ~ Ns(A,knots=seq(15,65,10)) + + Ns(P,knots=seq(1950,1990,10),ref=1970), + offset=log(y), family=poisson, data=testisdk ) > round( ci.exp( msps ), 3 ) exp(est.) 2.5% 97.5% (Intercept) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(A, knots = seq(15, 65, 10)) Ns(P, knots = seq(1950, 1990, 10), ref = 1970) Ns(P, knots = seq(1950, 1990, 10), ref = 1970) Ns(P, knots = seq(1950, 1990, 10), ref = 1970) Ns(P, knots = seq(1950, 1990, 10), ref = 1970) Rates models (crv-mod) 86/ 94

95 A spline period effect > Cp <- Ns( pp, knots=seq(1950,1990,10),ref=1970) > head( Cp, 4 ) [1,] [2,] [3,] [4,] > ci.exp( msps, subset="p" ) exp(est.) 2.5% 97.5% Ns(P, knots = seq(1950, 1990, 10), ref = 1970) Ns(P, knots = seq(1950, 1990, 10), ref = 1970) Ns(P, knots = seq(1950, 1990, 10), ref = 1970) Ns(P, knots = seq(1950, 1990, 10), ref = 1970) > matplot( pp, ci.exp( msps, subset="p", ctr.mat=cp ), + log="y", ylim=c(0.5,2), xlab="date", + ylab="testis cancer incidence RR", + type="l", lty=1, lwd=c(3,1,1), col="black" ) Rates models (crv-mod) 87/ 94

96 Period effect Testis cancer incidence RR Date > matplot( pp, ci.exp( msps, subset="p", ctr.mat=cp ), + log="y", ylim=c(0.5,2), xlab="date", + ylab="testis cancer incidence RR", + type="l", lty=1, lwd=c(3,1,1), col="black" ) > abline( h=1, v=1970 ) Rates models (crv-mod) 88/ 94

97 Period effect > par( mfrow=c(1,2) ) > matplot( aa, ci.pred( msps, newdata=data.frame(a=aa,p=1970,y=10^5) ), + log="y", xlab="age", + ylab="testis cancer incidence rate per 100,000 PY in 1970", + type="l", lty=1, lwd=c(3,1,1), col="black" ) > matplot( pp, ci.exp( msps, subset="p", ctr.mat=cp ), + log="y", xlab="date", ylab="testis cancer incidence RR", + type="l", lty=1, lwd=c(3,1,1), col="black" ) > abline( h=1, v=1970 ) Rates models (crv-mod) 89/ 94

98 Age period effect models Testis cancer incidence rate per 100,000 PY in Testis cancer incidence RR Rates Age(crv-mod) Date 90/ 94

99 Period effect > par( mfrow=c(1,2) ) > matplot( aa, ci.pred( msps, newdata=data.frame(a=aa,p=1970,y=10^5) ), + log="y", xlab="age", + ylim=c(2,20), xlim=c(15,65), + ylab="testis cancer incidence rate per 100,000 PY in 1970", + type="l", lty=1, lwd=c(3,1,1), col="black" ) > matplot( pp, ci.exp( msps, subset="p", ctr.mat=cp ), + log="y", xlab="date", + ylim=c(2,20)/sqrt(2*20), xlim=c(15,65)+1930, + ylab="testis cancer incidence RR", + type="l", lty=1, lwd=c(3,1,1), col="black" ) > abline( h=1, v=1970 ) Rates models (crv-mod) 91/ 94

100 Age period effect models Testis cancer incidence rate per 100,000 PY in Testis cancer incidence RR Rates Age(crv-mod) Date 92/ 94

101 Age period effect with ci.exp In rate models re is always one term with rate dimension usually age But it must refer to a specific reference value for all or variables (P). All parameters must be used in computing, at some reference value(s). For or variables, report RR relative to reference point. Only parameters relevant for variable (P) used. Contrast matrix is a difference between (splines at) prediction points reference point. Rates models (crv-mod) 93/ 94