Estimation and prediction in multistate models

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1 Estimation and prediction in multistate models Bendix Carstensen Steno Diabetes Center, Gentofte, Denmark & Department of Biostatistics, University of Copenhagen University of Aberdeen, 17 August / 38 Diabetologia DOI /s ARTICLE Years of life gained by multifactorial intervention in patients with type 2 diabetes mellitus and microalbuminuria: 21 years follow-up on the Steno-2 randomised trial Peter Gæde 1,2 & Jens Oellgaard 1,2,3 & Bendix Carstensen 3 & Peter Rossing 3,, & Henrik Lund-Andersen 3,,6 & Hans-Henrik Parving,7 & Oluf Pedersen 8 Received: 7 April 2016 /Accepted: 1 July 2016 # The Author(s) This article is published with open access at Springerlink.com Abstract Aims/hypothesis The aim of this work was to study the potential long-term impact of a 7.8 years intensified, multifactorial intervention in patients with type 2 diabetes mellitus and microalbuminuria in terms of gained years of life and years free from incident cardiovascular disease. Methods The original 17 (1.) intervention (mean treatment duration 16 (2.1) 1, years) 80 involved patients with type 80 2 diabetes 13 and microalbuminuria who were randomly assigned (using sealed 3 (3.2) 1 (6.7) envelopes) to receive either conventional therapy or intensified, multifactorial treatment including both behavioural and Peter Gæde and Jens Oellgaard contributed equally to this paper. 6 Electronic supplementary material The online version of this article 17 (12.9) 31 (1.7) (doi: /s ) contains peer-reviewed but unedited supplementary material, which is available to authorised users. * Oluf Pedersen oluf@sund.ku.dk (1.7) 13 (9.8) (11.2) Department of Cardiology and Endocrinology, Slagelse Hospital, Slagelse, Denmark 3 (12.1) D() Institute 2.7 for Regional Health Research, University of 67. Southern 3 Denmark, Odense, Denmark Steno Diabetes Center, Gentofte, Denmark Faculty of Health, Aarhus University, Aarhus, Denmark Faculty of Health Sciences, University of Copenhagen, Copenhagen, Denmark 17 (2.2) 1 (6.7) 11 (16.3) 1 (2) pharmacological approaches. After 7.8 years the study continued as an observational follow-up with all patients receiving 2/ 38 treatment as for the original intensive-therapy group. The primary endpoint of this follow-up 21.2 years after intervention start was difference in median survival time between the original treatment groups with and without incident cardiovascu- D(no lar CVD) disease. Non-fatal endpoints and causes of death were ad- 16 judicated by an external endpoint committee blinded for treatment allocation. Results Thirty-eight intensive-therapy patients vs conventional-therapy patients died during follow-up (HR 0. D(1 [9% CVD) CI 0.36, 3], p=0). The patients in the intensive- 1 therapy group survived for a median of 7.9 years longer than the conventional-therapy group patients. Median time before first cardiovascular event after randomisation was 8.1 years longer in the intensive-therapy group (p = 01). The hazard for all microvascular complications was decreased in the intensive- therapy 11 group in the range 0.2 to 7, except for peripheral neuropathy (HR 1.12). Conclusions/interpretation At 21.2 years of follow-up of 7.8 years of intensified, multifactorial, target-driven treatment of type 2 diabetes with microalbuminuria, we demon- D() strate 1a median of 7.9 years of gain of life. The increase in lifespan is matched by time free from incident cardiovascular disease. Trial registration: ClinicalTrials.gov registration 3/ 38 no. NCT Funding: The study was funded by an unrestricted grant from

2 1, (1.) (2.1) 16 3 (3.2) 1 (6.7) (9.8) (6.7) 1 17 (12.9) 31 (1.7).7 Models used 7 (1.7) (11.2) One model for the mortality rates One model for the 3 CVD rates 3 (12.1) both modelsd(3+ assume: CVD) (2.2) 11 (16.3) 1 (2) 11 D() 1 proportional hazards between CVD states (0, 1, 2(, 3) CVD events) proportional hazards between groups (conventional, intervention) proportional hazards between levels of sex and age Which just means: multiplicative effects of the covariates: time since baseline, CVD state, group, sex and age Proportional hazards means: no interaction with the time scale / 38 / 38 Hazard ratios Mortality CVD event HR, Int. vs. Conv. 3 (0.; 1.30) 0. (0.39;0.77) H 0 : PH btw. CVD groups p=38 p=61 H 0 : HR = 1 p=2 p=01 HR vs. 0 CVD events: 0 (ref.) (1.82;.19) 2.3 (1.67;3.2) 2.2 (2.36; 8.29) 3.8 (2.1;.6) (.11;1.6) Then use fitted rates to estimate the probabilities of being in each state at all times. (This is immensely complicated). 6/ 38

3 since baseline (years) Diabetologia7/ 38 a Cumulative mortality (%) between groups (HR 3 [9% CI 0., 1.30], p=3). Thus, the reduced mortality was primarily due to reduced risk of CVD. The patients in the intensive group experienced a total of 90 cardiovascular events vs 19 events in the conventional group. Nineteen intensive-group patients (2%) vs 3 conventional-group patients (3%) experienced more than one cardiovascular event. No significant between-group difference in the distribution of specific cardiovascular firstevent types was observed (Table 2 and Fig. ) Years since randomisation Number at risk b Death or CVD event (%) Years since randomisation Number at risk Microvascular complications Hazard rates of progression rates in microvascular complications compared with baseline status are shown Fig. 3. Sensitivity analyses showed a negligible effect of the random dates imputation. Progression of retinopathy was decreased by 33% in the intensive-therapy group (Fig. ). Blindness in at least one eye was reduced in the intensive-therapy group with an HR of 7 (9% CI 3, 0.98, p = ). Autonomic neuropathy was decreased by 1% in the intensive-therapy group (Fig. ). We observed no difference between groups in the progression of peripheral neuropathy (Fig. ). Progression to diabetic nephropathy (macroalbuminuria) was reduced by 8% in the intensive-therapy group (Fig. ).Tenpatientsinthe conventional-therapy groups vs five patients in the intensivetherapy group progressed to end-stage renal disease (p =61). Discussion Fig. 2 Cumulative mortality (a) and cumulative incidence of the composite cardiovascular or death endpoint (b). Solid lines, patients in the In the current report from the Steno-2 study we demonstrate intensive-therapy group; dashed lines, patients in the conventional-therapy that intensified treatment for 7.8 years was associated with a group; vertical dotted lines, end of trial and start of intensification of 7.9 years longer median lifespan over a period of 21.2 years conventional-therapy group patients treatment; horizontal dashed lines intersect with survival curves at median survival time (a) and median follow-up. Furthermore, the increased lifespan was matched CVD-free survival time (b). The median survival time in the original by the years gained free from incident CVD. The reduced intensive-therapy group was at least 7.9 years longer than in the conventional-therapy mortality was caused by a decreased risk of incident CVD group (8% of patients in the intensive-therapy group died and cardiovascular mortality. during follow-up, so formally this might be an underestimate, since 0% mortality is required to calculate the median). The median difference in Absolute risk and RR reductions for all endpoints were survival before first CVD event was 8.1 years in favour of the original well in line with earlier reported findings, confirming the durability intensive-therapy group of the intensified, multifactorial approach [13]. The frequency of recurrent events was high in both groups, CVD. Transitions between, and time in, different states of CVD but patients in the original conventional-therapy group experienced (0, 1, 2 or 3 or more events since randomisation, respectively) are shown in ESM Fig.. Twenty-eight patients (3%) in the more than twice as many cardiovascular events per person than patients from the original intensive-therapy group. intensive group vs 13 (16%) in the conventional group completed the entire follow-up without any incident macrovascular none of these have been exclusively in patients with type 2 Only a few studies have reported results on recurrent events; events; HR for CVD event in the intensive-therapy group 0. (9% CI 0.39, 0.77; p <01). Patients in both groups with one post-baseline cardiovascular event had a higher mortality rate than patients without; HR 3.08 (9% CI 1.82,.19) and an almost linear increase in mortality of 2.08 (9% CI 1.73, 2.1) per extra event. A similar pattern was seen for further CVD events. When the hazard for mortality was adjusted for CVD status, there was no difference in mortality diabetes [2, 3] and the follow-up periods were much shorter, hence direct comparison is difficult. In the Steno-2 study, we observed high rates of progression for microvascular complications with the vast majority of patients progressing in one or more complication types. Yet, we found significant and clinically relevant risk reductions for autonomic neuropathy, nephropathy, and retinopathy, as well as blindness, and a trend towards reduced risk for end-stage renal since baseline (years) Same allocation HR for Death and CVD between CVD levels 9/ 38 8/ 38

4 since baseline (years) Different allocation HRs for Death and CVD between CVD levels 10/ 38 Expected lifetime and YLL (well, gained) Expected lifetime (years) in the Steno 2 cohort during the first 20 years after baseline by treatment group and CVD status. State where Int. Conv. Int. Conv. Alive under black line No CVD green area Any CVD orange area What does expected mean? Expectation w.r.t. age and sex-distribution in the Steno 2 study! Computed as areas under survival curves 11/ 38 Men Women Age since entry (years) 12/ 38

5 Men Women Age 0 13/ 38 Expected lifetime (years) during the first 20 years after baseline by sex, age, treatment group and CVD status. sex Men Women state age Int. Conv. Int. Conv. Int. Conv. Int. Conv. Alive No CVD / Multistate models in practice: since entry (years) Representation: 17 (1.) 16 (2.1) 1, States Transitions Sojourn times Rates Analysis of rates: Cox-model Poisson model Reporting Rates HRs Probabilities Expected lifetime 3 (3.2) (12.9).7 7 (1.7) (9.8) (11.2) 3 (12.1) 13 D() 3 1 (6.7) (1.7) (2.2) (6.7) 11 (16.3) 1 (2) 1 11 D() 1 1/ 38

6 Representation of multistate FU: Lexis Allowing multiple time scales time-scale variables the starting point on each time scale sojourn time variable lex.dur risk time, exposure the same on all time scales Allowing multiple states requires state variables: lex.cst the state in which follow-up (lex.dur) is lex.xst the state to which transition occur 16/ 38 Representation of multistate FU: Lexis Multiple records per person: One record for each transient state (i.e. state with FU-time) lex.id per age dur tsb lex.dur lex.cst lex.xst allocation sex M M M 17/ 38 Likelihood for transition through states A B C given start of observation in A at time t 0 transitions at times t B and t C survival in C till (at least) time t x : L = P{survive t 0 t B in A} P{transition A B at t B alive in A} P{survive t B t C in B entered B at t B } P{transition B C at t C alive in B} P{survive t C t x in C entered C at t C } Product of likelihood contributions for each transition each one as for a survival model 18/ 38

7 Likelihood contributions reflected in Lexis object L = P{survive t 0 t B in A} P{transition A B at t B alive in A} P{survive t B t C in B entered B at t B } P{transition B C at t C alive in B} P{survive t C t x in C entered C at t C } lex.id time lex.dur lex.cst lex.xst 1 t_0 t_b-t_0 A B 1 t_b t_c-t_b B C 1 t_c t_x-t_c C C constant rate in interval log-likelihood term is Poisson: dlog(λ) λy = (lex.xst! =lex.cst) log(λ) λ lex.dur 19/ 38 Likelihood for multiple states Product of likelihoods for each state each one as for a survival model conditional on being alive at (observed) entry to current state Risk time is the risk time in the Current (lex.cst) state Events are transitions to the exit state (lex.xst) All other transitions out of lex.cst are treated as censorings (but they are not) Fit models separately for each transition... or jointly for all or some may require restructuring of data 20/ 38 Analysis of rates in multistate models Each transition modeled: Cox model Poisson model with log-py as offset either one requires that you decide on a time-scale: age / time since study start / time since current state... Poisson model allows smooth baseline hazards requires that follow-up is split in smaller pieces (so small that the assumption of constant rates is reasonable) also allows modeling of several time scales simultaneously simple to access baseline hazard without further ado 21/ 38

8 Representation of multistate FU: Lexis I Using splitlexis to obtain: lex.id per age dur tsb lex.dur lex.cst lex.xst allocation sex M M M M M M M M M M M M M M 22/ 38 Representation of multistate FU: Lexis II M M M M M M M M M M M M M M M M M M M 23/ 38 Representation of multistate FU: Lexis III M M M M M M M M M M M M M M M M M M M 2/ 38

9 Representation of multistate FU: Lexis IV M M M M M M M M M 2/ 38 Representation of multistate FU: Lexis lex.id per age dur tsb lex.dur lex.cst lex.xst allocation sex M M M M M M M M M M M M 26/ 38 1, (1.) (2.1) 16 3 (3.2) 1 (6.7) (9.8) (6.7) 1 17 (12.9) 31 (1.7).7 (11.2) (16.3) 11 7 (1.7) 17 (2.2) (12.1) D() (2) D() 1 27/ 38

10 Modeling mortality rates in Lexis objects > dlev <- c("", "", "", "D()") > m0 <- glm( (lex.xst %in% dlev ) & (lex.xst!=lex.cst) ~ + Ns( tsb, knots=d.kn ) + lex.cst + allocation + sex + age, + offset = log(lex.dur), + family = poisson, + data = S1 ) > m0i <- update( m0,. ~. + allocation:lex.cst ) > m0n <- update( m0,. ~. + allocation:tsb ) Test for interactions (PropHaz) > anova( m0i, m0, m0n, test="chisq" ) 28/ 38 1, (1.) (2.1) 16 3 (3.2) 1 (6.7) (9.8) (6.7) 1 17 (12.9) 31 (1.7).7 (11.2) (16.3) 11 7 (1.7) 17 (2.2) (12.1) D() (2) D() 1 29/ 38 Mortality rate per 1000 PY / Hazard ratio vs Rate ratio Population since baseline (years) / 38

11 Modeling CVD rates in Lexis objects > clev <- c("","","") > c0 <- glm( ( (lex.xst %in% clev) & (lex.xst!=lex.cst) ) ~ + Ns( tsb, knots=d.kn ) + lex.cst + allocation + sex + age, + offset = log(lex.dur), + family = poisson, + data = subset( S1, lex.cst!="" ) ) > c0i <- update( c0,. ~. + allocation:lex.cst ) > c0n <- update( c0,. ~. + allocation:tsb ) Test for interactions (PropHaz) > anova( c0i, c0, c0n, test="chisq" ) 31/ 38 From rates to probabilities There is a one-to-one correspondence between: all rates between states (by time) + initial state distribution state distribution by time Model for rates probability of being in a given state at any given time Analytically this is a nightmare Simulation is the answer 32/ 38 From rates to probabilities: simlexis Assume a person is in initially Simulate a time of death (transition to ) Simulate a time of CVD (transition to ) Choose the smaller as the transition If transition is to simulate death /, etc. Repeat for, say, 10,000 persons simulated cohort study simlexis does this for you, provided you have initial state and covariates for all persons models to predict (cumulative) rates Count how many is in each state at each time: state occupancy probabilities nstate and pstate does this for you 33/ 38

12 since baseline (years) 3/ 38 Using the Lexis machinery Allows estimation of fully parametric rate function Simple test for proportional hazards State occupancy probabilities requires simulation: simlexis see vignette in Epi package Access to other measures such as expected residual lifetime. similar machinery available in Stata: multistate Crowther, M. J. & Lambert, P. C.: Parametric multi-state survival models: flexible modeling allowing transition-specific distributions with application to estimating clinically useful measures of effect differences. Under review for Stats in Medicine Only one timescale, however... 3/ 38 History Epi package grew out of Statistical Practice in Epidemiology with R, annually since 2002 in Tartu Estonia Lexis machinery conceived by Martyn Plummer, IARC Naming originally by David Clayton & Michael Hills, stlexis in Stata, later renamed stsplit David Clayton wrote a lexis function for the Epi package. Obsolete now. 36/ 38

13 37/ 38 Summary of Lexis Proper representation of multistate data essential: States, transitions, risk time, occupancy Readable modeling code but standard models Calculation of state probabilities requires simulation in any realistic situation Examples of practical multistate modeling in: Worked example in the simlexis vignette in Epi package Thanks for your attention 38/ 38