Instantaneous geometric rates via generalized linear models Andrea Discacciati Matteo Bottai Unit of Biostatistics Karolinska Institutet Stockholm, Sweden andrea.discacciati@ki.se 1 September 2017
Outline of this presentation 1 September 2017 1 / 19 • Geometric rates • Instantaneous geometric rates • Models for the instantaneous geometric rates • Instantaneous geometric rates via generalized linear models • Example: survival in metastatic renal carcinoma • Final remarks
Geometric rates 1 September 2017 2 / 19 1.0 0.8 Survival probability 0.6 0.4 0.2 0.0 0 1 2 3 4 5 6 Follow-up time (years)
Geometric rates t 1 September 2017 3 / 19 • The geometric rate represents the average probability of the event of interest per unit of time over a specifjc time interval (0 , t ) 1 g (0 , t ) = 1 − S ( t )
Geometric rates 1 September 2017 4 / 19 1.0 1.0 0.8 0.8 Survival probability Annual rate 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0 1 2 3 4 5 6 Follow-up time (years) Survival Geometric rate
Geometric rates 1 September 2017 5 / 19 1.0 1.0 0.8 0.8 Survival probability Annual rate 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0 1 2 3 4 5 6 Follow-up time (years) Survival Geometric rate
Geometric rates 1 September 2017 6 / 19 1.0 1.0 0.8 0.8 Survival probability Annual rate 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0 1 2 3 4 5 6 Follow-up time (years) Survival Geometric rate
Instantaneous geometric rates instantaneous probability of the event of interest per unit of time 1 September 2017 7 / 19 • The instantaneous geometric rate (IGR) represents the
Instantaneous geometric rates lim 1 September 2017 (1) lim 8 / 19 lim gives the instantaneous geometric rate (Bottai, 2015) The limit of the geometric rate over shrinking time intervals ( t , t + ∆ t ) g ( t ) ≡ ∆ t → 0 + g ( t , t + ∆ t ) ] 1 [ S ( t + ∆ t ) ∆ t = ∆ t → 0 + 1 − S ( t ) { log S ( t + ∆ t ) − log S ( t ) } = ∆ t → 0 + 1 − exp ∆ t { ∂ log S ( t ) } = 1 − exp ∂ t { − f ( t ) } = 1 − exp S ( t ) = 1 − exp {− h ( t ) }
Models for the instantaneous geometric rate (2) (3) (GLM) framework by using two nonstandard link functions 1 September 2017 9 / 19 • Proportional instantaneous geometric rate model g i ( t | x i ) = g 0 ( t ) exp { x T i β } • Proportional instantaneous geometric odds model g i ( t | x i ) g 0 ( t ) 1 − g i ( t | x i ) = 1 − g 0 ( t ) exp { x T i β } • These models can be estimated within the generalized linear model
Instantaneous geometric rates via GLM Let’s focus on the proportional instantaneous geometric rate model By taking the logarithm of both sides of (2) we get and by equation (1) we write (4) polynomials or splines. 1 September 2017 10 / 19 log [ g i ( t | x i )] = log [ g 0 ( t )] + x T i β log [1 − exp {− h i ( t ) }| x i ] = log [ g 0 ( t )] + x T i β where log [ g 0 ( t )] (baseline log-IGR) is modelled using for example
Instantaneous geometric rates via GLM t ij 1 September 2017 assumed to follow a distribution of the exponential family where: 11 / 19 time into very short intervals ( stsplit ) • To model the baseline log-IGR, we split each individual’s follow-up • Given equation (4) we use the following link function [ { − µ ij }] η ij ≡ k ( µ ij ) = log 1 − exp • t ij is the length of the j th interval relative to the i th subject • µ ij is the expected value of d ij (the event/censoring indicator), which is
Instantaneous geometric rates via GLM instantaneous geometric rate ratios, whereas in model (3) they are interpreted as instantaneous geometric odds ratios populations, the instantaneous geometric odds are not, and vice-versa user-defjned link programs: log_igr and logit_igr 1 September 2017 12 / 19 • In model (2) the exponentiated coeffjcients exp { β } are interpreted as • If the instantaneous geometric rates are proportional across difgerent • Link functions for models (2) and (3) are implemented in two
Example: survival in metastatic renal carcinoma renal carcinoma oral medroxyprogesterone (MPA) www.imm.ki.se/biostatistics 1 September 2017 13 / 19 • Data from a clinical trial on 347 patients diagnosed with metastatic • The patients were randomly assigned to either interferon- α (IFN) or • A total of 322 patients died during follow-up • Stata code to reproduce the worked-out example is available at:
Example: survival in metastatic renal carcinoma 0.53 0.681 0.41 0.86 1.31 _rcs2 | 1.74 0.894 4.72 -0.13 0.29 0.96 _rcs1 | | 0.96 0.36 _rcs3 | 0.009 0.06 1 September 2017 ------------------------------------------------------------------------------ 0.86 0.61 0.000 -3.63 0.72 0.90 _cons | 1.51 0.54 0.693 -0.39 0.24 0.73 -2.62 . qui use http://www.imm.ki.se/biostatistics/data/kidney, clear log pseudolikelihood = log pseudolikelihood = -1959.6083 rescale: log pseudolikelihood = -4804.4455 feasible: (could not be evaluated) -<inf> initial: 0.06 nolog search eform baselevel noheader > . qui rcsgen _t, df(3) if2(_d == 1) gen(_rcs) . qui generate risktime = _t - _t0 . qui stsplit click, every(`=1/52') . qui stset survtime, failure(cens) id(pid) scale(365.24) ------------------------------------------------------------------------------ | Robust _d | 0.84 | IFN (base) 1.00 | MPA trt | -------------+---------------------------------------------------------------- [95% Conf. Interval] P>|z| z Std. Err. exp(b) 14 / 19 . glm _d i.trt c._rcs?, family(poisson) link(log_igr risktime) vce(robust) /// • IGRR comparing IFN versus MPA patients: 0.84 (0.73–0.96)
Example: survival in metastatic renal carcinoma 1 September 2017 15 / 19 0.75 MPA IFN 0.65 Instantaneous geometric rate (per year) 0.55 0.45 0.35 0.25 0 1 2 3 4 5 6 Years from randomization
Example: survival in metastatic renal carcinoma 0.66 1.57 | IFN trt#c._rcs2 | | 2.08 -0.52 0.238 1.18 0.78 1.14 | IFN trt#c._rcs1 | | 0.77 -0.54 0.727 0.35 0.33 1.38 0.256 _rcs3 | | 1 September 2017 ------------------------------------------------------------------------------ -0.09 -0.44 0.003 -3.02 0.09 -0.26 _cons | 0.47 -1.14 -1.70 0.267 -1.11 0.55 -0.62 | IFN trt#c._rcs3 | | 4.29 0.12 1.32 . glm _d i.trt##c._rcs?, family(poisson) link(log_igr risktime) vce(robust) /// | trt | -------------+---------------------------------------------------------------- [95% Conf. Interval] P>|z| z Std. Err. Coef. _d | Robust ------------------------------------------------------------------------------ | log pseudolikelihood = -1959.6083 rescale: log pseudolikelihood = -4804.4455 feasible: (could not be evaluated) -<inf> log pseudolikelihood = initial: nolog search baselevel noheader > MPA 0.00 -1.91 0.36 0.722 -0.36 0.82 -0.29 _rcs2 | 0.40 -1.02 0.389 -0.86 -0.31 (base) _rcs1 | | 0.01 -0.74 0.054 -1.92 0.19 -0.37 | IFN 16 / 19
Example: survival in metastatic renal carcinoma 1 September 2017 17 / 19 Instantaneous geometric rate (per year) 0.25 0.35 0.45 0.55 0.65 0.75 0 1 2 Years from randomization 3 4 5 IFN MPA 6 0.70 0.84 1.00 Instantaneous geometric rate ratio
Final remarks of covariates on the IGR nonstandard link functions 1 September 2017 18 / 19 • Instantaneous geometric rates are easy to interpret • Instantaneous geometric rates ̸ = hazard rates • Proportional instantaneous geometric rate/odds models for the efgect • These models can be estimated within the GLM framework by using
References Methods Med Res . 2015 Sep 18. generalized linear models. Stata J . 2017;17(2):358–371. 1 September 2017 19 / 19 • Bottai M. A regression method for modelling geometric rates. Stat • Discacciati A, Bottai M. Instantaneous geometric rates via
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