Adjusted Survival Curves with Multivariable Laplace Regression - online Appendix
Study population
The Cohort of Swedish Men (COSM) and the Swedish Mammography Cohort (SMC), largely
described elsewhere1, were combined. Briefly, the SMC started in 1987, when a questionnaire
on lifestyle and health was sent to all women residing in two regions of central Sweden and
born between 1914 and 1948. The questionnaire was replicated in 1997 with the inclusion of
additional items. 39,227 women responded to this questionnaire. COSM was established in
1997, and comprised 48,850 men from central Sweden, aged 45-79 at baseline, who
responded a self-administrated questionnaire similar to the one sent to women. The final
population of 50,594 participants used in this example was obtained after excluding
participants with missing/incorrect personal identification number, pre-baseline death, or prebaseline cancer, diabetes, or CVD. Former smokers were also excluded to simplify the
presentation.
During 16 years of follow-up (1 January 1998 – 31 December 2013) 12,134 deaths were
ascertained through linkage with the National Register of Death and Death Causes.
Methods and Results
The presented method calculates adjusted survival curves from estimates obtained with
Laplace regression. Given a set of survival percentiles, Laplace regression directly estimates
differences in survival according to level of a given exposure of interest, possibly adjusting
for potential confounders.2 When the exposure is the only covariate of the model, survival
percentiles estimated with Laplace regression and Kaplan-Meier are similar.3 For each
percentile in the observed range (1st-25th), unadjusted differences in survival between
smokers and non-smokers are reported in the first column of Table 1, and correspond to those
plotted in Figure 1A of the letter.
When multivariable adjustment is required, confounders can be included in the
Laplace regression model. In the example we presented we chose to include one dichotomous
covariate (gender, codified as 0 for women and 1 for men), one continuous (body mass index,
kg/m2), and one categorical (baseline age, with 7 5-years categories codified from 0 to 6) that
was included in the model by creating 6 dummy variables. For a given percentile p the
following model was estimated:
𝑇𝑝 (𝑡𝑖𝑚𝑒 𝑡𝑜 𝑑𝑒𝑎𝑡ℎ|𝑠𝑚𝑜𝑘𝑒, 𝑎𝑔𝑒, 𝑏𝑚𝑖, 𝑔𝑒𝑛𝑑𝑒𝑟)
= 𝛽𝑝 0 + 𝛽𝑝 1𝑠𝑚𝑜𝑘𝑒 + 𝛽𝑝 2𝑔𝑒𝑛𝑑𝑒𝑟 + 𝛽𝑝 3𝑏𝑚𝑖 + 𝛽𝑝 4𝑎𝑔𝑒1 + 𝛽𝑝 5𝑎𝑔𝑒2
+ 𝛽𝑝 6𝑎𝑔𝑒3 + 𝛽𝑝 7𝑎𝑔𝑒4 + 𝛽𝑝 8𝑎𝑔𝑒5 + 𝛽𝑝 9𝑎𝑔𝑒6
where age1-age6 are the dummy variables representing the original age-categories covariate.
𝛽𝑝 1 is interpreted as the multivariable adjusted difference in the pth percentile
of survival between smokers and non-smokers. This coefficient is reported, for each p, in the
second column of Table 1.
The adjusted survival curves for smokers and non-smokers presented in Figure
1B of the letter are calculated with linear predictors of these coefficients, by assigning the
mean value to all the included covariates and 0 and 1 to the smoking status exposure,
respectively.
Table 1: Difference in survival percentiles (PDs), with 95% confidence interval (CI), between
smokers and non-smokers
Survival Differences (years), PD (95% CI)
Percentile
Unadjusteda
Adjustedb
1
1,07 (0,8 ; 1,33)
1,85 (1,48 ; 2,23)
2
1,43 (1,15 ; 1,71)
2,68 (2,21 ; 3,15)
3
1,54 (1,24 ; 1,83)
3,17 (2,74 ; 3,6)
4
1,77 (1,47 ; 2,08)
3,72 (3,26 ; 4,17)
5
1,95 (1,62 ; 2,27)
3,88 (3,51 ; 4,25)
6
1,95 (1,64 ; 2,26)
3,89 (3,51 ; 4,26)
7
2,12 (1,78 ; 2,46)
3,85 (3,49 ; 4,21)
8
2,04 (1,73 ; 2,35)
3,84 (3,48 ; 4,2)
9
2,12 (1,82 ; 2,42)
3,76 (3,43 ; 4,08)
10
2,19 (1,86 ; 2,53)
3,69 (3,41 ; 3,97)
11
2,18 (1,86 ; 2,5)
3,8 (3,52 ; 4,07)
12
2,19 (1,88 ; 2,5)
3,73 (3,45 ; 4,01)
13
2,24 (1,89 ; 2,58)
3,75 (3,49 ; 4,01)
14
2,17 (1,84 ; 2,5)
3,72 (3,45 ; 3,98)
15
2,18 (1,87 ; 2,5)
3,7 (3,43 ; 3,97)
16
2,14 (1,83 ; 2,45)
3,66 (3,39 ; 3,93)
17
2,12 (1,8 ; 2,43)
3,66 (3,41 ; 3,92)
18
2,03 (1,68 ; 2,38)
3,69 (3,44 ; 3,95)
19
2,01 (1,65 ; 2,38)
3,68 (3,41 ; 3,95)
20
1,98 (1,66 ; 2,3)
3,66 (3,38 ; 3,93)
21
1,96 (1,63 ; 2,3)
3,62 (3,34 ; 3,9)
22
1,98 (1,66 ; 2,29)
3,59 (3,34 ; 3,83)
23
1,88 (1,57 ; 2,2)
3,58 (3,35 ; 3,82)
24
1,64 (1,3 ; 1,99)
3,61 (3,38 ; 3,83)
25
1,48 (1,16 ; 1,79)
3,61 (3,39 ; 3,84)
a
Estimates were obtained by fitting Laplace regression models with smoking status as the only
coviariate
b
models were further adjusted for age (categorical), body mass index (continuous), and gender
(dicothomous)
Marginal adjusted curves
Researchers may often be interested in computing marginal, rather than conditional, adjusted
survival curves. A possible approach to marginalize survival curves is to use inverse
probability weights (IPW), as described in Cole & Hernan, 2004.4 Instead of stratifying or
adjusting the model for a covariates vector (conditional approach), the inverses of the
probability of being exposed and non-exposed given the covariates are estimated. These
values are then used to weight the survival curves. A simple way to estimate these weights is
by fitting a logistic regression to compute the probability of the exposure given the covariates.
In our example data we fit a logistic regression to predict the individual probabilities of being
smoker or non-smokers given the predictors (baseline age, BMI, and gender). Next, we
estimated IPW Laplace regression models on percentiles 1st-25th using the inverse of these
probabilities as weights. Finally, survival curves were computed with the Stata command
laplace_surv, and are presented in Figure A1. These curves correspond to those we would
obtain by simply weighting the Kaplan-Meier curve.
However, Laplace regression can be extended to include more covariates – that means – to
produce survival curves that are marginalized over a set of predictors but still conditioning on
a second set. This is easily done by using the first covariates to calculate the weights, and
fitting a IPW Laplace regression further adjusted for the second set of covariates. For
example, we calculated inverse probability weights by fitting a logistic regression for smoking
conditioned on age and gender. IPW Laplace regression was then estimated further adjusting
for BMI, and survival curves are presented for BMI=30 Kg/m2 (Figure A2)
Figure A1
100
95
90
85
80
Current Smokers
Never Smokers
75
0
Figure A2
2
4
6
8
10
Follow-up time (years)
12
14
16
References
1.
2.
3.
4.
Harris H, Håkansson N, Olofsson C, Julin B, Åkesson A, Wolk A. The Swedish
mammography cohort and the cohort of Swedish men: Study design and
characteristics of 2 population-based longitudinal cohorts. OA Epidemiology 2013 Oct
01; 1 (2): 16.
Orsini N, Wolk A, Bottai M. Evaluating percentiles of survival. Epidemiology. Sep
2012;23(5):770-771.
Bottai M, Orsini N. A command for Laplace regression. Stata J. 2013;13(2):302-314.
Cole SR, Hernan MA. Adjusted survival curves with inverse probability weights.
Comput Methods Programs Biomed. Jul 2004;75(1):45-49.