Flexible Parametric Alternatives to the Cox Model - University of ...

Flexible Parametric Alternatives to the Cox Model Paul C Lambert1,2 and Patrick Royston3 1 Department

2 Medical

of Health Sciences, University of Leicester, UK Epidemiology & Biostatistics, Karolinska Institutet, Stockholm, Sweden 3 MRC Clinical Trials Unit, London, UK

UK Stata User Group 2009 London, 11th September 2009

Paul C Lambert

Flexible Parametric Survival Models

UK Stata User Group 2009, London

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Outline 1

The stpm2 command

2

Why We Need Flexible Models

3

Time-Dependent Effects

4

Quantifying Differences

5

Average Survival Curve

6

Attained Age as the Time-Scale

7

Relative Survival

8

Crude Mortality

Paul C Lambert


stpm2: A brief history Patrick Royston wrote stpm in 2001(Royston, 2001). Chris Nelson extended the methodology in stpm to relative survival(Nelson et al., 2007) in strsrcs. Time-dependent effects could be incorporated, but they tended to be over parameterised. I wrote stpm2 (Lambert and Royston, 2009) to Improve the modelling of time-dependent effects. Combine the methods for standard and relative survival. Make it easier to obtain useful predictions.

stpm2 is much faster than stpm, especially with large datasets.

Paul C Lambert



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How stpm2 works In Royston-Parmar models the linear predictor is

Linear Predictor ηi = s (ln(t)|γ, k0 ) + xβ For models on the log cumulative hazard scale.

Survival and hazard functions S(t) = exp (− exp (ηi ))

h(t) =

ds(ln(t)|γ, k0 ) exp (ηi ) dt

Feed these into the likelihood. ln Li = di ln [h(ti )] + ln [S(ti )] Paul C Lambert



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How stpm2 works A simplified version of the ml program is as follow,

stpm2 ml hazard.ado program stpm2 ml hazard version 10.0 args todo b lnf g negH g1 g2 tempvar xb dxb mleval ‘xb’ = ‘b’, eq(1) mleval ‘dxb’ = ‘b’, eq(2) local st exp(-exp(‘xb’)) local ht ‘dxb’*exp(‘xb’) mlsum ‘lnf’ = _d*ln(‘ht’) + ln(‘st’) /** then deal with late entry, first and *** *** second derivatives etc **/ end Paul C Lambert



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Run Rotterdam Example

Paul C Lambert



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England & Wales Breast Cancer Data

Women diagnosed with breast cancer in England and Wales 1986-1990 with follow-up to 1995(Coleman et al., 1999). As an example I will investigate the effect of deprivation (in five groups) on all-cause mortality in women who were diagnosed under the age of 50 years. Follow-up will be restricted to 5 years. Due to their age, most of the women who die within 5 years will die due to their cancer.

Paul C Lambert



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Kaplan-Meier Graphs for Breast Cancer Data Kaplan−Meier Survival Estimates Survival Proportion

1.0 0.9 0.8

Deprivation Group Least Deprived 2 3 4 Most Deprived

0.7 0.6 0

Paul C Lambert

1

2 3 Years from Diagnosis


4

5


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Why We Need Flexible Models

Mortality Rate

0.09 0.08 0.07 0.06 0.05 0.04 Smoothed hazard function Hazard (Gamma) Hazard (stpm2)

0

Paul C Lambert

1



4

5


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Time-Dependent Effects

The difference between two hazard rates may not be proportional. We can choose to, 1 2 3

Ignore. Model on a different scale. Fit an interaction between the covariate and time.

Paul C Lambert



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Time-Dependent Effects A proportional hazards model can be written ln [Hi (t|xi )] = ηi = s (ln(t)|γ, k0 ) + xi β With D time-dependent effects we write, ln [Hi (t|xi )] = s (ln(t)|γ, k0 ) +

D X

s (ln(t)|δ j , kj )xij + xi β

j=1

There is a set of spline variables for each time-dependent effect. For any time-dependent effect there is an interaction between the covariate and the spline variables. The number of spline variables for a particular time-dependent effect will depend on the number of knots, kj Paul C Lambert



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stpm2 and Time-Dependent Effects Non-proportional effects can be fitted by use of the tvc() and dftvc() options.

Non-proportional hazards models .

stpm2 dep5, scale(hazard) df(5) tvc(dep5) dftvc(3)

There is no need to split the time-scale when fitting time-dependent effects. When time-dependence is a linear function of ln(t) and N = 50, 000, 50% censored and no ties. stcox using tvc() - 28 minutes, 24 seconds. stpm2 using dftvc(1) - 0 minutes, 2.5 seconds.

Paul C Lambert



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Predicted Hazard Rates . . . .

stpm2 dep5, scale(hazard) df(5) tvc(dep5) dftvc(3) range temptime 0 5 200 predict h1, hazard timevar(temptime) at(dep5 0) per(1000) predict h5, hazard timevar(temptime) at(dep5 1) per(1000)

Mortality Rate (per 1000 person years)

120 100 80 60

Deprivation Group 40

Least Deprived Most Deprived 0

Paul C Lambert

1

2 3 4 Years from Diagnosis


5


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Predicting Hazard Ratios stpm2 dep5, scale(hazard) df(5) tvc(dep5) dftvc(3) predict hr tvc, hrnumerator(dep5 1) hrdenominator(dep5 0) ci

Most Deprived vs. Least Deprived 4.0 Mortality Rate Ratio

. .

3.0 2.0 1.5 1.0 0

Paul C Lambert

1



4

5


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Quantifying Differences

A key advantage of using a parametric model over the Cox model is that we can transform the model parameters to express differences between groups in different ways. The hazard ratio is a relative measure and a greater understanding of the impact of an exposure can be obtained by also looking at absolute differences. The predict command of stpm2 makes the predictions easy. They work in a similar way as the hrnumerator() and hrdenominator() commands.

Paul C Lambert



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Difference in Hazard Rates predict hdiff, hdiff1(dep5 1) hdiff2(dep5 0) ci

Most Deprived vs. Least Deprived Difference in Mortality Rates (per 1000 person years)

.

100 75 50 25 0 0

Paul C Lambert

1



5


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Difference in Survival Proportions predict sdiff, sdiff1(dep5 1) sdiff2(dep5 0) ci

Difference in Survival Probability

.

Paul C Lambert

Most Deprived vs. Least Deprived 0.00 −0.02 −0.04 −0.06 −0.08 0

1



5


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More than one time-dependent effect As we are modelling on the log cumulative hazard scale, we are essentially modelling non-proportional cumulative hazards. So far we have just considered one time-dependent factor. If we have two time-dependent effects (e.g. deprivation group and year of diagnosis) then the time-dependent hazard ratio for deprivation group may be different at different levels of year of diagnosis. Modelling on the log hazard scale would not have this problem.

Two time-dependent effects . stpm2 dep5 yeardiag, scale(hazard) df(5) tvc(dep5 yeardiag) dftvc(3) . predict hr_early, hrnum(dep5 1 yeardiag 1985) /// hrdenom(dep5 0 yeardiag 1985) /// timevar(timevar) ci . predict hr_late, hrnum(dep5 1 yeardiag 1990) /// hrdenom(dep5 0 yeardiag 1990) /// timevar(timevar) ci Paul C Lambert



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Time-dependent hazard ratios for deprivation group Hazard Ratio for Deprivation Group 6

1985 1990

5 4

hazard ratio

3

2

1

0

Paul C Lambert

1

2 3 Time from Diagnosis (years)


4

5


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Average Survival Curves It can be useful to summarise the average survival curve. The “easy” method is at the mean of the covariates. ˆ 0 (t) exp ¯ Sˆind (t) = exp −H xβˆ Prediction for an individual who happens to have the mean values of each covariate. Problem with binary covariates, e.g. a person of average sex.

This is what stcurve does. A different concept is the mean survival for a population with a particular covariate distribution. N X ˆ ˆ Spop (t) = exp −H0 (t) exp xβˆ i=1

These are not equivalent. Paul C Lambert



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Adjusted/Standardised Survival Curve We can extend the ideas of average survival curves to obtain adjusted survival curves. The key is to obtain the predicted mean population survival curves for two or more groups, while allowing the distribution of other covariates (e.g. age) to be the same for the two groups. The most common method is to use the covariate distribution in the study population as a whole, but other covariate distributions can also be used. The basic idea is similar to the “correct group prognostic method”(Nieto and Coresh, 1996). Using flexible parametric survival models we can allow for time-dependent covariates, continuous covariates etc. Paul C Lambert



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Kaplan-Meier Curves - Renal Replacement Therapy

Survival Function

1.0

Unadjusted HR = 0.62 (0.41, 0.94) Age adjusted HR = 1.14 (0.73, 1.79)

0.8 0.6 0.4 0.2 Non−Asian Asian

0.0 0

Paul C Lambert

2

Mean Age = 62.9 Mean Age = 55.5

4 6 Survival Time (years)


8


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Predictions for Adjusted Survival Curves The meansurv option stpm2 asian age, df(3) scale(hazard) /* Age distribution for study population as a whole */ predict meansurv pop0, meansurv at(asian 0) predict meansurv pop1, meansurv at(asian 1) /* Age distribution for non-asians */ predict meansurv pop0b if asian == 0, meansurv at(asian 0) predict meansurv pop1b if asian == 0, meansurv at(asian 1) /* Age distribution for asians */ predict meansurv pop0c if asian == 1, meansurv at(asian 0) predict meansurv pop1c if asian == 1, meansurv at(asian 1)

Survival curve calculated for each subject in the study population and then averaged. In large studies use the timevar() option. Paul C Lambert



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Adjusted Survival Curve 1 Age Distribution in Whole Study Population

Survival Function

1.0 0.8 0.6 0.4 0.2 Non−Asian Asian

0.0 0

Paul C Lambert

2



8


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Adjusted Survival Curve 2 Age Distribution in Non−Asians

Survival Function

1.0 0.8 0.6 0.4 0.2 Non−Asian Asian

0.0 0

Paul C Lambert

2



8


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Adjusted Survival Curve 3 Age distribution in Asians

Survival Function

1.0 0.8 0.6 0.4 0.2 Non−Asian Asian

0.0 0

Paul C Lambert

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8


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Example of Attained Age as the Time-scale Study from Sweden(Dickman et al., 2004) comparing incidence of hip fracture of, 17,731 men diagnosed with prostate cancer treated with bilateral orchiectomy. 43,230 men diagnosed with prostate cancer not treated with bilateral orchiectomy. 362,354 men randomly selected from the general population.

Outcome is for femoral neck fractures. Risk of fracture varies by age. Age is used as the main time-scale. Alternative way of “adjusting” for age. Gives the age specific incidence rates. Paul C Lambert



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Estimates from a PH Model stset using age as the time-scale . stset dateexit,fail(frac = 1) enter(datecancer) origin(datebirth) /// id(id) scale(365.25) exit(time datebirth + 100*365.25)

.

stcox noorc orc

Cox Model Incidence rate ratio (no orchiectomy) Incidence rate ratio (orchiectomy) .

= 1.37 (1.28 to 1.46) = 2.10 (1.93 to 2.28)

stpm2 noorc orc, df(5) scale(hazard)

Royston-Parmar Model Incidence rate ratio (no orchiectomy) Incidence rate ratio (orchiectomy) Paul C Lambert


= 1.37 (1.28 to 1.46) = 2.10 (1.93 to 2.28) UK Stata User Group 2009, London

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Incidence Rate (per 1000 py’s)

Proportional Hazards 75 50 25

Control No Orchiectomy Orchiectomy

10 5 1

.1 40

60

80

100

Age

Paul C Lambert



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Incidence Rate (per 1000 py’s)

Non Proportional Hazards 75 50 25

Control No Orchiectomy Orchiectomy

10 5 1

.1 40

60

80

100

Age

Paul C Lambert



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Incidence Rate Ratio

Incidence Rate Ratio

Orchiectomy vs Control

20 10 5 2 1 50

60

70

80

90

100

Age horizontal lines from piecewise Poisson model

Paul C Lambert



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Incidence Rate Difference

Difference in Incidence Rates (per 1000 person years)

Orchiectomy vs Control 30

20

10

0 50

60

70

80

90

100

Age

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Relative Survival/Excess Mortality 1

Relative Survival is used in population-based cancer studies. Growing interest in other disease areas: HIV (Bhaskaran et al., 2008), CHD (Nelson et al., 2008). Relative Survival is used to measure mortality associated with a particular disease. Avoids needing information on cause of death. Important as cause of death may not be recorded or may be inaccurately recorded. We use expected mortality (from routine data sources).

Paul C Lambert



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Relative Survival/Excess Mortality 1 The total mortality (hazard) rate is the sum of two components. Observed Expected Excess = + Mortality Rate Mortality Rate Mortality Rate h(t)

=

h∗ (t)

+

λ(t)

If we transform to the survival scale, Relative Survival =

Paul C Lambert

Observed Survival Expected Survival


R(t) =

S(t) S ∗ (t)


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Likelihood for Relative Survival Models Relative Survival Models ln Li = di ln(h∗ (ti ) + λ(ti )) + ln(S ∗ (ti )) + ln(R(ti )) S ∗ (ti ) does not depend on the model parameters and can be excluded from the likelihood. Merge in expected mortality rate at time of death, h∗ (ti ). This is important as many of other models for relative survival involve fine splitting of the time-scale and/or numerical integration. With large datasets this can be computationally intensive. Relative survival models can be fitted in stpm2 by specifying the bhazard() option that gives the expected mortality rate at death. Paul C Lambert



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Fitting Relative Survival Models using stpm2

Analyse all 115,331 women diagnosed with breast cancer. Compare 5 age groups.

All Cause Survival .

stpm2 agegrp2-agegrp5, df(5) scale(hazard)

For relative survival models, just add the bhazard() option.

Relative Survival .

stpm2 agegrp2-agegrp5, df(5) scale(hazard) bhazard(rate)

Paul C Lambert



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Hazard Ratios vs Excess Hazard Ratios All Cause Survival (Hazard Ratio) < 50 50-59 1.12 (1.08 to 1.15) 60-69 1.28 (1.25 to 1.32) 70-79 1.98 (1.92 to 2.04) 80+ 4.15 (4.02 to 4.28)

Relative Survival (Excess Hazard Ratio) 1.05 (1.02 to 1.09) 1.07 (1.04 to 1.11) 1.41 (1.36 to 1.46) 2.65 (2.55 to 2.75)

The excess hazard ratios come from a poor fitting model. The effect of age is nearly always time-dependent. The inclusion of time-dependent effects is the same as for standard survival models. Relative and standard survival are now analysed within the same framework. Paul C Lambert



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Crude Mortality

Patient survival is the most important single measure of cancer patient care (the diagnosis and treatment of cancer) and is of considerable interest to clinicians, patients, researchers, politicians, health administrators, and public health professionals (Dickman and Adami, 2006). Little attention has been paid to the fact that each of these consumers of survival statistics have quite different needs. The standard approach of estimating net survival (relative survival or cause-specific survival) is useful for comparing populations but not necessarily relevant to individual patients.

Paul C Lambert



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Interpreting Relative Survival The cumulative relative survival ratio can be interpreted as the proportion of patients alive after t years of follow-up in the hypothetical situation where the cancer in question is the only possible cause of death. Same interpretation for cause-specific survival. None of us live in this hypothetical world. An individual should understand their personal risk, which includes their risk of dying of other causes. To calculate “real world” probabilities we need to borrow ideas from competing risks theory.

Paul C Lambert



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Net and Crude Mortality Probability of death due to cancer Net Probability in a hypothetical world where the of Death = cancer under study is the only Due to Cancer possible cause of death Probability of death due to cancer Crude Probability in the real world where you may die of Death = of other causes before the Due to Cancer cancer kills you Some people refer to the crude probability as cumulative incidence. Paul C Lambert



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Life table calculation of crude mortality

Cronin and Feuer(Cronin and Feuer, 2000) showed how crude mortality due to cancer and due to other causes can be calculated from life tables. Available in Paul Dickman’s strs command. Calculated separately in age groups. Time-scale split into large (yearly) time intervals. No individual level prediction using continuous covariate.

Paul C Lambert



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Crude Mortality in Relative Survival Models

Crude mortality can be estimated after fitting a relative survival model. The fitting of the relative survival model is not any different, but we do some tricky calculations postestimation. The flexible parametric models allow individual level covariates to be modelled. See Lambert et al. (2009) for details.

Paul C Lambert



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Brief Mathematical Details h∗ (t) - Expected mortality rate λ(t) - Excess mortality rate ∗ h(t) = h (t) + λ(t) - All-cause mortality rate S ∗ (t) - Expected Survival R(t) - Relative Survival ∗ S(t) = S (t)λ(t) - Overall Survival Z t Net Prob of Death = 1 − R(t) = 1 − exp − λ(u)du 0

Z Crude Prob of Death (cancer) =

t

S ∗ (u)R(u)λ(u)du

0

Z Crude Prob of Death (other causes) =

t

S ∗ (u)R(u)h∗ (u)du

0 Paul C Lambert



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Integrating The integration is performed numerically by splitting the time-scale into a large number, n, of small intervals (e.g. 1000). The predicted value of the integrand at each of the n values of t is obtained. The crude probability of death is the sum of the these predicted values. The variance is a bit trickier, as the observation-specific derivatives need to be obtained. These are calculated numerically (Stata’s predictnl command). The approach is similar to that used by Carstensen when calculating survival functions from Poisson based survival models(Carstensen, 2006) Paul C Lambert



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Example 28,943 men diagnosed with prostate cancer aged 40-90 in England and Wales between 1986-1988 inclusive and followed up to 1995. Restricted cubic splines are used to Model the baseline excess hazard (6 knots). Model the main effect of age (4 knots). Model time-dependence of age (4 knots).

Splines, Splines, Splines . .

rcsgen agediag, gen(agercs) df(3) orthog stpm2 agercs1-agercs3, scale(h) df(5) bhazard(rate) /// tvc(agercs1-agercs3) dftvc(3)

Paul C Lambert



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The stpm2cm Command

stpm2cm is a post estimation command. It will calculate the crude probability of death due to cancer and other causes with associated confidence intervals.

stpm2cm .

stpm2cm using uk popmort, /// mergeby( year sex region caquint age) maxt(10) /// diagage(‘agediag’) diagyear(1986) attyear( year) /// attage( age) diagsex(1) /// at(agercs1 ‘a1’ agercs2 ‘a2’ agercs3 ‘a3’) /// stub(ci‘agediag’) tgen(ci t‘agediag’) /// mergegen(region 1 caquint 1) nobs(1000) ci

Paul C Lambert



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Net Probability 1.0 0.8 0.6 0.4 0.2 0.0 0

2 4 6 8 10 Years from Diagnosis 45 years 75 years

Paul C Lambert

Crude Probability of Death due to Cancer

Net Probability of Death due to Cancer

Net and Crude Probability of Death Crude Probability 1.0 0.8 0.6 0.4 0.2 0.0 0

2 4 6 8 10 Years from Diagnosis

55 years 85 years


65 years


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Predictions for a 75 year old man 1.0 Probability of Death

P(Alive)

0.8 P(Dead − Other Causes)

0.6 0.4 0.2 0.0

P(Dead − Prostate Cancer)

0

2


Dead (Prostate Cancer)

Paul C Lambert


Dead (Other Causes)

8

10 Alive


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Predictions for a 55 year old man

Probability of Death

1.0 0.8 0.6 0.4 0.2 0.0

0

2



Paul C Lambert


8

Dead (Other Causes)

10 Alive


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Predictions for a 85 year old man

Probability of Death

1.0 0.8 0.6 0.4 0.2 0.0

0

2



Paul C Lambert


Dead (Other Causes)

8

10 Alive


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Further Extensions

Update to Stata 11. Univariate and shared frailty models. Multiple Events. Competing Risks. Survey options? Cure models. Estimation of loss in expectation of life. Enhance ability to model multiple time-scale.

Paul C Lambert



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References I Bhaskaran, K., Hamouda, O., Sannes, M., Boufassa, F., Johnson, A. M., Lambert, P. C., Porter, K., and Collaboration, C. A. S. C. A. D. E. (2008). Changes in the risk of death after hiv seroconversion compared with mortality in the general population. JAMA, 300(1):51–59. Carstensen, B. (2006). Demography and epidemiology: Practical use of the lexis diagram in the computer age or: Who needs the cox-model anyway? Technical report, Department of Biostatistics, University of Copenhagen. Coleman, M., Babb, P., Damiecki, P., Grosclaude, P., Honjo, S., Jones, J., Knerer, G., Pitard, A., Quinn.M.J., Sloggett, A., and De Stavola, B. (1999). Cancer survival trends in England and Wales, 1971-1995: deprivation and NHS Region. Office for National Statistics, London. Cronin, K. A. and Feuer, E. J. (2000). Cumulative cause-specific mortality for cancer patients in the presence of other causes: a crude analogue of relative survival. Statistics in Medicine, 19(13):1729–1740. Dickman, P. W. and Adami, H.-O. (2006). Interpreting trends in cancer patient survival. J Intern Med, 260(2):103–117. Dickman, P. W., Adolfsson, J., Astrm, K., and Steineck, G. (2004). Hip fractures in men with prostate cancer treated with orchiectomy. Journal of Urology, 172(6 Pt 1):2208–2212. Lambert, P. C., Dickman, P. W., Nelson, C. P., and Royston, P. (2009). Estimating the crude probability of death due to cancer and other causes using relative survival models. Statistics in Medicine, (in press). Lambert, P. C. and Royston, P. (2009). Further development of flexible parametric models for survival analysis. The Stata Journal, 9:265–290. Nelson, C. P., Lambert, P. C., Squire, I. B., and Jones, D. R. (2007). Flexible parametric models for relative survival, with application in coronary heart disease. Statistics in Medicine, 26(30):5486–5498. Nelson, C. P., Lambert, P. C., Squire, I. B., and Jones, D. R. (2008). Relative survival: what can cardiovascular disease learn from cancer? European Heart Journal, 29(7):941–947. Nieto, F. J. and Coresh, J. (1996). Adjusting survival curves for confounders: a review and a new method. Am J Epidemiol, 143(10):1059–1068. Royston, P. (2001). Flexible parametric alternatives to the Cox model, and more. The Stata Journal, 1:1–28.

Example of different knots for baseline hazard

Predicted Mortality Rate (per 1000 py)

100

75

50

1 df: AIC = 53746.92, BIC = 53788.35 2 df: AIC = 53723.60, BIC = 53771.93 3 df: AIC = 53521.06, BIC = 53576.29

25

4 df: AIC = 53510.33, BIC = 53572.47 5 df: AIC = 53507.78, BIC = 53576.83 6 df: AIC = 53511.59, BIC = 53587.54 7 df: AIC = 53510.06, BIC = 53592.91

0

8 df: AIC = 53510.78, BIC = 53600.54 9 df: AIC = 53509.62, BIC = 53606.28 10 df: AIC = 53512.35, BIC = 53615.92

0

Paul C Lambert

1



4

5


Effect of number of knots on hazard ratios Deprivation Group 2

9 7 5 3 1 1

1.1

1.2

1.3

1.4

1.3

1.4

1.3

1.4

1.3

1.4

df for Splines

Deprivation Group 3

9 7 5 3 1 1

1.1

1.2 Deprivation Group 4

9 7 5 3 1 1

1.1

1.2 Deprivation Group 5

9 7 5 3 1 1

1.1

1.2

Hazard Ratio Paul C Lambert



Where to place the knots?

The default knots positions tend to work fairly well. Unless the knots are in stupid places then there is usually very little difference in the fitted values. The graphs on the following page shows for 5 df (4 internal knots) the fitted hazard and survival functions with the internal knot locations randomly selected.

Paul C Lambert



Baseline hazard - random knots

Predicted Mortality Rate (per 1000 py)

100

75

50

13.7 55.8 60.5 64.3 6.1 10.9 61.8 68.4 4.5 25.5 55.5 87.1

25

42.4 52.2 84.1 89.8 21.1 26.5 56.4 94.8 11.8 27.7 40.8 72.2 42.2 46.1 87.2 89.4

0

5.8 67.6 69.9 71.5 9.8 23.2 35.3 59.5 10.2 10.9 57.7 80.7

0

Paul C Lambert

1



4

5


Baseline survival - random knots 1 .9

Predicted Survival

.8 .7 13.7 55.8 60.5 64.3 6.1 10.9 61.8 68.4 4.5 25.5 55.5 87.1 42.4 52.2 84.1 89.8 21.1 26.5 56.4 94.8 11.8 27.7 40.8 72.2 42.2 46.1 87.2 89.4 5.8 67.6 69.9 71.5 9.8 23.2 35.3 59.5 10.2 10.9 57.7 80.7

0

Paul C Lambert

1



4

5


Sensitivity to the number of knots A potential criticism of these models is the subjectivity in the number and the location of the knots. A small sensitivity analysis was carried out where the following models were fitted.

Model Model Model Model Model Model

(a) (b) (c) (d) (e)

Paul C Lambert

Baseline dfb 5 8 5 3 8

Time-dependent dft 3 5 5 3 8


age dfa 3 5 3 3 8

No. of Parameters 18 39 24 16 81

AIC

BIC

97250.11 97059.30 97235.68 97447.35 97105.8

97399.02 97381.95 97434.23 97579.72 97775.92


Knot sensitivity analysis Age 45

Age 55

.6 .4 .2

1 Crude Probability

.8

0

.8 .6 .4 .2 0

0

2 4 6 8 Years from Diagnosis

10

.8 .6 .4 .2 0

0


Age 75

10

0


10

Age 85 1 Crude Probability

1 Crude Probability

Age 65

1 Crude Probability

Crude Probability

1

.8 .6 .4 .2 0

.8 .6 .4 .2 0

0


Model (a)

Paul C Lambert

10

0


Model (b)


Model (c)

10

Model (d)

Model (e)


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