Welcome and Introduction to Flexible Parametric Survival Models ...

Welcome and Introduction to Flexible Parametric Survival Models Paul C Lambert1,2 1 Department

2

of Health Sciences, University of Leicester, UK Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden

Workshop on Applications and Developments of Flexible Parametric Survival Models Stockholm 10/11/2011

Welcome to the workshop! This is a satellite meeting to the the Nordic and Baltic Stata Users Group meeting to be held tomorrow. Thanks to Nicola Orsini, Matteo Bottai and Peter Hedstr¨om, for allowing us to attach this workshop to the Stata meeting.

Aims To raise awareness of the models and software. To present and discuss current applications and developments. To discuss potential extensions and limitations. Please ask questions and contribute to the discussion!

Paul C Lambert

Flexible Parametric Models

Stockholm 10/11/2011

2

Timetable (morning) 09:00 Paul Lambert

Welcome and introduction to flexible parametric survival models

09:45 Camille Maringe

Using flexible parametric survival models for international comparisons of cancer survival.

10:10 Coffee 10:40 Edoardo Colzani

11:05 Patrick Royston 11:30 Paul Dickman 12:00-13:15 Lunch

Paul C Lambert

Prognosis of Patients With Breast Cancer: Causes of Death and Effects of Time Since Diagnosis, Age, and Tumor Characteristics Restricted mean survival time: computation and some applications Discussion of morning session.



3

Timetable (afternoon) 13:15 Anna Johansson

Estimation of absolute risks in case-cohort studies.

13:40 Therese Andersson

Cure models within the framework of flexible survival models

14:05 Sally Hinchliffe

Flexible parametric models for competing risks.

14:30 Sandra Eloranta

Partitioning of excess mortality associated with a diagnosis of cancer using flexible parametric survival models.

14:55 Coffee 15:25 Mark Clements

Fitting flexible parametric survival models in R

15:50 Michael Crowther

Flexible parametric joint modelling of longitudinal and survival data

16:15 Patrick Royston

Discussion of afternoon session

Paul C Lambert



4

Why Parametric Models We have the Cox model so why use parametric models? Parametric Models have advantages for Prediction. Extrapolation. Quantification (e.g., absolute and relative differences in risk). Modelling time-dependent effects. Understanding. Complex models in large datasets (time-dependent effects / multiple time-scales) All cause, cause-specific or relative survival.

The estimates we get from flexible parametric survival models are incredibly similar to those obtained from a Cox model.

Paul C Lambert



5

Pregnancy Associated Breast Cancer (Johansson 2011)

Paul C Lambert



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The Cox Model I Web of Science: over 23,300 citations (October 2008). Has an h-index of 13 from repeat mis-citations1 . hi (t|xi ) = h0 (t) exp (xi β) Estimates (log) hazard ratios. Advantage: The baseline hazard, h0 (t) is not estimated from a Cox model. Disadvantage: The baseline hazard, h0 (t) is not estimated from a Cox model.

Paul C Lambert



7

Quote from David Cox (Reid 1994 [1]) Reid “What do you think of the cottage industry that’s grown up around [the Cox model]?” Cox “In the light of further results one knows since, I think I would normally want to tackle the problem parametrically. . . . I’m not keen on non-parametric formulations normally.” Reid “So if you had a set of censored survival data today, you might rather fit a parametric model, even though there was a feeling among the medical statisticians that that wasn’t quite right.” Cox “That’s right, but since then various people have shown that the answers are very insensitive to the parametric formulation of the underlying distribution. And if you want to do things like predict the outcome for a particular patient, it’s much more convenient to do that parametrically.” Paul C Lambert



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Splines

Flexible mathematical functions defined by piecewise polynomials. Used in regression models for non-linear effects The points at which the polynomials join are called knots. Constraints ensure the function is smooth. The most common splines used in practice are cubic splines. However, splines can be of any degree, n. Function is forced to have continuous 0th , 1st and 2nd derivatives.

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Piecewise hazard function Interval Length: 1 week

Excess Mortality Rate (1000 py’s)

200 150

100

50

25

0

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1

2 3 Years from Diagnosis


4

5


10

No Continuity Corrections No Constraints


200 150

100

50

25

0

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4

5


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Function forced to join at knots Forced to Join at Knots


200 150

100

50

25

0

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4


5

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Continuous first derivative Continuous 1st Derivatives


200 150

100

50

25

0

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4

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13

Continuous second derivative Continuous 2nd Derivatives


200 150

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0

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Restricted Cubic Splines

Restricted cubic splines are splines that are restricted to be linear before the first knot and after the last knot [2]. Fitted as a linear function of derived covariates. For knots, k1 , . . . , kK , a restricted cubic spline function can be written s(x) = γ0 + γ1 z1 + γ2 z2 + . . . + γK −1 zK −1 Issue is to choose the number and location of the knots.

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Flexible Parametric Models: Basic Idea Consider a Weibull survival curve. S(t) = exp (−λt γ ) If we transform to the log cumulative hazard scale. ln [H(t)] = ln[− ln(S(t))] ln [H(t)] = ln(λ) + γ ln(t) This is a linear function of ln(t) Introducing covariates gives ln [H(t|xi )] = ln(λ) + γ ln(t) + xi β Rather than linearity with ln(t) flexible parametric models use restricted cubic splines (Roston & Parmar 2002 [3]). Paul C Lambert



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Flexible Parametric Models: Incorporating Splines We thus model on the log cumulative hazard scale. ln[H(t|xi )] = ln [H0 (t)] + xi β This is a proportional hazards model. Restricted cubic splines with knots, k0 , are used to model the log baseline cumulative hazard. ln[H(t|xi )] = ηi = s (ln(t)|γ, k0 ) + xi β For example, with 4 knots we can write ln [H(t|xi )] = ηi = γ0 + γ1 z1i + γ2 z2i + γ3 z3i {z } | log baseline cumulative hazard Paul C Lambert


+

xi β |{z} log hazard ratios


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Survival and Hazard Functions We can transform to the survival scale S(t|xi ) = exp(− exp(ηi )) The hazard function is a bit more complex. h(t|xi ) =

ds (ln(t)|γ, k0 ) exp(ηi ) dt

This involves the derivatives of the restricted cubic splines functions. These are easy to calculate.

Survival and hazard function used to maximize the likelihood. No need for numerical integration or time-splitting. Paul C Lambert



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Fitting a Proportional Hazards Model Example: 24,889 women aged under 50 diagnosed with breast cancer in England and Wales 1986-1990. Compare five deprivation groups from most affluent to most deprived. No information on cause of death, but given their age, most women who die will die of their breast cancer.

Proportional hazards models . stcox dep2-dep5 . stpm2 dep2-dep5, df(5) scale(hazard) eform The df(5) option implies using 4 internal knots and 2 boundary knots at their default locations. The scale(hazard) requests the model to be fitted on the log cumulative hazard scale. Paul C Lambert



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Comparison of Hazard Ratios Cox Proportional Hazards Model . stcox dep2-dep5, _t

Haz. Ratio

dep2 dep3 dep4 dep5

1.048716 1.10618 1.212892 1.309478

Std. Err.

z

P>|z|

[95% Conf. Interval]

.0353999 .0383344 .0437501 .0513313

1.41 2.91 5.35 6.88

0.159 0.004 0.000 0.000

.9815786 1.03354 1.130104 1.212638

P>|z|

[95% Conf. Interval]

0.158 0.004 0.000 0.000 0.000 0.012 0.000 0.010 0.030

.9816125 1.033513 1.130085 1.212639 2.087361 .9668927 1.048715 1.001288 1.000218

1.120445 1.183924 1.301744 1.414051

. stpm2 dep2-dep5, df(5) scale(hazard) eform exp(b)

Std. Err.

z

xb dep2 dep3 dep4 dep5 _rcs1 _rcs2 _rcs3 _rcs4 _rcs5

Paul C Lambert

1.048752 1.10615 1.212872 1.309479 2.126897 .9812977 1.057255 1.005372 1.002216

.0354011 .0383334 .0437493 .0513313 .0203615 .0074041 .0043746 .0020877 .0010203

1.41 2.91 5.35 6.88 78.83 -2.50 13.46 2.58 2.17



1.120483 1.183893 1.301722 1.414052 2.167182 .9959173 1.065863 1.009472 1.004218

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Sensitivity to choice of knots

Hazard Ratios are generally insensitive to the number and location of knots. Too many knots will overfit baseline hazard with local ‘humps and bumps’. Too few knots will underfit. In most situations the choice of knots is not crucial. We can use the AIC and BIC to help us select how many knots to use, but a simple sensitivity analysis is recommended.

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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

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1

2 3 Time from Diagnosis (years)


4

5


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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



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Where to place the knots?

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

Paul C Lambert



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Random knot positions for baseline hazard

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

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4

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Effect of location of knots on baseline survival 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

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1



4


5

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Why We Need Flexible Models There are a number of parametric models available, so why can’t we just use these? For proportional hazards only ‘simple’ models available: Exponential, Weibull, Gompertz. More complex models such as generalized gamma only available in accelerated failure form. These models still may not capture the underlying shape of the data. In Stata the most complex parametric survival distribution available is the generalized gamma.

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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

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1



4

5


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Comparison with Cox model Simulation where true baseline hazard is complex. ‘Truth’ is a mixture of two Weibull distributions. E.g.

Hazard Function

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

1 2 3 4 Time from Diagnosis (years)

5

Model with dichotomous covariate effect, β = −0.5. Simulate 1000 datasets each with sample size = 3000. Paul C Lambert



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Agreement between parameter estimates

Cox Model

-.4 -.45 -.5 -.55 -.6 -.6 Paul C Lambert

-.55 -.5 -.45 Flexible Parametric Model



-.4 30

Agreement between standard errors .0394

Cox Model

.0392 .039 .0388 .0386 .0384 .0384 .0386 .0388 .039 .0392 .0394 Flexible Parametric Model Paul C Lambert



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Time dependent effects

An important feature of flexible parametric models is the ability to model time-dependent effects, i.e., there are non-proportional hazards Time-dependent effects are modelled using splines, but will generally require fewer knots than the baseline. This is because we are now modelling deviation from the baseline hazard rate. Also possible to split time to estimate hazard ratio in different intervals.

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Time-Dependent Effects A proportional cumulative hazards model can be written ln [Hi (t|xi )] = ηi = s (ln(t)|γ, k0 ) + xi β New set of spline variables for each time-dependent effect [4] If there are D time-dependent effects then ln [Hi (t|xi )] = s (ln(t)|γ, k0 ) +

D X

s (ln(t)|δ j , kj )xij + xi β

j=1

The number of spline variables for a particular time-dependent effect will depend on the number of knots, kj Interaction between the covariate and the spline variables. 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.

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Example of Attained Age as the Time-scale Study from Sweden[5] 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 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.

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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)

Flexible Parametric 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)


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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

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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

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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

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Incidence Rate Difference

Difference in Incidence Rates (per 1000 person years)

Orchiectomy vs Control 30

20

10

0 50

60

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100

Age

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Relative Survival Three of today’s talks fit relative survival models. Relative survival is a measure used in population based cancer studies. Used as unreliable (or missing) cause of death information. Incorporates expected mortality, Excess Expected Observed + = 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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Modelling Relative Survival

Flexible parametric survival models extended to relative survival[6]. When using these models we estimate the excess hazard (mortality) rate rather than the hazard rate. the relative survival function rather than the survival function. excess hazard ratios and excess hazard differences.

All cause, cause-specific and relative survival analysed within same framework.

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Breast Cancer Data Comparison of breast cancer survival in England and Norway [7, 8]. The data consists of 303,657 women from England. 24,919 women from Norway. Year of Diagnosis was between 1996 and 2004.

Model includes Baseline hazard (splines) Age (splines) Country Age Country Interaction. Time-dependent effects for age & country (splines).

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Relative Survival Age 35

Age 45

0.6

1.0

Relative Survival

0.8

0.4

0.8

0.6

0.4 0

2 4 6 Years from Diagnosis

8


8

0

0.6

0.4

0.8

0.6

0.4 8

8

1.0

Relative Survival

Relative Survival

0.8

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Age 85

1.0


0.6

Age 75

1.0

0

0.8

0.4 0

Age 65

Relative Survival

Age 55

1.0

Relative Survival

Relative Survival

1.0

0.8

0.6

0.4 0



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8


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50 25

10 2 4 6 Years from Diagnosis

400 200 100 50 25

10 0


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10

8

Age 65

8

Excess Mortality Rate (per 1000 py)

100

Age 45

400

0


Age 75

400 200 100 50 25

10 0



8

Age 55

400 200 100 50 25

10

8

0 Excess Mortality Rate (per 1000 py)

200

0 Excess Mortality Rate (per 1000 py)


Age 35

400



Excess Mortality Rates


8

Age 85

400 200 100 50 25

10 0



8

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Excess Mortality Rate Ratios Age 35

3

2

Excess Mortality Rate Ratio

Age 45

3

2

1

2

1 0

2

4

6

1

8

Age 65

3

0

2

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Age 75

3

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Age 85

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Age 55

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Years from Diagnosis Paul C Lambert



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Differences in Excess Mortality (b) Age 35

Difference in Excess Mortality (1000pys)

50

Age 45

50 40

40

30

30

30

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20

20

10

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10

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0

0

−10

−10

−10

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2

4

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Age 65

50

0

2

4

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Age 75

300

0

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250

30

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200

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150

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−10

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Age 85

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Age 55

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Years from Diagnosis Paul C Lambert



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Software To install stpm2 using . ssc install stpm2

Some examples in Lambert, P. C., Royston, P. Further development of flexible parametric models for survival analysis. The Stata Journal 2009;9:265-290. [9]

Many more examples plus downloadable do files in

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Other Issues

Topics we may discuss today Number and locations of knots. Many time-dependent effects. Hazard vs cumulative hazard scale. Model selection (in large studies). Absolute vs relative effects. When to use flexible parametric models. Other alternatives.

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References [1] Reid N. A conversation with Sir David Cox. Statistical Science 1994;9:439–455. [2] Durrleman S, Simon R. Flexible regression models with cubic splines. Statistics in Medicine 1989;8:551–561. [3] Royston P, Parmar MKB. Flexible parametric proportional-hazards and proportional-odds models for censored survival data, with application to prognostic modelling and estimation of treatment effects. Statistics in Medicine 2002;21:2175–2197. [4] Lambert PC, Dickman PW, Nelson CP, Royston P. Estimating the crude probability of death due to cancer and other causes using relative survival models. Statistics in Medicine 2010;29:885 – 895. [5] Dickman PW, Adolfsson J, Astrm K, Steineck G. Hip fractures in men with prostate cancer treated with orchiectomy. Journal of Urology 2004;172:2208–2212. [6] Nelson CP, Lambert PC, Squire IB, Jones DR. Flexible parametric models for relative survival, with application in coronary heart disease. Statistics in Medicine 2007; 26:5486–5498. [7] Lambert PC, Holmberg L, Sandin F, Bray F, Linklater KM, Purushotham A, et al.. Quantifying differences in breast cancer survival between England and Norway. Cancer Epidemiology in press 2011;.

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References 2

[8] Møller H, Sandin F, Bray F, Klint A, Linklater KM, Purushotham A, et al.. Breast cancer survival in England, Norway and Sweden: a population-based comparison. International Journal of Cancer 2010;127:2630–2638. [9] Lambert PC, Royston P. Further development of flexible parametric models for survival analysis. The Stata Journal 2009;9:265–290.

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Welcome and Introduction to Flexible Parametric Survival Models ...

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