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EPI 5344:Survival Analysis in

EpidemiologyQuick Review and Intro to Smoothing Methods

March 4, 2014

Dr. N. Birkett,Department of Epidemiology & Community

Medicine,University of Ottawa

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Objectives (for entire session)

• Primary goal is to address two key concepts:– Hazard

• estimation

• role in survival methods

– Methods to compare two survival curves using non-

parametric methods

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Objectives (for entire session)

• Review– Survival concepts

– Hazard

• Smoothing methods

• Methods for estimation of hazard

• Proportional hazards

• Non-regression comparison of survival curves– Log-rank test

– Variations of log-rank test

• Relate Hazard/ID to person-time

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Review Material, Session #1

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Time ‘0’ (1)

• Time is usually measured as ‘calendar time’

Patient #1 enters on Feb 15, 2000 & dies on Nov 8, 2000

Patient #2 enters on July 2, 2000 & is lost (censored) on April 23, 2001

Patient #3 Enters on June 5, 2001 & is still alive (censored) at the end of the follow-up period

Patient #4 Enters on July 13, 2001 and dies on December 12, 2002

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Study course for patients in cohort

2001 2003 2013

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t

dxxftF0

)()(

Histogram of death time- Skewed to right- pdf or f(t)- CDF or F(t)

- Area under ‘pdf’ from ‘0’ to ‘t’

t

F(t)

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Survival curves (3)

• Plot % of group still alive (or % dead)

S(t) = survival curve

= % still surviving at time ‘t’

= P(survive to time ‘t’)

Mortality rate = 1 – S(t)

= F(t)

= Cumulative incidence

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Deaths CI(t)

Survival S(t)

t

S(t)

1-S(t)

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Essentially, you are re-scaling S(t) so that S*(t0) = 1.0

Conditional Survival Curves

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S*(t) = survival curve conditional on surviving to ‘t0‘

CI*(t) = failure/death/cumulative incidence at ‘t’ conditional on surviving to ‘t0‘

Hazard at t0 is defined as: ‘the slope of CI*(t) at t0’

Hazard (instantaneous)Force of MortalityIncidence rateIncidence density

Range: 0 ∞

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Some relationships

If the rate of disease is small: CI(t) ≈ H(t)If we assume h(t) is constant (= ID): CI(t)≈ID*t

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DEAD

DEAD

DEAD

p1

1-p1

p2

1-p2

p3

1-p3

Year 0 Year 1 Year 2 Year 3

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Actuarial Method

A B C D E F G H

Year # people under follow-up

# lost # people dying in this year

Effective # at risk

Prob die in year

Prob survive this year

S(t)

0-1 10 0 0 10 0 1 1

1-2

2-3

3-4

4-5

5-6

6-7

7-8

8-9

9-10

A B C D E F G H

Year # people under follow-up

# lost # people dying in this year

Effective # at risk

Prob die in year

Prob survive this year

S(t)

0-1 10 0 0 10 0 1 1

1-2 10 1 1 9.5 0.105 0.895 0.895

2-3

3-4

4-5

5-6

6-7

7-8

8-9

9-10

A B C D E F G H

Year # people under follow-up

# lost # people dying in this year

Effective # at risk

Prob die in year

Prob survive this year

S(t)

0-1 10 0 0 10 0 1 1

1-2 10 1 1 9.5 0.105 0.895 0.895

2-3 8 0 1 8 0.125 0.875 0.783

3-4

4-5

5-6

6-7

7-8

8-9

9-10

A B C D E F G H

Year # people under follow-up

# lost # people dying in this year

Effective # at risk

Prob die in year

Prob survive this year

S(t)

0-1 10 0 0 10 0 1 1

1-2 10 1 1 9.5 0.105 0.895 0.895

2-3 8 0 1 8 0.125 0.875 0.783

3-4 7 2 1 6 0.167 0.833 0.652

4-5 4 0 0 4 0 1 0.652

5-6 4 0 1 4 0.25 0.75 0.489

6-7 3 1 0 3.5 0 1 0.489

7-8 2 1 0 2.5 0 1 0.489

8-9 1 1 0 1.5 0 1 0.489

9-10 0 0 0 0 0 1 0.489

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Kaplan-Meier method

‘i' time # deaths

# in risk set

Prob die in interval

Prob survive interval

S(t1)

0 0 --- --- --- 1.0 1.0

1 22 1 9 0.111 0.889 0.889

2

3

4

‘i' time # deaths

# in risk set

Prob die in interval

Prob survive interval

S(t1)

0 0 --- --- --- 1.0 1.0

1 22 1 9 0.111 0.889 0.889

2 29 1 8 0.125 0.875 0.778

3

4

‘i' time # deaths

# in risk set

Prob die in interval

Prob survive interval

S(t1)

0 0 --- --- --- 1.0 1.0

1 22 1 9 0.111 0.889 0.889

2 29 1 8 0.125 0.875 0.778

3 46 1 5 0.200 0.800 0.622

4

‘i' time # deaths

# in risk set

Prob die in interval

Prob survive interval

S(t1)

0 0 --- --- --- 1.0 1.0

1 22 1 9 0.111 0.889 0.889

2 29 1 8 0.125 0.875 0.778

3 46 1 5 0.200 0.800 0.622

4 61 1 4 0.250 0.750 0.467

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END OF REVIEW MATERIAL

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Smoothing methods

• Naïve non-parametric regression• ‘windows’• Sliding windows• Local averaging• Kernel estimation

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Sliding windows (1)

• The divisions we used created five ‘windows’ into the data.– Within each window, we computed the mean ‘X’

and ‘Y’ and plotted that point for the regression line• Why do we need to make the windows ‘fixed’?

– Define the width of a window– Slide it from left to right– Compute the ‘window-specific data point’ and plot

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Sliding windows (2)

• The size of the window is a ‘tuning parameter’.– Fixed number of neighboring data points– Fixed width

• include all points inside

• Large windows tend to ‘over-smooth’• Small windows do little smoothing and

show the random noise.

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Window-specific data point (1)

• Many ways to compute the representative data point for

the window:– X-value

• Mean of the x’s in window

• Median of the x’s in window

• Define window around a specific data point and use that x-value

– Y-value• Mean of the y’s in the window

• Median of the y’s in the window

• Do a regression (linear, quadratic or cubic) of data points in window– use the predicted ‘y’ for the selected ‘x’

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Window-specific data point (2)

• Can ‘weight’ data points– Points closer to the middle should provide

more information about the true (x,y) than those further away.

• The weights are called a ‘kernel’. The method is called ‘Kernel Smoothing’

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Window-specific data point (3)

• Many weight functions (kernels) can be used.

• A common one is the tricube weight

• Select an ‘xi’

– Define the window around xi to get points inside window

– For each point inside the window• let ‘zij’ measure how far the point ‘xij’ is from the left boundary of the window towards

the right boundary– -1 means on the left boundary

– +1 means on the right boundary

– Then the weight for that point is given by:

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LOWESS• LOWESS = LOcally WEighted Scatterplot Smoothing

• Use above procedure but compute a linear regression of ‘x’ on ‘y’ and

use the regression equation to estimate ‘yi’ for given ‘xi’

• Implemented in SAS as a PROC (LOESS)– Available through ODS Graphics and elsewhere

• Can use a higher order polynomial regression instead of the linear

model– Linear model is usually OK

• ‘Tuning’ done by varying the percentage of the data set included in the

window.– Empirical/feel are ‘best’ for choosing tuning

– Some statistics are available (e.g. residuals) but that is advanced material

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