eha: more on plots and interpreting hazards sociology 229a: event history analysis class 9 copyright...

EHA: More On Plots and Interpreting Hazards

Sociology 229A: Event History AnalysisClass 9

Copyright © 2008 by Evan SchoferDo not copy or distribute without permission

Announcements

• Final paper assignment due next week• Questions?

• Class topics: • More on interpreting hazard & cumulative hazard

functions• More multilevel models…

Hazard Plots: Smoothing

• Issue: Stata heavily smooths hazard plots• “Raw” hazard plots are very spiky… smoothing can

help with interpretation• Issue: Too much smoothing obscures the detail within

your data

– Simplest way to control smoothing: • Set the “width” of the kernel smoother in Stata• EX: sts graph, haz width(3)• Lower width = less smoothing; try different values.

Hazard Smoothing• Environmental Law Data: Default smoothing

0.0

2.0

4.0

6.0

8.1

1970 1980 1990 2000analysis time

Smoothed hazard estimate

Hazard Smoothing• Environmental Law Data: width (1)

0.0

5.1

.15

.2

1970 1980 1990 2000analysis time


Hazard Smoothing• Environmental Law Data: width (.2)

0.1

.2.3

1970 1980 1990 2000analysis time


Hazard Smoothing• Don’t make width too small!: width (.001)

010

2030

4050

1970 1980 1990 2000analysis time

Smoothed hazard estimateStata’s default smoother amplifies peaks in data if width is too small!

Hazard Smoothing: Remarks

• Stata default smoothing is quite aggressive• Obscures detail in your data

– Stata default smoothing “width” is ~4 in this case• Smoothing of 1-2 works much better

• In addition to removing detail, smoothing but lowers the peaks…

• Highest peak = .1 (width 4)• Highest peak = .3 (width .2)• Also: REALLY narrow width exaggerates peaks

– Hightest peak = 50 (width .0001)

Survival Plot Problem: noorigin• Issue: Stata always likes to include t=0…

0.00

0.25

0.50

0.75

1.00

0 500 1000 1500 2000analysis time

Kaplan-Meier survival estimate

Survival Plot Problem: noorigin• Solution: sts graph, noorigin

0.00

0.25

0.50

0.75

1.00

1970 1980 1990 2000analysis time


Plots: Confidence Intervals

• Confidence intervals are a good idea• Especially useful when comparing groups

– Stata• sts graph, ci• sts graph, haz ci

– Issue: Adding CIs tends to compress the Y axis to make room for the confidence bands

• Makes the hazard look less variable over time• Watch for that…

– Issue: CIs can make charts “busy” / hard to read.

Hazard Plot with 95% CI0

.1.2

.3

1970 1980 1990 2000analysis time

95% CI Smoothed hazard function


Hazard plot with 95% CI0

.1.2

.3.4

1970 1980 1990 2000analysis time

95% CI 95% CIwest2 = 0 west2 = 1

Smoothed hazard estimates

Survivor plot with 95% CI0

.25

.5.7

51

1970 1980 1990 2000analysis time

95% CI 95% CIwest2 = 0 west2 = 1

Kaplan-Meier survival estimates

Other sts graph options

• Options to show # of lost, entered, or censored cases

• Lost: puts a number above plots showing cases lost• Atrisk: shows # of cases at risk

– Actually, it shows risk per interval– EX: if unit = nation, it shows nation-years in an interval

• Censored: shows number of cases censored

Sts graph: atrisk

8684

16581 166 247 82 81 80

7874 73 72

6866

63 6262

5755 53

4540

3330

2420

0.00

0.25

0.50

0.75

1.00

1970 1980 1990 2000analysis time


Interpreting Hazard & Cumulative Haz

• The survivor plot has a clear interpretation: The proportion of cases that have not experienced the event

• Assuming non-repeated events– If events repeat frequently, survivor falls to 0, stays there…

• Assuming the risk-set stays more-or-less constant– Survivor never goes back up, even if more cases enter the risk

set…

• But, hazard rates & cumulative hazard rates are harder to understand intuitively…

• So, I made some illustrative examples

Hazard Example 1

• Start with 10 people• Let’s put them in the risk set sequentially• All cases start at time t=0• One case fails at each point in time

Start End Failed?0 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 1

Example 1: Survivor Plot0.

000.

250.

500.

751.

00

0 2 4 6 8 10analysis time


Example 1: Hazard Plot.1

.2.3

.4.5



Events occur at an even interval… but rate goes up because the risk set dwindles…

Example 1: Integrated Hazard0.

001.

002.

003.

00


Nelson-Aalen cumulative hazard estimate

Example 2

• Let’s figure out what’s really going on…• Again, start with 10 people• Imagine each enters the risk set sequentially, and fails

after 1 time unit– So only 1 case at risk in any period of time– And, 1 event per each point in time

Start End Failed?0 1 11 2 12 3 13 4 14 5 15 6 16 7 17 8 18 9 19 10 1


000.

250.

500.

751.

00



Survivor drops to zero when first case fails… doesn’t go back up when additional cases enter

NOT very informative…


.92

.94

.96

.98

1



Hazard basically sits at 1.0. Variations = due to smoothing issues…

That’s because for every time unit at risk there is event

Interpreting Hazards

• Let’s run an exponential model• We’ll estimate the constant only… the baseline hazard

. streg , dist(exponential) nohr

Exponential regression -- log relative-hazard form

No. of subjects = 10 Number of obs = 10No. of failures = 10Time at risk = 10 LR chi2(0) = 0.00Log likelihood = 5.1044126 Prob > chi2 = .

------------------------------------------------------------------------------ _t | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- _cons | 0 .3162278 0.00 1.000 -.619795 .619795------------------------------------------------------------------------------

Why is the base rate zero?

Answer: We need to exponentiate!

Exp(0) = 1

The model estimates the baseline hazard to be 1.0!


002.

004.

006.

008.

0010

.00



Integrated Hazard reaches 10

Same number of events as previous example… but less time-at-risk… so overall cumulated risk was higher

Example 3

• Let’s keep those same cases but add 10 more• Each in risk for 1 time-unit; all of which are censored

Start End Failed?0 1 11 2 12 3 13 4 14 5 15 6 16 7 17 8 18 9 19 10 10 1 01 2 02 3 03 4 04 5 05 6 06 7 07 8 08 9 09 10 0


5.4

6.4

7.4

8.4

9.5



The risk set is doubled, but # events stays the same…

So, hazard drops by half… to .5


• Let’s run an exponential model• We’ll estimate the constant only… the baseline hazard

. streg , dist(exponential) nohr

Exponential regression -- log relative-hazard form

No. of subjects = 20 Number of obs = 20No. of failures = 10Time at risk = 20 LR chi2(0) = 0.00Log likelihood = -1.8270592 Prob > chi2 = .

------------------------------------------------------------------------------ _t | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- _cons | -.6931472 .3162278 -2.19 0.028 -1.312942 -.0733521------------------------------------------------------------------------------

Exp(-.693) = .5

The baseline hazard rate is .5…


001.

002.

003.

004.

005.

00



Likewise, integrated hazard is only half as big…

Example 4

• What about when events occur in clumps?• Example: two dense clusters of events

– Between times 1-2 and 4-5

Start End Failed?0 1 10 1.25 10 1.5 10 1.75 10 2 10 3 10 4 10 4.25 10 4.5 10 4.75 10 5 1


000.

250.

500.

751.

00



Here we see the two “clumps” of events…


.4.6

.8

1 2 3 4 5analysis time


Second “clump” has much higher hazard because the risk set is much smaller…

Default smoothing pretty much wipes out the first clump

Example 4: Hazard Plot, less smoothing

0.5

11.

52



Hazard with “width(.3)”

Now both clumps of events are clearly visible…


001.

002.

003.

00



Note how events with small risk set affect the cumulative hazard more (2nd clump)…


• The hazard rate reflects the rate of events per unit time at risk

• A constant hazard of .1 for one time-unit means that 10% of at-risk cases will have events

– But, things are often more complex than that when hazards are computed in continuous time

• The rate may vary within the interval depending on how the events are concentrated

• The risk set may change over the interval… esp. if cases leave the risk set.

Interpreting Integrated Hazards

• Integrated hazards represent the total amount of risk that has accumulated

• If the hazard is constant at .1, the integrated hazard would reach 10 after one hundred time-units…

eha: more on plots and interpreting hazards sociology 229a: event history analysis class 9 copyright...

Documents

hazard plotsraw hazard

hazard smoothingdont

hazard plotevents

narrow width

haz width3lower width

spiky smoothing

remarksstata default

risk setbut