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Introduction
Web site for slides
www.geos.ed.ac.uk/homes/graab
SLS synthetic survival data
SLS synthetic data
Derived from a real SLS extract
But contains no real people
See back page of notes for details
Start by looking at one table from it
Notes page 5
Deaths by age group
agegroup
Total20-29 30-39 40-49 50-59 60-69 70-75year of
death
1991 8 12 41 94 218 165 538
1992 15 42 72 221 470 367 1187
1993 14 54 109 256 620 426 1479
1994 28 64 120 304 727 490 1733
1995 25 53 145 317 854 528 1922
1996 36 47 155 341 803 523 1905
1997 41 75 172 407 888 522 2105
1998 33 67 166 427 904 510 2107
1999 45 79 214 413 847 513 2111
2000 45 85 169 407 866 447 2019
2001 42 93 202 450 899 441 2127
2002 52 115 215 474 837 459 2152
2003 39 101 246 515 821 383 2105
2004 49 102 226 515 896 384 2172
2005 37 117 255 494 831 362 2096
2006 52 105 252 521 860 311 2101
2007 52 112 242 440 744 280 1870
2008 55 119 270 544 780 258 2026
Total 668 1442 3271 7140 13865 7369 33755
Age at death by birth cohort
Birth Cohort 1910 1920 1930 1940 1950 1960 1970
Mean age at death
Answers
Birth Cohort 1910 1920 1930 1940 1950 1960 1970
Mean age at death 81 76 67 57 48 38 30
Number of deaths 3858 15051 8475 3822 1663 786 100
Youngest ages
age 1991 Census Birthday
From To Age end 2008
20 Apr-70 Mar-71 38 37
21 Apr-69 Mar-70 39 38
22 Apr-68 Mar-69 40 39
23 Apr-67 Mar-68 41 40
Only these three shaded groups born in 1970sFollowed up from age 20/21 to 37/38Hence mean age at death 30
Oldest ages
age 1991 Census Birthday
From To Age end 2008
71 Apr-19 Mar-20 89 88
72 Apr-18 Mar-19 90 89
73 Apr-17 Mar-18 91 90
74 Apr-16 Mar-17 92 91
Only these three shaded groups born in 1910sFollowed up from age 71-74 to 88-91Hence mean age at death 81
Extra information
Birth Cohort 1910s 1920s 1930s1940s1950s1960s1970s
Mean age at death 81 76 67 57 48 38 30
Number of deaths 3858 15051 8475 3822 1663 786 100
Most important for youngest group
Data are CENSORED, strictly RIGHT CENSORED
meaning that the follow up ended before the event
happened as in the two lower examples
1991 2008Moved to France 20011
Time origin at birth – oldest group
Age 0 Age 40Age 20Age 80
Age 60
The middle person died before we started the follow up, so we will not capture them in the study at all. This is called left truncation.
Summarising and plotting
survival data
Questions about practical 1?
Now continue
Survival data with no
censoring (fake)
Kaplan Meir calculations for SLS data
Time
months
At risk at
startDied Exited
Proportion
died
Proportion
died
Proportion
survived is
then P(t)
S(t) KM
.50 175059 25 0 25/175059 = 0.000143 0.999857191 0.999857
1.50 175034 44 19 63/17503 = 0.000251 0.99974862 0.999605
2.50 174971 53 11 64/174971 = 0.0003029 0.999697093 0.999303
3.50 174907 66 16 82/174907 = 0.0003773 0.999622657 0.998925
4.50 174825 0 11 0/174825 = 0 1 0.998925
5.50 174814 70 12 82/174814 = 0.0004004 0.999599574 0.998526
Table 3: Product limit calculations for the first few months of the synthetic SLS data
continued
Plot of KM for whole SLS sample
Can we get mean or median survival time?
Example of survival summaries from a study
of survival after surgery
No tissue
loss
AP>=50
Tissue loss
AP>=50
AP <50
Median
5-year
survival
Restricted
mean to 7
years
Answers - approximate
No tissue
loss
AP>=50
Tissue loss
AP>=50
AP <50
Median >8 yrs 5.2 yrs 3.6 yrs
5-year
survival
75% 42% 30%
Restricted
mean to 7
years
6.8 yrs 4.3 years 3.4 years
Afternoon session
Going over practical 2
Modelling survival
Practical 2 answers
Age group in
1991
20-29 30-39 40-49 50-59 60-69 70-74
Median survival
years - - - - 16.7 10.5
Restricted mean
years 17.6 17.4 17.0 15.8 13.3 10.6
5 year survival 99.8% 99.4% 98.5% 95.5% 88.2% 78.0%
19.4% died during follow upMedian survival >17.7 yearsTests by sexten9Chisquared tests all give values >500 1 degree of freedom 3.96 is 95% significance level
Survival by sexten9
Survival by Higher
education
By economic activity
Social class
With origin changed
Note Y axis
Confounding?
sexten9 Sex recorded at 91 Census Women survive longer
agegroup 10 year age groups Big effect
Hed Higher education qualification
at 1991 Census
Big effect
econpo9 Economic activity at 1991
Census
Retired and perm sick
poorest survival
sclas9 Social class at 1991 Census III non manual seem to
be best?
Out of work worst
urbgro9 6 fold urban rural classification Most rural best
Modelling survival
Models of survival could use mean survival
time – and some do
But censoring can make this tricky
Commonest method of modelling survival
is via the HAZARD function
Notes page 20
Hazard rate
Rate of failure for those still at risk
Higher hazard means shorter survival
The exponential distribution has a constant
hazard rate
But that doesn’t mean a linear decline in
survival
Because the number at risk keeps falling
Exponential survival
Survival and cumulative
hazard plots
Cumulative hazard plots
For exponential distributions?
As people age?
Following a major operation
Cox 1972 “Regression
models and life tables”
Time in
months
At risk at
startDied Exited
Proportion
died
0.50 175059 25 0 25/175059
1.50 175034 44 19 63/17503
2.50 174971 53 11 64/174971
The proportion at risk who die in the interval approximates the hazard.
The hazards can be compared across groups
Cox Proportional Hazard
models
h(group1) = h0(t) terms
h(group2)= K h0(t)
where K is the hazard ratio.
In terms of logs
log( h(group1)) = log(h0(t))
log( h(group2)) =log(K) + log(h0(t))
Cox PH model for age
group
Compared to age group 20-29
Hazard age 30-39 is 2.25 times
Hazard age 40-49 is 5.81 times
……..
Hazard age 70-74 is 84.0 times
Adjusting for age in PH model of effect
of education
------------------------------------------------------------------------------
_t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
Hed | 0.4231624 0.0088321 -41.20 0.000 .4062011 .440832
------------------------------------------------------------------------------
------------------------------------------------------------------------------
_t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
agegroup |
30 | 2.295549 .1080288 17.66 0.000 2.093288 2.517354
40 | 5.851488 .2496112 41.42 0.000 5.382153 6.361751
50 | 16.3997 .6668447 68.79 0.000 15.14343 17.76018
60 | 42.73684 1.702482 94.26 0.000 39.52697 46.20736
70 | 81.4729 3.314068 108.18 0.000 75.22963 88.2343
|
Hed | .6087054 .0127545 -23.69 0.000 .5842134 .6342241
------------------------------------------------------------------------------
Plots for checking PH
models
Adjusted analysis for social
class = Class I as baseline
Category Hazard
relative to
baseline
Hazard
adjusted for
agegroup
From delayed
entry model
(part 2)
Class II 0.99
Class III NM 0.88
Class III M 1.55
Class IV 1.49
Class V 1.82
Not worked
10 years
4.31
Adjusted analysis for social
class = Class I as baseline
Category Hazard
relative to
baseline
Hazard
adjusted for
agegroup
From delayed
entry model
(part 2)
Class II 0.99 0.94
Class III NM 0.88 0.96
Class III M 1.55 1.52
Class IV 1.49 1.48
Class V 1.82 1.40
Not worked
10 years
4.31 1.70
Adjusted analysis for social
class = Class I as baseline
Category Hazard
relative to
baseline
Hazard
adjusted for
agegroup
From delayed
entry model
(part 2)
Class II 0.99 0.94 0.94
Class III NM 0.88 0.96 0.96
Class III M 1.55 1.52 1.55
Class IV 1.49 1.48 1.47
Class V 1.82 1.40 1.42
Not worked
10 years
4.31 1.70 1.72
Extensions of the Cox
Model
Time dependent covariates
The effect of a factor changes over the
follow-up period
Delayed entry
Involves changing the origin of time
Data are left truncated so that their follow
up time does not start at zero
Time-dependent Cox modelThe effect of a factor can change over the follow-
up time, or can only come into effect over certain
times. Examples:
Comparison of types of surgery may vary over time,
e.g. a higher risk at operation followed by longer
term improvements
People who experience widowhood may have
increased risk of death compared to those
remaining married
Periods when an elderly person is visiting a day
centre may reduce their hazard of being taken into
hospital
Time dependent example
Start1991
2008
How does a young person’s hazard of a first diagnosis of a mental illness change while they are at university?
Time dependent example
Start1991
2008
A covariate for University by time gets switched on for certain time periods. The coefficient for this measures the difference in hazards between those at university and others not at university.
Time dependent example
Start1991
2008
Programs differ in how you define time-dependent covariates. Some require you to split up the follow-up records into two or more periods for each individual (R and Stata), while others (SPSS and SAS) require you to write some code to define the periods.
Delayed entry models
Age 0 Age 40Age 20Age 80
Age 60
For each person calculate their age when they come in to follow-up and their age at the terminal event or censoring. Then define your survival data with an entry time and an exit time. Sometimes called a counting process method.
Cohort effects from a counting
process model
Age 0 Age 40Age 20Age 80
Age 60
For each person calculate their age when they come in to follow-up and their age at the terminal event or censoring. Then define your survival data with an entry time and an exit time. Sometimes called a counting process method.
Delayed entry models
Age 0 30Age 20
Allow modelling of cohort effects by comparing survival of different cohorts at the same age.
Cohort 1970s
Cohort 1960s
Cohort 1950s
40 50 60
Cohort 1940s
Cohort 1930s
Cohort 1920s
70 80
Cohort 1910s
Cohort effect from delayed
entry model
Cohort Frequency Ln(hazard ratio) std. error Hazard ratio
1910 4792 1.04 0.12 2.84
1920 25405 0.81 0.12 2.24
1930 28279 0.56 0.12 1.75
1940 33051 0.24 0.12 1.27
1950 36309 -0.01 0.11 0.99
1960 40003 -0.15 0.11 0.86
1970 7220 0.00 1.00
Males survive even worse
in the 21st century
Variables in the Equation
B SE Sig. Exp(B)
sexten9 .473 .016 .000 1.605
agegroup 0.000agegroup(1) -4.480 .041 0.000 .011
agegroup(2) -3.673 .029 0.000 .025
agegroup(3) -2.724 .021 0.000 .066
agegroup(4) -1.667 .017 0.000 .189
agegroup(5) -.681 .014 0.000 .506
sex21st .123 .022 .000 1.131
