LITERATUUR WEEK 5
MACHINI & GARDNER
TIME TO EVENT STUDIES
It is common in follow-up studies to be concerned with the survival time between the time of entry to the
study and a subsequent event. The event may be death in a study of cancer, the disappearance of pain in a
study or the return of ovulation. These studies often generate some so-called ‘censored’ observations of
survival time. Such an observation would occur on any patient who is still alive at the time of analysis in a RCT
where death it the end point. The Kaplan-Meier product limit technique is the recognised approach for
calculating survival curves in such studies.
In some circumstances, the comparison between groups is adjusted for prognostic variables by means of Cox
regression. In this case the confidence interval describing the difference between the groups is adjusted for the
relevant prognostic variable.
SURVIVAL PROPORTIONS
SINGLE SAMPLE
Suppose that the survival time after entry to the study of a group of n subjects are t 1 , t2 , t3 … tn . The
proportion of subjects surviving beyond any follow-up time t, often referred to as S(t) but here denoted as p, is
estimated by the Kaplan-Meier technique.
Where ri is the
number of subjects alive just before time ti. di denotes the number who died at time ti , and indicates
multiplication over each time a death occurs up to and including time t. the standard error (SE) of p is given by:
Where neffective is the ‘effective’ sample size at time t. when there are no censored survival times, n effective will be
equal to n, the total number of subjects in the study group. When censored observations are present, the
effective sample size is calculated each time a death occurs :
Neffective = n- number of subjects censored before time t
The 100(1-α)% confidence interval for the population value of the survival proportion p at time t is then
calculated as:
Where z1-a/2 is the appropriate value from the standard Normal distribution for the 100 (1-α/2) percentile. Thus
for a 95% confidence interval , α = 0.05 and the table gives z1-a/2 = 1.96.
, MEDIAN SURVIVAL TIME
SINGLE SAMPLE
If there are no censored observations, then the median survival time, M, is estimated by the middle
observation of the ordered survival times if the number of observations n is odd, and by the average of t n/2 and
tn/2=1 if n is even. So:
In the presence of censored survival times the median survival is estimated by first calculating the Kaplan-Meier
survival curve, then finding the value of t that satisfies the equation: p=S(t)=0.5. this can be done by extending
a horizontal line from p=0.5 on the vertical axis of the Kaplan-Meier survival curve, until the actual curve is met,
then moving vertically down from that point to cut the horizontal time axis at t=M, which is the estimated
median survival time.
The expression for the standard error of the median includes SE(p) but evaluated at p=S9M) =0.5 when p=0.5 it
gives:
The standard error of the median is given by:
Where tsmall is the smallest observed survival time from the Kaplan-Meier curve for which p is less than or equal
to 0.45, while tlarge is the largest observed survival time from the Kaplain-Meier curve for which p is greater than
0.55. should psmall = plarge then the two values will need to be chosen wider than 0.45 and 0.55. The
100(1-α)% confidence interval for the population value of the median survival M is calculated as:
In the figure the M=30 months, the effective sample at
30 months is neffective = 14 so that
Psmall = 0.3852 < 0.45 gives tsmall = 30 months also, and
for plarge = 0.5870 > 0.55 gives tlarge = 20 months thus,
The 95% confidence interval therefore :
30 – (1.96 x 6.6230 ) to 30 + (1.96+6.6230 )
TWO SAMPLES
The difference between survival proportions at any time t in two study groups of sample sizes n 1 and n2 is
measured by p1-p2, where p1= S1(t) and p2=S2(t) are the survival proportions at time in groups 1 and 2
respectively.
MACHINI & GARDNER
TIME TO EVENT STUDIES
It is common in follow-up studies to be concerned with the survival time between the time of entry to the
study and a subsequent event. The event may be death in a study of cancer, the disappearance of pain in a
study or the return of ovulation. These studies often generate some so-called ‘censored’ observations of
survival time. Such an observation would occur on any patient who is still alive at the time of analysis in a RCT
where death it the end point. The Kaplan-Meier product limit technique is the recognised approach for
calculating survival curves in such studies.
In some circumstances, the comparison between groups is adjusted for prognostic variables by means of Cox
regression. In this case the confidence interval describing the difference between the groups is adjusted for the
relevant prognostic variable.
SURVIVAL PROPORTIONS
SINGLE SAMPLE
Suppose that the survival time after entry to the study of a group of n subjects are t 1 , t2 , t3 … tn . The
proportion of subjects surviving beyond any follow-up time t, often referred to as S(t) but here denoted as p, is
estimated by the Kaplan-Meier technique.
Where ri is the
number of subjects alive just before time ti. di denotes the number who died at time ti , and indicates
multiplication over each time a death occurs up to and including time t. the standard error (SE) of p is given by:
Where neffective is the ‘effective’ sample size at time t. when there are no censored survival times, n effective will be
equal to n, the total number of subjects in the study group. When censored observations are present, the
effective sample size is calculated each time a death occurs :
Neffective = n- number of subjects censored before time t
The 100(1-α)% confidence interval for the population value of the survival proportion p at time t is then
calculated as:
Where z1-a/2 is the appropriate value from the standard Normal distribution for the 100 (1-α/2) percentile. Thus
for a 95% confidence interval , α = 0.05 and the table gives z1-a/2 = 1.96.
, MEDIAN SURVIVAL TIME
SINGLE SAMPLE
If there are no censored observations, then the median survival time, M, is estimated by the middle
observation of the ordered survival times if the number of observations n is odd, and by the average of t n/2 and
tn/2=1 if n is even. So:
In the presence of censored survival times the median survival is estimated by first calculating the Kaplan-Meier
survival curve, then finding the value of t that satisfies the equation: p=S(t)=0.5. this can be done by extending
a horizontal line from p=0.5 on the vertical axis of the Kaplan-Meier survival curve, until the actual curve is met,
then moving vertically down from that point to cut the horizontal time axis at t=M, which is the estimated
median survival time.
The expression for the standard error of the median includes SE(p) but evaluated at p=S9M) =0.5 when p=0.5 it
gives:
The standard error of the median is given by:
Where tsmall is the smallest observed survival time from the Kaplan-Meier curve for which p is less than or equal
to 0.45, while tlarge is the largest observed survival time from the Kaplain-Meier curve for which p is greater than
0.55. should psmall = plarge then the two values will need to be chosen wider than 0.45 and 0.55. The
100(1-α)% confidence interval for the population value of the median survival M is calculated as:
In the figure the M=30 months, the effective sample at
30 months is neffective = 14 so that
Psmall = 0.3852 < 0.45 gives tsmall = 30 months also, and
for plarge = 0.5870 > 0.55 gives tlarge = 20 months thus,
The 95% confidence interval therefore :
30 – (1.96 x 6.6230 ) to 30 + (1.96+6.6230 )
TWO SAMPLES
The difference between survival proportions at any time t in two study groups of sample sizes n 1 and n2 is
measured by p1-p2, where p1= S1(t) and p2=S2(t) are the survival proportions at time in groups 1 and 2
respectively.
