In short, in Time to Event analysis, the analysis relates not just to
whether an event occurs but also when.
The planning (for example sample
size estimation) and analysis of the time to event study, several important
concepts and their relationship are important. These concepts and their
relationships are explained below: Time to Event is just the measure from the start of an intervention to the time when an event occurs. The start of an intervention could be the randomization, start of the treatment, date of surgery,…
Event Rate is the proportion of subjects or patients
in a group in whom an event is observed.
Event rate is usually measured for a period of the time from t to
t + Dt. For example, if the Dt = 12 months, the event rate
will be for one year. Event rate is also given as the event rate for the entire
study period.
Hazard Rate is the probability of
an event occurring given that it hasn’t occurred up to the current point in
time. Hazard rate is the instantaneous risk of a patient experiencing a
particular event at each specified time. The instantaneous rate with which an
event occurs at a single point in time. It is the probability that the event
occurs between time t and time t+delta given that it has not yet occurred by
time t, divided by delta, as delta becomes vanishingly
small. Note that rates, unlike probabilities, can exceed 1.0 because they are
quotients.
Hazard Ratio is a measure of effect
produced by a survival analysis. This represents the
increased risk with which one group is likely to
experience the outcome of interest. For example, if
the hazard ratio for death for a treatment is 0.5, then we can say that treated
patients are likely to die at half the rate of untreated patients.
Hazard ratio is calculated as the
ratio of hazard rates at a single time t, for two groups of subjects (treatment
versus control group). Hazard ratios are in the interval [0, infinity), and
they are frequently good ways to summarize the relative effects of two
treatments at a specific time t. Like odds ratios, hazard ratios can apply to
any level of outcome probability for the reference group. Note that a hazard
ratio is distinct from a risk ratio, the latter being the ratio of two simple
probabilities and not the ratio of two rates.
The Median Event Time is calculated as the smallest
even time for which the event function is less or equal to 0.5.
When the event is
death, the median event time is called the median survival time. The median
survival time is calculated as the smallest survival time for which the
survivor function is less than or equal to 0.5. In oncology study, median
survival time the time from either diagnosis or
treatment at which half of the patients with a given disease are found to be,
or expected to be, still alive. In a clinical trial, median survival time is
one way to measure the effectiveness of a treatment to see how well a new
treatment works. Median survival time may be called median overall survival or
simply median survival.
Censoring is a form of missing
data problem which is common in survival analysis and time to event analysis. In
clinical trials, we usually have to deal with the right censoring. In the situation
of the right censoring, the event did not occur when subjects are lost to
followup or when the study ends. A patient might
be known not to have had the event only up to a particular point in time, so ‘time
to event’ or ‘survival time’ is censored at this point.Lost to Followup refers to patients who at one point in time were actively participating in a clinical trial, but have become lost (either by error in a computer tracking system or by being unreachable) at the point of followup in the trial. These patients can become lost for many reasons. Without properly informing the investigator associated with the clinical trial, they may have opted to withdraw from the clinical trial, moved away from the particular study site during the clinical trial, or become ill and unable to communicate or are deceased.
Attrition: The loss of participants during the course of a study. Participants that are lost during the study are often call dropouts.
Accrual time or accrual period is recruitment period during
which subjects are being enrolled (recruited) into a study.
Followup time or followup period is the period after
the last subject entered the study until the end of the study. The followup
defines the phase of a study during which subjects are under observation and no
new subjects enter the study.
If T is the total duration of a study, and R is the
accrual period of the study, then followup period f is equal to T – R.
Event Rate = 1  Non Event Rate
Mortality Rate = 1  Survival Rate
Given the MET (median event time), we can calculate the hazard rate and the event rate, and hazard ratio.
If METc is the Median event
time for control group and METt is the Median event time for treatment group, HAZARDc
and HAZARDt are Hazard rates for control group and treatment groups, we will
have:
HAZARDc = log(2)/METc HAZARDt = log(2)/METt
METc = log(2)/HAZARDc
METt = log(2)/HAZARDt
Event rate at month 12 for
treatment group is
Et = 1  exp(12*HAZARDt);
HAZARD rate can be calculated from Event rate (for example event rate at month 12)
HAZARDc = ln(1Ec) / t (for example t=12)HAZARDt = ln(1Et) / t (for example t=12)
The hazard ratio is:
HAZARDt/HAZARDc
If the given parameter is event rate over the entire course of the study
(for example, 5 years), the event rate for one year can be calculated using the
following formula:
1  (1  event rate)^(1/t)
where t=5
where t=5
The formula above can also be used to convert the loss follow up rate from the entire treatment period to one year or vice versa.
References:

Clinical trial glossary 1
 Clinical trial glossary 2
 Janet Wittes (2002) Sample Size Calculations for Randomized Controlled Trials
 Lakatos (1988) Sample Sizes Based on the LogRank Statistic in Complex Clinical Trial
 Lachin and Foulkes (1986) Evaluation of Sample Size and Power for analyses of survival with allowance for nonuniform patient entry, losses to followup, noncompliance, and stratification
 Lakatos and Lan (1992) A comparison of sample size methods for the logrank statistic
 An introduction to survival analysis by Maarten Buis
 Survival models
1 comment:
Hi Dr. Deng:
Do you have references (papers or book) for the formula:
Hazard Rate = (ln(1Events/N)) / t
Also, if total events in experimental are Eexp and lost to followup is Nexplft then is there a way to estimate Hazard RATE in exp arm?
If not, then if evenets in control arm is Ectl and lost to follow up is Nctllft then is there a way to get Hazard RATIO?
You can email at peru2007@yahoo.com
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