Question

 

Part 1. What is the primary difference between the life-table and Kaplan-Meier approaches to constructing
survival curves?

Part 2. Provide an example of comparing survival curves using log-rank tests.

Part 3. Contrast the odds ratio with the hazard ratio.

Part 4. Is failing to reject the null the same as accepting the null? Explain.

Answer 1

Part(1)

The main difference is the time intervals, i.e., with the actuarial life table approach we consider equally spaced intervals, while with the Kaplan-Meier approach, we use observed event times and censoring times. The calculations of the survival probabilities are detailed in the first few rows of the table.

Part(2)

This procedure computes the sample size and power of the logrank test for equality of survival distributions under very general assumptions. Accrual time, follow-up time, loss during follow up, noncompliance, and timedependent hazard rates are parameters that can be set. A clinical trial is often employed to test the equality of survival distributions for two treatment groups. For example, a researcher might wish to determine if Beta-Blocker A enhances the survival of newly diagnosed myocardial infarction patients over that of the standard Beta-Blocker B. The question being considered is whether the pattern of survival is different. The two-sample t-test is not appropriate for two reasons. First, the data consist of the length of survival (time to failure), which is often highly skewed, so the usual normality assumption cannot be validated. Second, since the purpose of the treatment is to increase survival time, it is likely (and desirable) that some of the individuals in the study will survive longer than the planned duration of the study. The survival times of these individuals are then said to be censored. These times provide valuable information, but they are not the actual survival times. Hence, special methods have to be employed which use both regular and censored survival times. The logrank test is one of the most popular tests for comparing two survival distributions. It is easy to apply and is usually more powerful than an analysis based simply on proportions. It compares survival across the whole spectrum of time, not just at one or two points. This module allows the sample size and power of the logrank test to be analyzed under very general conditions.

Part(3)

Hazard ratios are often treated as a ratio of death probabilities. For example, a hazard ratio of 2 is thought to mean that a group has twice the chance of dying than a comparison group. In the Cox-model, this can be shown to translate to the following relationship between group survival functions: (where r is the hazard ratio). Therefore, with a hazard ratio of 2, if S0(t) = 0 (20% survived at time t), (4% survived at t). The corresponding death probabilities are 0.8 and 0.96. It should be clear that the hazard ratio is a relative measure of effect and tells us nothing about absolute risk.

While hazard ratios allow for hypothesis testing, they should be considered alongside other measures for interpretation of the treatment effect, e.g. the ratio of median times (median ratio) at which treatment and control group participants are at some endpoint. If the analogy of a race is applied, the hazard ratio is equivalent to the odds that an individual in the group with the higher hazard reaches the end of the race first. The probability of being first can be derived from the odds, which is the probability of being first divided by the probability of not being first:

HR = P/(1 − P); P = HR/(1 + HR).

Odd Ratio

An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due to symmetry), the ratio of the odds of B in the presence of A and the odds of B in the absence of A. Two events are independent if and only if the OR equals 1, i.e., the odds of one event are the same in either the presence or absence of the other event. If the OR is greater than 1, then A and B are associated (correlated) in the sense that, compared to the absence of B, the presence of B raises the odds of A, and symmetrically the presence of A raises the odds of B. Conversely, if the OR is less than 1, then A and B are negatively correlated, and the presence of one event reduces the odds of the other event.