Abstract
In many areas of science, interest centers on the time it takes for an event to occur. For example, product engineers monitor how long a device lasts before it fails; developmental psychologists study the emergence of language; and educators track how long it takes students to master some task. Historically, issues of this nature were investigated by researchers studying mortality, so the name “survival analysis” is used as an umbrella term to cover any sort of “time-to-event” analysis, even when the event has nothing to do with life or death. For example, the time it takes an athlete to reach maximum heart rate on a treadmill (see final section of Chap. 11) can be modeled using survival analysis.
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Notes
- 1.
To be precise, the form of censoring being discussed is termed “right censoring.” Left censoring occurs when the starting point (not the end point) is missing. In this chapter, we will consider only right censoring as it is the more common of the two types.
- 2.
The data are fabricated, but the percentage of teens who have tried drugs in our fictitious sample matches estimates of recreational drug use (including alcohol) among American teenagers.
- 3.
The ℛ code that accompanies this section includes a second estimator known as the Fleming-Harrington estimator. We will discuss this procedure later in this chapter.
- 4.
If no survival rate falls below this standard, the median is not reported.
- 5.
Bear in mind that the accuracy of the test requires a large sample, which is not characteristic of our example.
- 6.
If the predictor is a two-group categorical variable, the score test is asymptotically equivalent to the log rank test discussed earlier.
- 7.
The significance of the test can be assessed by referring the absolute value of the obtained value to a normal distribution or by referring the squared test statistic to a χ2distribution with 1 df.
- 8.
We will concern ourselves only with a single predictor, but when multiple predictors are present, there will be separate Schoenfeld residuals for each predictor.
- 9.
Because the test is used with multiple predictors, it will not be illustrated with our single predictor data set. For those wishing a more complete treatment, the test is implemented in ℛ using the function: cox.zph(mod), with the survival times from a Kaplan-Meier estimate used as the measure of time, and the scaled residuals found by dividing each Schoenfeld residual by its respective variance. Additional information appears in Therneau and Grambsch (2002, pp. 130–134.)
- 10.
The comparable value using Breslow ties is b = − .3678, p = .04, and the interpretation is similar to the one using Efron ties.
- 11.
In some cases, the Weibull distribution includes a location parameter. When it is not specified, as is true in this text, it is assumed to be 0.
- 12.
The ℛ code that accompanies this section adds a weighting scheme to update σ more gradually.
- 13.
The survreg function in ℛ uses a different parameterization than the one described here. The conversion is as follows: survreg’s scale σ = 1/γ; survreg’s intercept = log(λ).
- 14.
If you look back to Eq. (12.13), you will see that the Fleming-Harrington (FH) estimator can be obtained from the Nelson-Aalen (NA) estimator: FH = exp − (NA)
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Brown, J.D. (2018). Survival Analysis. In: Advanced Statistics for the Behavioral Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-93549-2_12
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