Survival Analysis: Applications to Ophthalmic Research
Article Outline
The purpose of this Editorial is to consider statistical analysis techniques when the outcome is time to an event. In another Editorial, Lemeshow and Hosmer described the logistic regression model and stressed that its estimated odds ratios are relevant only to the patient's status at the end of the study.1 A survival analysis, more appropriately called a time to event analysis in the current setting, considers not only if disease occurs, but also the length of time from the initial (baseline) observation of the subject to its occurrence or the patient being lost to follow-up. Before discussing regression modeling of time-to-event data, we present a few basic methods that are used for descriptive purposes in any time-to-event analysis. (A more detailed discussion of any of the concepts presented in this Editorial may be found in Hosmer and associates.2)
As in Lemeshow and Hosmer, consider a study where patients are at risk for the recurrence of an adverse ophthalmic outcome and are assigned randomly to 1 of 2 treatments, called A and B, and are then followed up until the disease recurs or the study ends, whichever comes first.1 In this Editorial, we are concerned with modeling the rate of occurrence over time, whereas Lemeshow and Hosmer considered modeling the presence or absence of disease recurrence at the end of the study.1
The most frequently used descriptive statistic of time to event is the Kaplan-Meier estimator of the survival function, possibly for each treatment. The survival function gives the proportion of subjects who have not yet experienced the event of interest at each time point in the period of follow-up. The estimator of the survival function for the 2 treatments in the example from the previous article is shown in the Figure.
FIGURE.
Kaplan-Meier estimator of the survival function for the 2 treatment groups.
In the Figure, the estimated curve for treatment A lies above that of treatment B from approximately 4 months to the end of the study at 20 months. This indicates that fewer subjects receiving treatment A have had a recurrence at each time than subjects receiving treatment B. The next question is whether the difference seen in the 2 estimated survival curves in the Figure is statistically significant. There are many related tests, but the one most frequently used is called the log-rank test. This test compares the observed number of recurrences in, say, treatment A at each observed event time to that expected if there were no difference in the 2 survival curves. In the current example, the value of the log-rank test is 25.13. Under the hypothesis that the 2 survival curves are the same and assuming that certain statistical assumptions hold, the distribution of the log-rank test is Chi-square with 1 degree of freedom and in this example, P < .001.
The most commonly used estimator of the effect of study covariates is the hazard ratio (HR) as typically obtained from fitting the proportional hazards regression model. Cox proposed this regression model, and its application is now widespread.3 The estimator of the HR for treatment obtained from fitting this model (results not shown) is 0.28 with a 95% confidence interval [CI] (0.16 to 0.48). Hence, we see that treatment A is associated with a 72% reduction in the rate of event occurrence throughout the 20 months of follow-up.
Lemeshow and Hosmer, when fitting a logistic regression model to recurrence or not by the end of the study, believed that the subject's age would have an effect.1 The results of fitting a proportional hazards model that includes both age and treatment are shown in the Table. The coefficient for treatment is highly significant, and age is significant at the 10% level but not the 5% level. The estimated HR for treatment obtained from the fit in the Table is nearly identical to that from the model not containing age, namely 0.28 with a 95% CI (0.16 to 0.50). The estimator of the HR is obtained by exponentiating the coefficient for treatment (ie, e−1.251 = 0.28). Age, although marginally significant, seems to offer no important adjustment to the effect of treatment on recurrence. In a setting with a richer model with more complicated associations between the predictors, there can be important differences in estimates of treatment effect between the simple and multivariable model.
TABLE. Fitted Proportional Hazards Containing Treatment and Age
| Coefficient | Standard Error | z | P value | 95% Confidence Interval | |
|---|---|---|---|---|---|
| Treatment | −1.251 | 0.2826 | 4.42 | <.001 | −1.804 to−0.697 |
| Age | 0.018 | 0.0010 | 1.91 | .056 | −0.001to0.037 |
In settings such as the current example, where the goal is to estimate the effect of treatment adjusting for other covariates, it often is useful to provide a plot of the model-based covariate-adjusted survival function for the 2 treatment groups. These covariate-adjusted survival curves may be used to estimate quantiles of time to response as well as to estimate the proportion of patients with recurrent disease at set values of follow-up.
In this Editorial, we have demonstrated that the Cox proportional hazards regression model is appropriate for modeling the rate of disease recurrence throughout the period of follow-up. The estimator of effect, the HR, is based on exponentiating the model-based estimated coefficients. The interpretations apply at each point in time throughout the follow-up and depend on the assumptions of the model.
The authors indicate no financial support or financial conflict of interest. Both authors were involved in design and conduct of study; data collection; analysis and interpretation of data; and preparation and review of the manuscript.
References
- . Logistic regression analysis: applications to ophthalmic research. Am J Ophthalmol. 2009;147:766–767
- . Applied survival analysis: regression modeling of time to event data. 2nd ed.. New York, New York: John Wiley & Sons; 2008;
- . Regression models and life tables (with discussion). J R Stat Soc Ser B Stat Methodol. 1972;34:187–220
PII: S0002-9394(08)00608-9
doi:10.1016/j.ajo.2008.07.040
© 2009 Elsevier Inc. All rights reserved.
