Analysis of time-to-event data, otherwise known as survival analysis, is a common investigative tool in ophthalmic research. For example, time-to-event data is useful when researchers are interested in investigating how long it takes for an ocular condition to worsen or whether treatment can delay the development of a potentially vision-threatening complication. Its implementation requires a different set of statistical tools compared to those required for analyses of other continuous and categorial outcomes. In this installment of the Focus on Data series, we present an overview of selected concepts relating to analysis of time-to-event data in eye research. We introduce censoring, model selection, consideration of model assumptions, and best practice for reporting. We also consider challenges that commonly arise when analyzing time-to-event data in ophthalmic research, including collection of data from two eyes per person and the presence of multiple outcomes of interest. The concepts are illustrated using data from the Laser Intervention in Early Stages of Age-Related Macular Degeneration study and statistical computing code for Stata is provided to demonstrate the application of the statistical methods to illustrative data.

^{1}In this installment of the Focus on Data series, we present an overview of the concepts and best practice for reporting and interpreting the results of survival analyses in eye research. We start with a description of our illustrative example, taken from the Laser Intervention in Early Stages of Age-Related Macular Degeneration (LEAD) study. Then we discuss censoring, survival probability, and the hazard function. An overview of model selection will be presented with a focus on the assumptions required for valid interpretation, before briefly introducing competing risks and dealing with data from two eyes per person. These concepts will then be applied to our illustrative data. A simulated dataset, along with statistical computing code for Stata, has been provided in the Supplementary Material.

^{2}Participants were randomized to either laser or sham treatment, which was applied to only one eye per person every six months for up to 30 months. The outcome of interest was the time to develop late AMD, either of the atrophic or neovascular type, and both eyes were assessed for progression every six months up to 36 months. Below, as an illustrative example, we explore the association between pigmentary abnormalities of the retinal pigment epithelium (hypopigmentation or hyperpigmentation) detected on color fundus photography at baseline, and the time from baseline to the date that late AMD was first detected. This association was explored among participants in the sham treatment group only to avoid effect modification by the laser intervention.

*time-to-event*data.

*Time-at-risk*usually begins at randomization in clinical trials and at the time of exposure in observational studies. In observational studies, time of exposure may be defined as the date of diagnosis or treatment, or at the baseline visit in a cohort study. Time-at-risk ends when the outcome is reached or at the time of

*censoring*.

*right censoring*occurred for participants who did not progress to late AMD before withdrawal, loss to follow-up, or the end of the study (participants B-F in Fig. 1). This is referred to as

*right-censored*data because only the lower bound of the time-at-risk is known. Censored participants are still included in the analyses and are counted among the number of people at risk for the outcome until the last date that their status is known. This is in contrast to analyses of binary outcome data observed at a fixed timepoint (e.g., the proportion of participants with late AMD at three years). In analyses that do not use time-to-event data, outcomes from participants who have not experienced the event of interest before being lost to follow-up are treated as missing, introducing a potential source of bias.

*interval censoring.*When late AMD was detected, the exact date of onset was usually not known, other that it occurred between the previous and current study visits. In the presence of interval-censored data, the date at which the outcome is first detected or a date midway between the two study visits may be treated as the event date (known as right-point and mid-point imputation, respectively). As with any imputation method, there is an associated risk of bias. Statistical techniques are available to model this uncertainty when estimating the survival function but are seldom used.

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*noninformative*censoring) is required, or statistical techniques should be used to adjust for this association.

^{4}Noninformative censoring may be a valid assumption among people who withdraw from a study after moving away from the study area. However, if people were unable to attend study visits due to poor vision secondary to the outcome of interest, the assumption of non-informative censoring would not be valid. It can be difficult to assess whether censoring caused by loss to follow-up is related to the outcome because outcome status is usually unknown among participants with right censoring. It is important to consider this potential source of bias and whether drop out could be related to the intervention or exposure of interest.

^{5}

*nonparametric*approach to estimating the survival function. Nonparametric methods do not require any assumptions to be made about the changes in the rate of the outcome over time. The Kaplan-Meier estimate is derived from the number of events (such as progression to late AMD), the number of people at risk (i.e., those who had not been censored or reached the endpoint before that time), and the survival probability immediately before that time (see Supplementary Material for formulae). At baseline, time-at-risk equals zero for all participants and the survival function equals one (because all participants have not yet experienced the outcome). The survival function decreases as time progresses and more participants experience the outcome of interest. The failure function is equal to one minus the survival function and is interpreted as the cumulative proportion of participants who

*have*experienced the outcome.

^{6}The log-rank test is commonly used because it is easily implemented. However, this test does not provide an estimate of the magnitude of difference between groups such as a HR or incidence rate ratio.

*semiparametric*approach to assessing time-to-event data. The term semiparametric refers to the fact that researchers do not need to make any assumptions about the shape of the underlying hazard function. Estimates of HRs from Cox regression models are valid under the

*proportional hazards*assumption, as discussed below. Unadjusted Cox regression models provide similar statistical power to detect an intervention effect to that of the log-rank test. However, additional power is obtained when strong predictors of the rate of events are included as covariates in a Cox model.

^{7}Power is diminished for Cox regression and the log-rank test if hazards are nonproportional.

^{8}It should be noted that

*joint models*of longitudinal and survival data may provide more efficient and less biased estimates of the effect of an intervention on time to event in the presence of correlation between the time-dependent variable and the outcome of interest.

^{9}

^{,}

^{10}Researchers are urged to seek statistical advice when considering this approach.

^{11}

^{,}

^{12}This is particularly useful in long-term cohort studies with a wide range of ages at baseline.

*fully-parametric*approach to assess time-to-event data. These models have not been as widely used as log-rank or Cox models in the past because assumptions are required to be made about the shape of the survival function (i.e., whether the survival curve approximates a known distribution such as Weibull or gamma, as seen in Fig. 2). These assumptions can be tested, as demonstrated in the illustrative example. AFT modes can be used to estimate the ratio of time to event between exposure groups. A negative time coefficient indicates decreased time to event (i.e., a higher incidence rate) on average among the intervention group compared to the reference group, whereas a positive coefficient indicates the time to event will be greater (i.e., a lower incidence rate) among intervention group compared to the reference group. Weibull and exponential models are examples of AFT models, which also can be used to estimate HRs. A more flexible model, known as the Royston−Parmar model, has become more common in recent years and is particularly suited to predicting time to disease progression.

^{13}AFT models can facilitate estimation of survival times and incidence beyond the range of the observed study period (whereas Cox models cannot).

^{14}Small degrees of nonproportionality will have minimal impact interpretation of the estimates. However, strategies such as stratification or inclusion of time-varying coefficients should be considered in the case of obvious violations of the assumption, for example, in the case of a delayed treatment effect.

*competing risk*for atrophic AMD. An inferior approach to investigating the effect of an exposure on the time to atrophic AMD is to censor participants at the time of neovascular AMD detection. This censoring is

*informative*, that is, the participants who are censored because of the detection of neovascular AMD are likely to have poorer ophthalmic health than those who do not have neovascular AMD, and this is a potential source of bias. Therefore it is recommended that competing risk regression be used, although interpretation of HRs from these models may not be intuitive.

^{15}

^{,}

^{16}In Fine and Gray's subdistribution hazard model,

^{15}participants who experience the competing event are still counted among those

*at-risk*for the event of interest, even though these participants can no longer be observed to experience the event of interest.

^{16}Therefore the HR from this model is interpreted as the relative difference in the effect on the cumulative incidence function (or the event rate for the outcome of interest) between exposure categories among participants who are either event free or have experienced the competing event.

^{16}

*multistate model*when each state is distinctly defined.

^{17}Researchers are urged to seek statistical advice when considering this approach.

^{18}On the other hand, eye-specific data on interventions and outcomes may be available in clinical trials. Collecting data from two eyes per person may be less resource intensive than collecting data on one eye per person with double the number of participants. However, given the likely similarity between the two eyes of one person, data from these two eyes (which share the same environment and genes) may not contribute as much statistical information as would two eyes from two separate people. The methods so far presented provide valid estimates and confidence intervals under the assumption that the outcome from each eye is independent. In the presence of correlated outcomes (i.e., outcomes from two eyes of the same person), these methods are likely to provide confidence intervals that are narrower than they should be. Methods for analyzing data with clustered observations (e.g., eyes within people or patients within hospitals) include the use of shared frailty models and the use of robust (sandwich) standard errors to account for this clustering.

^{19}The correlation between clustered individuals is modeled when shared frailty models are applied. When clustering is accounted for using robust standard errors, the estimates will be the same as those from the equivalent model fitted with regular standard errors, but the confidence intervals and

*P*values will change.

*Time-at-risk*is rarely distributed symmetrically around the mean, so the median (i.e., the time required for the outcome to be observed for half the participants) is often used as a summary measure. Incidence rates and HRs should be reported with 95% confidence intervals (CI) to allow readers to assess the precision of the estimates. When graphing survival or failure plots, it is good practice to include a risk table that gives the number of people or eyes at risk at selected timepoints below the plot and to plot confidence intervals around each curve (as seen in Fig. 4). Compliance with reporting guidelines such as STROBE (observational studies), CONSORT (trials), and ARRIVE (animal research) is recommended to facilitate transparent and reproducible research, regardless of the statistical approach used.

^{20}

^{–}

^{22}

*P*= 0.978 for pigmentary abnormality status, p-value between 0.301 and 0.767 for covariates).

*e*

^{−1.35}) was 0.26 (95% CI 0.13, 0.53) which is interpreted as a 74% decrease in the time to late AMD among those with pigmentary abnormalities compared to those without. The shape parameter was greater than one, indicating that the hazard of progressing to late AMD increased with time from baseline. Estimates from this model suggest it would take 13.7 years for 50% of the participants without pigmentary abnormalities to progress to late AMD (median survival time, 95% CI 3.1, 24.3 years), whereas it would only take 3.9 years for 50% of the participants with pigmentary abnormalities to progress (95% CI 2.5, 5.3).

**M.B. McGuinness**, None;

**J. Kasza**, None;

**Z. Wu**, None;

**R.H. Guymer**, Advisory board Roche, Genentech, Bayer, Novartis, Apeliis

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