Optimising assessment of dark adaptation data using time to event analysis

Abstract

In age-related macular degeneration (AMD) research, dark adaptation has been found to be a promising functional measurement. In more severe cases of AMD, dark adaptation cannot always be recorded within a maximum allowed time for the test (~ 20-30 min). These data are recorded either as censored data-points (data capped at the maximum test time) or as an estimated recovery time based on the trend observed from the data recorded within the maximum recording time. Therefore, dark adaptation data can have unusual attributes that may not be handled by standard statistical techniques. Here we show time-to-event analysis is a more powerful method for analysis of rod-intercept time data in measuring dark adaptation. For example, at 80% power (at α = 0.05) sample sizes were estimated to be 20 and 61 with uncapped (uncensored) and capped (censored) data using a standard t-test; these values improved to 12 and 38 when using the proposed time-to-event analysis. Our method can accommodate both skewed data and censored data points and offers the advantage of significantly reducing sample sizes when planning studies where this functional test is an outcome measure. The latter is important because designing trials and studies more efficiently equates to newer treatments likely being examined more efficiently.

Figure 1

Empirical curves for the original (top) and the transformed (bottom) data, scaled to illustrate RITs that surpass a cut off time. The vertical dashed line acts as a marker, representing this capping limit of 20-min. Note that the actual underlying distribution of RIT values is depicted in the figure, so no censoring is shown here. However, the survival model is fitted considering all values beyond 20 min as censored. For the scaled data, both the fitted survival curves from the mode (dashed curves) and the median values (vertical solid lines) were calculated from capped data. However, the time-to-event model fits the data well even beyond the cut off time. For the scaled AMD data, the time-to-event model correctly predicts a median value beyond the capping limit. Figure generated using the ggplot2 package.

Figure 2

Power curves as a function of sample size for the three methods considered with and without capping for the scaled data. The power curves for the original data (not scaled) are identical to the scaled uncapped data and are therefore not reported (10,000 bootstrap simulations for each step in the sample size). Figure generated using the ggplot2 package.

Figure 3

The change in Log₁₀ sensitivity for three control subjects and three patients with AMD is plotted in the top panel (filled dots) with the corresponding Rod Intercept Time values (vertical strokes) provided by the device. The horizontal dashed line represents the 3 log-step change in sensitivity used by the device to define the event (recovery from bleaching). The RIT time for each recovery “event” is used to build the survival curves (bottom panels). In this case, the vertical coloured lines also identify the same RIT values recorded for the curves plotted in the top panels. Notice how each vertical line corresponds to a downward step change in the survival curve (in black). The same process is applied for all RIT values in the dataset to calculate the other step changes that make up the rest of the survival curve. A marker demonstrating the cut off time has been added. Figure generated using the ggplot2 package.