Bias Associated with the Use of Maximum, Minimum, or Range of a Number of Intraocular Pressure Measurements

We congratulate Drs. Koozekanani, Covert, and Weinberg ¹ for their excellent article on the systematic bias associated with the use of the best of several visual acuity measurements as an outcome variable in clinical studies. 

In general terms, the authors showed that the expected value for the maximum of a series of measurements increases with the number of measurements, using mathematical derivation from the cumulative distribution function of the normal distribution and Monte Carlo simulation. 

We would like to point out that this insight also has implications in the field of glaucoma. Intraocular pressure (IOP) cannot be measured continuously, and so inferences have to be made from a few “random” samples. 

In clinical studies, the maximum of several IOP measurements is frequently reported. A bias might be introduced when the maximum of many preoperative IOP readings is compared to few postoperative values. 

The situation is even more complicated when fluctuation of IOP is reported as the range of several measurements—that is, the maximum minus the minimum value. It can be shown that the IOP range measured is highly dependent on the number of determinations. The theoretical considerations are quite similar to those mentioned by Koozekanani et al., ¹ and so they do not have to be repeated here. 

We performed a Monte Carlo simulation using R (R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria). Figure 1 shows a single simulation of 50 IOP measurements. It demonstrates that the range of IOP fluctuation is also largely dependent on chance when there is a distribution with a low probability of extreme values (like the normal distribution). Figure 2 illustrates the association of the expected range of IOP and the number of measurements for a normally distributed IOP with a given standard deviation. 

Figure 1.

 

A set of 50 normally distributed random numbers with a mean of 16 and a standard deviation of 2.3 was generated by the computer. The diagram shows the individual numbers, as well as the mean, the range, and the fourfold standard deviation for the first n readings. The range remains constant until the nth is higher than any number before. Then, it changes dramatically (arrows). The fourfold standard deviation shows much more stable behavior.

Figure 2.

 

For standard deviations of 1.2, 2.3, and 3.5, sets of random numbers (simulated IOP measurements) were generated as in Figure 1 (100,000 sets for each standard deviation). The mean of the range and the standard deviation of the random numbers after n measurements over all 100,000 sets are diagrammed. While the standard deviation remains relatively constant after the first couple of measurements, the mean range still increases beyond 50 simulated IOP measurements.

Two assumptions were made for the simulation: (1) normal distribution and (2) independence of measurements. To our knowledge, there are few data concerning the distribution of IOP measurements within an individual under different circumstances (while there are ample data on the distribution of IOP within populations). However, departures from the first assumption do not seem to have great consequence for our model. 

On the other hand, the second condition is quite important. Whether the IOP measurements under consideration are independent depends on the type of fluctuation studied and the study design. 

Fluctuation of IOP is caused by rhythmic changes, sporadic influences, and measurement inaccuracies. Short-term fluctuation within 24 hours is generally distinguished from long-term fluctuation, ² because it is supposed to represent different aspects: while short-term fluctuation is presumably dominated by biological circadian and environmental rhythms, long-term fluctuation may be caused by a variety of sporadic or rhythmic factors. 

Circadian fluctuation of IOP follows a pattern that can be approximated by a cosine curve. ³ Thus, measurements are not independent. If sporadic influences and measurement inaccuracies are small compared with the rhythmic fluctuation, the range will approximate the twofold amplitude regardless of the number of measurements. 

However, when intervals are larger and not synchronized to a biological or environmental rhythm, individual measurements are more likely to be independent from each other—for example, several IOP determinations in the morning over several visits. Consequently, authors analyzing IOP data from AGIS (Advanced Glaucoma Intervention Study) and EMGT (Early Manifest Glaucoma Trial) have reported the empiric standard deviation of IOP measurements instead of range. ^4,5  

Unfortunately, in contrast to maximum IOP or IOP range, the empiric standard deviation of IOP is less convenient for the clinician because of the lack of an intuitive approach. Assuming a standard distribution, this may be overcome by reporting the fourfold standard deviation, which comprises 96% of measurements and is about the same magnitude as the range in many situations (see Fig. 2). 

In conclusion, reporting the empiric standard deviation (or the fourfold empiric standard deviation) instead of the range of IOP measurements may be warranted, especially when long-term fluctuation of IOP is analyzed. Similar biases have to be taken into account in many situations and therefore the importance of the article by Koozekanani et al. extends beyond its original scope.