**Purpose**:
To compare parametric models for fitting published distributions of visual field progression rates (in dB/yr) for glaucoma.

**Method**:
We fitted a modified Gaussian model, a modified Cauchy model and a modified hyperbolic secant model to previously published distributions of visual field progression rates from Canada, Sweden, and the United States. The modification allowed the shape of the model's distribution either side of the mode to be independently varied to allow for the asymmetric tails seen in visual field progression rate distributions.

**Results**:
Summing likelihoods across datasets, the modified hyperbolic secant was strongly favored (by 26.7 log units) compared with the next best-fitting model, the modified Cauchy. The modified hyperbolic secant was not the best fit for the Canadian dataset, however. Best-fitting modified hyperbolic secant parameters were broadly similarly between datasets, with parameter variances being less than those expected to negate the benefits of a previously described Bayesian method for improving individual visual field progression rate estimates in glaucoma.

**Conclusions**:
Although the optimum model differed depending upon the particular dataset, a modified hyperbolic secant performed well for all distributions investigated and was strongly favored when evidence was summed across datasets.

**Translational Relevance**:
Despite differences in the progression rate distributions between studies, the use of an “average” distribution may still be of benefit for improving individual visual field progression rate estimates in glaucoma using Bayesian methods.

^{1}or by averaging the individual rates of loss at each point in the visual field.

^{2}It is a common finding that the distribution of glaucoma progression rates is skewed, with a longer tail for negative rates of progression (i.e., worsening visual fields over time) than for positive rates.

^{2–5}The ability to quantify such distributions allows rates of progression to be compared between different glaucoma types

^{3}or between different groups with the same glaucoma type (e.g., study populations versus general clinical populations).

^{5}More specifically, fully quantifying distributions allows for differences other than those in central tendency (e.g., mean or median rate) or spread (e.g., the interquartile range) to be explored. This may be particularly important when trying to examine differences within the tails of distributions: for example, the number of very fast progressors in a group.

^{5}Quantifying progression distributions is also needed for Bayesian methods that use population data to constrain estimates of rates in individuals

^{6–9}and so help to reduce the wide variation in rates seen when the amount of longitudinal data available is limited.

^{10}

^{6}which allows for the slopes of the upper and lower tails to be specified, along with the mode of the distribution. This model was selected because it fitted the general shape of the distribution well enough for the simulation studies then performed, and so no attempt was made to see whether other models might provide substantially better fits.

^{6}There has been an increased use of nonparametric methods to quantify aspects of the visual field,

^{11,12}particularly given the ubiquitous access to fast computing required to generate these analyses. Therefore, it may be questioned why parametric models are required at all, at least for statistical quantification purposes. We believe that parametric models are useful for several reasons. Firstly, simple parametric models for progression rate distributions efficiently summarizes the entire shape of the distribution in a way that simple metrics such as the mode and interquartile range cannot. Secondly, the frequency of rapid rates of progression is relatively small and so it is not uncommon for histograms of empirical progression data to have some bins with a zero frequency in one range, yet have nonzero frequencies at lower and higher ranges. This can occur despite large cohort sizes (e.g., 583 patients in Heijl et al.

^{13}). Presuming that the distribution of glaucoma is a continuous function, these zero frequencies reflect sampling variability rather than that glaucoma never produces progression rates within the missing range. Conventional nonparametric bootstrapping procedures, where new datasets are generated by sampling with replacement from the original data, will also always have zero frequencies at the same ranges, and so do not solve the problem. It is possible to smooth the data using a spline curve or moving average

^{5}and so fill in these gaps, although the degree of smoothing selected is commonly ad hoc and such techniques necessarily widen the frequency distribution and flatten its peak. Modelling the distribution parametrically avoids intermediate zero frequency bins. There is also no inherent reason for such models to bias a distribution in a particular direction (e.g., always widen or narrow) or to systematically reduce the peak of the distribution, although a specific model may indeed create a bias if it is always found to be either too compact or too broad relative to the empirical data. Bootstrap data may still be generated from fitted distributions in order to determine probability limits, a procedure known as a parametric bootstrap. Finally, empirical data is often presented in histograms with markedly differing binning strategies (e.g., regular 0.1-dB/yr wide bins

^{5}vs. 0.5-dB/yr bins with open ended-bins for distribution tails

^{2}) making a comparison of distributions between different studies challenging. Parametric models of these data can be presented using a common binning strategy and so avoid distortions introduced by different histogram binning strategies.

*t*distribution, the former having many degrees of freedom and the later having one degree of freedom,

^{14}thereby representing a compact and a wide distribution, respectively. A Cauchy distribution is sometimes used in robust regression methods to account for outliers in the data not well modelled by a Gaussian distribution.

^{15}

^{16}and as previously used to model glaucoma progression rate distributions.

^{6}Its parameters were defined as per the modified Gaussian distribution, with

*mode*being equivalent to parameter

*t*in the original equation by King-Smith et al.

^{16}

*A*was set to give an area of one under each function, as calculated by summing the bin heights at 0.1-dB/yr wide intervals between −10 and +10 dB/yr. All three distributions therefore had only three free parameters (i.e.,

*mode*,

*B*,

*C*).

^{5}587 eyes with treated glaucoma from the New York Glaucoma Progression Study (≥8 SITA-Standard 24-2 Humphrey Field Analyzer II fields, tested each 6 to 12 months; data from progressing and non-progressing patients combined),

^{2}and 583 patients with primary open-angle glaucoma or pseudoexfoliation glaucoma in Sweden (≥5 Humphrey SITA-Standard fields over ≥5 years).

^{13}The Canadian and Swedish studies determined progression rates from linear regression of the summary index MD, whereas the US study determined the overall rate of visual field loss from the average of pointwise linear regression slopes. The variability in each dataset was estimated using a bootstrap procedure, with 10,000 new random datasets produced by sampling (with replacement) from the original distribution and 2.5% and 97.5% limits for histogram bin frequencies then calculated.

^{14}which are in turn equivalent to the version of the AIC corrected for finite samples (AIC

_{C}) given that both the number of data points and the number of free parameters are identical for the models being compared. We also quantified absolute goodness of fit by calculating the coefficient of determination

*R*(i.e., the fraction of the total variance in the

^{2}*y*-direction that is explained by our nonlinear models, similar to the

*r*

^{2}value used in linear regression) for the maximum likelihood fit.

^{14}

^{17}

_{10}likelihoods for the modified Gaussian, modified Cauchy and modified hyperbolic secant models were −3036.8, −2891.1, −2893.9, respectively, indicating that the modified Cauchy model fitted best and was 583 times (−2891.1 minus −2893.9 = 2.8 log units) more likely than the next best fitting model, the modified hyperbolic secant. While the shape of the modified Cauchy distributions well captures the extended tails of the empirical data, the modified Gaussian distribution must substantially increase its spread in order to encompass these tails, resulting in a comparatively poor representation of the more central portion of the distribution. The modified hyperbolic secant improves the representation of the central portion of the fit while retaining broad tails, although still underestimates the peak of the data.

^{6}times, compared with the modified Cauchy distribution). For the US data (Fig. 3), log-likelihoods were −412.3, −412.1, and −402.8, indicating the modified hyperbolic secant gave the best fit (by 2.8 × 10

^{9}times). Summing likelihoods across datasets, the hyperbolic secant was strongly favored (by 26.7 log units) compared with the next best-fitting model, the modified Cauchy. For all three datasets, the modified Gaussian distribution fit had the lowest likelihood and the lowest goodness of fit

*R*values.

^{2}*R*

^{2}never less than 0.96 for the three datasets analyzed. The best-fitting model differed depending upon the dataset fitted, but was either the modified Cauchy and modified hyperbolic secant. In each situation, these two models returned fits with

*R*

^{2}of at least 0.94. In contrast, the modified Gaussian provided a comparatively poor fit to the Canadian data (Fig. 1) and was never the best-fitting model. When evidence was summed across datasets, the modified hyperbolic secant model was strongly favored.

^{2,5}can make the comparison of empirical distributions difficult. By applying a common binning strategy to our parametric models, the differences between datasets can be readily appreciated however (Fig. 4). The Canadian dataset had a comparatively narrow distribution and a high modal progression rate. The high mode may reflect that this distribution includes glaucoma suspects

^{5}who by definition would show little or no progression and so should shift the distribution's mode to the right. In addition, a robust regression technique was used to determine progression rates,

^{5}which would be expected to reduce the number of very extreme rates arising from statistical variability alone and so narrow the distribution of rates for the population. In comparison, the Swedish dataset contained over one-third of patients with pseudoexfoliation glaucoma.

^{13}This likely contributes to the long tail of rapid progression rates in this distribution, given that extremely rapid progression can occur in pseudoexfoliation glaucoma.

^{3}Analyzing glaucomatous progression rates using linear regression of MD may also be influenced by other factors such as cataract, learning, or physiological aging at a rate other than that predicted by average values

^{18}: while such nonglaucomatous factors might be well controlled in clinical trials, their influence may be expected to be more marked in data collected from general clinical populations.

^{6}as such methods cannot return a progression rate within this bin as the a priori probability of such an event is zero (hence, the posterior probability given via Bayes theorem must also be zero). This problem is avoided when a parametric model of the empirical data, rather than empirical data itself, is used to estimate glaucoma progression rates in the population.

^{8}and so is therefore very well matched to the particular cohort of patients analyzed in terms of such factors as glaucoma type, ethnicity, and testing procedure. While producing very encouraging results, empirical Bayes probably represent an upper boundary on what performance benefits might be expected, as any commercially developed, widely-implemented progression tool will almost certainly assume a population distribution that differs from that of the patients the tool is eventually applied to. Indeed, outside of a research environment, the precise distribution of glaucoma rates in a given clinical population is almost certainly unknown and so recourse to some form of “average” distribution is required, at least initially. Our data gives some idea of what variation might be expected between populations of different geography and patient inclusion criteria (Fig. 4, Table 1), and it has been argued that at least one of our hospital-based datasets does not represent a selected subgroup of glaucoma patients distinct from that in the wider community.

^{13}Previous work suggests that modest performance benefits of Bayesian methods still persists despite discrepancies between the lower tail of the prior distribution and the patient data (each modelled with a modified hyperbolic secant) that were slightly greater than those seen in Table 1, suggesting that use of our distribution defined by average parameters (Fig. 4, thick line) may still have some benefits in improving individual visual field progression rate estimates. An average prior could then be made to better reflect the particular patient group to which it is applied by progressively updating the prior to incorporate information from each patient's test result, in an approach similar to that proposed for determining perimetric thresholds.

^{19}Improvements from Bayesian methods are likely to be modest, however, owing to the fact that the distribution of progression rates in the population is quite broad and so does not constrain individual progression rate estimates much.

^{6}Even if a parametric model were found that fitted the population distribution perfectly (as is very nearly the case for the modified Cauchy distribution fit to the Canadian data, and the modified hyperbolic secant fit to the Swedish data), the broad nature of the fit remains: hence, the goodness of fit of a model is not a guarantee of its use in a Bayesian method.

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