On alternative methods for measuring visual field decay: Tobit linear regression.

We read with interest the article by Caprioli et al. in the June 2011 issue. In this article, the authors propose a novel method of measuring and predicting the rate of visual field (VF) decay in glaucoma patients. This is an important and clinically relevant topic. Of particular interest was the authors’ application of an exponential fit for threshold sensitivity (in decibels) against time, which was found to be the best-fitting model compared against a linear and a quadratic regression—according to the Akaike information criterion (AIC)—in 96.8% of the 21,006 of the data series investigated. This contradicts the idea that point-wise VF decay is best modeled and predicted using simple ordinary least-squares linear regression (OLSLR); previous research investigating other more complex models has shown these to be less accurate than OLSLR for predicting VF progression. In response to this article, we would like to suggest possible statistical reasons why the exponential regression is shown to be a better-fitting model than simple OLSLR. Standard VF threshold measurements from the Humphrey Visual Field Analyzer (Carl Zeiss Meditec, Dublin, CA) are limited to the range of 0 to 50 dB. Consequently, VF thresholds are vulnerable to floor (left-censoring) and ceiling (right-censoring) effects. Caprioli et al. state that “exponential fits for individual locations seem to work well for advanced damage because values usually approach 0 dB in an asymptotic fashion”; conversely, we argue that VF progression may occur linearly throughout the disease but the exponential model is more robust than OLSLR to the floor effects of VF threshold data. Floor effects are pertinent to the VF data investigated by Caprioli et al., since the models were applied to patients with moderate to advanced glaucomatous damage. Furthermore, in Figure 9 of their study, it is apparent that for “actual thresholds” above approximately 20 dB, the predicted thresholds from the exponential regression greatly underestimate the VF thresholds. In circumstances in which the dependent variable is censored (such as threshold sensitivity), the assumptions of the OLSLR model are not strictly valid. However, there are statistical prescriptions that may provide a solution. The Tobit linear regression model employs a latent (unobservable) dependent variable that respects leftand/or right-censoring and predicts this variable only within the specified range. To illustrate the usefulness of Tobit regression as well as support the hypothesis that VF progression may occur linearly, even in advanced disease, we synthetically created the example shown in Figure 1, which is akin to Figure 2 in Caprioli et al. The Tobit linear regression model is fitted to the synthetic data in Figure 1 (blue line) and compared with the exponential regression (red line) and OLSLR (black line). Different models can be contrasted using the AIC, with the lowest AIC indicating the best model. Specifying appropriate candidate models is imperative, because if only poor models are compared, then the AIC will simply select the best of a bad bunch. Models can be further evaluated using the formula: