Sensitivity to light reduces with increasing eccentricity from the center of the visual field. Kinetic perimetry assesses the location at which moving light of a fixed size/intensity can be seen and is used to quantify visual field sensitivity. The location of each test point is recorded as a polar coordinate, and points are joined to form an isopter, denoting a line of differential light sensitivity, within which light of a particular size/intensity can be perceived.
Constructing robust normative perimetry standards provides an evidence base for interpretation of clinical/research findings—aiding detection of visual field defects. Analysis of normative kinetic perimetry data requires an understanding of anatomical and physiological characteristics, and the correct application of appropriate statistical methods. However, the analytical methods currently employed in the literature do not take into account the repeated-measures structure of a kinetic isopter, nor do they adequately address the lack of normality in the empirical distribution of data points.
Niederhauser and Mojon (
2002) summarized kinetic isopters at each meridian with the mean and 95% confidence intervals. Similar methods have been used by other researchers (Egge,
1984; Wilscher, Wabbels, and Lorenz,
2010) though they assume symmetric distributions along test meridians. There are two issues with this approach: It can define poorly fitted regions of uncertainty that may fall outside the edge of the perimeter test surface or cause crossing of confidence bands; also, as it is not model-based, it cannot be used for inferential purposes (e.g., to compare goodness-of-fit or to determine whether the resulting normative curves and confidence regions need further adjustment by relevant covariates, such as age). Other techniques for summarizing normative fields, such as total isopter area (Bjerre, Codina, & Griffiths,
2014; Quinn, Fea, & Minguini,
1991) and visual field extent along meridians (Wilson, Quinn, Dobson, & Breton,
1991) do not provide reference values that are easily interpretable in a clinical setting.
Isopter data may show conditional distributions that do not conform to normality assumptions, such as symmetry and constant variance (homoscedasticity), either on the observed or the transformed (e.g., logarithmic) scale of the outcome. For these reasons, modeling based on mean (normal) regression can lead to incorrect inference. Even when approximate normality is achieved after transformation, back-transformation of conditional expectations is troublesome as it may lead to estimates and/or confidence regions outside the admissible range of the outcome. Moreover, isopter data that are collected repeatedly on the same subject are correlated by design. While mixed-effects models for the mean account for the clustered design, they are still subject to strong distributional assumptions and back-transformation issues.
As an alternative to mean regression, we consider quantile regression (QR; Koenker,
2005), which introduces weak assumptions on the distribution of the error and therefore is robust to deviations from normality. For this reason, no transformation is required—note, however, that even when transformations are introduced to achieve linearity, back-transforming quantiles is a simple task (see for example, Geraci & Jones,
2015). More specifically, in this article we address the analytical challenges of modeling isopter data using mixed-effects models for conditional quantiles called linear quantile mixed models (LQMMs; Geraci & Bottai,
2014). The inclusion of subject-specific effects in mixed-effects models allows for within-subjects correlation resulting from repeated measurements. This approach yields clinically appropriate estimates within expected clinical ranges and correct inference from kinetic perimetry data.
Here we report the application of LQMMs to normative kinetic isopter data obtained from healthy children (Patel, Cumberland, Walters, Russell-Eggitt, Cortina-Borja et al.,
2015) using the R (The R project for Statistical Computing, version 3.1.2; R Core Team,
2016) package kineticF (Patel & Cortina-Borja,
2015). The kineticF package is a collection of functions for cleaning, processing, visualization, and analysis of manual (Goldmann) and automated (Octopus 900) kinetic visual field data, which depends on the package lqmm (Geraci,
2014) used to fit QR models with random effects.