Pointing Data Analysis

The quantitative characterization of both patient and control subject pointing errors was carried out with non-linear mixed-effect (ME) analyses (nlme package with R version 2.14.1). ME analyses are based on multilevel (hierarchical) modeling (Snijders and Bosker, 1999; Pinheiro and Bates, 2006; Hox, 2010). As such, they are the best statistical method to reveal significant differences between groups (controls vs. patients), between individuals within groups, and between hands or hemifields within individuals. These differences may concern both fixed and random effects. Fixed effects indicate significant differences between group averages. Random effects indicate significant differences between within-group inter-individual variability. Therefore, ME analyses can reveal between-group differences for both the average value and the inter-individual variability of model parameters. Another advantage of ME analyses is that they allow residual errors to be modeled as a function of model parameters. This is especially interesting because the variability in the residual errors may increase with target distance and may depend on groups or hemifields. In general, residual error models involved 2 or 3 parameters. As a result, ME models had 2 or 3 more parameters than the corresponding individual models.

Mixed effects analyses have two main drawbacks. First, they may yield excessive type I error rates when the sample size is smaller than 30 (Maas and Hox, 2005; Paccagnella, 2011; Vindras et al., 2012). As it was impossible to increase the sample size due to the scarcity of patients with Optic Ataxia, we used a low significance threshold (10^-4 instead of the usual 0.05), and we checked the results of the ME analyses by carrying out individual linear analyses and testing their outcome using three complementary methods. First, to confirm whether a fixed or random effect was significant, we tested the samples of p-values provided by matching individual linear models using a new method based on the Kolmogorov–Smirnov test (Vindras et al., 2012). This test yields a significant outcome not only if the inter-individual average is significantly different from zero (ME Fixed effect), but also if an abnormally large number of individual analyses indicate significant negative or positive effects (ME Random Effect). Second, we confirmed significant between-group differences revealed by ME analyses by using Mann–Whitney U tests to assess whether individual parameter values differed between groups. Third, when ME analyses revealed between-hand or between-hemifield differences, we used non-parametric Wilcoxon signed-rank tests to assess the hand-specific or hemifield-specific parameters provided by the individual models.

The second drawback of ME analyses is their complexity, and because of this, stepwise forward modeling is strongly recommended (Pinheiro and Bates, 2006). Thus, throughout these data-driven analyses we introduced parameters one by one into the models, keeping the most significant ones at each step. This process means that significant parameters cannot be described in advance but are presented throughout the results section. Because our goal was to check for differences between controls and patients and hand- and hemifield- specific differences within patients, the data set was progressively enriched in four stages. We first modeled the errors made by control subjects. Second, we fit the errors made by patients pointing in the ipsilesional hemifield with their ipsilesional hand by introducing new parameters in a stepwise manner to test whether and how these errors differed from control subjects’ errors. Third, we fit the errors made by patients pointing in the ipsilesional hemifield with their contralesional hand to test whether and how these errors differed from the previous data set (including errors made both by control subjects and patients with their ipsilesional hand in their ipsilesional hemifield). Finally, we modeled the additional errors specifically associated with the targets in the contralesional hemifield which consisted exclusively of large errors directed toward the point of ocular fixation (Blangero et al., 2010) that depended non-linearly on target eccentricity. These were modeled using one or two-parameter equations. These equations were standard modeling functions (polynomial, logarithmic, or power functions), as well as complex logarithms based upon those used in the literature to describe the collicular and cortical visual mappings (Schwartz, 1980; Ottes et al., 1986; Van Gisbergen et al., 1987; Polimeni et al., 2006; Schira et al., 2007; Wu et al., 2012). Section “Results” progressively describes the outcome of these four stages.