Recent development of the quick contrast sensitivity function (qCSF) method has made it possible to obtain accurate, precise, and efficient contrast sensitivity function (CSF) assessment. To improve statistical inference on CSF changes in a within-subject design, we developed a hierarchical Bayesian model (HBM) to compute the joint distribution of CSF parameters and hyperparameters at test, subject, and population levels, utilizing information within- and between-subjects and experimental conditions. We evaluated the performance of the HBM relative to a non-hierarchical Bayesian inference procedure (BIP) on an existing CSF dataset of 112 subjects obtained with the qCSF method in three luminance conditions (Hou, Lesmes, Kim, Gu, Pitt, Myung, & Lu, 2016). We found that the average *d*′s of the area under log CSF (AULCSF) and CSF parameters between pairs of luminance conditions at the test-level from the HBM were 33.5% and 103.3% greater than those from the BIP analysis of AULCSF. The increased *d*′ resulted in greater statistical differences between experimental conditions across subjects. In addition, simulations showed that the HBM generated accurate and precise CSF parameter estimates. These results have strong implications for the application of HBM in clinical trials and patient care.

*d′*s of CSF changes between luminance conditions for each subject, and improve statistical inference across subjects.

*y*= (

_{ijkm}*f*,

_{ijkm}*c*,

_{ijkm}*r*), where

_{ijkm}*r*, either correct or incorrect, is individual

_{ijkm}*i*’s response in trial

*m*of test

*k*in experimental condition

*j*tested with a stimulus of spatial frequency

*f*and contrast

_{ijkm}*c*. The BIP consists of four components (Hou, et al., 2016): (1) a log-parabola model of the contrast sensitivity function with several parameters, (2) a likelihood function that specifies the probability of making a correct or incorrect response in each stimulus condition, (3) a Bayesian procedure to infer the posterior distribution of the CSF parameters for each subject in each test, and (4) inference based on statistics computed from posterior distributions either at the subject level or aggregated across subjects. In this section, we first provide a brief review of the BIP, and then introduce the HBM.

_{ijkm}*S*(

*f*, θ

_{ijkm}_{ijk}) at spatial frequency

*f*is modeled with a log parabola function with three parameters, \({\theta _{ijk}} = ( {\gamma _{ijk}^{max},f_{ijk}^{max},{\beta _{ijk}}} )\) (Lesmes et al., 2010; Rohaly & Owsley, 1993; Watson, & Ahumada Jr, 2005):

_{ijkm}^{1}

_{ijk}is the bandwidth (octaves) at half of the peak sensitivity. The probability of making a correct response is described with a psychometric function (Hou et al., 2015):

*g*is the guessing rate, λ, usually set to 0.04 (Lesmes et al., 2010; Wichmann & Hill, 2001), is the lapse rate, Φ is the standard cumulative Gaussian function, and σ determines the steepness of the psychometric function. The probability of making an incorrect response is:

*p*(θ

_{ijk}|

*Y*) — the distribution of the CSF parameters θ

_{ijk}_{ijk}given the experimental data

*Y*= {

_{ijk}*y*}, for

_{ijkm}*m*=1, …,

*M*, where

*M*is the total number of trials in a test. This can be accomplished using Bayes’ rule:

*p*

_{0}(θ

_{ijk}) is the prior probability distribution of the CSF parameters for individual

*i*in test

*k*of experimental condition

*j*, which is usually uninformative and the same for all subjects and experimental conditions, and the denominator is the integral across all possible values of θ

_{ijk}, and is a constant for a given dataset and BIP.

*p*(

*r*|θ

_{ijkm}_{ijk},

*f*,

_{ijkm}*c*) of response

_{ijkm}*r*is determined only by the CSF parameters θ

_{ijkm}_{ijk}in that test (Equations 2, 3).

*J*experimental conditions at the population level,

*p*(η), is modeled as a mixture of 3 ×

*J*-dimensional Gaussian distributions \({\cal N}\) with mean μ and covariance

**Σ**, which have distributions

*p*(μ) and

*p*(

**Σ**):

_{i,1: J}of individual

*i*across all experimental conditions 1:

*J*at the individual level,

*p*(τ

_{i,1: J}|η), is modeled as mixtures of three-dimensional Gaussian distributions with mean ρ

_{ij}and covariance

**ϕ**

_{j}, which have distributions

*p*(ρ

_{i,1: J}|η) and

*p*(

**ϕ**

_{j}):

*p*(ρ

_{i,1: J}|η) denotes that ρ

_{i,1: J}is conditioned on η, and

**ϕ**

_{j}is a 3 × 3 covariance matrix in experimental condition

*j*. Finally, at the test level,

*p*(θ

_{ijk}|τ

_{ij}), the joint distribution of the CSF parameters, θ

_{ijk}, is conditioned on τ

_{ij}.

*X*= (θ

_{1: I, 1: J, 1: K}, ρ

_{1: I, 1: J},

**ϕ**

_{1: J}, μ,

**Σ**) are all the parameters and hyperparameters in the HBM.

*X*(Kruschke, 2015; Lee, 2006; Lee, 2011; Rouder & Lu, 2005; Wilson et al., 2020):

*X*and is a constant for a given dataset and HBM;

*p*

_{0}(μ),

*p*

_{0}(

**Σ**), and

*p*

_{0}(

**ϕ**

_{j}) are the prior distributions.

*K*= 1) in three luminance conditions (low = 2.62 cd/m

^{2}, medium = 20.4 cd/m

^{2}, and high = 95.4 cd/m

^{2}) with the qCSF method (Hou et al., 2016). Each test consisted of 150 trials. Three test trials were presented in each display consisting of three filtered letters of the same size, randomly sampled with replacement from 10 SLOAN letters (C, D, H, K, N, O, R, S, V, and Z), with the center spatial frequency and contrasts of the letters determined by qCSF. Subjects were asked to verbally report the identity of the letters on the screen.

*g*to 0.1, and σ to 0.1485 in Equation 2 (Foley & Legge, 1981; Hou et al., 2015; Legge, Kersten, & Burgess, 1987; Lesmes et al., 2010; Lu & Dosher, 1999). Following the qCSF procedure (Hou et al., 2015; Lesmes et al., 2010), we defined a three-dimensional CSF parameter space with 60 log-linearly spaced \(\gamma _{ijk}^{max}\) values between 1.05 and 1050, 40 log-linearly spaced \(f_{ijk}^{max}\) values between 0.1 and 20 cycles/degree, and 27 log-linearly spaced β

_{ijk}values between 1 and 9 octaves. The weakly informative prior,

*p*

_{0}(θ

_{ij1}), identical across all the tests, subjects, and experimental conditions, was defined by a hyperbolic secant function (Lesmes et al., 2010):

_{ijk}for

*a*= 1, 2, and 3, respectively, θ

_{a, confidence}= (0.5, 0.5, 0.5), and θ

_{a, mode}= (100, 1, 3).

*p*(θ

_{ij1}|

*Y*

_{ij1}) was computed using Equation 4. Convergence of the BIP solutions was quantified by the half-width of 68.2% credible interval (HWCI: Clayton & Hills, 1993; Edwards, Lindman, & Savage, 1963), equivalent to the standard deviation of the distribution if it is normal. With sufficient number of trials in the qCSF, the HWCI can reach its asymptotic minimum (Hou et al., 2015; Lesmes et al., 2010).

*p*

_{0}(μ), was a nine-dimensional uniform distribution:

_{0,min}and μ

_{0,max}of the three parameters in the three luminance conditions specified in Table 1.

**Σ**,

*p*

_{0}(

**Σ**), was specified by a 9 × 9 precision matrix

**Ω**with a Wishart distribution:

**Σ**

_{BIP}

^{−1}, was based on the covariance matrix of the estimated CSF parameters

**Σ**

_{BIP}across all the subjects and luminance conditions from the BIP procedure.

**ϕ**

_{j},

*p*

_{0}(

**ϕ**

_{j}), was specified with a 3 × 3 precision matrix

**Λ**

_{j}with a Wishart distribution:

_{j}= 3, and the expected mean, \({\boldsymbol\phi} _{BIP,j}^{ - 1}\), was based on the average covariance matrix

**ϕ**

_{BIP,j}computed from the estimated CSF parameters across all the subjects in luminance condition

*j*from the BIP procedure.

*in JAGS (Plummer, 2003) was used to compute representative samples of the posterior distributions of θ*

**autorun.jags**_{ij1}(3 parameters/condition × 3 conditions × 112 subjects = 1008 parameters), ρ

_{i,1: J}(9 parameters × 112 subjects = 1008 parameters),

**ϕ**

_{j}(6 parameters/condition × 3 conditions = 18 parameters), μ (9 parameters), and

**Σ**(45 parameters) in three Markov Chain Monte Carlo (MCMC) chains. The MCMC is an algorithm used to efficiently sample the joint posterior distribution (Kruschke, 2015). It started at a randomly selected position in the 2088-dimensional parameter space. In each step, one of the 2088 parameters was selected randomly. The one-dimensional conditional posterior probability distribution of the selected parameter was evaluated by fixing the values of all the other 2087 parameters at the current position. A new value of the selected parameter was chosen based on the one-dimensional conditional probability distribution (Equation 8). By reiterating this process, the probability of visiting a location in the random walk approximated the joint posterior distribution of all the 2088 parameters in Equation 8. These steps were re-iterated until the convergence criterion was reached.

*X*in subsequent analysis.

_{ij1 }from the HBM and BIP.

*d′*quantifies the signal (mean separation) to noise (variability) ratio of two probability distributions. We used the difference distribution between conditions to compute

*d′*. Each sample in the difference distribution represented the difference between two randomly drawn samples from the corresponding distributions.

*d′*is defined as (Green & Swets, 1966):

*d′*is defined as (Ashby & Townsend, 1986):

*cov*(Δ) are the mean separation and covariance matrix of the difference distribution,

*cov*(Δ)

^{−1}is the inverse of

*cov*(Δ), Δ

^{T}is the transpose of Δ, and * represents matrix multiplication.

_{ij1}from the HBM and BIP using Hotelling's T-squared test (Anderson, 2003) in R (R Core Team, 2020; Nordhausen, Sirkia, Oja, & Tyler, 2018). We also compared the correlation coefficients of pairs of CSF parameters from the two methods with paired

*t*-test.

_{ij1}and AULCSF between pairs of experimental conditions from each method with Hotelling's T-squared test and paired

*t*-test, respectively.

_{1: I, 1: J, 1}of the simulated tests were a random sample from the posterior distribution of τ

_{1: I, 1: J}obtained from the HBM fit to the real data. Each qCSF test consisted of 150 trials, identical to the real experiment (Hou et al., 2016), with the trial-by-trial responses determined by the CSF parameters of the simulated subject (Equations 1 to 3). Both the HBM and BIP were fit to the simulated dataset. The mean of the posterior distribution of θ

_{ij1}was used as the best estimate for each test. The bias, root mean square error (RMSE), variance,

*d*′, and

*t*statistics were computed based on the posterior distributions of θ

_{ij1}from both methods.

_{i,1: J}for one individual, and θ

_{i,1: J, 1}for one individual in one test from the HBM.

*d*′s of η between the three pairs of experimental conditions. In the HBM, the posterior distributions of η constrained τ

_{i,1: J}. The large

*d*′s of the posterior distributions of η between different experimental conditions indicated that the posterior distributions of η provided strong constraints on τ

_{i,1: J}.

_{1: I, 1: J}across all 112 individuals and experimental conditions. Figure 5(b) illustrates the three-dimensional posterior distributions of τ

_{i,1: J}for one individual in all three luminance conditions. Table 3 shows the average

*d*′s of τ

_{ij}between the three pairs of experimental conditions. In the HBM, the posterior distributions of τ

_{i,1: J}constrained θ

_{i,1: J, 1}. The large

*d*′s of the posterior distributions of τ

_{i,1: J}between different experimental conditions indicated that the posterior distributions of τ

_{i,1: J}provided strong constraints on θ

_{i,1: J, 1}.

_{1: I, 1: J, 1}in the three luminance conditions from the HBM and compared them with the results from the BIP.

_{ij1}from the HBM and BIP were significantly different (

*t*

^{2}(9,103) = 5.34,

*p*< 0.001), and the average variance of the estimated CSF parameters from the HBM (mean = 0.00139 log10 units; range = 0.00030 to 0.00739 log10 units) was 65.8% less than that from the BIP (mean = 0.00407 log10 units; range = 0.00035 to 0.11893 log10 units) (

*t*

^{2}(9,103) = 109,

*p*< 0.001), consistent with the well-known variance shrinkage effect of the HBM (Kruschke, 2015).

_{ij1}, and the standard deviation (SD = \(\sqrt {variance} \)) of θ

_{ij1}from the BIP and HBM. Whereas most of the differences between the expected values of θ

_{ij1}from the two methods were small (mean absolute difference = 0.027 log10 units), there were thirteen instances (out of a total of 3 parameters × 3 conditions × 112 subjects = 1008) in which the absolute difference was greater than 0.2 log10 units (range = 0.200 to 0.676 log10 units). The discrepancies were associated with large variances of the BIP estimates in those instances: their average variance of 0.065 log10 units (range = 0.027 to 0.119 log10 units) was 16 times the mean variance (0.00407 log10 units) of θ

_{ij1}in the BIP procedure, suggesting that BIP did not converge well in those cases. On the other hand, the HBM generated more precise estimates with on average a 93.7% reduction of variance (mean = 0.00407 log10 units; range = 0.00139 to approximately 0.00681 log10 units) compared to the BIP in the 13 cases by incorporating data from all the subjects and conditions in a single model.

_{ij1}in pairs of luminance conditions across subjects from the HBM and BIP procedures. All correlations were negative, with the strongest between

*f*and β. Across all the subjects, 97.4% and 97.9% of the correlation coefficients from the BIP and HBM were statistically significant, respectively. Although the paired

^{max}*t*-test showed that the correlation coefficients between γ

^{max}and β in the high luminance condition (

*p*= 0.003) and between

*f*and β in all three luminance conditions (

^{max}*p*< 0.001) from the two procedures were significantly different, the magnitudes of the differences were very small and probably not of practical importance.

*d*′s of θ

_{ij1}and AULCSF between pairs of luminance conditions across all the subjects. Averaged across the three pairs, the AULCSF

*d*′ from the HBM was 33.5% greater than that from the BIP. Compared to AULCSF, incorporating information from the three-dimensional joint distributions of θ

_{ij1}led to an average

*d*′ increase of 66.6% for the BIP and 51.7% for the HBM. Compared to the AULCSF

*d*′ from the BIP, using θ

_{ij1}in the HBM increased

*d*′ by 103.3% across the three pairs of luminance conditions.

_{ij1}and AULCSF across individuals

*t*(111) of the means of θ

_{ij1}and AULCSF among the three pairs of experimental conditions. The HBM generated larger

*t*values than the BIP for both θ

_{ij1}and AULCSF in all pairs of experimental conditions. Averaged across the three pairs,

*t*(111) of AULCSF and \(\sqrt {{t^2}( {3,109} )} \) of θ

_{ij1}from the HBM were 51.2% and 49.6% greater than those from the BIP.

_{ij1}in the simulation, with very small bias (γ

^{max},

*f*, β: 0.0028, −0.0091, and 0.0023 log10 units), RMSE (0.0373 log10 units), and average variance (0.00149 log10 units). In comparison, the BIP exhibited lower accuracy and precision (bias = γ

^{max}^{max},

*f*, β: 0.0147, −0.0395, and 0.0118 log10 units; RMSE = 0.0673 log10 units; average variance = 0.00428 log10 units).

^{max}*d*′s of AULCSF and CSF parameters between different experimental conditions at the test level for each subject, and bigger statistical differences across subjects. Relative to the BIP, the HBM increased the average

*d*′s of AULCSF and θ

_{ij1}between conditions at the test level by 24.5% and 20.5%, and the corresponding

*t*(111)and \(\sqrt {{t^2}( {3,109} )} \) by 51.2% and 49.6%, respectively. Simulations also showed that the HBM generated accurate and precise CSF parameter estimates.

*d*′ and

*t*statistics at the test level because it reduced the variance of θ

_{ij1}by 65.8% relative to the BIP (0.00139 vs. 0.00407 log10 units). In addition, the 13 instances in which the absolute difference of θ

_{ij1}from the HBM and BIP were greater than 0.2 log10 units further demonstrated the benefit of incorporating information across tests, subjects, and conditions in the HBM (Kruschke, 2015; Rouder & Lu, 2005; Rouder, Sun, Speckman, Lu, & Zhou, 2003). In those cases, the variances of the BIP estimates were very large (16 times the mean variance), suggesting that the BIP did not converge well. On the other hand, the HBM generated much more precise CSF estimates for each test by incorporating data across subjects and conditions in a single model. The ability of the HBM to generate more precise estimates from insufficient or poor-quality data can be quite valuable in clinical trials.

*d*′s of AULCSF and θ

_{ij1}between conditions at the test level by 24.5% and 20.5 %, and the corresponding

*t*statistics by 51.2% and 49.6%, respectively. Future research will further evaluate the potential value of the HBM for analyzing clinical changes in contrast sensitivity, whether in individual patients or groups in clinical trials.

*Journal of Neuroscience, Psychology, and Economics*, 4(2), 95–110. [CrossRef] [PubMed]

*An introduction to multivariate analysis*. Hoboken, N.J.: Wiley-Interscience.

*Biometrika*, 94, 443–458. [CrossRef]

*American Journal of Mathematical and Management Sciences*, 31, 13–38. [CrossRef]

*Psychological Review*, 93(2), 154–179. [CrossRef] [PubMed]

*Graefe's Archive for Clinical and Experimental Ophthalmology*, 241, 968–974. [CrossRef] [PubMed]

*Journal of Cataract & Refractive Surgery*, 31, 712–717.

*Journal of Clinical Epidemiology*, 60(12), 1234–1238. [PubMed]

*Ophthalmic and Physiological Optics*, 11, 218–226. [PubMed]

*Optometry & Vision Science*, 83, 290–298.

*PLoS One,*9(3), e90579.

*Vision Research*, 42, 2137–2152. [PubMed]

*Statistical models in epidemiology*. Oxford, UK: Oxford University Press.

*Journal of the American Statistical Association*, 94(448), 1254–1263.

*Psychological Review*, 70(3), 193–242.

*BMC Medical Research Methodology*, 14, 49. [PubMed]

*Vision Research*, 21, 1041–1053. [PubMed]

*Statistical Science*, 7, 457–511.

*Clinical applications of visual psychophysics*(pp. 70–106). Cambridge, UK: Cambridge University Press.

*International Ophthalmology Clinics*, 43(2), 5–15. [PubMed]

*Current Opinion in Ophthalmology*, 17, 19–26. [PubMed]

*Signal Detection Theory and Psychophysics*. New York, NY: John Wiley & Sons.

*Journal of Vision*, 16(6), 15. [PubMed]

*Investigative Ophthalmology & Visual Science*, 47, 2739–2745. [PubMed]

*Clinical applications of visual psychophysics*(pp. 11–41). Cambridge, UK: Cambridge University Press.

*Journal of the American Statistical Association*, 91, 1461–1473

*Graefe's Archive for Clinical and Experimental Ophthalmology*, 245,1805–1814. [PubMed]

*Investigative Ophthalmology & Visual Science*, 51, 5365–5377. [PubMed]

*Journal of Vision*, 15(9):2, 1–18.

*Journal of Vision*, 16(6), 18. [PubMed]

*Journal of Vision*, 14(13):9, 1–14.

*Vision Research*, 114, 135–141. [PubMed]

*Journal of Cataract and Refractive Surgery*, 15(2), 141–148. [PubMed]

*Investigative Ophthalmology & Visual Science*, 58, BIO277–BIO290. [PubMed]

*Proceedings of the National Academy of Sciences, USA*, 111, 2035–2039.

*Perception & Psychophysics*, 14, 313–318.

*Neural Computation*, 26(11), 2465–2492. [PubMed]

*Archives of Ophthalmology*, 106, 55–57. [PubMed]

*Frontiers in Psychology*, 10, 1675. [PubMed]

*Vision Research*39(16), 2729–2737. [PubMed]

*Doing Bayesian data analysis: a tutorial with R, JAGS, and Stan*. San Diego, CA: Academic Press.

*Psychonomic Bulletin & Review,*25(1), 178–206.

*Journal of Vision*, 5(5), 8.

*Cognitive Science*, 30(3), 1–26. [PubMed]

*Journal of Mathematical Psychology*, 55(1), 1–7.

*Journal of the Optical Society of America A -Optics Image Science and Vision*, 20(7), 1434–1448.

*Journal of the Optical Society of America A: Optics, Image Science, and Vision*, 4, 391–404.

*Drug Discovery Today: Therapeutic Strategies*, 10(1), e43–e50. [PubMed]

*Journal of Vision*, 10(3):17, 1–21. [PubMed]

*Investigative Ophthalmology & Visual Science*, 54, 2762. [Abstract]

*ARVO Meeting Abstracts*, 53, 4358.

*Philosophical Transactions of the Royal Society B: Biological Sciences*, 364, 399–407.

*Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Volume 2*, pp. 524–531.

*Translational Vision Science & Technology*, 7(2), 22. [PubMed]

*Archives of Ophthalmology*, 102, 1303–1306. [PubMed]

*Journal of the Optical Society of America A: Optics, Image Science, and Vision*, 16, 764–778.

*Visual Psychophysics: From Laboratory to Theory*. Cambridge, MA: MIT Press.

*Journal of Modern Optics,*44(1), 127–148.

*British Journal of Ophthalmology*, 70, 553–559.

*Journal of Mathematical Psychology*, 55(1), 57–67.

*Investigative Ophthalmology & Visual Science*, 38, 469–477. [PubMed]

*Computational Brain & Behavior*, 1(2), 184–213.

*Journal of Cognitive Neuroscience*, 31(12), 1976–1996. [PubMed]

*Package ‘ICSNP’ in CRAN repository*. Retrieved from: https://cran.r-project.org/package=ICSNP.

*American Journal of Ophthalmology,*226, 148–155.

*Vision Research*, 23, 689–699. [PubMed]

*Journal of Mathematical Psychology*, 84, 20–48.

*British Journal of Ophthalmology*, 88, 11–16.

*Proceedings of the 3rd international workshop on distributed statistical computing*. Retrieved from: https://www.r-project.org/nosvn/conferences/DSC-2003/Drafts/Plummer.pdf.

*Journal of Vision*, 13(7), 3. [PubMed]

*R: A language and environment for statistical computing*. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

*ARVO Annual Meeting Abstracts*, 56, 2225.

*Journal of Vision*, 14(10), 1428.

*Marine Biology*, 162(2), 469–478.

*Journal of the Optical Society of America A, Optics and Image Science*, 10, 1591–1599. [PubMed]

*ARVO Annual Meeting Abstracts*, 56, 2224.

*Journal of Vision*, 14(8):3, 1–10.

*Psychonomic Bulletin & Review*, 12(4), 573–604. [PubMed]

*Psychometrika*, 68(4), 589–606.

*Vision Research*, 122, 105–123. [PubMed]

*Journal of Vision*, 17(12), 12. [PubMed]

*Sensors*, 20(14), 3874.

*Multiple Sclerosis Journal - Experimental, Translational & Clinical*, 1, 1–8.

*Evolution*, 56(1), 154–166. [PubMed]

*Journal of Cataract & Refractive Surgery*, 34, 570–577.

*Statistics in Medicine*, 22(5), 763–780. [PubMed]

*Journal of VitreoRetinal Diseases,*5(4), 313–320.

*Publications of the Astronomical Society of Australiam*36, e010.

*Vision Research*, 35, 2503–2522. [PubMed]

*NeuroImage*, 72, 193–206. [PubMed]

*Journal of Cataract & Refractive Surgery*, 35, 47–56.

*Journal of Clinical Medicine,*10(13), 2768.

*British Journal of Ophthalmology*, https://doi.org/10.1136/bjophthalmol-2020-318494. [e-pub ahead of print].

*Statistical Methods in Medical Research*, 29(4), 1112–1128. [PubMed]

*Journal of Vision*, 17(3):10, 1–27.

*Optics Express*, 6(1), 12–33. [PubMed]

*Journal of Vision*, 5(9), 717–740. [PubMed]

*IEEE International Conference on Image Processing*. 3, 41–44.

*Perception & Psychophysics*, 33(2), 113–120. [PubMed]

*Perception & Psychophysics*, 63, 1293–1313. [PubMed]

*Ecology*, 84(6), 1382–1394.

*Modern Statistics for the Social and Behavioral Sciences: A Practical Introduction*(pp. 101–102). Boca Rato, Florida, USA: CRC Press.

*Computational Brain & Behavior*, 3, 384–399

*Scientific Reports*, 7, 5045. [PubMed]

*Bayesian Analysis*, 11(3), 649–670. [PubMed]

*Translational Vision Science & Technology,*10(12), 18.