Clinical trials typically analyze multiple endpoints for signals of efficacy. To improve signal detection for treatment effects using the high-dimensional data collected in trials, we developed a hierarchical Bayesian joint model (HBJM) to compute a five-dimensional collective endpoint (CE_{5D}) of contrast sensitivity function (CSF) and visual acuity (VA). The HBJM analyzes row-by-row CSF and VA data across multiple conditions, and describes visual functions across a hierarchy of population, individuals, and tests. It generates joint posterior distributions of CE_{5D} that combines CSF (peak gain, peak frequency, and bandwidth) and VA (threshold and range) parameters. The HBJM was applied to an existing dataset of 14 eyes, each tested with the quantitative VA and quantitative CSF procedures in four Bangerter foil conditions. The HBJM recovered strong correlations among CE_{5D} components at all levels. With 15 qVA and 25 qCSF rows, it reduced the variance of the estimated components by 72% on average. Combining signals from VA and CSF and reducing noises, CE_{5D} exhibited significantly higher sensitivity and accuracy in discriminating performance differences between foil conditions at both the group and test levels than the original tests. The HBJM extracts valuable information about covariance of CSF and VA parameters, improves precision of the estimated parameters, and increases the statistical power in detecting vision changes. By combining signals and reducing noise from multiple tests for detecting vision changes, the HBJM framework exhibits potential to increase statistical power for combining multi-modality data in ophthalmic trials.

_{5D}) that consists of two parameters of the visual acuity behavioral function (VABF) (Lesmes & Dorr, 2019) and three parameters of contrast sensitivity function (Watson & Ahumada, 2005). CE

_{5D}is considered as a “collective endpoint” because it combines information from two endpoint assessments. We demonstrated that the high dimensional collective endpoint significantly improved the statistical power in a study with a small sample size (

*N*= 14 eyes).

_{5D}) of CSF and VA by modeling row-by-row VA and CSF data simultaneously, and (2) exploit the statistically reciprocal relationships between VA and CSF parameters to combine signals from VA and CSF tests and reduce the variability of the estimated endpoints and therefore improve the power of statistical inference (Figure 1).

_{5D}as a five-dimensional vector consisting of VA threshold, VA range, PG, PF, and BH.

_{5D}from the row-by-row data of human observers in all the tests and make statistical inferences at the population, individual, and test levels.

_{5D}distribution for each eye in each condition. We quantified the uncertainty, sensitivity, and discriminability of the estimated CE

_{5D}, and compared them with metrics from Bayesian inference procedure (BIP) analyses of the tests, VA

_{1D}and AULCSF

_{1D}.

*M*= 45), each with three equal-size high-contrast optotypes (Figure 3A) randomly sampled without replacement from the 10 Sloan letters (C, D, H, K, N, O, R, S, V, and Z). The size of the optotypes was determined by qVA (Lesmes & Dorr, 2019; Zhao, Lesmes, Dorr, Bex, et al., 2021). Each qCSF test consisted of 50 rows (

*K*= 50), each with three equal-size bandpass-filtered optotypes (Figure 3B) randomly sampled with replacement from the ten Sloan letters (C, D, H, K, N, O, R, S, V, and Z). The sizes (and therefore center spatial frequencies) and contrasts of the optotypes were determined by qCSF (Hou et al., 2015; Lesmes et al., 2010). In both tests, the stimuli stayed on the screen until the subject verbally reported the identities of the letters presented on the screen, which were entered into the computer by the experimenter.

_{5D}), comprised of parameters of the two generative models, across all tests and individuals and at the population level. Finally, we describe the computational and statistical inference procedures. In the current development, we treat each unique combination of eye and Bangerter foil condition as an individual.

*dʹ*= 2 performance level, and \(\theta _{ij,Range}^{VA}\) is VA range defined as the difference between VA thresholds at

*dʹ*= 1 and

*dʹ*= 4 performance levels, the generative model of visual acuity (Figure 4A) specifies the probability of obtaining

*r*(=0, 1, 2, or 3) correct responses from individual

_{ijm}*i*tested with optotype size

*os*in row

_{ijm}*m*of the

*j*

^{th}test (Figure 2A) (Lesmes & Dorr, 2019; Zhao, Lesmes, Dorr, Bex, et al., 2021):

*i*in all

*M*rows in the

*j*

^{th}test is the product of \(p( {r_{ijm}^{VA}{\rm{|}}\theta _{ij}^{VA},o{s_{ijm}}} )\) in all rows:

*i*in the

*j*

^{th}test with the BIP (Lesmes et al., 2010; Zhao, Lesmes, Dorr, Bex, et al., 2021; Zhao, Lesmes, Dorr, & Lu, 2021):

*f*for individual

_{ijk}*i*in row

*k*of the

*j*

^{th}test using a log parabola function with three parameters \(\theta _{ij}^{CSF} = ( {\theta _{ij,PG}^{CSF},\theta _{ij,PF}^{CSF},\theta _{ij,BH}^{CSF}} )\) (Figure 2C) (Lesmes et al., 2010; Rohaly & Owsley, 1993; Watson & Ahumada, 2005; Zhao, Lesmes, Hou, et al., 2021):

*c*are the response to and contrast of the

_{ijkl}*l*

^{th}(=1, 2, 3) optotype in row

*k*with a total of

*L*optotypes;

*f*is the spatial frequency for all three optotypes in the row;

_{ijk}*g*is the guessing rate, λ, set to 0.04, is the lapse rate (Hou et al., 2015; Lesmes et al., 2010; Wichmann & Hill, 2001), Φ is the standard cumulative Gaussian function, and σ determines the steepness of the psychometric function. The probability of an incorrect response is:

*i*in all

*K*rows in the

*j*

^{th}test is the product of \(p( {r_{ijkl}^{CSF}{\rm{|}}\theta _{ij}^{CSF},{f_{ijk}},{c_{ijkl}}} )\) in all rows:

*i*in the

*j*

^{th}test with the BIP (Hou et al., 2015; Hou et al., 2016; Lesmes et al., 2010):

_{5D}, comprised of the parameters of the generative models of all the endpoint assessments. In the current development, the five-dimensional collective endpoint \(\theta _{ij}^{{\rm{CE}}} = ( {\theta _{ij,{\rm{VA}}}^{{\rm{CE}}},\;\theta _{ij,{\rm{CSF}}}^{{\rm{CE}}}} )\;\)for individual

*i*in the

*j*

^{th}test consists of two-dimensional \(\theta _{ij,{\rm{VA}}}^{{\rm{CE}}} = ( {\theta _{ij,Threshold}^{{\rm{CE}}},\theta _{ij,Range}^{{\rm{CE}}}} )\) and three-dimensional \(\theta _{ij,CSF}^{CE} = ( {\theta _{ij,PG}^{CE},\theta _{ij,PF}^{CE},\theta _{ij,BH}^{CE}} )\) components. The CE

_{5D}hyperparameters at the population and individual levels are five-dimensional random variables η

^{CE}and \(\tau _i^{{\rm{CE}}}.\) In the HBJM, the distribution at a higher level sets the prior for the means of the distributions at a lower level, i.e., the distribution of the population hyperparameter η

^{CE}sets the prior for \(\rho _i^{{\rm{CE}}}\), the mean of the distribution of the hyperparameter \(\tau _i^{{\rm{CE}}}\) of individual

*i*, which in turn sets the prior for the mean of the distribution of parameter \(\theta _{ij}^{{\rm{CE}}}\) of individual

*i*in the

*j*test.

^{th}*p*(η

^{CE}) of the CE

_{5D}hyperparameter \({\eta ^{{\rm{CE}}}} = (\eta _{VA}^{{\rm{CE}}},\) \(\eta _{CSF}^{{\rm{CE}}})\) with \(\eta _{VA}^{{\rm{CE}}} = ( {\eta _{Threshold}^{{\rm{CE}}},\eta _{Rang{\rm{e}}}^{{\rm{CE}}}} )\) and \(\eta _{CSF}^{{\rm{CE}}} = ( {\eta _{PG}^{{\rm{CE}}},\eta _{PF}^{{\rm{CE}}},\eta _{BH}^{{\rm{CE}}}} )\) at the population level is modeled as a mixture of five-dimensional Gaussian distributions \(\mathcal{N}( {{\eta ^{{\rm{CE}}}},{\mu ^{{\rm{CE}}}},\Sigma } )\):

^{CE}is the mean of η

^{CE}, with distribution

*p*(μ

^{CE}), and Σ is the covariance of η

^{CE}, with distribution

*p*(Σ).

_{5D}hyperparameter \(\tau _i^{\rm CE} = (\tau _{i,VA}^{\rm CE},\) \(\tau _{i,CSF}^{\rm CE})\) of individual

*i*is modeled as the product of two-dimensional \(\tau _{i,VA}^{\rm CE} = ( {\tau _{i,Threshold}^{\rm CE},\tau _{j,Range}^{\rm CE}} )\) and three-dimensional \(\tau _{i,CSF}^{\rm CE} = ( {\tau _{i,PG}^{\rm CE},\tau _{i,PF}^{\rm CE},\tau _{i,BH}^{\rm CE}} )\) Gaussian distributions:

^{CE}, ϕ

_{VA}and ϕ

_{CSF}are 2 × 2 and 3 × 3 covariances of \(\tau _{i,{\rm{VA}}}^{{\rm{CE}}}\;\)and \(\tau _{i,{\rm{CSF}}}^{{\rm{CE}}}\), with distributions

*p*(ϕ

_{VA}) and

*p*(ϕ

_{CSF}).

_{5D}parameter \(\theta _{ij}^{{\rm{CE}}}\) for individual

*i*in the

*j*

^{th}test is the product of two distributions conditioned on \(\tau _i^{{\rm{CE}}}\):

*I*is the total number of individuals, and

*J*is the total number of tests on each individual.

^{CE}, Σ, ϕ

_{CSF}, and ϕ

_{VA}.

^{−1},\(\;\phi _{VA}^{ - 1}\), and \(\phi _{CSF}^{ - 1}\) are the inverse of covariances Σ, ϕ

_{VA}, and ϕ

_{CSF}; \(\mathcal{W}( {\frac{{\rm{Y}}}{{{{\rm{\nu }}_{\rm{Y}}}}},{{\rm{\nu }}_{\rm{Y}}}} )\) denotes a Wishart distribution with mean precision matrix Y and degrees of freedom ν

_{Y}, with ν = 5, ν

_{VA}= 2 , and ν

_{CSF}= 3;\(\;\phi _{qVA}^{ - 1}\) and \(\phi _{qCSF}^{ - 1}\) are the inverse of the average covariance matrices ϕ

_{qVA}and ϕ

_{qCSF}computed from the BIP posterior distributions across all qVA and qCSF tests; and Σ

^{−1}is the inverse of the covariance matrix Σ computed from the estimated \(\theta _{ij}^{VA}\) and \(\theta _{ij}^{CSF}\) across all qVA and qCSF tests.

*X*and is a constant for a given dataset and HBJM.

*run.jags*in JAGS (Plummer, 2003) to compute representative samples of the joint posterior distribution of \(\theta _{i1}^{{\rm{CE}}}\;\)(5 parameters × 56 individuals=280 parameters), \(\rho _i^{{\rm{CE}}}\) (5 parameters × 56 subjects = 280 parameters), ϕ

_{CSF}and ϕ

_{VA}(9 parameters), μ

^{CE}(5 parameters), and Σ (15 parameters) in three Markov Chain Monte Carlo (MCMC) chains. The MCMC efficiently samples the joint posterior distribution in a random walk (Kruschke, 2014). It started at a position randomly selected in the 589-dimensional parameter space. In each step, one of the 589 parameters was randomly selected. The values of all the other 588 parameters at the current position were fixed and the one-dimensional conditional probability distribution of the selected parameter was evaluated, from which a new value of the selected parameter was generated. The process reiterated until a pre-defined convergence criterion was reached so that the probability of visiting a position in the random walk approximated the joint posterior distribution of all the 589 parameters in Equation 14.

*M*= 5,

*K*= 15), medium (the first 15 qVA and first 25 qCSF rows;

*M*= 15,

*K*= 25), and long (all qVA and qCSF rows;

*M*= 45,

*K*= 50), separately.

*in JAGS (Plummer, 2003) to compute representative samples of the posterior distribution of \(\theta _{i1}^{VA}\) and \(\theta _{i1}^{CSF}\) for each qVA and qCSF test independently using their respective BIP. Five thousand and 5000 steps were used for adaptation and burn-in in each MCMC chain based on pilot studies. The random walk continued until 10,000 were stored in each MCMC chain with thinning ratio = 150, 50, and 50 for short, medium, and long test lengths, respectively.*

**run.jags***HotellingsT2*(Nordhausen, Sirkia, Oja, & Tyler, 2018; R Core Team, 2003).

_{1D}), area under the log CSF from the BIP (AULCSF

_{1D}), and the collective endpoint (CE

_{5D}) from the HBJM. AULCSF

_{1D}was constructed by computing the AULCSF between 1.5 and 18 cycles per degree.

_{1D}, AULCSF

_{1D}, and CE

_{5D}between a condition pair,

*p*(

*x*|

*G*) and

_{null}*p*(

*x*|

*G*), where

_{diff}*x*= VA

_{1D}, AULCSF

_{1D}, or CE

_{5D}, were constructed from the corresponding individual test-level distributions in the two conditions: (1) For each eye, a pair of random samples were drawn from the test-level distributions in the two conditions. The difference between the two samples was computed. (2) Step (1) was repeated for all 14 eyes. (3) The mean difference across the 14 eyes was added as a new sample in

*p*(

*x*|

*G*). (4) By repeating (1) ∼(3),

_{diff}*p*(

*x*|

*G*) was constructed. And (5) the baseline distribution

_{diff}*p*(

*x*|

*G*) was constructed by shifting the mean of

_{null}*p*(

*x*|

*G*) to the origin.

_{diff}_{1D}, AULCSF

_{1D}, and CE

_{5D}for sample sizes

*N*= 1∼14 between each condition pair,

*p*(

*x*|

*G*) and

_{null}*p*(

*x*|

*G*), where

_{diff}*x*= VA

_{1D}, AULCSF

_{1D}, or CE

_{5D}, were constructed from the corresponding individual test-level distributions in the two conditions: (1)

*N*eyes were randomly selected with replacement from the 14 eyes. (2) For each selected eye, a pair of random samples were drawn from the test-level distributions in the two conditions. The difference between the two samples was computed. (3) Step (2) was repeated for the

*N*randomly selected eyes. (4) The mean difference across the

*N*eyes was added as a new sample in

*p*(

*x*|

*G*). (5) By repeating (1) ∼(4) 10,000 times,

_{diff}*p*(

*x*|

*G*) was constructed. (6) the baseline distribution

_{diff}*p*(

*x*|

*G*) was constructed by shifting the mean of

_{null}*p*(

*x*|

*G*) to zero (the origin).

_{diff}*fitgmdist*was used to fit GMMs with one to six Gaussian components. The Bayes information criterion (BIC) was computed for each GMM fit. The best fitting GMM was determined from the elbow of the BIC versus number of Gaussian component function.

*p*(

*x*|

*B*) and a treatment distribution

*p*(

*x*|

*T*) (Figure 5A), specificity is the probability of correctly identifying a random sample

*x*

_{s,B}from

*p*(

*x*|

*B*), and sensitivity is the probability of correctly identifying a random sample

*x*

_{s,T}from

*p*(

*x*|

*T*) (Green & Swets, 1966; Swets & Pickett, 1982; US Department of Health and Human Services et al., 2007). We used the likelihood ratio \(\frac{{pdf( {x{\rm{|}}B} )}}{{pdf( {x{\rm{|}}T} )}}\) to define the criterion and computed sensitivity and specificity over a wide range of criteria to construct the receiver operating characteristic curves (ROC): (1-specificity) versus sensitivity. Accuracy is quantified by the area under ROC (AUROC), which is a criterion-free measure in signal detection theory (Figure 5B). (Green & Swets, 1966; Swets & Pickett, 1982; US Department of Health and Human Services et al., 2007) For a given pair of

*p*(

*x*|

*B*) and

*p*(

*x*|

*T*), the

*p*value is defined as the probability of a random sample

*x*

_{s,B}from

*p*(

*x*|

*B*) being incorrectly identified as from

*p*(

*x*|

*T*) with \(\frac{{pdf( {x{\rm{|}}B} )}}{{pdf( {x{\rm{|}}T} )}} \le 1\) (Figure 5A):

*p*(

*x*|

*B*) and

*p*(

*x*|

*T*) are test-level distributions in two conditions for individual eyes and

*p*(

*x*|

*G*) and

_{null}*p*(

*x*|

*G*) at the group level. We also computed the p values (Equation 15) between \(\theta _{i1}^{VA}\) and the VA component of \(\theta _{i1}^{{\rm{CE}}}\), and between \(\theta _{i1}^{CSF}\) and the CSF parameter components of \(\theta _{i1}^{{\rm{CE}}}\) to quantify the agreement between BIP and HBJM estimates in each test, with BIP as the baseline.

_{diff}^{CE}at the population level. We found large correlations among η

^{CE}components (Table 1). Specifically, \(\eta _{Threshold}^{{\rm{CE}}}\) was positively correlated with \(\eta _{Range}^{{\rm{CE}}}\) (

*r*= 0.516), but negatively correlated with \(\eta _{PG}^{{\rm{CE}}}\) (

*r*= −0.486), \(\eta _{PF}^{{\rm{CE}}}\) (

*r*= −0.583), and \(\eta _{BH}^{{\rm{CE}}}\) (

*r*= −0.288); \(\eta _{range}^{{\rm{CE}}}\) was negatively correlated with \(\eta _{PF}^{{\rm{CE}}}\) (

*r*= −0.494); \(\eta _{BH}^{{\rm{CE}}}\) was positively correlated with \(\eta _{PG}^{{\rm{CE}}}\) (

*r*= 0.577), but was negatively correlated with \(\eta _{PF}^{{\rm{CE\;}}}\)(

*r*= −0.536). The positive correlation between \(\eta _{Threshold}^{{\rm{CE}}}\) and \(\eta _{Range}^{{\rm{CE}}}\) is consistent with our previous observation that worse VA thresholds were associated with wider ranges (shallower slope of VABF) across individuals (Zhao, Lesmes, Dorr, Bex, et al., 2021; Zhao, Lesmes, Dorr, & Lu, 2021). The negative correlations between \(\eta _{VA}^{{\rm{CE}}}\) and \(\eta _{CSF}^{{\rm{CE}}}\) components were expected because increased VA threshold is associated with worse vision, and increased CSF is associated with better vision. The correlations among \(\eta _{CSF}^{{\rm{CE}}}\) components indicated that individuals with wider CSF bandwidths tended to have higher peak gains but lower peak frequencies.

*r*= −0.757 ± 0.028) and \(\tau _{i,BH}^{{\rm{CE}}}\) (

*r*= −0.900 ± 0.010), whereas \(\tau _{i,PG}^{{\rm{CE}}}\) was positively correlated with \(\tau _{i,BH}^{{\rm{CE}}}\;\)(

*r*= 0.551 ± 0.047). The correlations among \(\tau _{i,CSF}^{{\rm{CE}}}\) components resulted from the properties of the generative model of CSF: The three CSF model parameters jointly determine contrast sensitivities at all the spatial frequencies and different combinations of parameter values may result in similar probabilities of correct responses in some experimental conditions. On the other hand, correlations between \(\tau _{i,VA}^{{\rm{CE}}}\) and \(\tau _{i,CSF\;}^{{\rm{CE}}}\)were small because VA and CSF assessments were conducted separately.

*r*= −0.730 ± 0.210) and \(\theta _{i1,BH}^{{\rm{CE}}}\) (

*r*= −0.930 ± 0.024), whereas \(\theta _{i1,PG}^{{\rm{CE}}}\) was positively correlated with \(\theta _{i1,BH\;}^{{\rm{CE}}}\)(

*r*= 0.520 ± 0.271). \(\theta _{i1,PF}^{CSF}\) was negatively correlated with \(\theta _{i1,PG}^{CSF}\) (

*r*= −0.762 ± 0.271) and \(\theta _{i1,BH}^{CSF}\) (

*r*= −0.931 ± 0.026), whereas \(\theta _{i1,PG}^{CSF}\) was positively correlated with \(\theta _{i1,BH}^{CSF}\) (

*r*= 0.562 ± 0.355). These patterns are consistent with those at the individual level and our previous publications (Zhao, Lesmes, Dorr, Bex, et al., 2021; Zhao, Lesmes, Dorr, & Lu, 2021; Zhao, Lesmes, Hou, et al., 2021).

*p*values for the difference between the posterior distributions of \(\theta _{i1}^{VA}\) and \(\theta _{i1}^{CSF}\) and those of the corresponding \(\theta _{i1}^{{\rm{CE}}}\) components were 0.20 to 0.41 and 0.20 to 0.45. The average variance of the estimated \(\theta _{i1}^{{\rm{CE}}}\) components (mean = 0.0091 log10 units; range 0.00026∼0.059 log10 units) was 71.5% less than that of the corresponding \(\theta _{i1}^{{\rm{BIP}}}\) components (mean = 0.032 log10 units; range 0.00026∼0.30 log10 units) (

*t*

^{2}(5, 51) = 46,

*p*< 0.001), consistent with the well-known variance shrinkage effect of HBMs (Kruschke, 2014). Figure 10 shows histograms of the variance of \(\theta _{i1}^{{\rm{BIP}}}\) and \(\theta _{i1}^{{\rm{CE}}}\) components.

_{5D}combines signals from VA and CSF. The increased signal and reduced noise of CE

_{5D}improved signal detection. At the test level for individual eyes, median

*p*values only for CE

_{5D}from the five-dimensional HBJM analysis were all significant for short (

*p*< 0.02), medium (

*p*< 0.002), and long testing (

*p*< 0.0004) (Figure 12). The advantage provided by the HBJM was stronger when less information was available with the short and medium test lengths relative to those with the long test length.

*p*value

*p*values were close to 0 for VA

_{1D}, AULCSF

_{1D}, and CE

_{5D}at both the individual and group levels. CE

_{5D}exhibited advantage over VA

_{1D}and AULCSF

_{1D}between condition pairs with smaller differences (F1-F2, F2-F3, F1-F3), and the advantage was more profound when less information was collected with shorter test lengths or smaller numbers of eyes (see results for the short and long test lengths in Supplementary Materials B and C).

_{5D}exhibited significantly higher sensitivity at 95% specificity than VA

_{1D}between F1 and F2 (

*t*(13) = 2.9,

*p*= 0.006), between F1 and F3 (

*t*(13) = 1.9,

*p*= 0.04), and between F2 and F3 (

*t*(13) = 3.5,

*p*= 0.002), and AULCSF

_{1D}between F1 and F2 (

*t*(13) = 2.7,

*p*= 0.009), and between F2 and F3 (

*t*(13) = 2.3,

*p*= 0.02). CE

_{5D}exhibited significantly higher accuracy than VA

_{1D}between F1 and F2 (

*t*(13) = 2.6,

*p*= 0.01), and between F2 and F3 (

*t*(13) = 3.2,

*p*= 0.004), and AULCSF

_{1D}between F1 and F2 (

*t*(13) = 2.2,

*p*= 0.02), and between F2 and F3 (

*t*(13) = 2.0,

*p*= 0.03). Table 7 shows that larger number of eyes reached

*p*< 0.01 with CE

_{5D}relative to VA

_{1D}and AULCSF

_{1D}between F1 and F2, between F1 and F3, and between F2 and F3, indicating highest discriminative power for CE

_{5D}.

_{1D}, AULCSF

_{1D}, and CE

_{5D}were all 100% and the

*p*values were all <0.001, except for those of VA

_{1D}between F2 and F3 with sensitivity and

*p*values of 99% and 0.03.

_{5D}exhibited much more statistical power than VA

_{1D}and AULCSF

_{1D}(Figure 13). Only five, two, and five eyes were needed for CE

_{5D}to reached 95% sensitivity at 95% specificity between F1 and F2, F1 and F3, and F2 and F3. Many more eyes were needed for VA

_{1D}and AULCSF

_{1D}to reach the same level of specificity. Similar results were obtained in terms of accuracy and

*p*values.

_{5D}) from multiple assessment modalities by modeling data from two test modalities. Using information across all individuals and test modalities, the framework explicitly quantifies the statistical relationships between multiple test modalities and generates a multidimensional collective endpoint that can improve statistical power. We applied the framework to compute the five-dimensional CE

_{5D}, consisting of VA threshold, VA range, peak gain, peak spatial frequency, and bandwidth, from trial-by-trial qVA and qCSF data. The HBJM revealed strong correlations among CE

_{5D}components at the population, individual and test levels. Compared with the one-dimensional endpoints from BIP analysis (VA

_{1D}and AULCSF

_{1D}), the five-dimensional CE

_{5D}from the HBJM generated the highest sensitivity at 95% specificity, the highest accuracy, and the most significant p values at both the test and group levels. The higher sensitivity and accuracy and more significant p values of CE

_{5D}resulted mainly from the HBJM, which optimally combines information from test modalities, individuals, and tests to mutually constrain the estimates to reduce the uncertainties of the parameters and hyperparameters of the VA and CSF models (Figures 11 and 12).

_{5D}at the population level, the HBJM can be used to generate informative priors for qCSF based on qVA assessment or priors for qVA based on qCSF assessment by treating the to-be-observed data as missing data in the model. We previously developed the hierarchical adaptive design optimization (Gu et al., 2016; Kim et al., 2014) algorithm to simultaneously categorize each new patient into existing groups and provide the most informative prior for them based on their group membership for qCSF. A combination of hierarchical adaptive design with the collective endpoint analysis developed in this article may further improve the efficiency of adaptive tests in multiple assessment modalities.

_{5D}of VA and CSF. Generative models of other endpoints including clinical events (Yin et al., 2018) and structural endpoints can be developed and incorporated into the HBJM framework. They can be either statistical models that capture the probability of certain aspects of a human condition or mechanistic models based on human biology or physiology. For example, optical coherence tomography (OCT) (Huang et al., 1991) uses light to generate cross-section retina images. Anatomical changes (e.g., thickness of a retina layer) identified with OCT have been related to the severities of eye diseases (Vujosevic et al., 2022). A generative model can be constructed by specifying the probability of an eye disease and its severity as a function of the anatomical features of the retina, such as the thickness of retinal layers in different locations. The HBJM can be extended to compute CE from OCT, VA, and CSF and further facilitate disease staging and treatment monitoring and improve statistical evaluation in clinical trials.

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