Accounting for Standard Errors of Vision-Specific Latent Trait in Regression Models

Abstract

Purpose.: To demonstrate the effectiveness of Hierarchical Bayesian (HB) approach in a modeling framework for association effects that accounts for SEs of vision-specific latent traits assessed using Rasch analysis.

Methods.: A systematic literature review was conducted in four major ophthalmic journals to evaluate Rasch analysis performed on vision-specific instruments. The HB approach was used to synthesize the Rasch model and multiple linear regression model for the assessment of the association effects related to vision-specific latent traits. The effectiveness of this novel HB one-stage “joint-analysis” approach allows all model parameters to be estimated simultaneously and was compared with the frequently used two-stage “separate-analysis” approach in our simulation study (Rasch analysis followed by traditional statistical analyses without adjustment for SE of latent trait).

Results.: Sixty-six reviewed articles performed evaluation and validation of vision-specific instruments using Rasch analysis, and 86.4% (n = 57) performed further statistical analyses on the Rasch-scaled data using traditional statistical methods; none took into consideration SEs of the estimated Rasch-scaled scores. The two models on real data differed for effect size estimations and the identification of “independent risk factors.” Simulation results showed that our proposed HB one-stage “joint-analysis” approach produces greater accuracy (average of 5-fold decrease in bias) with comparable power and precision in estimation of associations when compared with the frequently used two-stage “separate-analysis” procedure despite accounting for greater uncertainty due to the latent trait.

Conclusions.: Patient-reported data, using Rasch analysis techniques, do not take into account the SE of latent trait in association analyses. The HB one-stage “joint-analysis” is a better approach, producing accurate effect size estimations and information about the independent association of exposure variables with vision-specific latent traits.

Introduction

There is increasing recognition that patient-reported outcomes (commonly referred to as questionnaires or instruments) are important to assess the impact of vision loss in ophthalmic research. ^1–9 For vision-specific instruments, vision functioning is one of the key latent traits. Recently, Rasch analysis has been used to estimate and evaluate latent traits, producing Rasch-scaled scores in preference to raw scores from questionnaire data. Numerous vision-specific instruments have been rigorously validated using Rasch analysis for daily living activities dependent on vision ^10,11 and for a spectrum of eye conditions, such as cataract, ^8,12,13 diabetic retinopathy, ⁶ AMD, ^4,14 and glaucoma. ^7,15,16 A literature review for instruments for the assessment of vision-specific quality of life identified 22 instruments. ¹⁷  

However, the handling of latent variables and their interpretation can be challenging, as they are not observed scores and so estimation of latent variables comes with uncertainty, commonly termed as SE. Subsequent applications of traditional statistics, such as t-tests and linear regressions, are not strictly appropriate for association analysis of latent data as they require dependent/outcome variables to be known. The analysis of latent variables requires its associated SE to be accounted for, as failure to do so may lead to biased estimates of correlations, associations, and statistical inferences. ^18,19 Multilevel item response theory (MLIRT) models have been shown to allow for a better estimation of relationships between predictor variables and MLIRT latent traits. ^20,21 However, appropriate handling of SEs of latent dependent variables in multilevel models are only more often practiced in some fields, such as in educational ^22,23 and psychometric ^24,25 research. 

The purpose of our study was to demonstrate the effectiveness of a modeling framework that integrates Rasch and regression models using the Hierarchical Bayesian (HB) approach that accounts for SEs of latent trait. This modeling is appropriate for the assessment of association effects related to the vision-specific latent trait, with proper treatment of its associated SEs for a more accurate and contemporary estimates of association effects. We compared the one-stage “joint-analysis” and two-stage “separate-analysis” model results using real data and assessed the performance of the methods in a simulation study based on the frequently used Andrich rating scale model. ²⁶  

Materials and Methods

Literature Review

We systematically reviewed publications that used Rasch analysis by searching the electronic databases of PubMed in the top four Ophthalmic journals (Ophthalmology, American Journal of Ophthalmology [AJO], British Journal of Ophthalmology [BJO], and IOVS) for relevant papers published up to July 2013, with the following search terms (formatted for PubMed search): 

1.  
   (rasch[All Fields] AND ("analysis"[Subheading] OR “analysis"[All Fields])) AND “Ophthalmology"[Journal]
2.  
   (rasch[All Fields] AND ("analysis"[Subheading] OR “analysis"[All Fields])) AND ("Am J Ophthalmol"[Journal] OR “american journal of ophthalmology"[All Fields])
3.  
   (rasch[All Fields] AND ("analysis"[Subheading] OR “analysis"[All Fields])) AND ("Br J Ophthalmol"[Journal] OR “british journal of ophthalmology"[All Fields])
4.  
   (rasch[All Fields] AND ("analysis"[Subheading] OR “analysis"[All Fields])) AND ("Invest Ophthalmol Vis Sci"[Journal] OR “investigative ophthalmology and visual science"[All Fields])

The strategy identified 70 articles and the full texts were reviewed (by WLW and XL) to identify studies having performed Rasch analysis on visual functioning questionnaire data. Of the 70 articles identified, the following were excluded: two letters, ^27,28 one study that applied the Rasch model to investigate inter-reader agreement, ²⁹ and another study that focused on the genetic components of the optic nerve head. ³⁰ The remaining 66 articles related to visual functioning data were reviewed for choice of Rasch model, implementation software, and sample size of studies (Table 1). 

Table 1

 

Summary of Articles Reviewed, n = 66

	Major Ophthalmic Journals				Total
	Ophthalmology	AJO	BJO	IOVS
No. of articles	8 (12.1)	3 (4.6)	8 (12.1)	47 (71.2)	66 (100.0)
Method
Andrich	6 (75.0)	1 (33.3)	5 (62.5)	31 (66.0)	43 (65.1)
PCM	0 (0.0)	0 (0.0)	1 (12.5)	4 (8.5)	5 (7.6)
Not indicated	2 (25.0)	2 (66.7)	2 (25.0)	12 (25.5)	18 (27.3)
IRT software
Winsteps	5 (62.5)	2 (66.7)	5 (62.5)	33 (70.2)	45 (68.2)
RUMM2020	3 37.5)	0 (0.0)	1 (12.5)	10 (21.3)	14 (21.3)
BIGSTEPS	0 (0.0)	0 (0.0)	0 (0.0)	3 (6.4)	3 (4.5)
Facets	0 (0.0)	0 (0.0)	1 (12.5)	0 (0.0)	1 (1.5)
Not indicated	0 (0.0)	1 (33.3)	1 (12.5)	1 (2.1)	3 (4.5)
Sample size
Median	1992	411	360	192	240
Range	108–14,817	135–3400	16–3280	22–7363	16–14,817

 Values are n (%) unless otherwise indicated. IRT, item-response theory; PCM, partial credit model.

*  RUMM2020 (Rasch unidimensional measurement models; RUMM Laboratory, Perth, WA, Australia).

†  BIGSTEPS (Bigsteps; Mesa Press, Chicago, IL, USA).

Pitfalls in Observed Analysis Framework (Two-Stage “Separate-Analysis” Procedure)

All 66 articles reviewed performed Rasch analysis to evaluate the validity, reliability, and measurement characteristics of instruments (i.e., visual functioning) for their population sample data. Most (86.4%) performed further statistical analysis on the Rasch-scaled score (i.e., vision-specific latent trait), such as performing correlations or linear regressions with visual acuity, demographic, or clinical data to assess the impact of visual impairment and other patients' characteristics or factors on visual functioning. 

However, none of the articles mentioned or discussed the potential bias in estimation of association effects and the underestimation of their SEs ³¹ having ignored the associated uncertainties involved in the estimation of the latent trait when used naively in subsequent association analysis. In the first stage, the vision-specific latent trait (i.e., vision functioning) was modeled and estimated given a set of item responses using a polytomous Rasch model (e.g., Andrich rating scale model when item response options are more than dichotomous, that is, three or more). In the second stage, relationships between the estimated Rasch-scaled data (treated as known outcome variable) and risk factors were analyzed using regression techniques. Ignoring the uncertainty regarding the abilities within the regression model may lead to biased estimation of association effects. ³¹ Underestimation of SEs may also result in false identified positive factors and hence mislead statistical inferences. ³¹  

Moreover, a key assumption in the Rasch model states that a change in the latent variable is completely described by the item characteristic functions (the relationship of the latent trait and responses of the items) and hence any association analysis on the latent trait with other covariates performed in the second stage can violate and contradict the key assumptions in the first stage of Rasch analysis (having assumed that vision-specific latent trait only depends on item response data). Such estimation procedure can cause serious underestimation of the SEs of the model parameters. ³¹  

Our literature review also observed that studies assessing visual-functioning traits were often conducted for moderately small sample sizes (median 240 with interquartile range of 497) and, together with response data (and Rasch-scaled scores) that are typically non-normally distributed, it is precarious to rely on asymptotic approximations and the properties of conditional maximum likelihood estimates obtained from analysis software without showing them to be accurate. ³² Similarly, validation inference based on correlations may not be accurate. 

Our study analysis and discussion focuses on the Andrich rating scale ²⁶ because it is the most frequently used polytomous Rasch model for vision-specific instruments. 

Andrich Rating Scale Model

The Andrich rating scale²⁶ model is an extension of the Rasch model³³ for polychotomous responses (i.e., three or more categories). The natural log of the likelihood ratio of adjacent response category probabilities is given by  

 

where Display Formula is the probability of person Display Formula on encountering item Display Formula would be observed in category Display Formula , Display Formula is ability trait for the Display Formula^th person (i.e., i = 1, …, n) , Display Formula is the item Display Formula difficulty parameter (i.e., k items means k = 1, …, k), and Display Formula is the threshold for category Display Formula (e.g., items with five categories means y is in the range of integers 1 to 5), which is constant across items.  

Proposed Analysis Framework (One-Stage “Joint-Analysis”)

A rigorous alternative is to combine the observed two-stage analysis procedure to overcome the problematic issues described above. Item response data structure are hierarchical, as item responses are nested within respondents and respondents also may be nested (e.g., patients nested in hospitals). Such relationships can be adequately explained by the multilevel Rasch model using the HB approach. Model parameters also can be incorporated and estimated from the item response data without having to condition on estimated person ability parameter (i.e., latent trait). In Supplementary Figure S1, a path diagram of this multilevel Rasch model is depicted and explained. 

For example, the combined model for linear regression (i.e., continuous latent trait outcome) can be written as  

 

where Display Formula is the beta coefficient from linear regression, Display Formula is the observed covariates of person Display Formula , and Display Formula is the residual random error. The HB approach provides an elegant execution of the multilevel Rasch modeling framework that allows the incorporation of explanatory variables or covariates at different levels of hierarchy by specifying parameters to come from a specific distribution with parameters and possibly hyper-parameters that are, themselves, estimated prior information.³¹ All model parameters can then be estimated simultaneously using the Monte Carlo Markov Chain method with the JAGS program.^34–36 The proposed procedure enables direct estimation of beta coefficients for association effects without having to explicitly know the latent trait measurements (i.e., personal ability). The JAGS codes used to fit our example model in one-stage “joint-analysis” HB approach are provided in the Supplementary Material, and can be altered readily to conform to different data structure.  

Comparison of Methods Using Real Data

Both the HB one-stage “joint-analysis” and the two-stage “separate-analysis” methods were performed to assess the relationship of reading and writing literacy on visual functioning (measured by a modified VF-9 questionnaire) using data from the Singapore Malay Eye Study (SiMES), ³⁷ a population-based cross-sectional study of 3280 Singaporean Malays older than 40. Previous studies suggest an association of inadequate literacy with systemic health and hence the influence of literacy on vision functioning (in addition to visual impairment), another aspect contributing to vision-specific quality of life, is important. ^38,39 Association of reading and writing literacy with visual functioning were adjusted for age, sex, language of interview, body mass index (BMI), occupation, marital status, income, housing type, education, smoking status, and presenting visual acuity in the better-seeing eye. 

Simulation Study

As there is no “gold standard” in the comparison of methods using real data, we conducted a simulation study to demonstrate the performance of our proposed HB one-stage “joint-analysis” approach as compared with the observed two-stage “separate-analysis” procedure. Two independent covariates ( Display Formula , Display Formula ), a continuous variable data, such as standardized age, and a binary variable, such as sex, were simulated with prespecified association effects ( Display Formula , Display Formula ) for the impact of these two covariates with the latent visual functioning ability parameter and, hence, these were considered as the “true” association effects or “gold standard” for reference when we re-run analysis on our simulated data using both analytical methods. Association estimates and their SEs from both approaches were computed to assess their performance, where estimates closer to the “gold standard” indicate higher accuracy and smaller SEs suggest greater precision. The calibration of nine item difficulty parameters, Display Formula was fixed according to table 3 of a study conducted by Lamoureux et al.⁴⁰ that performed a systematic evaluation of the reliability and validity of the visual functioning questionnaire (VF-11) using Rasch analysis that was later modified to nine items (VF-9) to tailor fit to the Asian population. We also investigated the empirical power for both approaches. We provide a detailed description of our simulation study in the Supplementary Material.  

Results

Table 1 shows the summary of articles reviewed in the four major ophthalmic journals that have performed Rasch analysis for visual functioning–related instrument data. Most (65.1%) performed Rasch analysis by using the Andrich rating scale model and most (68.2%) conducted Rasch analysis by using Winsteps software (Winsteps, Beaverton, OR, USA). The median sample size of these studies was 240 with interquartile range of 497. 

Associations of inadequate reading and writing literacy with visual functioning adjusted for potential confounding variables (models 1 and 2, respectively) derived from both approaches analyzed on our simulated data are shown in Table 2. Comparison of both approaches for Model 1 assessing reading literacy showed no difference in statistically significant factors identified but results were not consistent for writing literacy in Model 2. Inadequate writing was statistically significant based on the HB one-stage “joint-analysis” but was not significant in the two-stage “separate-analysis” approach. Smaller association effects also were estimated in the two-stage “separate-analysis” approach, which is consistent with the attenuation bias observed in the simulation study. 

Table 2

 

Comparison Between Approaches Using Real Data*

Approach	Model 1 (Reading Literacy)			Model 2 (Writing Literacy)
	2-Stage	P	1-Stage	2-Stage	P	1-Stage Approach
Age, y	0.000 (−0.008 to 0.008)	0.906	−0.001 (−0.020 to 0.014)	0.001 (−0.007 to 0.009)	0.892	−0.002 (−0.018 to 0.014)
Sex
Male vs. female	0.012 (−0.161 to 0.184)	0.894	0.073 (−0.330 to 0.474)	0.010 (−0.163 to 0.182)	0.913	0.053 (−0.265 to 0.399)
Language of interview
English vs. Malay	−0.227 (−0.501 to 0.046)	0.103	−0.442 (−1.026 to 0.142)	−0.229 (−0.502 to 0.045)	0.101	−0.482 (−1.001 to 0.027)
Others vs. Malay	0.659 (−0.685 to 2.002)	0.336	6.389 (−0.604 to 26.143)	0.649 (−0.696 to 1.994)	0.343	5.657 (−0.515 to 17.086)
BMI	0.002 (−0.010 to 0.014)	0.739	0.003 (−0.021 to 0.027)	0.002 (−0.010 to 0.014)	0.733	0.005 (−0.021 to 0.028)
Occupation
Office work	Reference		Reference	Reference		Reference
Service work	−0.060 (−0.263 to 0.143)	0.559	−0.127 (−0.633 to 0.380)	−0.062 (−0.265 to 0.141)	0.549	−0.133 (−0.655 to 0.343)
Factory work	−0.186 (−0.451 to 0.078)	0.166	−0.402 (−1.013 to 0.226)	−0.181 (−0.446 to 0.083)	0.178	−0.396 (−0.890 to 0.212)
Homemaking	−0.118 (−0.376 to 0.141)	0.371	−0.210 (−0.809 to 0.420)	−0.104 (−0.363 to 0.155)	0.431	−0.273 (−0.797 to 0.304)
Unemployed/others	−0.091 (−0.334 to 0.152)	0.463	−0.197 (−0.750 to 0.424)	−0.090 (−0.333 to 0.153)	0.468	−0.227 (−0.775 to 0.339)
Marital status
Never married	Reference		Reference	Reference		Reference
Married	0.072 (−0.165 to 0.309)	0.552	0.227 (−0.304 to 0.772)	0.074 (−0.164 to 0.311)	0.543	0.232 (−0.216 to 0.727)
Separated/divorced	−0.218 (−0.527 to 0.091)	0.167	−0.305 (−0.866 to 0.304)	−0.213 (−0.522 to 0.097)	0.177	−0.272 (−0.828 to 0.283)
Widowed	0.082 (−0.237 to 0.400)	0.615	0.307 (−0.348 to 0.990)	0.093 (−0.227 to 0.414)	0.568	0.324 (−0.253 to 0.929)
Income
> SGD$1000/mo	Reference		Reference	Reference		Reference
< SGD$1000/mo	0.049 (−0.114 to 0.212)	0.556	0.106 (−0.219 to 0.475)	0.049 (−0.115 to 0.212)	0.557	0.082 (−0.257 to 0.406)
Retirement income	0.065 (−0.127 to 0.256)	0.508	0.105 (−0.338 to 0.570)	0.067 (−0.125 to 0.259)	0.495	0.135 (−0.278 to 0.491)
Current housing status
1/2-room flat	Reference		Reference	Reference		Reference
3/4-room flat	−0.259 (−0.416 to −0.101)	0.001	−0.580 (−0.944 to −0.182)	−0.255 (−0.413 to −0.098)	0.002	−0.526 (−0.843 to −0.214)
5-room/private house	−0.281 (−0.477 to −0.085)	0.005	−0.617 (−1.147 to −0.194)	−0.278 (−0.474 to −0.082)	0.006	−0.547 (−0.926 to −0.192)
Education
No formal education	Reference		Reference	Reference		Reference
Primary education	−0.164 (−0.394 to 0.066)	0.162	−0.360 (−0.930 to 0.135)	−0.160 (−0.393 to 0.074)	0.180	−0.342 (−0.789 to 0.115)
Secondary education	−0.152 (−0.411 to 0.107)	0.249	−0.318 (−0.976 to 0.237)	−0.151 (−0.414 to 0.112)	0.261	−0.297 (−0.794 to 0.201)
Poly/University	−0.124 (−0.427 to 0.180)	0.424	−0.178 (−0.965 to 0.508)	−0.120 (−0.426 to 0.187)	0.443	−0.147 (−0.75 to 0.506)
Smoking status
Past or never	Reference		Reference	Reference		Reference
Current	−0.203 (−0.359 to −0.048)	0.010	−0.461 (−0.802 to −0.141)	−0.201 (−0.356 to −0.045)	0.012	−0.411 (−0.706 to −0.159)
PVA of better eye	−0.934 (−1.165 to −0.703)	0.000	−1.568 (−2.112 to −1.085)	−0.950 (−1.180 to −0.720)	0.000	−1.468 (−1.999 to −1.022)
Read
Yes vs. no	0.297 (0.026 to 0.568)	0.032	0.578 (0.051 to 1.193)
Write
Yes vs. no				0.245 (−0.010 to 0.500)	0.060	0.462 (0.0496 to 0.966)

 Model 1 adjusted for factors in table (excluding writing literacy); model 2 adjusted for factors in table (excluding reading literacy). Data represented are difference in Rasch-scaled score with 95% credible interval in parentheses. P value less than 0.05 suggests evidence of associations in two-stage “separate-analysis” approach. The 95% credible intervals not including 0 suggest evidence of associations in one-stage “joint-analysis” approach. PVA, presenting visual acuity; SGD, Singapore dollar.

*  Based on SiMES data.

Simulation results (based on sample size of 300) on the association effects and their SEs of a continuous measurement, such as standardized age ( Display Formula ) and that for a binary factor, such as sex ( Display Formula ), with the vision-specific latent trait compared with the “gold standards” for both approaches are depicted in the Figure. There is greater inaccuracy (average of 5-fold increase in bias) in effect size estimations from the frequently used two-stage “separate-analysis” procedure compared with the proposed HB one-stage “joint-analysis” approach. We also observed an attenuation bias in estimations (shrunk toward zero) from the two-stage procedure. Smaller SEs for estimates were expected for the two-stage procedure having assumed no uncertainty in the latent trait measurements, but the slightly larger SEs (average of one-tenth–fold difference) from the HB approach suggest comparable precision for a more accurate estimation of associations. The impact of sample size on the level of bias tabulated in Supplementary Table S1 shows marked decrease in bias for estimations from HB one-stage “joint-analysis” with increasing sample size (from 0.011 to 0.001 and −0.010 to −0.003 for betas 0.2 and 0.5, respectively, when sample size increases from 50 to 600). Bias is negligible with sufficiently large sample size, such as 100 or more, in our study. However, the two-stage “separate-analysis” approach consistently produced negative bias regardless of sample size (little difference in bias with sample sizes from 50 to 600). Hence, bias cannot be eliminated/reduced even with large sample size. The negative bias reflects the attenuation bias observed in the Figure. Furthermore, the level of power in our proposed HB approach was comparable to the two-stage approach (no difference for beta ≥ 0.5 at 5% significance level and less than one-twelfth–fold difference for beta at 0.2 for both continuous and categorical variables), despite taking into account uncertainty in the latent trait (Supplementary Table S2).  

Figure

 

Association effects and SEs: comparison of proposed one-stage HB and observed two-stage analysis framework from simulation results (attenuation bias: red dots were shrunk toward zero and greater bias observed from the greater distance away from the black dots, the true value as compared with blue dots from the one-stage approach).

Discussion

This is the first study to assess the performance of two regression models for visual functioning, comparing the frequently used two-stage “separate-analysis” method (ignoring SE of the dependent latent trait) and our proposed one-stage “joint-analysis” approach in terms of estimation accuracy of association effects, precision of their SEs, and power. Association effect sizes from our real data analysis were observed to be smaller in the two-stage “separate-analysis” approach with slightly tighter intervals and the identification of significant factors between approaches were different. Our simulation study results (assessing methods performance) provided support for these observations. Attenuation bias, the shrinking of estimations toward zero, was found using the two-stage procedure (a phenomenon expected from ordinary least-squares regression of explanatory variables with measurement errors) that explains the (artificial) smaller association effects observed in our real-data analysis. This attenuation bias exists regardless of sample size. 

The one-stage “joint-analysis” approach allows the estimation of all model parameters simultaneously and hence integrates visual functioning in the regression model accounting for its SEs. Simulation results also showed that the one-stage “joint-analysis” method produced highly accurate estimations (average of 5-fold decrease in bias), with comparable precisions and power as compared to the commonly used two-stage “separate-analysis” procedure. The magnitude of SE also affects the size of attenuation bias. Accurate estimation of effect size and its variance are both critical to statistical significance testing results that directly influence our interpretation of risk factors. Hence, moving forward, the one-stage “joint-analysis” approach is preferred when we perform regression analysis with dependent outcome that is essentially a latent variable being derived from some prior analyses. Such analytical frameworks were already adopted to assess impact on longitudinal vision outcome in terms of vision-related quality of life using multilevel item response models, ^41,42 and similarly for some other diseases. ^43,44  

Our model codes provided in the Supplementary Material can be altered readily to analyze data from any instruments/questionnaires with any choice of Rasch or item response theory models (e.g., Partial Credit model, Graded Response model) and complexity of multilevel regression models as depicted in Supplementary Figure S1 to adequately describe the hierarchical data structure, incorporate different sources of uncertainty, and inclusion of explanatory covariates at different levels. Furthermore, latent variables also can be analyzed as independent covariates (as required) instead of an outcome variable used in our simulation example. 

In our literature review, visual functioning rating scale instruments/questionnaires were mainly validated using Rasch analysis for their well-known scaling and measurement properties. Without a realist interpretation of latent variables, actual Rasch-scaled scores do not have straightforward meaning and its interpretations are based on relative comparisons of the scaled scores (i.e., relative difference tells us how much more of persons' visual functioning ability compared with another). Many reviewed articles provide ready-to-use spreadsheets that convert raw scores entered to Rasch-scaled scores for their respective instruments to benefit clinicians and researchers unfamiliar with Rasch analysis who may wish to use its scoring benefits. It is important to note that the population of respondents plays an important part in the probability model for each response and so the personal ability and item parameters always will be estimated with respect to a population. Hence, ready-to-use spreadsheets should be used only on different samples of individuals from the same population as validated in the article and that Rasch-scaled scores are not comparable between studies from different populations unless it happens (rarely) that both populations have identical item characteristic functions. Researchers also were unaware of the SEs associated with Rasch-scaled scores and performed further analysis directly with simple statistical tests, such as independent t-tests to examine between-group differences in the instrument scores across various sociodemographic variables and levels of vision impairment. 

The strength of our study includes the simulating datasets with various prespecified association effects that act as “ground truth” to enable the comparison between methods. The simulating conditions in our study were, however, limited to only the Andrich Rating Scale model with linear regression analysis for covariates at the respondents' level. Statistical computations are necessary for applying our proposed analysis using the HB approach and some background in statistics and programming skills are needed to alter codes to conform to other conditions. 

In conclusion, there is a need to account for SEs associated with vision-specific latent trait in association analysis. We demonstrated that our HB one-stage “joint-analysis” approach is a better method that produces greater accuracy with comparable power and precision in estimation of association effects compared with the frequently used two-stage procedure, despite taking into account greater uncertainty due to the latent trait. The study finding has direct implications in our inference drawn from statistical significance of risk factors. 

Supplementary Materials

pdf S1 

Acknowledgments

Supported by Grant 0796/2003 from the National Medical Research Council, Singapore. 

Disclosure: W.L. Wong, None; X. Li, None; J. Li, None; T.Y. Wong, None; C.-Y. Cheng, None; E.L. Lamoureux, None