**Purpose.**:
Evidence is growing that dry eye represents a common disease process resulting from a number of underlying pathologies that impact the ocular surface and that clinical estimates of dry eye severity reflect the magnitude of a single dry eye disease state variable, Θ. A theory for estimating Θ from scaled clinical observations is developed, and the hypothesis is tested that Θ exists.

**Methods.**:
The theory is developed around three assumptions: (1) a monotonic function unique to each person and indicator maps the indicator onto Θ, (2) between-person differences in mapping functions are random, and (3) observed indicator values include random perturbations. Data recently published by Sullivan and his colleagues were digitized from scatter plots of seven different indicators versus a composite severity score (square root of summed weighted squared indicator scores).

**Results.**:
The data were analyzed with a model derived under the specific assumptions that between-person variance in mapping functions is independent of the indicator value and random perturbations in observed indicator values are normally distributed. Tear osmolarity was the most sensitive indicator, and tear breakup time was the least. The distribution of residuals (squared difference between observed and predicted indicator values) agreed with model expectations for all indicators except tear osmolarity, which had larger residuals than expected, and the composite severity score, which had smaller residuals than expected.

**Conclusions.**:
The results are consistent with the existence of a single latent dry eye disease state variable. Only tear osmolarity does not appear to map monotonically and/or unidimensionally onto the latent variable.

^{ 1 }Dry eye appears to be the consequence of many different types of disorders but also appears to have a common set of clinical characteristics that progress through similar stages of severity.

^{ 2 }and the International Dry Eye Workshop (DEWS)

^{ 3 }recommended two major pathogenic classes: tear-deficient dry eye, which both workshops further subdivided into Sjogren and non-Sjogren tear deficiency, and evaporative dry eye, which DEWS subdivided into “intrinsic” causes (e.g., Meibomian gland dysfunction) and “extrinsic” causes (e.g., contact lens wear).

^{ 4 }suggested a different approach to classifying dry eye, which they recommended be called “dysfunctional tear syndrome” (DTS). The panel subdivided DTS into groups based on appropriate therapeutic strategies and concurred that DTS should be classified as having or not having clinically evident inflammation, with or without lid margin disease, and with or without abnormal tear distribution and clearance. They also recommended that four levels of severity of DTS could be discriminated clinically for the purpose of choosing different treatment algorithms. The DEWS analysis endorsed this concept that treatment algorithms were most appropriately based upon stratification of patients according to disease severity.

^{ 5 }

^{ 6 }employed cluster analysis on the results of a battery of clinical tests. They concluded from the statistical patterns in the data that dry eye patients could be sorted into nine different disease classes based on a tree structure that starts with the presence or absence of Meibomian gland drop out, followed by average lipid viscosity, then evaporation rate or Schirmer test value (depending on average lipid viscosity), and finally, lipid volume.

^{ 7 }But, consistent with the idea of a common underlying dry eye disease state variable, the Delphi panel and DEWS agree that irrespective of cause, clinicians can estimate the magnitude of dry eye severity from observed clinical signs and symptoms.

^{ 4,5 }To reconcile such diverging lines of evidence, Baudouin proposed that despite a plethora of disorders that result in dry eye, it appears that once started, the progression of the dry eye syndrome follows a common cascading and reciprocating course that can be linked to hyperosmolarity of the tear film.

^{ 8 }Several investigators have proposed that tear film hyperosmolarity is the common denominator for dry eye,

^{ 9 }should be considered the dry eye diagnostic gold standard,

^{ 10 }and is the best single measure of dry eye disease severity.

^{ 11,12 }

^{ 13 }To construct this variable, an expert panel of clinicians employed a modified version of the DEWS severity scale to rate the strength of evidence of dry eye disease on a scale of 0 to 1 for a range of scores for each of six physiologic tests and for a dry eye symptom summary score (ocular surface disease index [OSDI]

^{14}). Monotonic functions were fit to the trends in panel members' disease severity ratings versus observed physiologic sign or symptom scores. The fitted functions were inverted, and all the dry eye sign and symptom data were transformed to the hypothetical continuous dry eye disease severity scale. Finally, weighting coefficients were estimated to compensate for correlations between sets of scores, and a composite score was generated for each person as the square root of the sum of squared weighted transformed scores. All of these operations are monotonic and treated as independent, and therefore, the calculated composite dry eye disease severity variable must be monotonic with the raw observation score for each physiologic sign and with the OSDI symptom score. Correlations of raw observation scores with the composite dry eye disease severity score varied across signs and symptoms from 0.17 for the Schirmer test to 0.55 for tear film osmolarity.

^{ 11,12 }

^{ 13 }This ordering implies the existence of a pattern in clinical observations on dry eye patients that is congruent with a latent disease state variable.

^{ 2,3 }To avoid confusing the disease with its signs and symptoms, we will call this latent disease state variable Θ, a single variable that is assumed to underlie the clinically observed DTS that formally defines dry eye disease.

^{ 4 }For the purpose of theory development, we define Θ to be continuous and unbounded (i.e., has infinite limits). The clinician cannot directly observe Θ in a patient; rather, the clinician observes the indicator variables that clinically constitute DTS. The indicator variables are labeled

*I*for the

_{j}*j*indicator. Indicator variables are signs and symptoms that can be ordered in severity and expressed on a dichotomous, polytomous, or continuous scale. Indicator variables act as surrogates for Θ. Examples of dry eye indicator variables are millimeters of wetting for the Schirmer test, tear breakup time (TBUT), tear osmolarity, and OSDI summary score. We assume that the value of each indicator variable carries information about the magnitude of Θ. However, the indicator variable also reflects other patient-specific physiologic and environmental traits unrelated to Θ that idiosyncratically distort or mask the information it carries about the magnitude of Θ.

^{th}*for patient*

_{n}*n*. Each indicator variable maps onto Θ

*by way of the person-specific and indicator-specific monotonic and, most likely, nonlinear mapping function, Θ*

_{n}*=*

_{n}*u*(

_{nj}*I*).

_{j}*û*(

_{j}*I*) =

_{j}*E*{

*u*(

_{nj}*I*)}.

_{j}*j*across a representative sample of

*N*persons, The person-specific error function is the difference between the mapping function for person

*n*and the expected mapping function for the population,

*η*(

_{nj}*I*) =

_{j}*u*(

_{nj}*I*) −

_{j}*û*(

_{j}*I*).

_{j}*I*but has a variance across persons that can be a function of

_{j}*I*, that is,

_{j}*VAR*{

*η*(

_{nj}*I*)} =

_{j}*E*{

*η*(

_{nj}*I*)

_{j}^{2}}.

*and*

_{n}*I*, but within each person unknown forces, such as physiologic and environmental variables that are unrelated to Θ

_{j}*can perturb that relationship. A second type of observation error arises from the operations themselves that define the observation. Patient self-reports, ratings by judges, and physical measurement instruments are inherently unreliable and can be biased. Given the fixed relationship between Θ*

_{n},*and*

_{n}*I*, the expected value of indicator variable

_{j}*j*for person

*n*with disease state Θ

*is*

_{n}*Î*(

_{j}*n*), but the observed value,

*I*(

_{j}*n*), may not agree. Thus, the observation error in

*I*for person

_{j}*n*, irrespective of its source, is

*e*=

_{nj}*I*(

_{j}*n*) −

*Î*(

_{j}*n*).

*j*to be observed having the discrete value

*x*for person

*n*, we assume that where

*C*is the criterion value of indicator variable

_{jx}*j,*on a theoretically continuous observation scale, that must be exceeded to assign the discrete value

*x*to the observation and

*C*

_{jx}_{+1}is the criterion value of indicator variable

*j*that must be exceeded to assign the discrete value

*x*+1 to the observation. There are

*m*+1 intervals with

*m*criterion values (interval boundaries) ranging from

*C*

_{j}_{1}to

*C*[

_{jm}*C*

_{j}_{0}and

*C*

_{jm}_{+1}are the scale boundaries; if

*I*(

_{j}*n*) is unbounded, then

*C*

_{j}_{0}= −∞ and

*C*

_{jm}_{+1}= +∞]. To make the stochastic nature of the observation explicit, we re-express the preceding observation rule in terms of the observation error, where

*Î*(

_{j}*n*) is a fixed variable and

*e*is a random variable.

_{nj}*n*. To make between-person random variability explicit, this expression for transforming indicator variable units can be re-expressed in terms of the expected population mapping functions and person-specific error functions If we define a new random variable,

*ε*=

_{nj}*û*(

_{j}*Î*(

_{j}*n*) +

*e*) −

_{nj}*û*(

_{j}*Î*(

_{j}*n*)), then expression 2a can be re-expressed as

*δ*=

_{njx}*η*(

_{nj}*C*) −

_{jx}*η*(

_{nj}*Î*(

_{j}*n*) +

*e*) –

_{nj}*ε*, expression 2c can be simplified to Expressions 1b and 2d constitute the axiomatic theory for the relation of clinical indicator variables to a single latent disease state variable.

_{nj}*δ*. An open-ended model of the covariance matrix with respect to persons, items, and interval boundaries is likely to be overdetermined. To constrain the model, we must impose requirements on the covariance matrix that have to be met in order for observations of indicator variables to be considered valid surrogates from which measurements of the latent Θ variable can be estimated.

_{njx}*η*is constant within an indicator (i.e., independent of the value of the argument of the function). In this case, all variance in

_{nj}*δ*can be attributed to variance in

_{njx}*ε*. When these conditions are satisfied,

*VAR*{

*δ*} is independent of

_{njx}*x*, and expression 2d becomes or, from the definition of

*ε*, expression 3a is identical to We will refer to the class of models that include this constraint as “added independent noise” (i.e., added

_{nj}*ε*) models. We say the added noise is independent because it does not depend on the magnitude of the indicator variable. This class of models would include Samejima's graded response model in psychometrics

^{15}and proportional odds ordinal logistic regression models in statistics.

^{16}(See the Appendix for more detail on the form of added noise models.)

*η*(…) depends on the argument of the error function. When these conditions are satisfied, the joint probability density function for

_{nj}*δ*depends on the response category threshold, the indicator variable, and the sample of persons. We will refer to the class of models that incorporate this constraint as “mapping noise” models. Mapping noise refers to between-person differences in the shapes of the mapping functions. This class of models includes Masters's partial credit

_{njx}^{ 17 }and Muraki's generalized partial credit

^{ 18 }latent variable models in psychometrics. (See the Appendix for additional detail on mapping noise models.)

*value for person*

_{n}*n*. Therefore, if there were no mapping noise or added noise,

*E*{Θ

*} =*

_{n}*û*(

_{i}*Î*(

_{i}*n*)) =

*û*(

_{j}*Î*(

_{j}*n*)) for all combinations of indicators

*i*and

*j*observed in person

*n*. If

*J*different indicators are observed for a sample of

*N*persons, we can define a summary variable for each person,

*S*, that is a monotonic function,

_{n}*v*(…), of the vector of

*J*indicator variables for person

*n*, that is, Because the indicator variables have added noise,

*e*, we can re-express the summary variable in terms of the expected values of

_{nj}*I*and an added noise term for person

_{j}*n*,

*s*, that is, where the expected value of

_{n}*s*across persons is zero. We have constrained

_{n}*S*to be a monotonic function of each of the indicator variables, and therefore, we can posit a mapping function that monotonically transforms

_{n}*S*to Θ

_{n}_{n}, and, as for each of the indicator variables,

^{19}

*w*(…), between the summary score,

_{j}*S*, and the expected values of the indicator variables, that is, where

*n*(

*S*) refers to person

*n*who has a summary score of

*S*and

*N*refers to the total number of persons with a summary score of

_{S}*S*. One can estimate

*w*(…) for each indicator variable from observations of indicator variable values for each person, an acceptably defined summary variable function,

_{j}*v*(…), and regression analysis employing equation 7.

*w*(…) functions have been estimated from the sample data, the next step is to transform each observed indicator variable value to a summary variable value for each person/indicator combination, that is, This summary variable estimated for each person/indicator pair includes random noise. So at this point, it is necessary to make assumptions about variances and covariances and the form of the joint density function for δ

_{j}*to reduce the general theory to a specific measurement model. When the measurement model has been defined, maximum likelihood estimation procedures can be applied to the matrix of*

_{njx}*S*values to estimate Θ

_{nj}*for each person and*

_{n}*û*(

_{j}*C*)for each discrete value of each indicator,

_{jx}*C*,

_{jx}^{20}which in turn leads to an estimate of each mapping function,

*û*(

_{j}*Î*).

_{j}^{ 13 }They obtained measures of seven different DTS indicator variables from each of 299 eye clinic patients and volunteer staff members in a prospective multicenter study; the DTS indicator variables were tear osmolarity (mOsm/liter), Schirmer test (millimeters), TBUT (seconds, average of three repeated observations), OSDI score,

^{ 14 }Meibomian gland score (using the Bron/Foulkes scoring system

^{21}), corneal staining with sodium fluorescein dye under cobalt blue illumination (using the NEI/Industry Workshop scale

^{2}), and conjunctival staining with sodium lissamine green dye (using the NEI/Industry Workshop scale

^{2}). Thus, they obtained a set of observed DTS indicator variable values for each participant. Using expert clinician ratings of disease severity and independent component analysis on the sets of indicator values to correct for correlations between measures, Sullivan and his colleagues then computed a disease summary score, the composite disease severity score, for each participant.

^{ 13 }We digitized the displayed data for each indicator variable. Because of overlapping data points, some data were missed. We were able to obtain 267 osmolarity points (89%), 277 Schirmer's test points (93%), 237 Meibomian score points (79%), 258 TBUT points (86%), 248 OSDI score points (83%), 214 conjunctiva stain score points (72%), and 190 cornea stain score points (64%). It was not possible to match the participants with the digitized points.

**Figure 1.**

**Figure 1.**

_{n}=

*u*(

_{nj}*I*(

_{j}*n*)) =

*û*(

_{j}*Î*) +

_{j}*η*(

_{nj}*Î*+

_{j}*e*) +

_{nj}*ε*for all indicator variables and for the composite disease severity score, and testing the hypothesis that all variables map onto Θ, as predicted by equation 6. Because of the constraints we are working under with the digitized data, we employ the added independent noise model, which is built on the assumption that

_{nj}*VAR*{

*δ*} ≅

_{njx}*VAR*{

*ε*} and that expression 2a is a sufficient description of the observations. We also make the simplifying assumption that the probability density function for

_{nj}*ε*is a logistic, approximately normal, with an expected value of 0 and a standard deviation of

_{nj}*σ*, that is, Monotonicity of

_{j}*û*(

_{j}*Î*) requires that

_{j}*P*(

*ε*<

_{nj}*û*(

_{j}*C*) −

_{jx}*û*(

_{j}*Î*(

_{j}*n*)) =

*P*(

*e*<

_{nj}*C*−

_{jx}*Î*(

_{j}*n*)), where The logit of equation 9 is or in terms of the logit for the indicator variable, which must be the same as the logit for equation 9, and using the definition introduced earlier,

*I*(

_{j}*n*) =

*Î*(

_{j}*n*) +

*e*, where

_{nj}*σ*is the standard deviation of the

_{j}*ε*distribution and

_{nj}*is the value of Θ, that is,*

_{j}*û*(

_{j}*Î*(

_{j}*n*)), at the median value of the indicator variable

*j*for the sample of persons. We can conclude from equation 10 that the mapping functions for the indicator variables are expected by the model to be linear transformations of the logit of

*P*(

*I*(

_{j}*n*) <

*C*). (N.B., the derivation of equation 10 employs sign notation that implies the monotonic relationship between the indicator variable and Θ is expected to be positive. In some cases, the monotonic relationship will be negative, e.g., Meibomian gland drop-out, in which case the sign notation would have to be modified accordingly.)

_{jx}*P*(

*I*(

_{j}*n*) <

*C*), as a function of

_{jx}*C*for each indicator variable and for the composite disease severity score (filled circles). The solid curve in each panel is the least squares fit of an atheoretical third-order polynomial (the estimated equations are displayed in their respective panels). The ordered pair of indicator variable score and composite disease severity score, (

_{jx}*I*(

_{j}*n*),

*S*), for each digitized data point from the study of Sullivan et al.

_{nj}^{ 13 }was then transformed to the ordered pair (

*û*(

_{j}*I*(

_{j}*n*)),

*û*(

_{s}*S*)) using the estimated polynomial equations from the regressions illustrated in Figure 1. Scatter plots of these transformed ordered pairs are illustrated in Figure 2.

_{nj}**Figure 2.**

**Figure 2.**

*E*{

*û*(

_{j}*I*(

_{j}*n*))} =

*E*{

*û*(

_{s}*S*)} =

_{nj}*E*{Θ

*} and substituting*

_{n}*I*(

_{j}*n*) or

*S*for

_{nj}*C*in equation 10, leads to the conclusion or if we rescale Θ

_{jx}*to*

_{n}*and*

_{s}*σ*, then where the rescaled Θ

_{s}*is*

_{n}*σ*)(Θ

_{s}*−*

_{n}*). Figure 2 illustrates scatter plots of*

_{s}*û*(

_{s}*S*) versus

_{nj}*û*(

_{j}*I*(

_{j}*n*)) for each indicator (filled circles). The solid lines are bivariate regression lines that represent the principal component (i.e., orthogonal fit with unit variances). The Pearson correlations between the logits are indicated in each panel of the figure. These correlations are higher than those reported by Sullivan et al.

^{13}for the raw indicator observations versus the composite severity score (

*P*values for differences between correlations range from <0.0001 for TBUT to 0.06 for cornea staining). The slope of the regression line for each indicator variable in Figure 2 corresponds to

*σ*and the intercept corresponds to (1.8/

_{j/}σ_{s}*σ*)(

_{s}*−*

_{j}*) in equation 10.*

_{s}*= 0 and*

_{s}*σ*= 1.8. The intercept can be interpreted as an index of the sensitivity of the indicator to the disease state variable relative to the composite disease severity score. That is,

_{s}*û*(

_{s}*S*) =

*at the median value of*

_{j}*I*. So, if

_{j}*is low, the indicator variable is identifying more people as being in a low dry eye disease state than indicator variables for which*

_{j}*is high. Thus, TBUT is the least sensitive indicator of dry eye disease and osmolarity is the most sensitive indicator. As might be expected from any central tendency variable, the composite disease severity score is less sensitive than four of the indicators but more sensitive than the other three indicators.*

_{j}**Table 1.**

**Table 1.**

Indicator Variable | SD | |

Osmolarity | 1.07 | 2.32 |

Cornea stain score | 0.53 | 2.12 |

Conjunctiva stain score | 0.48 | 2.29 |

OSDI score | 0.18 | 2.27 |

Composite severity score | 0.00 | 1.80 |

Meibomian score | −0.10 | 2.23 |

Schirmer test | −0.72 | 1.76 |

TBUT | −4.93 | 2.43 |

*σ*, relative to the standard deviation of the added noise for the composite disease severity score,

_{j}*σ*= 1.8, also are compared in Table 1. The estimated standard deviations are similar for all indicator variables; however, they are somewhat lower for the Schirmer test and the composite disease severity score. The standard deviation of the added noise determines the standard error of the estimate of Θ, which, relative to the standard deviation of the estimated measures, defines measurement reliability. Because of limitations imposed by the data digitization, we cannot combine information from all the indicators to estimate a single value of Θ for each person, so at this time, we cannot offer meaningful estimates of the standard errors.

_{s}*û*(

_{j}*I*(

_{j}*n*)),

*û*(

_{s}*S*)) that defines each data point in Figure 2. Figure 3 is a reproduction of the top left panel in Figure 2 for osmolarity but with all except one data point removed. The regression line was fit to minimize the perpendicular distances of the data points from the line. Thus, in this example, the estimated value of Θ, using the transformed value of the observed osmolarity and the transformed value of the composite disease severity score for that person, is the point that falls on the regression line where the perpendicular from the data point (filled circle) to the line intersects the line (open circle). The corresponding ordered pair for the estimated point is (

_{nj}*û*(Î

_{j}*(*

_{j}*n*)),

*û*(

_{s}*Ŝ*)), the values of which are indicated by the dashed lines. The coordinates of the estimated point for each person were computed from the horizontal distance of the observed data point from the line, which is the difference between the logit for the composite disease severity score and the logit for the osmolarity value in osmolarity logit units (Δ

_{nj}*x*in the figure), the vertical distance of the observed data point from the line, which is the same difference between logits, but in composite disease severity score logit units (Δ

*y*in the figure), and the slope of the regression line, that is,

*σ*.

_{j/}σ_{s}**Figure 3.**

**Figure 3.**

*for each indicator were linearly transformed to composite disease severity score logit units using the appropriate regression coefficients. These estimated values of Θ*

_{n}*were substituted for*

_{n}*û*(

_{j}*Î*(

_{j}*n*)) in equation 9 and the probability of observing the indicator value

*C*was calculated for all possible values of the indicator variable in its defined range. These probabilities then were used to estimate the expected value of the indicator variable for each person

_{jx}*E*{

*} using equation 12 but substituting*

_{n}*C*multiplier in the numerator. These expected values were used to calculate a squared residual for each observed indicator value, (

_{jx}*I*(

_{j}*n*) −

*E*{

*I*|Θ

_{j}*})*

_{n}^{2}, and the squared residual expected by the model,

*E*{

*} −*

_{n}*E*{

*I*|Θ

_{j}*}*

_{n}^{2}. If the observations,

*I*(

_{j}*n*), adhere to the assumptions of the measurement model, then the ratios of these squared residuals across persons are expected to be distributed as χ

^{2}normalized to its degrees of freedom,

^{ 20 }with one degree of freedom for each indicator (because two observations for each indicator are used to estimate Θ

*).*

_{n}*was estimated from a pairing of the composite disease severity score with each of the indicator variables (see legend). The solid curve is the expected quantized (Δratio = 0.1) normalized χ*

_{n}^{2}distribution for one degree of freedom. The histogram for each pairing of the composite severity score with the indicator variable is not consistent with the expected normalized χ

^{2}distribution. The expected value of the χ

^{2}distribution is 1. The means of the ratios of squared residuals for the composite severity scores range from 0.11 when paired with tear film osmolarity to 0.49 when paired with conjunctiva stain score. Mean squared residual ratios of much less than 1 indicate that the observed composite severity scores are in better agreement with predicted scores than would be expected by the model. This situation can occur when the distributions of indicator values that make up the composite severity score are biased toward the ceiling or floor of the scale (the distribution of composite disease severity scores is biased toward the normal limit of 0; mean = 0.3, skew = 0.8 ).

**Figure 4.**

**Figure 4.**

^{2}distribution for one degree of freedom (N.B. the vertical scale changes across panels). With the exception of osmolarity, the histogram for each indicator variable agrees with the expected normalized χ

^{2}distribution. For six of the indicator variables, the mean squared residual ratio is close to or less than 1 (Schirmer test = 0.94; Meibomian score = 0.31; TBUT = 0.38; OSDI = 0.50; conjunctiva stain = 0.81; cornea stain = 0.48). However, in the case of osmolarity, the distribution of squared residual ratios does not match expectations. Besides the difference in the shape of the distribution, the observed squared residuals for osmolarity are larger than those expected by the model (mean = 1.28). This result indicates that confounding variables or operations in the person sample make contributions to observed osmolarity that distort the estimates of Θ.

**Figure 5.**

**Figure 5.**

^{ 11,22 }But implicit in the testing and in the goals of consensus workshops is the premise that dry eye disease can be represented by a single disease severity variable that can vary in magnitude between patients and over time and can be used by clinicians to guide treatment or to measure response to therapeutic interventions. The concept of a single dry eye disease severity variable is a theoretical construction. However, simply postulating its existence is insufficient; it is necessary to also postulate its relationship to surrogate clinical indicator variables. In the theory presented here, we postulated that the relationship between indicator variables and the disease severity variable is monotonic and perturbed by random factors. More explicit models of the relationship devolve from explicit assumptions about the sources and probability densities of the random perturbations. The assumption of monotonicity is a requirement of measurement. If the relationship for an indicator variable is not monotonic, the variance in the squared residuals will exceed expectation and that variable will be flagged as possibly not making a valid contribution to the measurement (e.g., osmolarity in Figure 5). In the analysis presented here, we assumed that the perturbations can be characterized as logistically distributed (i.e., approximately normal) independent added noise. Within the framework of monotonicity and the explicit added noise assumption, and within the limitations imposed on the analysis by the data recovery from the publication of Sullivan et al.,

^{ 13 }we tentatively conclude that a single dry eye disease severity variable that is monotonically related to clinical indicators exists.

^{ 13 }agrees with the prediction of the model and, therefore, would be considered a valid indicator variable, it does not enjoy any special status among the other indicator variables. Because each indicator variable was paired with the composite severity score, it was possible to perform the analysis described here and estimate values of the latent dry eye disease variable. However, because we could not identify which set of indicator variable data points belonged to each person, we were unable to explore the role of mapping noise in the relationship between indicator variables and the dry eye variable, we were unable to calculate meaningful standard errors of dry eye variable estimates for each person and thereby assess measurement reliability, and we were unable to explore the possible multidimensional factor structure of the univariate latent dry eye variable, as suggested by the cluster analysis of Mathers and Choi.

^{ 6 }

*I*) from the effects of treatments that alter the underlying disease state (i.e., global changes in Θ). Treatment-related changes in an indicator variable(s) that are not the result of changes in underlying disease state would manifest as increased mean square residuals posttreatment for indicator variable

_{j}*j,*with no effect on the distributions of mean square residuals for other indicators. Second, because the theory is axiomatic and derivative models require explicit assumptions, it is possible to separate the evaluation of the measurement accuracy of the underlying latent disease state variable from the evaluation of measurement precision. Measurement accuracy refers to the agreement of the observations with the expectations of the theoretical assumptions. Measurement precision is evaluated by comparing the expected variance in the estimate of Θ to the observed variance in the distribution of Θ estimates for the sample. Third, because the theory assumes the existence of a single underlying latent disease state variable, that is, Θ, it is possible to test the hypothesis that Θ is a composite variable with identifiable factors. In other words, various methods for analyzing the covariances of residuals (e.g., principal components analysis and independent components analysis) could be used to evaluate the factor structure of Θ.

^{ 13 }that a single latent dry eye disease state variable exists and can be constructed from clinically observed physiologic and patient symptom indicators. Although tear film osmolarity appears to be the most sensitive dry eye indicator, the observed behavior relative to the other indicators suggests that the mapping function might not be monotonic and/or might not be univariate. If confirmed, that conclusion would suggest that tear film osmolarity, although possibly playing a central role in the disease pathway, would not function as a surrogate for measuring dry eye severity.

*n*, indicator variable

*j*, and interval boundary

*x*is where

*e*is a between-person and between-indicator random variable and

_{nj}*η*(…) is a between-person and between-indicator random function. If the variance of

_{nj}*η*is constant within an indicator (i.e., independent of the value of the argument of the function), then it is safe to assume that and With this single constraint and resulting approximations, the between-person variance in

_{nj}*δ*is determined almost entirely by

_{njx}*ε*, that is,

_{nj}*VAR*{

*δ*} ≅

_{njx}*VAR*{

*ε*}. When these conditions are satisfied,

_{nj}*VAR*{

*δ*} is independent of

_{njx}*x*, and expression 2d in the paper becomes or from the definition of

*ε*, expression A1a is identical to

_{nj}*f*(

_{j}*e*) is defined to be the probability density function of the random error in the observed variable for indicator

_{nj}*j*, then the probability of observing

*C*<

_{jx}*Î*(

_{j}*n*) +

*e*<

_{nj}*C*

_{jx}_{+1}is Similarly, if

*g*(

_{j}*ε*) is defined to be the probability density function of the random error in expression A1a for indicator

_{nj}*j*, then the probability of

*û*(

_{j}*C*) <

_{jx}*û*(

_{j}*Î*(

_{j}*n*) +

*e*) <

_{nj}*û*(

_{j}*C*

_{jx}_{+1}) is Because

*û*(…)is a monotonic function and because the random variable in expression A1b is included in the argument of the function, the probability of

_{j}*û*(

_{j}*C*) <

_{jx}*û*(

_{j}*Î*(

_{j}*n*) +

*e*) <

_{nj}*û*(

_{j}*C*

_{jx}_{+1}) must be equal to the probability of

*C*<

_{jx}*Î*(

_{j}*n*) +

*e*<

_{nj}*C*

_{jx}_{+1}. Therefore, we conclude from the equivalence of expressions A1b and A1a that

*ε*), that differ only in their definitions of

*f*(

_{j}*e*) and

_{nj}*g*(

_{j}*ε*). We say the added noise is independent because it does not depend on the magnitude of the indicator variable. Since there is only one sample of

_{nj}*I*(

_{j}*n*) for each person or

*I*(

_{j}*n*) is an average of repeated samples from each person,

*ε*represents between-person differences in factors that add their effects to the observed value of

*I*. Thus,

_{j.}*g*(

_{j}*ε*) can be sample dependent. If

_{nj}*g*(

_{j}*ε*) is a normal or logistic density function, then equation A4 takes the same form as Samejima's graded response model in psychometrics

_{nj}^{ 15 }or a proportional odds ordinal logistic regression model in statistics.

^{ 16 }However, more generally,

*g*(

_{j}*ε*) can take any form, which can be different for different indicator variables,

_{nj}*j*, and for different samples of persons,

*n*= 1 to

*N*.

*η*(…) depends on the argument of the error function, then

_{nj}*VAR*{

*δ*} ≅

_{njx}*VAR*{

*η*(

_{nj}*C*)} +

_{jx}*VAR*{

*η*(

_{nj}*Î*(

_{j}*n*) +

*e*)} + VAR{

_{nj}*ε*} because the variance of

_{nj}*η*(

_{nj}*Î*(

_{j}*n*) +

*e*) depends on both the variance of the error function and the variance of

_{nj}*Î*(

_{j}*n*) +

*e*in the sample, the variance of

_{nj}*η*(

_{nj}*C*) depends only on the variance of the error function in the sample, and the variance of

_{jx}*ε*depends only on the variance of

*Î*(

_{j}*n*) +

*e*; therefore, all of the covariances are likely to be approximately zero. When these conditions are satisfied, the joint probability density function depends on

_{nj}*x*, the indicator variable j, and the sample of persons,

*n*= 1 to

*N*.

*û*(

_{j}*C*) +

_{jx}*δ*<

_{njx}*û*(

_{j}*Î*(

_{j}*n*)) in expression 2d in the paper is equal to the probability of

*δ*<

_{njx}*û*(

_{j}*Î*(

_{j}*n*)) −

*û*(

_{j}*C*), which is and the probability of

_{jx}*û*(

_{j}*Î*(

_{j}*n*)) <

*û*(

_{j}*C*

_{jx}_{+1}) +

*δ*

_{njx}_{+1}in expression 2d in the paper is equal to the probability of

*δ*

_{njx}_{+1}>

*û*(

_{j}*Î*(

_{j}*n*)) −

*û*(

_{j}*C*

_{jx}_{+1}), which is To the extent that the probability of the event described in equation A5a is independent of the probability of the event described in equation A5b, the probability of both events occurring is the product of the two probabilities. Generalizing to all of the terms in expression 2d in the paper, the probability of the observation satisfying expression 2d, so that we can infer that Θ

*=*

_{n}*x,*is If we impose the additional requirement that the latent interval boundaries for each indicator variable/person combination must be ordered for each observation, that is,

*û*(

_{j}*C*

_{j}_{1}) +

*δ*

_{nj}_{1}<

*û*(

_{j}*C*

_{j}_{2}) +

*δ*

_{nj}_{2}< … <

*û*(

_{j}*C*

_{jm}_{−1}) +

*δ*−1 <

_{njm}*û*(

_{j}*C*) +

_{jm}*δ*, then the conditional probability that Θ

_{njm}*=*

_{n}*x*is

*h*(

_{jx}*δ*) is a joint logistic density function with an equal variance diagonal covariance matrix that is independent of the choice of indicator variable,

_{njx}*j*, then equation A7 takes the form of the Masters partial credit latent variable model.

^{ 17 }If

*h*(

_{jx}*δ*) is a joint logistic density function with a diagonal equal variance covariance matrix but the variance depends on the choice of indicator variable,

_{njx}*j*, then equation A7 takes the form of Muraki's generalized partial credit latent variable model.

^{ 18 }However, more generally,

*h*(

_{jx}*δ*) can be different for each value of

_{njx}*x*and each indicator variable

*j*.

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