Measurement Error Models

Measurement Error Models correctly says, this is a rejection of the nonreductionist position. (See Horwich 1998). An obvious difficulty with this antireductionist approach concerns how to fit it into a naturalistic world picture. Naturalism has been understood by many to require that every fact is ultimately reducible to a fact of natural science. This general issue of naturalistic reducibility is an enormous one, which cannot be pursued here. Clearly the success of a anti reductive response to the rule-following worries will depend on the success of anti reductive approaches more generally. However, the worry can be somewhat lessened by pointing out that whilst anti reductionists reject the idea that meaning facts are reducible to nonmeaning facts, it is not obvious that they need reject the idea that they super
ene on natural facts. Reduction requires that, for every meaning fact, a corresponding nonmeaning fact can be given. Supervenience requires only the weaker theory that the class of meaning facts is somehow made true by the class of nonmeaning facts. That is, someone who embraces supervenience without reduction will accept that nonmeaning facts (in particular, facts about usage, belief, and the like) make true the meaning facts, in the sense that there could not be a change in the meaning facts without a change in the nonmeaning facts. Whether such a picture can really be made good, however, remains to be seen. (There is a gesture towards such a picture in McDowell 1984, p. 348; for a fuller discussion of supervenience of meaning on use, without reduction, see Williamson 1994, pp. 205–9.) See also: Reduction, Varieties of; Verstehen und Erklren, Philosophy of; Wittgenstein, Ludwig (1889– 1951); Word Meaning: Psychological Aspects

Bibliography Blackburn S 1984 The individual strikes back. Synthese 58: 281–301 Boghossian P 1989 The rule-following considerations 5. Mind 98: 507–49 Boghossian P 1991 The problem of meaning in Wittgenstein. In: Puhl K (ed.) Meaning Skepticism. De Gruyter, Berlin, Germany Chomsky N 1986 Knowledge of Language: Its Nature, Origin and Use. Praeger, New York, Chap. 4 Fodor J A 1990 A Theory of Content and Other Essays. MIT Press, Cambridge, MA Fogelin R 1987 Wittgenstein, 2nd edn. Routledge, London Goodman N 1973 Fact, Fiction, and Forecast, 3rd edn. BobbsMerrill, Indianapolis, IN Horwich P 1998 Meaning. Clarendon Press, Oxford, UK Kripke S A 1982 Wittgenstein on Rules and Pri
ate Language: An Elementary Exposition. Harvard University Press, Cambridge, MA Lewis D 1983 New work for a theory of universals. Australasian Journal of Philosophy 61: 343–77

McDowell J 1984 Wittgenstein on following a rule. Synthese 58: 325–63 McDowell J 1993 Meaning and intentionality in Wittgenstein’s later philosophy. In: French P, Uehling T, Wettstein H (eds.) Midwest Studies in Philosophy: The Wittgenstein Legacy 17. University of Notre Dame Press, Notre Dame, IN McGinn C 1984 Wittgenstein on Meaning: An Interpretation and E
aluation. Blackwell, Oxford, UK Millikan R 1990 Truth rules, hoverflies, and the Kripke– Wittgenstein paradox. Philosophical Re
iew 99: 323–53 Pettit P 1990 The reality of rule following. Mind 99: 1–21 Pettit P 1999 A theory of normal and ideal conditions. Philosophical Studies 96: 21–44 Quine W V 1960 Word and Object. MIT Press, Cambridge, MA Williamson T 1994 Vagueness. Routledge, London Wittgenstein L 1958 Philosophical In
estigations, 2nd edn. Blackwell, Oxford, UK Wittgenstein L 1978 Remarks on the Foundations of Mathematics, 3rd edn. rev. Blackwell, Oxford, UK Wright C 1990a Kripke’s account of the argument against private language. Journal of Philosophy 81: 759–78 Wright C 1990b Wittgenstein’s rule-following considerations and the central project of theoretical linguistics. In: George A (ed.) Reflections on Chomsky. Blackwell, Oxford, UK

R. Holton

Measurement Error Models The term measurement error model is used to denote a regression model, either linear or nonlinear, where at least one of the covariates or predictors is observed with error. If xj denotes the value of the covariate for the ith sample unit, then xi is unobserved and we observe instead Xi l f (xi, ui), where ui is known as the measurement error. The observed (or indicator) variable Xi is assumed to be associated to the unobserved (or latent) variable xi via the function f; the form of this association defines the different types of measurement error models described in the literature. For a comprehensive bibliography on this literature, see Fuller (1987) and Carroll (1995). In the simple additive measurement error model, measurement error has the effect of obscuring the relationship between the response variable and the covariate that is measured with error. This effect is sometimes known as attenuation, as in the linear regression case, the ordinary least squares estimator of the regression coefficient associated to the variable that is measured with error is biased towards zero (e.g., Fuller 1987). The next section describes the effect of additive measurement error in the simple regression case, and notes the relationship between this topic and those in other entries. Subsequent sections describe extensions 9435

Measurement Error Models to the multivariate and nonlinear regression cases. The article ends with a brief historical overview of the development of measurement error models.

1. Simple Linear Regression and Measurement Error

X l xju

The simple linear regression model (see Linear Hypothesis: Regression (Basics)) is given by Yi l β jβ xijei (1) ! " where Yi is the response observed for the ith sample item, xi is the explanatory variable (or covariate) measured for the ith sample item, β and β are " unknown regression coefficients, and !ei is random error typically assumed to be distributed as a normal random variable with mean 0 and variance σe#. Here we talk about functional models when the explanatory variable x is assumed to be a fixed value, and about structural models when instead x is assumed to be a random draw from some distribution. Typically, we choose x " N (µx, σ#x) (e.g., Fuller 1987). [Carroll (1995), however, makes a different distinction between functional and structural models. They refer to functional modeling when the variable x is either fixed or random but minimal assumptions are made about its distribution, and to structural modeling when a parametric form is chosen for the distribution of x. Carroll (1995) argues that when a consistent parameter estimate under the structural model formulation can be found, it tends to be robust to misspecification of the distribution of x.] It is well known that the ordinary least squares estimator (see Linear Hypothesis: Regression (Basics)) of β " n

βV l "



(YikYz )(xikx` )

i="



n

(xikx` )# i="

(2)

is unbiased and has smallest variance among linear unbiased estimators. If x " N (µx, σ#x) and Co
(x, e) l 0, then the estimator β# in (2) is the maximum likelihood estimator of β (see "Estimation: Point and Inter
al). " 1.1 Models for Measurement Error Suppose now that we cannot observe the explanatory variable xi directly. Instead, we observe the manifest or indicator variable Xi which is a noisy measurement of xi. We distinguish two different types of measurement error models depending on the relationship between Xi and xi: (a) Error models, discussed in this entry, where we model the distribution of X given x. (b) Regression calibration models, where the distribution of x given X is of interest. 9436

Within the class of error models, we distinguish two variants: classical additi
e error models, and error calibration models. The classical additi
e error model establishes that the observed variable X is an unbiased measure of x. That is (3)

where u " (0, σ#u). As an example, consider the problem of estimating the relationship between serum cholesterol levels and usual intake of saturated fat for a sample of individuals. Suppose further that intake of cholesterol for an individual is measured by observing the individual’s intake over a randomly chosen 24hour period. Because daily nutrient intake varies from day to day for an individual, the intake measured on any one day is a noisy measurement of the individual’s usual or habitual intake of the nutrient. We talk about error calibration models when the observed variable is a biased measurement of the variable of interest. In this case, we use a regression model to associate the two variables X l α jα xju (4) ! " where as before, E (u) l 0, but now E(X ) l α jα x. ! the " As an example, consider the problem of assessing habitual daily alcohol consumption for a sample of individuals. Typically, individuals in the sample are asked to record their alcohol consumption over a short period, for example over a week. The mean of the observed alcohol intakes over the seven days cannot be assumed to be an unbiased estimator of the usual daily alcohol consumption, as there is evidence (see Freedman et al. 1991; Carroll 1995) that individuals tend to under-report the amount of alcohol they drink. Suppose in addition that the underreporting is not correlated with alcohol consumption. In this case, the measurement X is biased for x and must be corrected. If information about relationship (4) is available, the measurement X can be calibrated by using α−" (Xkα ). " discuss ! regression calibration models later We briefly in this article. 1.2 The Effect of Measurement Error Consider the simple example of a linear regression with additive measurement error in the covariate, as in expressions (1) and (3). Further, assume that the measurement error is independent of the true measurements x and of the errors e. If X l xju, x " ( µx, σ#x) and u " (0, σ#u) then the regression coefficient estimated using the noisy measurements is biased towards zero. If γ# is the estimated regression coefficient " using X in the regression equation, then obtained from E(γV ) l λβ (5) " " where β is the regression coefficient associated to x and λ is" called a reliability ratio (e.g., Fuller 1987,

Measurement Error Models diamonds represent the noisy measurements. Note that the regression line fitted to the noisy data is attenuated towards zero. In fact, in this simulated example, the true attenuation coefficient or reliability ratio is only ". % of the slope is greatly dependent on the Attenuation simple additive error model. In other cases, the ordinary least squares estimator of the regression coefficient may o
erestimate the true slope. Consider, for example, the case where X is a biased measurement of x, and in addition, the measurement error is correlated with the equation error. That is, X is as in Eqn. (4) and Corr(e, u) l ρeu. It can be shown that Figure 1 Simulated example. Black circles represent observations without measurement error. Diamonds represent the observations subject to measurement error in the covariate. The steeper line is the fitted regression line in the absence of measurement error, while the flatter line shows the fitted regression line in the presence of measurement error.

Sect. 1.1; Carroll 1995, Sect. 2.2.1), computed as σ#x λl σ#xjσ#u

(6)

Clearly, λ 1, and approaches 1 as the variance of the measurement error goes to zero. Therefore, ordinary least squares estimation in the model with X instead of x produces an estimator of the regression coefficient that is biased towards zero, or is attenuated. Measurement error has in addition an effect on the variability of points around the estimated regression line. Note that under model (1) with measurement error as in (3), β#σ# σ#
ar (Y Q X ) l σ#e j " u x σ#xjσ#u

(7)

Thus measurement error has the effect of increasing the noise about the regression line (Carroll 1995). 1.3 A simple simulated example To illustrate the effect of measurement error on regression estimation, we constructed 10 responses Y l β jβ xje, where β l 0, β l 2, e " N (0, 1), " (1, 1). We then ! constructed " and x! " N a noisy measurement of x by adding measurement error u where u " N (0, 3). We fitted two simple linear regression models, using x and using X as the covariate in the model. The data, together with the two fitted regression lines are shown in Fig. 1. In the figure, the dark dots represent the true measurements, while the

β#α σ#jρeuσeσu E oγV q l " " x " α#σ#xjσ#u "

(8)

where, as in Eqn. (5), γ# denotes the estimated " regression coefficient obtained by using X in the model, and β is the true slope. Notice that, depending on the values" of α and of ρeu, γ# may actually over" " estimate β . "

2. Multiple Linear Regression and Measurement Error Extension of results presented in Sect. 1 to the case of multiple predictors is not straightforward, even if only one of the predictors is subject to measurement error and the measurement error model is as in Eqn. (3). Let Y l β jβ xjβhwwje (9) ! " where now w is a pi1 vector of predictors that are measured without error, βw is the vector of regression coefficients associated to w, and x is measured with simple additive measurement error as in Eqn. (3). Furthermore, let σ#x Q w denote the variance of the residuals in the regression of x on w. If model (9) is fitted using X in place of x, it can be shown (e.g., Gleser et al. 1987) that, as in Eqn. (5), the estimated regression coefficient for X is γ# l λβ , where " " λl

σ#x Q w . σ#x Q wjσ#u

(10)

Notice that when x and w are uncorrelated, λ in Eqn. (10) reduces to Eqn. (6). Gleser et al. (1987) show that when one of the covariates is measured with error, the regression coefficients associated to w that are measured without error are also biased, unless x and w are uncorrelated. If now x is a qi1 vector of covariates measured with error, and w is as above, then the ordinary least 9437

Measurement Error Models where for σ# #u we choose the smallest root of the quadratic expression above, so that σ# #u s#X. Fuller (1987, Theorem 1.3.1) shows that, for θ l (β , β , σ#u)h, ! n" _, and θ# l ( β# , β# , σ#u)h (as given in Eqn. (14)), as ! "

squares estimates have expectation A A

γV x E γV B w

C

j xx

l



D B 1

A

 xx

i 2 3 4

B

uu

wx

 xw



wx

ww

D

A

D

βx β B w

n"/#(θV kθ),-N(0, Γ ) 

C

xw



−"

 ww



C

A

C

j D



C

5 6

. (11)

ue

0 B

7 D 8

The ordinary least squares estimate of the regression coefficients will be unbiased only when uu l 0 and ue l 0.

3. Estimation in Measurement Error Models Consider first the simple model (1) with measurement errorasinEqn.(3).Assumethat(x,e,u)h " N[( µx, 0, 0)h, diag (σx#, σ#e , σ#u)], and suppose that σu# is known. Under this model, the vector, (Y, X) is distributed as a bivariate normal vector, and therefore, the sample firstand second-order moments form a set of sufficient statistics for the parameters. Fuller (1987, Sect. 1.2.1) and Carroll (1995, Sect. 2.3.1) derive method of moments (see Estimation: Point and Inter
al) parameter estimates that can be obtained from a one-toone mapping from the minimal sufficient statistics to the vector (µ# x, σ# #x, β# , β# , σ# #e ). For example, ! " Vβ l Yz kβV Xz ! " # kσ# )−"s (12) βV l (sX u XY " where YF and XF are the sample means of Y and X, and # and s are the sample variance of X and the sample sX XY covariance between X and Y, respectively. Fuller (1987) shows that when σ#u is known and as n _,  kβ ), (βV kβ )]h,-No(0, 0)h, T q ! ! " "

n"/#[(βV

(13)

where ( β# , β# ) are as in (12) and T is a positive definite ! Fuller " matrix (see 1987, Sect. 1.2.2). Suppose now that rather than assuming that σ#u is known, we consider the case where the relati
e sizes of the variances σ−u# σ#e are known. We use δ to denote the ratio of variances. Under the assumption of normality of (x, e, u)h as above, method of moments estimators for (β , β , σ#u) are ! " βV l Yz kβV Xz ! " # # j[(s# kδs#)#j4δs# ]"/# s kδs X Y x XY βV l Y " 2sXY σV #u l 9438

# ]"/# s#Yjδs#Xk[(sY# kδs#X)#j4δsXY 2δ

(14)

(15)

for Γ a positive definite covariance matrix. Asymptotic normality of (β# , β# ) holds even if the distribution of x is not normal. ! " The estimators above can be extended to the case where a vector x is measured with error. While in the simple linear regression case results apply in the functional and in the structural model framework, in the multivariate case we must distinguish between models in which x is assumed to be random and those in which x is assumed to be fixed. We consider first the structural model set-up, where x is taken to be random. In this case, the maximum likelihood estimators of model parameters derived under the normal model can be shown to be consistent and asymptotically normal under a wide range of assumptions. Let Y l β jxhβ je ! " X l xju

(16)

where E A

x e u B

A

C

A

µx 0 0 B

"N D

C

0

0

0

σ#e



0





ue

uu

xx

, D

F



B

C

G

eu

D

H

(17)

Here, uu and eu are known. The maximum likelihood estimators of ( β , β ) are ! " Vβ l Yz kXz βV ! " Vβ l E S k G −"(S k) (18) XX XY " F H uu

ue

where SXX is the sample covariance matrix of X and SXY is the covariance matrix of X and Y. A modified maximum likelihood estimator of (β , β ) for the case ! " be estimated where uu and eu are unknown and must from the data is given by Fuller (1987, Sect. 2.2). If the vector x is taken to be fixed, rather than random, the method of maximum likelihood fails to produce consistent estimators of the parameters in the model. This is because the vector of unknown model parameters (β , β , x , x , …, xn, σ#e ) is nj3-dimen" " size # is only n. In this case, then, sional, but the !sample the number of parameters increases with sample size.

Measurement Error Models Estimation of model parameters in these cases has been discussed by, for example, Morton (1981). Consider now model (16) with assumptions as in (17), and suppose that we do not know uu and eu as above. Instead, let ε l (e, u)h " N (0, εε), where εε l Ωεε σ#, and Ωεε is known. In this case, the maximum likelihood estimators of β and σ# are " βV l (Xz hXz kλV Ωuu)−"(Xz hYz kλV Ωue) " σV # l (qj1)−"λV

(19)

where XF hXF and XF hYF are averages taken over the n sample observations, and λ# is the smallest root of Q ZF hZF kλ Ωεε Q with Z l (Y, X). The estimator of β is the same whether x is fixed or random. Under" relatively mild assumptions, the estimator β# is distri" buted, in the limit, as a normal random vector (see Fuller 1987, Theorem 2.3.2).

4. Related Methods 4.1 Regression with Instrumental Variables In the preceding sections we have assumed implicitly that when the variance of the measurement error is not known, it can be estimated from data at hand. Sometimes, however, it is not possible to obtain an estimate of the measurement error variance. Instead, we assume that we have available another variable, which we denote Z, and which is known to be correlated with x. If Z is independent of e and u, and Z is correlated with x, we say that Z is an instrumental
ariable (see Instrumental Variables in Statistics and Econometrics). Fuller (1987) presents an in-depth discussion of instrumental variable estimation in linear models (Sects. 1.4 and 2.4). He shows that method of moments estimators for the regression coefficient can be obtained by noticing that the first and second sample moments of the vector (Y, X, Z) are the set of minimal sufficient statistics for the model parameters. The resulting estimators are "s βV l s−XY YZ " βV l Yz kβV Xz ! "

(20)

where sXZ and sYZ are the sample covariances of the covariate and the response variables with the instrumental variable, respectively. Because the sample moments are consistent estimators of the population moments, the estimators in (20) are consistent estimators of β and β . Notice, however, that if the !  0 "is violated, the denominator in assumption σ xZ

the expression for β# above estimates zero and the estimator is therefore "not consistent. Amemiya (1985, 1990a, 1990b), Carroll and Stefanski (1994) and Buzas and Stefanski (1996) discuss instrumental variable estimation in nonlinear models.

4.2 Factor Analysis Factor analysis is heavily used in psychology, sociology, business, and economics (see Factor Analysis and Latent Structure: O
er
iew). We consider a model similar to the model used for estimating an instrumental variable, with a few additional assumptions. The simplest version of a factor model is Y l β jβ xje " !" "" " Y l β jβ xje # !# "# # X l xju

(21)

Here, (Y , Y , X ) are observed. We assume that the " associated # coefficients to x are nonzero, and that (x, e , e , u)h " N[(µx, 0, 0, 0)h, diag (σ#x, σ#e , σ#e , σ#u)]. # " # x is In" factor analysis, the unobservable variable denoted common factor or latent factor. For example, x might represent intelligence, and the observed variables (Y , Y , X ) might represent the individual’s " #appropriate tests. The ratio of the scores in three variance of x to the variance of the response variable is called the communality of the response variable. For example, the communality of Y above is calculated as " β# σ#x " λ# l " β#σ#jσ# e" " x

(22)

Note that λ# is equal to the reliability ratio that was presented in" Sect. 1. Estimators for β and for β can be obtained using "# used for estimating the same methods"" that were instrumental variables in Sect. 4.1. For example, β# is the instrumental variable estimator of β under ""the "" model Y l β jβ xje using Y as an instrumental !" ""an estimator # for β is obtained variable." Similarly, using Y as an instrumental variable. "# " extensive discussion of factor analysis, refer For an to Jo$ reskog (1981) and Fuller (1987).

4.3 Regression Calibration Suppose that for each sample item we observe Y, X, and w, where X l xju, and w is a vector of covariates measured without error. The idea behind regression 9439

Measurement Error Models calibration is simple. Using information from X and w, estimate the regression of x on (X, w). Then fit the regression model for Y using Eox Q X, wq using standard methods. The standard errors of the parameters in the regression model for Y must be adjusted to account for the fact that x is estimated rather than observed. Estimation of the regression of x on (X, w) is straightforward when replicate observations are available. Suppose that there are m replicate measurements of x for each individual, and that XF is their mean. In that case, the best linear estimator of x is Eox Q XF , wq, where E ox Q Xz , wq l µxj A

i

F

E

σ#x wx

σ#xjσ#u\m hxw B

G

H

xw ww

C

D

−" F

E

Xz kµX wkµw G

1

f ( y Q x, w, Θ) l exp 2 3 4

yηkD(η) jc ( y, φ) φ

(23)

5. Nonlinear Models and Measurement Error Carroll (1995, Chap. 6) presents an in-depth discussion of the problem of estimating the parameters in a nonlinear model when one or more of the covariates in the model are measured with error. Most of their discussion assumes that the measurement error is additive and normal, but some results are presented for more general cases. Other references include Amemiya (1990), Gleser (1990), and Roeder et al. (1996). Carroll (1995) presents two approaches for estimating the parameters in the nonlinear model subject to measurement error, denoted conditional score and corrected score methods, that are based on an estimating equations approach. In either case, the resulting estimators do not depend on the distribution of x. The conditional score method is derived for the case of the canonical generalized linear model (McCullagh and Nelder 1989) (see Linear Hypothesis). Without

6

5

(24) 7 8

(expression 6.4 in Carroll 1995), where η l β jβ xjβ w is called a natural parameter, Θ l # is the unknown parameter to be esti(β! , β ", β , φ) ! " # mated, and where the first and second moments of the distribution of Y are proportional to the first and second derivatives of D(η) with respect to η. The form of η, D(η) φ and c( y, φ) depends on the model choice. If x were observed, then Θ is estimated by solving the following equations E

H

Moment estimators for all unknown parameters in the expression above can be obtained as long as replicate observations are available and an estimator for the unknown measurement error variance can be computed. If replicate observations are not available, it is possible to use an external estimate of the measurement error variance. Regression calibration was proposed by Gleser (1990). Carroll (1995, Chap. 3) discusses regression calibration in nonlinear models. An alternative approach called SIMEX (for simulation extrapolation) that is well suited for the additive measurement error case was proposed by Cook and Stefanski (1994).

9440

loss of generality we discuss the case where both x and w are scalar valued. The conditional density of Y given (x, w) is given by

n

 oYikD(")(ηi)q

i=" n

 i="

A

B

F

E

F

nk1 n

G

H

φk

1 wi xi

G

l0 H

(YikD(")(ηi))# D(#)(ηi)

C

l0

(25)

D

where D(k)(η) is the kth order partial derivative of D(η) with respect to η. We now assume that x is measured with normal additive measurement error with variance σ#u. A sufficient statistic ∆ can be found for x (Stefanski and Carroll 1987) so that the distribution of Y conditional on (∆, w) has the same form as the density given in (24). The form of the functions D(η), η, φ and c( y, φ) must be modified accordingly (see, for example, Carroll et al. 1995, p. 127). Therefore, this suggests that an estimator for Θ when x is observed with measurement error can be obtained by solving a set of equations similar to Eqn. (25). The approach described above can be used only in the case of canonical generalized linear models. An approach applicable to the broader class of generalized linear regression models is the corrected score method, also derived under an estimating equations framework. Suppose, first, that x is observable, and that an estimator of Θ is obtained by solving n

 ξ(Yi, xi, wi, Θ) l 0 i="

(26)

where ξ is a likelihood score from the model when x is observed without measurement error. Suppose now that we can find a function ξ*(Y, X, w, Θ, σu# ), such that E oξ*(Y, X, w, Θ Q Y, x, wq l ξ (Y, x, w, Θ) (27)

Measurement Error Models for all (Y, x, w). An estimate for the unknown parameter Θ can then be obtained by solving Eqn. (26) where the corrected score ξ* is used in place of ξ. Corrected score functions that satisfy Eqn. (27) do not always exist, and when they do exist they can be very difficult to find. Stefanski (1989) derived the corrected score functions for several models that are commonly used in practice.

6. A Brief Historical O
er
iew The problem of fitting a simple linear regression model in the presence of measurement error was first considered by Adcock (1877). Adcock proposed an estimator for the slope in the regression line that accounted for the measurement error in the covariate for the special case where σ#a l σ#e . The extension to the p-variate case was presented by Pearson (1901). It was not until the mid-1930s that the terms errors in
ariables models and measurement error models were coined, and a systematic study of these types of model was undertaken. The pace of research in this area picked up in the late 1940s, with papers by Lindley (1947), Neyman and Scott (1951), and Kiefel and Wolfowitz (1956). These authors addressed issues of identifiability and consistency in measurement error models, and proposed various approaches for parameter estimation in those models. Lindley (1947) presented maximum likelihood estimators for the slope in a simple linear regression model with additive measurement error for the functional model. He concluded that, when the two variance components are unknown, the method of maximum likelihood cannot be used for estimating β , as the resulting estimator must satisfy the relation" β# , σ# #u l σ# #e . Solari " (1969) later showed that this unexpected result could be explained by noticing that the maximum likelihood solution was a saddle-point and not a maximum of the likelihood function. Lindley (1947) also showed that the generalized least squares estimator of the slope is the maximum likelihood estimator when the ratio of variances is assumed known. Extensions of the simple linear regression model to the multiple regression model case, with one or more covariates measured with error were presented in the late 1950s, 1960s and 1970s. Excellent early in-depth reviews of the literature in measurement error models are given by Madansky (1959), and Morton (1981). In the early 1990s, Schmidt and Rosner (1993) proposed Bayesian approaches for estimating the parameters in the regression model in the presence of measurement error. More recently, Robins and Rodnitzky (1995) proposed a semi-parametric method that is more robust than the maximum likelihood estimator of the slope in structural models when the distributional assumptions are not correct. Carroll et al. (1997), however, showed that the Robins and Rodnitzky estimator can be very inefficient relative to the parametric estimation when the distributional

assumptions are correct. Roeder et al. (1996) and Carroll et al. (1999) have argued that the sensitivity of the maximum likelihood estimator to model specification can be greatly reduced by using flexible parametric models for the measurement error. In particular, the authors suggest that a normal mixture model be used as the flexible model. In the 1999 paper, Carroll et al. (1999) proceed from a Bayesian viewpoint, and use Markov chain Monte Carlo methods to obtain approximations to the marginal posterior distributions of interest. The literature on measurement error models is vast, and we have cited a very small proportion of the published books or manuscripts. In addition to the review articles listed above, the text books by Fuller (1987) and Carroll (1995) are widely considered to be the best references for linear and nonlinear measurement error models, respectively. The recent manuscript by Stefanski (2000) cites several papers that have appeared in the literature since 1995.

Bibliography Adcock R J 1877 Note on the method of least squares. The Analyst 4: 183–4 Amemiya Y 1990 Two stage instrumental variable estimators for the nonlinear errors in variables model. Journal of Econometrics 44: 311–32 Buzas J S, Stefanski L A 1996 Instrumental variable estimation in generalized linear measurement error models. Journal of the American Statistical Association 91: 999–1006 Carroll R J, Stefanski L A 1994 Measurement error, instrumental variables, and corrections for attenuation with applications to metaanalyses. Statistics in Medicine 13: 1265–82 Carroll R J, Freedman L S, Pee D 1997 Design aspects of calibration studies in nutrition, with analysis of missing data in linear measurement error. Biometrics 53: 1440–57 Carroll R J, Roeder K, Wasserman L 1999 Flexible parametric measurement error models. Biometrics 55: 44–54 Carroll R J 1995 Measurement Error in Nonlinear Models. Chapman and Hall, London Cook J, Stefanski L A 1994 Simulation extrapolation estimation in parametric measurement error models. Journal of the American Statistical Association 89: 1314–28 Freedman L S, Carroll R J, Wax Y 1991 Estimating the relationship between dietary intake obtained from a food frequency questionnaire and true average intake. American Journal of Epidemiology 134: 510–20 Fuller W A 1987 Measurement Error Models. Wiley, New York Gleser L J 1985 A note on Dolby G. R. unreplicated ultrastructural model. Biometrika 72: 117–24 Gleser L J 1990 Improvements of the naive approach to estimation in nonlinear errors-in-variables regression models. In: Brown P J, Fuller W A (eds.) Statistical Analysis of Measurement Error Models and Applications. Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference. American Mathematical Society, Rhode Island, RI, pp. 99–114 Gleser L J, Carroll R J, Gallo P P 1987 The limiting distribution of least squares in an errors-in-variable regression model. Annals of Statistics 15: 220–33 Jo$ reskog K G 1981 Structural analysis of covariance structures. Scandina
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Measurement, Representational Theory of The representational theory of measurement—RTM for short—is a sort of ‘interface’ between a qualitative, empirical, or material domain of research and the use of numbers to quantify the empirical observations and to draw inferences from the data. Roughly, the main tenets of RTM consist of the following steps (for definition of the main concepts see Ordered Relational Structures): (a) Describe the situation under scrutiny by a qualitative relational structure, consisting of a base set

A and a finite number of relations. Among these relations is usually an order relation and sometimes special elements such as units. These entities are the primitives of the structure. They must have an empirical identification. The properties of this qualitative relational structure are stated as conditions or axioms on its primitives, which are true statements about the structure, at least in the intended empirical identification. (b) Verify by empirical procedures (experiments, observations) the truth of those conditions that are testable (which should be almost all of them) and find arguments (plausibility, consequences of failure) for those that are not testable. (c) Find a structure with a numerical base set (a subset of  or n) and find homomorphism between both structures. Finding a homomorphism may consist of providing a representation theorem which, essentially, states the existence of at least one homomorphism. (d) Find all homomorphisms between the qualitative and the numerical structure. This step amounts to proving a uniqueness theorem, which in most cases yields a procedure to calculate any homomorphism from a given one. (e) Clarify on the basis of the uniqueness properties of the set of homomorphisms which numerical statements about the empirical domain are meaningful and which are not. Meaningful statements can be used for further analysis, such as statistical tests. This theory is thoroughly described in the three volumes Foundations of Measurement (see Krantz et al. 1971, Suppes et al. 1989, Luce et al. 1990. Other comprehensive expositions, differing slightly in emphasis and philosophical argumentation, are Pfanzagl (1971), Roberts (1979), and Narens (1985)

1. History and Contemporary Formulation of Measurement Within RTM The above kind of formalization of measurement goes back to Helmholtz (1887). Ho$ lder (1901) was the first to prove a representation theorem in the above sense. One version or other of Ho$ lder’s theorem lies at the heart of almost all representation theorems proved in the following years up to the present time. It did not, however, stimulate much research until the late 1950s. The 1930s and 1940s saw an ongoing debate about the measurability of sensory variables, which seemed to some a prerequisite to laying a firm foundation for psychology, not unlike that of most of the natural sciences. In this debate, the introduction of scale types by Stevens (1946) was a cornerstone. The search for a precise formulation led to the model-theoretic formulation of Scott and Suppes (1958). This framework was extended in the three volumes Foundations of

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International Encyclopedia of the Social & Behavioral Sciences

ISBN: 0-08-043076-7