Regression with measurement error
4.2 Regression with measurement error
Multiple regression analysis is probably the most widely used analysis tool to investigate the relation between educational performances and one or more background variables, such as gender or socio-economic status. In EER, the most interesting variables are usually constant within schools but vary across schools, while the basic measurements – the performances of the students – are collected at the student level, necessitating a two-level (or more generally, a multilevel) approach to the analysis of the data (see Chapter 11). In this chapter,
212 Different methodological orientations the discussion will be restricted to a simple linear two-level model with a random
intercept, which may be represented as follows:
Y ij =
z ∑ g β () g ij + u j + ε ij .
The dependent variable Y ij is some observed measure on student i in school j. The p regressors z (1) , . . ., z (p) are fixed constants that do or do not vary across students of the same school. Notice that the characterization of the model as
a two-level model does not depend on this variation, but on the presence of the latter two terms, u j and ij in the right-hand side of (21). Both are to be considered as realizations of two random variables: the random intercept at the school level (u j ) and the residual at the student level ( ij ). In most applications of multilevel modelling these two random variables are considered as statistically independent, and as being normally distributed. In the simplest case, these are the two assumptions:
(22) The estimation problem in multilevel analysis is then to obtain estimates of
ij ~ N(0, 2 ) and u
2 j ~ N(0, ).
the regression coefficients  1 , . . .,  p and of the two variance components 2 and 2 .
A conceptual problem is associated with the fact that the observed dependent variable Y is never a pure operationalization of the concept in which one is interested, but is in a sense polluted by measurement error, implying that at the
lowest level, true variance and error variance are subsumed in 2 , or equivalently, that the residual variance will increase as the reliability of the measurement decreases. Variation in the reliability has the following effects on the results of the multilevel regression analysis:
• There is no bias in the estimates of the regression coefficients (Cook and Campbell 1979, Chapter 4). • The standard errors of the regression coefficients increase with decreasing reliability. • The residual variance 2 increases if the reliability decreases. • The higher level variance 2 is not affected in a systematic way. Although no proof seems to be available for this, it appears to be the case in a significant number of simulation studies.
The latter two bullet points have consequences when the intra-class correlation is considered:
IRT models 213 Using the estimates of both variance components to estimate will not affect
the numerator of (23) but will systematically inflate the denominator as the reliability decreases, and this will affect a clear interpretation of the intra-class correlation, a variable of considerable importance in EER.
Latent regression As in IRT, where the concept to be measured is at the centre of the model, it
would be nice if one could carry out a regression analysis where the dependent variable is the latent variable. So the basic regression model given by (21) is changed to
() ∑ g β
z g ij + u j + ε ij . (24)
ij
The term latent regression is shorthand for ‘regression model where the dependent variable is not directly observed’. Basically, two approaches can be used to analyse model (24): either one substitutes the latent observation ij by
a proxy – an estimate of it – or one tries to carry out the analysis without using proxies. In the first approach, an estimation error is re-introduced, but not in the second approach. The use of the first approach is recommended here and is presented in the final section of this chapter. For the second approach, see Verhelst and Verstralen (2002).