—from Epidemiological Bulletin , Vol. 25 No. 1, march 2004—

A Glossary for Multilevel Analysis - Part III

Ana V. Diez Roux
Divisions of Medicine and Epidemiology, Columbia University
New York, New York, United States


Non-independence of observations
Refers to situations in which dependent variables for observations at a lower level nested within the same higher level unit (or cluster) are correlated, even after measured characteristics are taken into account. For example, two persons from the same neighborhood may tend to have more similar blood pressure levels than two persons from different neighborhoods, even after measured individual and neighborhood characteristics are taken into account. In the case of repeat measures on individuals over time, two blood pressure measurements on the same person may tend to be more similar than two measures on different persons even after relevant covariates are taken into account. One reason for this correlation may have to do with the omission of important higher level variables that observations within the same higher level unit share. This residual correlation violates the assumption of independence of observations underlying usual regression approaches. Ignoring this correlation may lead to incorrect inferences. Efficiency of estimation may also be reduced.40 Multilevel models account for potential residual correlation by modelling intercepts and regression coefficients as random (for example, by allowing for macro level errors, U0j and U1j in second level equations, see Multilevel models).

Population-average models
Models that account for correlation between lower level units within higher level units (or clusters) by modelling the correlations or covariances themselves rather than by allowing for random effects or random coefficients as Multilevel models do.40, 46 These correlations are taken into account in the estimation of regression coefficients and their standard errors. Different correlation structures (describing within cluster or within higher level unit correlations) can be specified. “Population-average models” are also referred to as “marginal models”40, 46 or “covariance pattern models”.26 Whereas multilevel models model the dependent variable conditional on the random effects (or random coefficients), population-average models model the marginal expectation of the dependent variables across the population (in a sense, “averaged “ across the random effects). For this reason, marginal models have also been called “population-average” models (as a way to contrast them with subject specific random effects models).46 The Generalized Estimating Equation (GEE) approach is one approach to fitting marginal models.46

Population-average models model the population-average response as a function of covariates without explicitly accounting for heterogeneity across higher level units.46 In contrast, multilevel models investigate and explain the source of group to group variation (and of the within group correlation) by modelling group specific regression coefficients as a function of group level variables plus random variation. Therefore, although population-average models account for the correlation between outcomes within higher level units, the source of this correlation is not directly investigated (the correlation, and sometimes higher level effects themselves, are viewed as nuisance parameters that must be taken into account but are not of direct interest). Therefore, population average models do not allow examination of group to group variation, of the group level or individual level variables related to it, or of the degree of variation present between and within groups, as multilevel models do (see variance components). Differences between both types of models also have consequences for the interpretation of regression coefficients: in the multilevel model, the regression coefficient estimates how the response changes as a function of covariates conditional on the random effects; in the marginal model, the coefficient expresses how the response changes as a function of covariates “averaged” over group to group heterogeneity (or group random effects).40, 46 In the case of continuous dependent variables these coefficients are mathematically equivalent, but in the case of non-normally distributed variables (for example, logistic models) the marginal parameter values will usually be smaller in absolute value than their random effects analogues.46, 47

Psychologistic fallacy
An inferential fallacy that may arise from the failure to consider group characteristics in drawing inferences regarding the causes of variability across individuals1, 2—that is, assuming that individual level outcomes can be explained exclusively in terms of individual level characteristics. Although the level at which data are collected may fit the conceptual model being investigated (that is, individual level), important facts pertaining to other levels (that is, group level) may have been ignored.1, 2 For example, a study based on individuals might find that immigrants are more likely to develop depression than natives. But suppose this is only true for immigrants living in communities where they are a small minority. A researcher ignoring the contextual effect of community composition might attribute the higher overall rate in immigrants to the psychological effects of immigration or to genetic factors, ignoring the importance of community level factors and thus committing the psychologistic fallacy.1 The term “psychologistic fallacy” is not entirely appropriate because the individual level factors used to explain the outcome are not always exclusively psychological.2 Although the term “individualistic fallacy”may appear more adequate, it has also been used as a synonym for the related but distinct atomistic fallacy.3, 4 See also sociologistic fallacy.

Random coefficient models
Term originally used for models in which the regression coefficients corresponding to covariates in the model are treated as random rather than fixed19, 26 (that is, models containing random coefficients, see for example b1j in the entry for multilevel models). Traditional random coefficient models do not include higher level (or group level) predictors in the group level equations for the covariate effects (that is, in a traditional random coefficient model, equation (3) would be b1j = 10 + U1j).19 Thus random coefficient models can be thought of as a particular case of the more general multilevel models. However, the term random coefficient models is sometimes used more broadly to refer to multilevel models generally. See also random effects models.

Random effects/random coefficients
Regression coefficients (intercepts or covariate effects) that are allowed to vary randomly across higher level units (that is, are assumed to be realizations of values from a probability distribution) (see multilevel models). For example, in the case of persons nested within neighborhoods, neighborhood effects can be assumed to vary randomly around an overall mean (random effect, see random effects models). Similarly, the effect of personal income on individual health may be allowed to vary randomly across neighborhoods (random coefficient, see random coefficient models). Although the terms “random effects” and “random coefficients” are sometimes distinguished as noted above, they are often used interchangeably. The use of random effects or random coefficients is especially appropriate when the higher level units (or groups) can be thought of as random samples from a larger population of units (or groups) about which inferences wish to be made. See also fixed effects/fixed coefficients.

Random effects models
Term originally used for models in which differences across groups (or other classification system) are treated as random rather than fixed19, 26 (that is, models containing random effects). For example, in the case involving individuals nested within neighborhoods, a model treating neighborhood differences as fixed would include all neighborhoods represented in the sample as a set of dummy variables in a regression equation with individuals as the units of analysis (see fixed coefficients). In contrast, a random effects model would treat neighborhood differences as realizations from a probability distribution—that is, neighborhood intercepts would be allowed to vary randomly across neighborhoods following a probability distribution (see multilevel models). An underlying assumption is that the neighborhoods in the study are a random sample from a larger population of neighborhoods about which inferences wish to be made. Random effects models can be thought of as a particular case of the more general multilevel models in which only intercepts are allowed to vary randomly across groups (that is, random intercept models). Sometimes, however, the term random effects models is used more broadly to refer to multilevel models generally (that is, models that allow for both random intercept and random covariate effects). See also random coefficient models.

Residual correlation
See non-independence of observations.

Sociologistic fallacy
An inferential fallacy that may arise from the failure to consider individual level characteristics in drawing inferences regarding the causes of variability across groups.1, 2 Although the level at which data are collected may fit the conceptual model being investigated (that is, group level), important facts pertaining to other levels (that is, the individual level) may have been ignored.1 Suppose a researcher finds that communities with higher rates of transient population have higher rates of schizophrenia, and he/she concludes that higher rates of transient population lead to social disorganization, breakdown of social networks, and increased risk of schizophrenia among all community inhabitants. But suppose that schizophrenia rates are only increased for transient residents (because transient residents tend to have fewer social ties, and individuals with few social ties are at greater risk of developing schizophrenia). That is, rates of schizophrenia are high for transient residents and low for non-transient residents, regardless of whether they live in communities with a high or a low proportion of transient residents. If this is the case, the researcher would be committing the sociologistic fallacy in attributing the higher schizophrenia rates to social disorganization affecting all community members rather than to differences across communities in the percentage of transient residents. See also psychologistic fallacy.

Structural variables
A type of group level variable that refers to relations or interactions between members of a group,13 for example, characteristics of social networks within the group or patterns of contacts or interactions between members of the group. Structural variables are sometimes considered a subtype of integral variables.12, 18

Subject specific models
Term used to refer to random effects/random coefficient models (or multilevel models generally) in order to contrast them with population-average models. “Subject specific” is used because the term was originally developed in the context of longitudinal data analysis,46 where individuals or subjects are the higher level units and repeat measures are the lower level units. In this case, the fixed effects coefficients derived from a random effect, random coefficient, or multilevel model are conditional on person level (or person specific) random effects, hence the term “subject specific”. More generally, they can be thought of as “higher level unit” specific (or cluster specific), because they are conditional or higher level unit (or cluster specific) random effects. For example, in the entry for multilevel models, the estimate of 01 is conditional on group level random effects (as reflected by the presence of Uoj and U1j).

Variance components
Using multilevel models the total variance in individual level outcomes (or lower level outcomes generally) can be decomposed into variance within and between groups (or higher level units generally). For example, the variance in blood pressure across individuals can be decomposed into variance within and between neighborhoods. These components are referred to as variance components. The ability to estimate the variance components (which provide important information on the variability in the outcome between and within groups) is a key feature of multilevel models, and what distinguishes multilevel models from traditional contextual effects models and population-average models. For this reason, multilevel models have also sometimes been referred to as variance component or covariance component models. See also multilevel models.

References:
NOTE: References 1-38 were included in Part I and II of the Glossary, in Vol. 24, No. 3 (2003) and Vol. 24, No. 4 (2003) of the Epidemiological Bulletin.
(39) Wong G, Mason W. The hierarchical logistic regression model for multilevel analysis. J Am Stat Assoc 1985;80:513–24.
(40) Diggle PJ, Liang KY, Zeger SL. Analysis of longitudinal data. New York:Oxford University Press, 1994.
(41) Laird NM, Ware H. Random effects models for longitudinal data. Biometrics 1982;38:963–74.
(42) Longford NT. Random coefficient models. Oxford: Clarendon, 1982.
(43) Dempster AP, Rubin DB, Tsutakawa RK. Estimation in covariance components models. J Am Stat Assoc 1981;76:341–56.
(44) Searle SR, Casella G, McCullogh CE. Variance components. New York: Wiley, 1992.
(45) Morris C, Christiansen C. Fitting Weibull duration models with random effects. Lifetime Data Anal1995;1:347–59.
(46) Zeger S, Liang K, Albert P. Models for longitudinal data: a generalized estimating equation approach. Biometrics 1988;44:1049–60.
(47) Burton P, Gurrin L, Sly P. Extending the simple linear regression model for correlated responses: an introduction to generalized estimating equations and multi-level mixed modeling. Stat Med1998;17:1261–91.

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Epidemiological Bulletin , Vol. 25 No. 1, march 2004