Abstract
Drawing on the contextual, evaluative, and summarizing studies of this book, this chapter explicates 11 uses for multilevel models and defines relevant vocabulary, concepts, and notational conventions. Because multilevel models are composed of both fixed and random components, statisticians refer to them as mixed models (Littell et al. 2006). Because multilevel models focus on data at different hierarchical levels, educational researchers refer to them as hierarchical models (Raudenbush and Bryk 2002). In the contextual analyses of data at one point in time, the level-1 response variable and its covariates are conceptualized as being grouped (or contained) within the level-2 units. In the analyses of data at several points in time, the level-1 response variable is an observation at a time point on an entity, and the repeated observations on that entity are said to be grouped or contained by that entity. The entity (e.g., a person, an organization, a country) is the level-2 unit. Ideally, multilevel models assess change on disaggregated data at several points in time (e.g., the scores on the repeated assessments of individual students who are grouped into classrooms). When the chapters of this book model aggregated data, it is because the disaggregated data are not available.1 By applying special cases of generalized linear mixed models—the Poisson and logit—some chapters model response variables that are not normally distributed. By applying multilevel models, all of the core chapters address the clustering of level-1 units when they are contained within level-2 units.
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Notes
- 1.
Cox and Wermuth (1996, 51) note that aggregating data: “over individuals at each of a number of time points, or over time for each of a number of individuals… can appreciably simplify analysis,… but care is needed in relating the properties of the aggregated data back to the behavior of individuals.”
- 2.
Littell et al. (2006, 7) state:
Of these [three procedures], ML is usually discouraged, because the variance component estimates are biased downward, and hence so are the standard errors that are computed from them. This results in excessively narrow confidence intervals whose coverage rates are below the nominal 1-α level, and upwardly biased test statistics whose Type I error rates tend to be well above the nominal α level. The REML procedure is the most versatile, but there are situations for which ANOVA procedures are preferable; Proc Mixed in SAS uses the REML approach by default, but provides optional use of ANOVA and other methods when needed.
- 3.
Claeskens and Hjort (2008, 271) state:
The maximum likelihood method for the estimation of variance components in mixed models results in biased estimators. This is not only a problem for mixed models, even in a simple linear model Y = Xβ + ε with ε ~ N N (0, \( \sigma_\varepsilon^2 \) I N ) the maximum likelihood estimator of \( \sigma_\varepsilon^2 \) is biased. Indeed \( \hat{\sigma }_\varepsilon^2 \)= N −1 SSE(\( \hat{\beta } \)), while an unbiased estimator is given by (N − r)−1SSE(\( \hat{\beta } \)), with r being the length(β). The method of restricted maximum likelihood produces unbiased estimators of the variance components.
- 4.
Holland (1986, 945) distinguishes the study of the “causes of effects” from “measuring the effects of causes.” The potential outcomes perspective of the evaluative research studies of Part 3 focus on the multiple effects of a cause, whereas the more basic research studies of Part 2 focus on the multiple causes of a response variable.
- 5.
Philip Gibbs of the SAS Institute offers this advice (personal communication, April 12, 2007):
You can only compare the likelihoods (and by extension the associated IC [information criteria] statistics) under REML if the models involved have the same set of fixed effects….The usual progression in MIXED usually goes something like this. Come up with a set of fixed effects (your MODEL statement) that makes sense to you. You do not have to be statistically rigorous when you do this. Then, under REML and NOT changing the MODEL statement in this phase, make changes to the RANDOM and/or REPEATED statement(s) until you find a model that has a “best” AIC value. Now, switch over to ML estimation and change the effects on the MODEL statement until you come up with a “best” AIC value. When that is done, switch back to REML to report the final model.
- 6.
The term “deviance” most often is used in the context of models with a categorical response variable. This book sometimes uses this word in the context of models with normal response variables, as done here. At other times it just reports the −2×residual log likelihood of REML estimation or the −2×log likelihood of ML estimation.
- 7.
The following Estimate statements ask SAS to calculate estimates of the predicted regional grand mean, intercept 1 | region (dichofre), and the 18 predicted regional means when the levels of the random regions are classified by full democracy. There are 18 spaces on the right-most part of each estimate statement, one for each region; a 1 in a space tells SAS to estimate that region’s parameter whereas a 0 tells SAS not to estimate that region’s parameter. For the grand mean, there is a 1 for each region so SAS estimates the mean across all 18 regions.
estimate ‘mean’ intercept 1 | region (dichofre) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;
estimate ‘region 01e’ intercept 1 | region (dichofre) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
estimate ‘region 01 m’ intercept 1 | region (dichofre) 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
estimate ‘region 01n’ intercept 1 | region (dichofre) 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
estimate ‘region 01s’ intercept 1 | region (dichofre) 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
estimate ‘region 01w’ intercept 1 | region (dichofre) 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
estimate ‘region 02’′ intercept 1 | region (dichofre) 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ;
estimate ‘region 03c’ intercept 1 | region (dichofre) 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ;
estimate ‘region 03ca’ intercept 1 | region (dichofre) 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ;
estimate ‘region 03sa’ intercept 1 | region (dichofre) 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ;
estimate ‘region 04e’ intercept 1 | region (dichofre) 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ;
estimate ‘region 04sc’ intercept 1 | region (dichofre) 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ;
estimate ‘region 04se’ intercept 1 | region (dichofre) 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ;
estimate ‘region 04w’ intercept 1 | region (dichofre) 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ;
estimate ‘region 05e’ intercept 1 | region (dichofre) 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ;
estimate ‘region 05n’ intercept 1 | region (dichofre) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ;
estimate ‘region 05s’ intercept 1 | region (dichofre) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ;
estimate ‘region 05w’ intercept 1 | region (dichofre) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ;
estimate ‘region 06’′ intercept 1 | region (dichofre) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ;
run;
- 8.
Singer and Willett (2003, 136) clarify the notion of borrowing strength as follows:
Because OLS trajectories differ markedly from person to person, the model-based trajectories differ as well, but their discrepancies are smaller because the population average trajectories are more stable. Statisticians use the term “borrowing strength” to describe procedures like this in which individual estimates are enhanced by incorporating information from others with whom he or she shares attributes. In this case, the model-based trajectories are shrunk toward the average trajectory of that person’s peer group (those with the same predictor values). This combination yields a superior, more precise estimate.
- 9.
When commenting on William Cochran’s many contributions to observational studies, Donald Rubin (2006, 10) states that he and Cochran agree that several comparison groups are better than only one:
A second theme in design is the need for a control group, perhaps several control groups (e.g., for a within-hospital treatment group, both a within-hospital control group and a general-population control group). The rationale for having several control groups is straightforward: If similar estimates of effects are found relative to all control groups, then the effect of the treatment may be thought large enough to dominate the various biases probably existing in the control groups, and thus the effect may be reasonably well estimated from the data.
- 10.
The ongoing controversies about the alleged causal effects of childhood vaccinations on engendering autism, and the ignoring of scientific evidence regarding global warming, suggest that ordinary citizens need practice in assessing evidence about causality.
- 11.
The present-day fragmentation of the social sciences leads to numerous interest groups, each one of which may have their own unique standards for judging the quality of a piece of work. Consequently, intra-disciplinary conflicts over styles of research are common; to advance cumulative knowledge, shared standards are needed. Their absence leads to personal attacks that inhibit progress rather than collegial peer reviews that improve the work.
- 12.
The logical structure of this explication is similar to that of a multilevel model: The review groups in the two different geographical areas, Europe and the USA., are level-2 variables that encompass the studies that they review, which are level-1 variables, as are the conclusions reached by these two review groups. In essence, the research summary of Chapter 15 exemplifies a qualitative meta-analysis.
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Smith, R.B. (2011). Uses for Multilevel Models. In: Multilevel Modeling of Social Problems. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9855-9_5
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