Multilevel model
[5] Multilevel models are able to analyze these experiments without the assumptions of homogeneity-of-regression slopes that is required by ANCOVA.
[2][4] When there are multiple level 1 independent variables, the model can be expanded by substituting vectors and matrices in the equation.
Second, the researcher must decide whether parameter values (i.e., the elements that will be estimated) will be fixed or random.
[5] In order to conduct a multilevel model analysis, one would start with fixed coefficients (slopes and intercepts).
[2] One such statistic is the chi-square likelihood-ratio test, which assesses the difference between models.
The likelihood-ratio test can be employed for model building in general, for examining what happens when effects in a model are allowed to vary, and when testing a dummy-coded categorical variable as a single effect.
The assumption of linearity states that there is a rectilinear (straight-line, as opposed to non-linear or U-shaped) relationship between variables.
[9][disputed – discuss] However, most statistical software allows one to specify different distributions for the variance terms, such as a Poisson, binomial, logistic.
[12] The type of statistical tests that are employed in multilevel models depend on whether one is examining fixed effects or variance components.
[16] To conduct research with sufficient power, large sample sizes are required in multilevel models.
In order to detect cross-level interactions, given that the group sizes are not too small, recommendations have been made that at least 20 groups are needed,[16] although many fewer can be used if one is only interested in inference on the fixed effects and the random effects are control, or "nuisance", variables.
[4] The issue of statistical power in multilevel models is complicated by the fact that power varies as a function of effect size and intraclass correlations, it differs for fixed effects versus random effects, and it changes depending on the number of groups and the number of individual observations per group.
In an educational research example, the levels for a 2-level model might be However, if one were studying multiple schools and multiple school districts, a 4-level model could include The researcher must establish for each variable the level at which it was measured.
As a simple example, consider a basic linear regression model that predicts income as a function of age, class, gender and race.
It might then be observed that income levels also vary depending on the city and state of residence.
A simple way to incorporate this into the regression model would be to add an additional independent categorical variable to account for the location (i.e. a set of additional binary predictors and associated regression coefficients, one per location).
In other words, a simple linear regression model might, for example, predict that a given randomly sampled person in Seattle would have an average yearly income $10,000 higher than a similar person in Mobile, Alabama.
A multilevel model, however, would allow for different regression coefficients for each predictor in each location.
Essentially, it would assume that people in a given location have correlated incomes generated by a single set of regression coefficients, whereas people in another location have incomes generated by a different set of coefficients.
In psychological applications, the multiple levels are items in an instrument, individuals, and families.
In sociological applications, multilevel models are used to examine individuals embedded within regions or countries.
In organizational psychology research, data from individuals must often be nested within teams or other functional units.
The problem with this approach is that it would violate the assumption of independence, and thus could bias our results.
The problem with this approach is that it discards all within-group information (because it takes the average of the individual level variables).
[18] This is known as ecological fallacy, and statistically, this type of analysis results in decreased power in addition to the loss of information.
This also allows for an analysis in which the slopes are random; however, the correlations of the error terms (disturbances) are dependent on the values of the individual-level variables.
[18] Multilevel modeling is frequently used in diverse applications and it can be formulated by the Bayesian framework.
Particularly, Bayesian nonlinear mixed-effects models have recently received significant attention.
A central task in the application of the Bayesian nonlinear mixed-effect models is to evaluate the posterior density:
Bayesian-specific workflow comprises three sub-steps: (b)–(i) formalizing prior distributions based on background knowledge and prior elicitation; (b)–(ii) determining the likelihood function based on a nonlinear function
