#### Multilevel regression analysis as hierarchical model. What is the advantage of this method over classical models and in what conditions does it have the best properties of estimation results?

In this post we will moot multilevel linear model issue, i.e. statistical analysis, which is used in case an assumption about independence of observations from each other is broken or there is a high risk the assumption will be broken. In the first stage we will present the methodological arguments for using this method, while in the second stage we will show when to use it and we will discuss the basic terms underlying the Hierarchical Linear Model: multilevel design of measurements and its interactions, fixed and random effects and components of variance. Structural equation modeling CB

# Multilevel Modeling

#### When to use a multilevel analysis?

From a methodological perspective independence of observations is one of the most important assumption of inference about relations between variables. This assumption means that variable measurement result in one observation is independent from variable measurement result in one another. Such an interdependence may appear due to:

• systematic changes over time in the same observation
• dependence of repeated measurement - occurs within the context of multiple measurement for the same observation in time.
• exploration of similar groups of observations e.g. school classes, groups of employees, daily measurements for the same person, diary interviews
• exploration of hierarchical structures e.g. structures consist of students nested in school classes, which are nested in schools, and schools nested in regions etc.

Problems with dependence between observations arise in several types of situations. Some of them are not under the experimental and statistical control and thankfully there is now a solution – the multilevel analysis of data performed by using hierarchical linear/nonlinear models. Multilevel regression analysis are used to eliminate distortive effect of data analysis performer in groups and hierarchical structures. Also with some changes, this method is excellent for analysis a repeated measurements in time and space.

## Multilevel Modeling

#### HLM – Hierarchical Linear Modeling

Multilevel analysis is used in research conducted among observations, which are nested in groups/cluster/within and between objects. There are various analysis methods for such data. First method is an analysis consisting in ignoring the fact that data is arranged into groups/clusters and taking action based on data analysis using method „as is” or „roller”. Second method is an aggregation of data from/by groups and treating those groups also as units of measurement. This way we get mean values of independent and dependent variables in groups. To approach an analysis in a more ambitious way we can skip these procedures and use multilevel analysis. This is the only statistical method, which verifies hypothesis about an influence of higher-order factors (second level predictors) on the lower-order measeurements (relations considered on the lower level of analysis). Structural equation modeling CB

#### Intraclass correlation –ICC coefficient as a statistical criteria (benchmark?) determining whether to apply multilevel analysis

The assessment of data aggregation is presented by ICC coefficients (intraclass corealation coefficient). ICC measure gives information on how similar are group members to each other (coefficient takes values between 0 (a total lack of similarity) and 1 (a total similarity) [ICC= within-object variance/total variance of base model]. It is worth to check this coefficient already at the level of initial calculations to make a decision for the right method of statistical analysis. However, in case of having a hierarchical structure of data, without thinking deeply we should apply a multilevel analysis. It is worth to calculate this coefficient already at the level of zero model, which tests the variance of dependent variable and random variance between intercepts. Multilevel Modeling

#### Model of random effects, fixed affects and mixed model. How to estimate a multilevel relationships between variables?

Multilevel hierarchical regression model is much more advanced than a linear regression model based on classical least squares method. Multilevel nature of the regression analysis controls a grouped/aggregated structure of data and relations between variables on the lower level of analysis, simultaneously. Usually in a multilevel regression model the aim is to explain relations on the lower level by predictors from the higher level (multi-level interaction / interaction between levels).

In a multilevel regression analysis model we are dealing with various types of coefficients, estimates and transformations, which can be put in the multilevel regression equation as an individual components of an equation.

• fixed effects model of intercepts (constant level of intercepts in all groups)
• random effects model of intercepts (random level of intercepts in all groups)
• fixed effects model of regression slopes (constant regression slope in all groups)
• random effects model of regression slopes (random regression slopes in all groups)
• multilevel interaction model (second-level factors influences on first-level results)
• error variance on the level of intercepts
• mixed model of intercepts and regression line slopes (various specification of variance components)
• correlations between random components (e.g. higher values of intercepts can be related to higher values of slopes)

Most commonly used programs for analysis of a multilevel structure data is SPSS (Heck, Thomas, & Tabata, 2010; Mayers, 2013), process macro SPSS, which allows to perform Multilevel Mediation Analysis (Hayes & Rockwood, 2020). Structural equation modeling in M PLUS allows to perform very advanced Path Modeling within and between groups (Byrne, 2012). Very popular software is HLM program, and of course the best tool for analysis of such data is R and “lme4” package.

If you have a problem with multilevel methodology, performing or interpreting statistical analysis results of mixed models? We kindly invite you to contact us. Structural equation modeling CB