From GLM to HLM Working with Continuous Outcomes EPSY 5245 Michael C. Rodriguez

Continuous Variables Review statistical procedures for continuous variables Consider options on Variables Chart Generalize options under the GLM approach

Statistical Paradigm Model Building Estimation of Parameters Testing Fit of the Model

Model Building Theory Model Specification Measurement Data Collection

Estimation of Parameters Conceptualizing Relevant Factors A General Approach to Data Analysis GLM Model Assumptions

General Approach to Data Analysis Univariate & bivariate descriptive analyses Specifying the model Testing interaction terms Removing insignificant terms Examining outliers Checking assumptions

General Linear Model Assumptions STRUCTURAL Independent observations Linear relationships Variable independence Errorless measurement Correct specification STOCHASTIC Independence Normality Mean of zero Homogeneity of variance Independence from explanatory variables

Testing Model-Data Fit Parsimony Indicators Correlation Simple Regression Multiple Regression Controlling Type-I error

Common Problems in the analysis of clustered (nested) data The “unit of analysis” problem – misestimated precision Testing hypotheses about effects occurring at each level and across levels Problems related to measurement of change or growth

Estimation Requirements Estimation of parameters requires some distributional assumptions. One requires the error term (the part of the outcome that is not explained by observed factors) to be independent and identically distributed. This is in contrast with the idea that people exist within meaningful relationships in organizations. Frank, K. (1998). Quantitative methods for studying social contexts. Review of Research in Education, 23, 171-216.

The Notation of Regression 𝑌 𝑖 = β 0 + β 1 𝑋 1𝑖 + β 2 𝑋 2𝑖 +…+ β 𝑄 𝑋 𝑄𝑖 + ε 𝑖 𝑌 𝑖 = β 0 + β 1 𝑋 1𝑖 + ε 𝑖 or

What’s in a name… Sociology: Multilevel Models Biometrics: Mixed-Effects Models or Random-Effects Models Econometrics: Random-Coefficient Regression Models Statistics: Covariance Components Models

When to use HLM Nested data: Dependent observations Do children of different gender, race, or exposure to different reading programs grow at the same rate in reading? Is the relationship between social status and achievement the same in schools of different size or sector (public v. catholic)?

Building Models in HLM Level One Within Units Level Two Between Units

Examples of Multiple Levels Students Classrooms Schools Teachers School Districts Children Families Neighborhoods Cities Nurses Wards/Units Hospitals Workers US-Based Firms Multinational Firms Juvenile Delinquents Social Workers Social Service Agencies Longitudinal Scores

Advantages of HLM Adjusting for nonindependence of observations within subjects Larger framework for real-life problems Unbalanced designs and missing data are accommodated

Alternatives to HLM Individual level Group level Just use regression

What do we gain through HLM? Improved estimation of effects within individual units. Example: Developing an improved estimate of a regression model for an individual school by borrowing strength from the fact that similar relationships exist for other schools.

What do we gain through HLM? Formulation and testing of hypotheses about cross-level effects. Example: How school size might be related to the magnitude of the relationship between social class and academic achievement within schools.

What do we gain through HLM? Partitioning variance and covariance components among levels. Example: Decomposing the correlation among a set of student-level variables into within- and between-school components. How much of the variance is within or between schools?

A relationship between 8th grade and 11th grade performance? Goldstein (1999). http://www.arnoldpublishers.com/support/goldstein.htm

Accounting for school Goldstein (1999). http://www.arnoldpublishers.com/support/goldstein.htm

When school is a meaningful organizational unit, relations may be a function of the unit. Achievement Socioeconomic Status

Example Model β 0𝑗 = γ 00 + γ 01 𝐶𝑎𝑡ℎ𝑜𝑙𝑖𝑐 𝑗 + 𝑢 0𝑗 β 1𝑗 = γ 10 + γ 11 𝐶𝑎𝑡ℎ𝑜𝑙𝑖𝑐 𝑗 + 𝑢 1𝑗