# By Zach Andersen Jon Durrant Jayson Talakai

## Presentation on theme: "By Zach Andersen Jon Durrant Jayson Talakai"— Presentation transcript:

By Zach Andersen Jon Durrant Jayson Talakai
Multilevel Analysis By Zach Andersen Jon Durrant Jayson Talakai

OUTLINE Jon – What is Multilevel Regression Jayson – The Model Zach – R code applications / examples

WHAT IS MULTILEVEL REGRESSION
Regression models at multiple levels, because of dependencies in nested data Not two stage, this occurs all at once

Employees in organizations Firms in various industries
EXAMPLES Students in schools Individuals by area Employees in organizations Firms in various industries Repeated observations on a person

WHEN TO USE A MULTILEVEL MODEL?
Individual units (often people), with group indicators (e.g. Schools, area). Dependent variable (level 1) More than one person per group Generally we need at least 5 groups, preferably more. (Ugly rule of thumb)

WHEN TO USE A MULTILEVEL MODEL?
Use a multilevel model whenever your data is grouped (or nested) into categories (or clusters) Allows for the study of effects that vary by group Regular regression ignores the average variation between groups and may lack the ability to generalize

DATA STRUCTURE AND DEPENDENCE
Independence makes sense sometimes and keeps statistical theory relatively simple. Eg; standard error(sample average) = s/n requires that the n observations are independent But data often have structure, and observations have things in common; same area, same school, repeated observations on the same person Observations usually cannot be regarded as independent

Multilevel Models

PROBLEMS CAUSED BY CORRELATION
Imprecise parameter estimates Incorrect standard errors

A SIMPLE 2-LEVEL HIERARCHY
School 1 School 2 Student 1 Student 2 Student 3 Student 1 Student 2 Student 3

A SIMPLE 2-LEVEL HIERARCHY
School 1 School 2 Level 2 Student 1 Student 2 Student 3 Student 1 Student 2 Student 3 Level 1

The first level of a hierarchy is not necessarily a person
PEOPLE ARE AT LEVEL 1?? The first level of a hierarchy is not necessarily a person

A SIMPLE 2-LEVEL HIERARCHY
Industry 1 Industry 2 Level 2 Firm 1 Firm 2 Firm 3 Firm 1 Firm 2 Firm 3 Level 1

A SIMPLE 2-LEVEL HIERARCHY
Person 1 Person 2 Level 2 Event 1 Event 2 Event 3 Event 1 Event 2 Event 3 Level 1

BRIEF HISTORY Problems of single level analysis, cross level inferences and ecological fallacy

DISCUSSION AS TO WHY A NORMAL REGRESSION CAN BE A POOR MODEL
Because Reality might not conform to the assumptions of linear regression (Independence) Because in nature observation tend to cluster A random person in Lubbock is more likely to be a student then a random person in another city (clustering of populations/not independent) Different clusters react differently

Also longitudinal, geographical studies
EXTENSIONS Focus was initially on hierarchical structures and especially students in schools Also longitudinal, geographical studies More recently moved to non hierarchical situations such as cross-classified models. (single level is part of more than one group)

INTRACLASS CORRELATION
Level 1 variance explained by the group (level 2) ICC is the proportion of group-level variance to the total variance Formula for ICC: Variance in group Overall variance

Random or Fixed Effects
MULTILEVEL MODELING Random or Fixed Effects What are random and fixed effects? When should you use random and fixed effects? Types of random effects models The Model Assumptions of the model Building a multilevel model

Fixed vs random effects
**Anytime that you see the word “population” substitute it with the word “processes.”

INTRODUCING THE MODEL

Types of Models: Random Intercepts Model
Intercepts are allowed to vary: The scores on the dependent variable for each individual observation are predicted by the intercept that varies across groups.

Types of Models: Random Slopes Model
Slopes are different across groups. This model assumes that intercepts are fixed (the same across different contexts).

Types of Models: Random intercepts and slopes model
Includes both random intercepts and random slopes Is likely the most realistic type of model, although it is also the most complex.

Assumptions for Multilevel Models
Modification of assumptions Linearity and normality assumptions are retained Homoscedasticity and independence of observations need to be adjusted. Observations within a group are more similar to observations in different groups. Groups are independent from other groups, but observations within a group are not.

Multilevel Model: Example

Multilevel Model: Level 1 Regression Equation

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