Multilevel Modelling

Question 1

What is multilevel data?

Answer 1

Data that has clustering at different hierarchical levels e.g. different pupils in different schools which are based in different areas.

Question 2

What are 3 types of multilevel data?

Answer 2

Geographic: – e.g data of country will have data from different regions

Relational- relationships between different factors e.g., Schools, classrooms, pupils
Countries, regions, towns, families
Company, office, team, individuals

TemporalLongitudinal: Different time points

Question 3

Where do we find multilevel data?

Answer 3

1. Naturally occurring e.g schools
2. Due to sample design e.g. clustered, multi-stage)
3. Repeated measures

Question 4

What is one way of differentiating between data that is multilevel and data that is not?

Answer 4

If can identify different hierarchies then that is a good example of multilevel data

If data is a single level with subgroups e.g. office with different ethnicities, this would not be considered multi level data

Question 5

What are two reasons for computing multilevel analysis?

Answer 5

1.Clustering as a problem - Sometimes we have data that is clustered and must take this into account

2. Clustering as a substantive interest- Answer multilevel hypotheses

Question 6

Why can clustering be a problem?

Answer 6

1. Clustered data violate the assumptions of simple random sampling.
2. Traditional multiple regression techniques treat the units of analysis (e.g. individuals) as independent observations.
3. If we fit hierarchical data with simple OLS models, (i.e. ignoring any clustering) the standard errors of our regression coefficient will be underestimated, leading to an overstatement of statistical significance.

Question 6

Why can clustering be a problem?

Answer 6

1. Clustered data violate the assumptions of simple random sampling.
2. Traditional multiple regression techniques treat the units of analysis (e.g. individuals) as independent observations.
3. If we fit hierarchical data with simple Ordinary least-squares (OLS) models, (i.e. ignoring any clustering) the standard errors of our regression coefficient will be underestimated, leading to an overstatement of statistical significance.

Question 7

Why can clustering be a substantive interest?

Answer 7

We are often interested in estimating the amount of variation between groups, and the extent to which it can be explained by group-level explanatory variables.

E.g., Is there between-school variability in students’ academic progress?
Do health outcomes vary across areas?
Are between-area variations in health explained by differences in access to health services?
Is the amount of variation between areas different for rural and urban areas?

Question 8

What are alternative data analysis strategies if clustering is not of substantive interest?

Answer 8

Fixed effects regression

Design-based modelling

Adjusted standard errors - Used most often when need to adjust for potential clustering

Question 9

According to Richard McElreath what should multilevel regression be the default approach?

Answer 9

1. Improved estimates for repeat sampling
2. Improved estimates for imbalance in sampling
3. Estimates of variation
4. Avoid averaging, retain variation

Question 10

What software can be used for multilevel analysis?

Answer 10

Stata

R

Julia

Python

Stan

brms

Question 11

What terms are classified under multilevel modelling?

Answer 11

Multilevel model
Random effects model
Mixed model
Random coefficient model
Random parameter models
Hierarchical model
Nested models
Split-plot designs
Subject specific models
Variance component models

Question 12

What is variance a measure of and how is it defined?

Answer 12

Variance is a measure of how spread out your data is.

Defined as the average of the squared differences from the mean.

Question 13

What can we think of a residual as?

Answer 13

As a measure of error, or how far off your predicted value was from the actual value.

Question 14

In a single level Ordinary least-squares regression what is the Bo represented by?

Answer 14

Bo - coefficient

Question 15

What does a least squares regression select?

What is the value called?

Answer 15

The line with the lowest total sum of squared prediction errors.

Sum of Squares of Error, or SSE.

Question 16

What does the notation yi = B0 +B1xi + Ei mean?

Answer 16

Presents single level model.
I represents number of individuals in sample

Question 17

What does the notation yij +B0 + B1xij+ Eij mean?

Answer 17

Two level notation, outcome y for individual(s) in cluster j(ranges from 1 to however many clusters we have. If there was a three level model we would include convention k to represent this.

Question 18

With clustered data, the residuals within each cluster are likely to be correlated. What does this mean?

Answer 18

Individuals within a cluster are likely to be more similar to one another than they are to individuals from another clusters

Question 19

Explain what the formula yij = B0 + uj + Eij mean?

Answer 19

Uj represents distance from overall intercept B0 to group intercept

Eij represents distance from group level mean to observed data point at individual level

Question 20

What is meant by partitioning variance?

Answer 20

Having two sources of variance - between groups variance and within group variance

Question 21

What does we assume about epsilons?

Answer 21

They are normally distributed with a mean of 0 and variance of sigma squared e

Question 22

What does the variance partition coefficient measure?

Answer 22

Proportion of total variance
is due to differences between groups

Question 23

What is the formulae for variance partition coefficient?

Answer 23

Between group variance divided by total variance(between group variance plus within group variance)

Question 24

What is the variance partition coefficient sometimes but not always equivalent to?

Answer 24

Intraclass correlation coefficient

Question 25

What are two caveats of fitting a two level model and what is a possible solution?

Answer 25

• The random intercepts model introduced additional complexity.
• Did it result in a significant improvement in model fit? - We can test this with a likelihood-ratio test

Question 26

What does a likelihood ratio test examine?

Answer 26

Group effects

Question 27

What is the formulae for a likelihood ratio test?

Answer 27

D = -2 Log L2 (Fit of multilevel model) - -2 Log L1(Fit of single level model)

The deviance statistic (D) is compared against a X2 distribution with D.F. degrees of freedom.

Question 28

What is the degrees of freedom?

Answer 28

The difference in the number of parameters in the multilevel vs. individual level model).