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MULTIPLE REGRESSION AND
MULTILEVEL MODELLING
1
2. 2
I NTRODUCTION
1. Brief overview of multiple regression analysis
2. Multiple regression using PISA data
3. Brief overview of multilevel modelling
4. Multilevel modelling using PISA data
5. Differences between the two types of analyses
4. 4
S IMPLE REGRESSION MODEL
Predicting the dependent variable using linear
relationship with independent variables
Regression analysis with one independent
variable:
β0 is the intercept (the value of Ŷi when Xi=0)
β1 is the slope of the line that minimises εi
9. 9
PISA DATA
P LAUSIBLE VALUES
Plausible values for cognitive performance
5 randomly drawn values from a student’s most
likely ability range (posterior distribution)
Unbiased populations estimates (even with one
PV)
Imputation variance (measurement error)
NEVER average the plausible values!
Instead, average 5 statistics (means, regression
coefficients, etc.)
10. 10
PISA DATA
S TUDENT WEIGHTS
Final student weight: the number of students
represented in the population by each student
The inverse of the probability to select the
student’s school times the probably of selecting
the student given that the school is selected
Non-response and post-stratification
adjustments and trimming
11. 11
PISA DATA
R EPLICATE WEIGHTS
80 BRR replicate weight with Fay’s k=0.5
Used to compute sampling variance
Computation of sampling variance using BRR
weights
Takes two-stage sampling method into account
Takes stratification into account
is identical for any statistic
12. 12
E RROR VARIANCE
Error variance is a combination of the sampling
variance and the imputation variance
(measurement error)
Imputation variance can only be estimated when
using a set of plausible values
Imputation variance is small compared to the
sampling variance
Standard error is the square root of the error
variance
13. 13
C OMPUTATION OF
STANDARD ERROR
Error variance
Sampling variance
Imputation variance
Standard error is the square root of the error variance
14. 14
SPSS REPLICATES ADD - IN
Password WI-FI: Hawking09+
mypisa.acer.edu.au
Public data & analysis
Software & manuals
Download and install replicates add-in
Start SPSS
Copy CD to C:Kiel and unzip file
15. 15
E XAMPLE IN SPSS
C:KielINT_Stu06_SCHWGT.sav
German data
Regress science performance on
Sex
Immigration status
ESCS
17. 17
E XAMPLE
For Japan in 2006:
Strong relationship between ESCS and science
(38.8)
Large intra-class correlation in performance
Small intra-class correlation in ESCS
For this example, only nine Japanese schools are
selected
18. 18
S INGLE LEVEL REGRESSION
Overall slope is 38.8
20. 20
M ULTILEVEL MODEL WITH
FIXED SLOPES
Average slope is 7.2
21. 21
I NTERPRETATION REGRESSION
COEFFICIENTS
Single level regression gives the overall relationship
between ESCS and performance in a country (38.8 in
Japan)
Multi-level regression takes the 2-level structure of
the data into account and
Estimates a unique slope within each school (or the
variance of the slopes) or
Estimates the average slope within schools (7.2 in
Japan)
Which type of analysis is more correct?
22. 22
N OTATION MLM
Random intercept model
Level 1: Yij 0 j 1 X ij rij
Level 2: 0 j 00 01W j u0 j
Random slopes and random intercept
Level 1: Yij 0 j 1 j X ij rij
Level 2: 0 j 00 01W j u0 j
1 j 10 11W j u1 j
23. 23
R ANDOM INTERCEPT
System of equations
Level 1: Yij 0 j 1 X ij rij
Level 2: 0 j 00 0 jW j u0 j
Mixed-effects model
Yij 00 0 jW j 1 X ij u0 j rij
Fixed part Random part
24. 24
R ANDOM INTERCEPT AND
RANDOM SLOPES
System of equations
Level 1: Yij 0 j 1 j X ij rij
Level 2: 0 j 00 0 jW j u0 j
1 j 10 11W j u1 j
Mixed-effects model
Yij 00 0 jW j u0 j 10 11W j u1 j X ij rij
00 0 jW j 10 X ij 11W j X ij u1 j X ij u0 j rij
Fixed part Random part
Cross-level interaction
25. 25
VARIANCE DECOMPOSITION
In single level regression analysis, the overall
variance of the dependent variable is estimated
and the amount of this variance that is explained
by the independent variables (R-squared)
In multilevel analysis, the variance is
decomposed in between-cluster (school) and
within-cluster variance
The independent variables can explain variance
at either level or at both levels
26. 26
VARIANCES
Total variance = within-cluster variance +
between-cluster variance
Average within-cluster variance
y y j
2
n( 2) n(1)
2 ij
n(2) n(1) 1
r
j 1 i 1
Between-cluster variance
y y
2
n( 2 )
r2
0 j
j
2
(2)
j 1 n n (1)
27. 27
I NTRACLASS CORRELATION
AND EXPLAINED VARIANCE
Null model: yij 00 u0 j rij
Intraclass correlation (rho)=
between-cluster variance / total variance
Explained variance (R-squared) of a model with
predictors:
Level 1: 1 - (var(W)p / var(W)n)
Level 2: 1 - (var(B)p / var(B)n)
28. 28
T HE STANDARD ERROR
One assumption of OLS is independence of
observations
In 2-stage sampling designs, observations within
clusters are often not independent
MLM allows for correlated errors and therefore gives
unbiased SEs
Generally, SEs estimated with OLS are too small
However, BRR replicate weights are designed to deal
with the dependence of observations within schools,
so OLS with BRR gives correct standard errors!
29. 29
W EIGHTING - 1
Single level regression: final students weights and
BRR replicate weights
How do we use PISA weights in MLM?
Data analysis manual: normalise final student
weights and replication weights and run the
analysis in SPSS or SAS
We now know this is not the best way
30. 30
W EIGHTING - 2
SPSS and SAS do not assume the weights to be
sampling weights (they are precision weights)
SPSS and SAS can only weight at the student level
MLM and BRR are both taking the multi-level
structure of the data into account, so this is done
twice in the PISA data analysis manual method
However, there is no final consensus about the
right way to use weights in MLM
31. 31
W EIGHTING - 3
In PISA school-level sampling is much more
informative than student-level sampling
(stratification is at school-level; students have
often very similar weights within schools )
Therefore, schools should be weighted by a
school-level weight
Students should be weighted by a conditional
student weight (inverse of the probability to be
selected given that the student’s school is
sampled)
32. 32
W EIGHTING - 4
Options for conditional student level weights:
Equal weights (weight=1)
Raw conditional student weights
Rescaled weights: Pfefferman method 1 when
student sampling is not informative
Rescaled weights: Pfefferman method 2 when
student sampling is informative
Differences are small when cluster sizes are
larger than 20 students
33. 33
R AW CONDITIONAL STUDENT
WEIGHTS
Raw conditional student weights:
W_FSTUWT
w
(1)
i| j
W_FSCHWT
School weight is included in the school
questionnaire data file
Not exactly correct, because some adjustments
are made independent of schools (e.g. non-
response adjustment)
Often leads to an overestimation of the
between-school variance
34. 34
P FEFFERMAN METHOD 1
When student sampling is not informative at
level 1
Conditional student weights are multiplied by the
sum of weights within cluster divided by the sum
of squared weights within cluster
n(1)
j
|j
wi(1)
PFEFF1 wi(1)
|j
i 1
n(1)
w
j
(1) 2
i| j
i 1
35. 35
P FEFFERMAN METHOD 2
When student sampling is informative
Conditional student weights are divided by the
average conditional student weight in school j or
n (1)
PFEFF 2 wi(1)
j
|j n(1)
j
w
i 1
(1)
i| j
This is the same as normalising full student
weights within schools
36. 36 L ET ’ S TRY IT OUT IN MLWI N
Australia, because they oversample indigenous
students who perform less than non-indigenous
students (positive correlation between
conditional student weights and performance)
C:Kiel INT_Stu06_SCHWGT.sav
I have added the full school weights
(W_FSCHWT) and the normalised school weights
(N_FSCHWT)
N_FSCHWT= W_FSCHWT*SAMPSIZE/POPSIZE
38. 38
C OMPARING CONDITIONAL
STUDENT WEIGHTS IN MLWI N – 2
Equal weights (=1) and Pfefferman methods 1
and 2 give similar results when using PISA data
Pfefferman method 2 most conservative:
recommended
Raw weights over-estimate the within-school
variance (I think this is MLwiN specific, similar
problem with unscaled school weights)
39. 39
W EIGHTS STANDARDISED BY
MLWI N
MLwiN’s standardisation of the weights:
At the school level, the full school weight is
normalised at country level
The student level weight is the Pfefferman 2
conditional student weight * the normalised
school weight * a factor to make the average
student weight equal to one
Odd that the school weight is included at both
levels, but results are the same as in HLM
40. 40 W HICH WEIGHTS ARE BETTER ?
In simulation study the differences in results
were minimal, but the differences were big when
using data from some real countries
We do not know which method is best
Probably safest in MLwiN to use standardised
weights, because we do not know how the
weights are built into their algorithm
Need to explore what other software packages
do (gllamm in STATA)
41. 41
R EFERENCES
Rabe-Hesketh, S. & Skondral, A. (2006). Multi-
level modelling of complex survey data. Journal
of Royal Statistical Society, 169, 805-827
Chantala, K., Blanchette, D. & Suchindran, C. M.
(2006). Software to compute sampling weights
for multilevel analysis.
http://www.cpc.unc.edu/restools/data_analysis/
ml_sampling_weights/Compute%20Weights%20f
or%20Multilevel%20Analysis.pdf
43. 43
E XERCISE
For MLwiN, data has to be sorted first by the
highest level ID variable, then by the second
highest, etc. (SCHOOLID in PISA)
MLwiN needs a constant in the data (compute
CONS=1.) to estimate the intercept
Start with data from Chile, where the intraclass
correlation in both science performance and
ESCS is high
Start MLwiN…
44. 44
WARNINGS
Definition of a school is not the same in each
country and not always that clear (campus)
Differences in educational systems between or
even within countries or cycles (tracked)
Risk of swimming and too complicated models to
interpret if MLM is more data driven than theory
driven
To interpret results carefully, you need to know
enough about the educational system in a
country or differences across countries
46. 46 C OMPARISONS
OLS with BRR MLM
Fixed effects Random effects and cross-
level interactions
Includes measurement Difficult to include
error measurement error
Takes stratification into I think it doesn’t take
account school stratification into
account
Output is SPSS data file for Output is often in text
easy editing format
47. 47
O PTIONS FOR FINAL PART OF
THE WORKSHOP
Try a MLM on data of your own country
Try school and student level variables
Try to add cross level interactions (free the
slopes)
Discuss MLMs that you have tried in the past or
would like to do in the future
Ask any PISA related data analysis questions