Post on 29-Jan-2023
Learn to Use the Multilevel Model
Test in SPSS With Data From the
Longitudinal Educational
Outcomes Dataset (2017–18)
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Learn to Use the Multilevel Model
Test in SPSS With Data From the
Longitudinal Educational
Outcomes Dataset (2017–18)
How-to Guide for IBM® SPSS® Statistics Software
Introduction
In this guide, you will learn how to produce a Multilevel Model (MLM) test in
IBM® SPSS® Statistics software (SPSS) using a practical example to illustrate
the process. You will find links to the example dataset, and you are encouraged
to replicate this example. An additional practice example is suggested at the end
of this guide. The example assumes that you have already opened the data file in
SPSS.
Contents
1. Multilevel Modelling
2. An Example in SPSS: Exploring the Differing Effect Graduates’ Gender
and Number of Years After Graduation Has on Median Annual
Earnings Between Universities Attended
2.1 The SPSS Procedure
2.2 Exploring the SPSS Output
2.3 How to Report the Findings
3. Your Turn
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1 Multilevel Modelling
An MLM test is a test used in research to determine the likelihood that a number
of variables have an effect on a particular dependent variable. This sounds very
similar to multiple regression; however, there may be a scenario where an MLM
is a more appropriate test to carry out. This is due to the nature of hierarchy that
at times can be found in a dataset. By hierarchy, we are referring to the levels
which each factor (or independent variable) is being measured. This is called a
clustering effect, due to the potential for Level 1 variables (i.e., Gender) to be
clustered via a Level 2 variable (i.e., University Studied). Level 3 variables could
also be recognized in a dataset (Location). It is important to recognize a potential
clustering effect in order to carry out the appropriate analysis.
2 An Example in SPSS: Exploring the Differing Effect Graduates’ Gender and
Number of Years After Graduation Has on Median Annual Earnings Between
Universities Attended
2.1 The SPSS Procedure
MLM tests are appropriate when certain assumptions have been met. To look
for normal distribution, we must carry out the appropriate analysis for each of
the variables we intend to use. Prior to running any statistical test, it is good
practice to examine each variable on its own, this is called univariate analysis.
This allows us an opportunity to describe the variable and get an initial “feel”
for our data. For categorical variables, frequency tables can show us whether
the number of cases in each group, which will show whether any groups are
significantly larger or smaller than others, could affect the results. For scale/
interval variables, measures of central tendency (MoCT) allow us to see whether
the data are skewed in any direction which can also affect results.
Univariate analysis can be carried out by selecting the following on SPSS:
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Analyze → Descriptive Statistics → Frequencies
By selecting the categorical variables, Gender and Years after graduation, we can
see that each variable is normally distributed with the groups all being equal in the
number of cases.
To carry out univariate analysis on scale interval variables, POLAR3 Quintile
1 Proportion and Median Annual Earnings, we must carry out the same steps
as with categorical variables and then select “statistics” to ensure we have the
information highlighted in Figure 1. A histogram is also useful as it allows us to
visual the distribution. This is done by selecting “charts” and “histogram” along
with “show normal curve on histogram” as shown in Figure 2.
Figure 1: Measures of Central Tendency.
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Figure 2: Selecting a Histogram.
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Table 1 and Figure 3 show that both variables are positively skewed with wider
range towards the higher end of each scale.
Table 1: MoCT for Scale/Interval Variables.
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POLAR3 Q1 Proportion Median Annual Earnings
N 7,216 14,453
Mean 11.6 22,801.63
Median 10.35 21,700
Standard deviation 6.75 6,802.74
Range 63 80,000
Minimum 1 5,500
Maximum 64 85,500
Figure 3: Histograms for Scale/Interval Variable.
As the data are skewed, recoding is necessary to provide normally distributed
variables for the MLM test. This is done by selecting the following
Transform → Recode into different variables
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It is important you select the correct variable and provide it with a new name and
appropriate label. Once you have done this, select “old and new values” to enter
the new values for your variables. Simply type the old value in the value box under
Old value and enter the new value you would like to change it to in the value box
under New value. Once you have entered all new values, select “All other values”
under Old value and select “System-missing” under New value. Figure 4 shows
how Years after graduation was recoded before being entered into the MLM.
Figure 4: Recoding Years After Graduation.
When recoding scale variables, it is more practical to select the ranges.
To reduce the wide range in the upper end of the scale for POLAR3 Q1 proportion,
the highest in the range was set at 34.6. This was done by entering 1, the lowest,
under the “range” box and entering 34.6 under the “through” box and selecting
“Copy old value(s)” box. Figure 5 shows what this looks like in SPSS.
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Figure 5: Recoding POLAR3 Q1.
To select a new range for Median annual earnings, the same process was applied
as with POLAR 3 Q1proportion with the minimum being 5,500 and the new highest
value as 51,200.
Along with univariate analysis, it is also important that you carry out parametric
assumptions before running an MLM test. It is good practice to ensure the data
have linearity, and it is good practice to run a Levene’s test for homogeneity.
Multicollinearity is also a common problem in MLMs due to the nature of the
clustering effect. We must therefore centre the scale measures of both Level 1
and levels above Level 1 variables.
POLAR3 Q1 Proportion was group mean centred as a Level 1 variable to be
appropriately placed into the MLM test. This was done by a number of steps, the
first aggregating the proportion according to university studied. This was done by
selecting the following
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Data → Aggregate
University studied was entered into the “Break variable” box, and POLAR3 Q1
Proportion was entered into the “Summaries of variables” box and then selecting
“OK.” Figure 6 shows what this looks like in SPSS.
Figure 6: Aggregating Data.
This step generated a new variable (POLARRecoded_mean) which was then
used to compute another new variable, which would in fact represent the group
mean centred variable for POLAR3 Q1 proportion. This was done by selecting:
Data → Compute Variable
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The variable consisted of the following numeric expression
“POLARRecoded_mean – POLARRecoded” to provide a reported group centred
value for each respondent. Figure 7 shows what this looks like in SPSS.
Figure 7: Computing New Variable.
To run an MLM in SPSS, select from the menu:
Analyze → Mixed Models → Linear
Figure 8 shows what this looks like in SPSS.
Figure 8: Mixed Models/Linear.
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Once the dialog box is open, ensure that the variable which will provide the
clustering effect (in this case, University (providerName) is placed in the subjects
box.
Figure 9 shows what this looks like in SPSS.
Figure 9: Linear Mixed Models Dialog Box.
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Click Continue.
You have now began to build your MLM. The first step we need to carry out
is to test whether there is any variance in Graduates Median Annual Earnings
(dependent variable) between universities (Level 2 variable). This model is the
unconditional model. This tests variance without any additional factors that
provide conditions, hence the name.
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Once the Linear Mixed Models dialog box is open, ensure you place your
dependent variable (in this case Median Annual Earnings) in the dependent list.
Figure 10 shows what this looks like in SPSS
Figure 10: Linear Mixed Models/Placing Dependent Variable.
To test the unconditional model, we must examine the random effects of both
the intercept and the Level 2 variable. This is because we assume the intercepts
across the groups (Universities) vary. In simpler terms, we expect the different
contexts (Universities) to experience the same effect (increase/decrease in
median annual salary depending on factors) but that effect would begin from
different starting points in median annual salary (they have different intercepts).
We can acknowledge the varying intercepts by clicking “Random.” We must then
tick the “Include Intercept” box and place the Level 2 variable (Provider) in the
“Combinations” box. Therefore, we are looking to see whether there is variance in
the intercepts of each University.
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Figure 11 shows what this looks like in SPSS.
Figure 11: Random Effects With “Include Intercept” Ticked and “providerName” in
Combinations.
In order to examine the effects of the unconditional model, we also have to
examine the “Parameter estimates and covariance parameters.” These can both
be found in the “statistics” dialog box.
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Figure 12 shows what this looks like in SPSS.
Figure 12: Statistics Dialog Box With “Parameter Estimates” and “Tests for
Covariance Parameters.”
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You are now ready to analyse the effects of your unconditional model. It’s time to
explore the SPSS Output.
2.2 Exploring the SPSS Output
Step 1: Examine the estimated Level 2 variance (L2V) and its significance in the
“Estimates of Covariance Parameters”
Step 2: Examine the estimated RV and its significance in “Estimates of
Covariance Parameters”
Step 3: Calculate the intra-class correlation coefficient (ICC) using the following
formula:
Intra-class correlation coefficient = Level 2 Variance (L2V)/Residual
Variance (RV)
ICC = L2V/(RV + L2V)
In the case of this example, the unconditional model reported the following:
Step 1: A statistically significant variance of 17,299,678.41 at the university level
was identified (p ≤ .001).
Step 2: A statistically significant residual variance of 31,009,294.65 was identified
(p ≤ .001).
Step 3: The ICC was computed to be .36 with the following formula:
ICC = 17,299,678.41/(31,009,294.65 + 17,299,678.41)
ICC = 17,299,678.41/48,303,973.06
ICC = .358
Figure 13 shows the output for the unconditional model in SPSS.
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Figure 13: Output for the Unconditional Model.
An example of how to report the unconditional model is below.
The unconditional model yielded a statistically significant University variance of
17,299,678.41 along with a statistically significant residual variance of
31,009,294.65 The ICC was calculated to be .358, indicating 36% of the total
variance of median annual earnings is associated with the university groupings.
The assumption that median annual earnings is independent of university studied
is therefore violated.
Moving On
We now have evidence to suggest that graduates’ median annual earnings
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depend on the university at which they studied. We can now test a conditional
model, which assesses the effects of Level 1 factors (Gender, Years after
Graduation, and proportion of students from POLAR3 Quintile 1) on median
annual earnings when also considering the universities at which the graduates
studied. However, in our first model, we will see whether those factors have an
effect on median annual earnings before determining whether those effects differ
between universities.
Running a Conditional Model With Gender, Number of Years After Graduation, and
Proportion of Students From POLAR3 Quintile 1 (Most Disadvantaged Area)
Since we already have an unconditional model built, we simply need to add to
the model. We therefore go back to our Linear Mixed Model dialog box and place
the variables we need to answer our hypothesis into the appropriate box. It is
important to remember that categorical variables are placed in the factor box (as
they usually consist of a small number of groups), and scale variables are placed
in the covariates box (as they usually consist of a wide range of groups/scores).
Figure 14 shows what this would look like in SPSS.
REMEMBER: Use the new recoded variables for the model. In this case, the
group mean centred variable for Proportion from POLAR3 Quintile 1 and Years
After Graduation.
Figure 14: Adding Level 1 Independent Variables to the Model.
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We then have to determine the tests for fixed effects. This is where we place the
factors we think will have the same rate of influence over the whole sample.
Figure 15 shows the fixed effects dialog box with both factors being added to the
model through assessing their rate of influence for the whole sample. This is done
by simply highlighting an individual variable and clicking “Add” (see figure below).
Figure 15: Fixed Effects.
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We are now ready to explore the output for our first model, which is assessing
the relationship between proportion of Students from POLAR3 Quintile 1 (Most
Disadvantaged) and median annual earnings, controlling the influence of both
Gender and Number of years since graduating.
Exploring the Output of the First Model on SPSS
Step 1: Examine the Estimates of Covariance Parameters.
Step 2: Determining whether the model is better than the unconditional model in
estimating the variance of graduate median annual earnings between universities.
Step 3: Examining the estimates of Fixed Effects of the Level 1 Independent
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Variables.
So,
Step 1: The ICC was calculated to be .531, estimating that 53% of the total
median annual earnings variance, up from 36% in the unconditional model, is
explained by the university groupings when controlling Gender, Number of Years
after graduation, and proportion of students from POLAR3 Quintile 1.
Step 2: The log likelihood of the first model (65,634.24) has also decreased from
the unconditional model (290,266.59) and is therefore a significantly better fit than
the unconditional model in estimating the variance of graduate median annual
earnings between universities.
Step 3: A statistical significant difference was found between Females and Males
in median annual earnings (B = −1067.42; SE = 133.98). A statistically significant
difference was found between years after graduation between 5 years (B =
7,470.48; SE = 168.08), 3 years (B = 4,380.92; SE = 160.17), and 1 year. A
statistically significant relationship was found between proportion of students from
POLAR3 Quintile 1 and median annual earnings (B = −61.95; SE = 15.9).
Figure 16 below displays the output of the first model in SPSS.
Figure 16: Output of the First Unconditional Model.
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An example of how to summarize the first conditional model will be provided below
after exploring the second and final model.
Running the Second Conditional Model
The second and final model to be ran in this guide will answer the following null
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hypothesis:
How the effect the proportion of students from POLAR3 Quintile 1 (most
disadvantaged) has on median annual earnings varies between the
Universities the graduates studied.
As we already have a conditional model built, we are simply adding to the model
we already have. In the case of answering this hypothesis, we do not need to
add any more variables. However, in order to identify if the effect a Level 1
variable has on the dependent variables varies by the Level 2 separator, we must
examine the random effects of the Level 1 variable. In other words, we must see
whether the significant effect (slope) and starting point (intercept) for graduates’
median annual earnings varies between the different universities the graduates’
studies. To do this, we simply click the “Random” dialog box and add the variable
to random effects (Figure 17) and change our covariance type to unstructured
(Figure 18).
Figure 17: Add the Variable to Random Effects.
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Figure 18: Change Covariance Type to Unstructured.
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We are now ready to explore the output for the second conditional model, which
will identify whether the effect proportion of students from POLAR3 Quintile 1
(most disadvantaged) on graduate median annual earnings varies between
universities the graduates studied.
Exploring the Output of the Second Model on SPSS
Step 1: Examine the Estimates of Covariance Parameters
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Step 2: Determining whether the model is better than the first model in estimating
the variance of graduate median annual earnings between universities
Step 3: Examining the estimates of Fixed Effects of the Level 1 Independent
Variables
Step 4: Examine the slope variance and slope intercept variance to identify
whether the effect the Level 1 variable has on the dependent variable varies
significantly as a result of the Level 2 separator.
So,
Step 1: The ICC was calculated to be .54, estimating that 54% of the total median
annual earnings variance, up from 53% in the first model, is explained by the
university groupings when controlling Gender, Number of Years after graduation,
and proportion of students from POLAR3 Quintile 1 varying between groups.
Step 2: The log likelihood of the second model (65,604.24) has also decreased
from the first model (65,634.24) and is therefore a significantly better fit than
the first model in estimating the variance of graduate median annual earnings
between universities.
Step 3: A statistically significant difference was found between Females and
Males in median annual earnings (B = −1,080.57; SE = 133.31). A statistically
significant difference was found between years after graduation between 5 years
(B = 7,486.77; SE = 167.19), 3 years (B = 4,394.63; SE = 159.16), and 1 year. A
statistically significant relationship was found between proportion of students from
POLAR3 Quintile 1 and median annual earnings (B = −89.71; SE = 24.91).
Step 4: The slope variance was 30,360.96 and was statistically significant (p ≤
.001). The slopes and intercept variance was −448,794.2 and was statistically
significant (p ≤ .000). It can therefore be assumed that there is variation in the
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effect (of proportion of students from POLAR3 Quintile 1 on graduate median
annual earnings) between both the intercepts (starting point) and the slopes
(effects) between universities.
Figure 19 below displays the output from SPSS for the second conditional model.
Figure 19: Output of the Second Conditional Model.
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Tables 2 and 3 provide an example of how to summarize the findings of all three
models. The paragraphs below are an example of how to report the summarized
findings for each model.
Table 2: Model Summaries.
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Empty model (random intercept only) Conditional Model 1 Conditional Model 2
Residual variance 31,009,294.7*** 15,076,192.37*** 14,710,127.75***
Intercept variance (university level) 17,299,678.4*** 17,101,812.45*** 17,202,605.75***
Slope variance 30,360.96**
Slopes and intercepts covariance −448,794.2**
Intra-class correlation .36 .53 .54
Log likelihood 290,266.59 65,634.24 65,604.24
*p < .05; **p < .01; ***p < .001.
Table 3: Estimates of Fixed Effects.
Predictors
Model 1 Model 2
Β SE Β SE
Intercept 17,813.25*** 410.3 17,810.2*** 411.23
Gender (female) −1,067.42*** 133.98 −1,080.57*** 133.31
Years after graduation:
5 7,470.48*** 168.08 6,486.77** 167.18
3 4,380.92*** 160.17 4,394.63** 159.16
Proportion from POLAR3 Q1 −61.95*** 15.9 −89.71** 24.91
*p < .05; **p < .01; ***p < .001.
2.3 How to Report the Findings
Model 1 Summary
The first model had three variables added. The ICC was calculated to be .531,
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estimating that 53% of the total median annual earnings variance, up from 36% in
the unconditional model, is explained by the university groupings when controlling
Gender, Number of Years after graduation, and proportion of students from
POLAR3 Quintile 1. The log likelihood of the first model (65,634.24) had also
decreased from the unconditional model (290,266.59). This is therefore a
significantly better fit than the unconditional model in estimating the variance of
graduate median annual earnings between universities.
Estimates of Fixed Effects
Level 1 Predictors
Females were found to be earn significantly less than males, with their median
annual earnings predicted to be approximately £1,067 less. Graduates of 5 years
were found to be earning approximately £7,470 more than those who had
graduated 3 or 1 year ago, and this was statistically significant. A significant
relationship was found between proportion of students from POLAR3 Quintile 1
(most disadvantaged) and graduate median annual earnings. More specifically, an
increase of 1 unit in proportion of students from POLAR3 Quintile 1 is associated
with a decrease of approximately £62. In other words, an increase of 1% in
intake of non-mature students who are most disadvantaged is associated with a
decrease of £62 pounds in graduates’ median annual salary. Therefore, it can be
estimated that an increase of 10% of most disadvantaged entering a university is
associated with a decrease of approximately £620 in median annual earnings.
Model 2 Summary
The second model consisted of the same number of variables as the first model,
with the level predictor “Proportion of students from POLAR3 Quintile 1” (most
disadvantaged) added to random effects in order to examine whether the
predictors effect varied between universities (the Level 2 separator). The ICC was
calculated to be .54, estimating that 54% of the total median annual earnings
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variance, up from 52% in the first model, is explained by the university groupings
when controlling Gender, Number of Years after graduation, and proportion of
students from POLAR3 Quintile 1. The log likelihood of the second model
(65,604.24) had also decreased from the first model (65,634.23). This is therefore
a significantly better fit than the first model.
Estimates of Fixed Effects
Level 1 Predictors
Females were found to earn significantly less than males, with their median annual
earnings predicted to be approximately £1,080 less. Graduates of 5 years were
to be earning approximately £7,487 more than those who had graduated 3 or 1
year ago, and this was statistically significant. A significant relationship was found
between proportion of students from POLAR3 Quintile 1 (most disadvantaged)
and graduate median annual earnings. More specifically, an increase of 1 unit in
proportion of students from POLAR3 Quintile 1 is associated with a decrease of
approximately £90. In other words, an increase of 1% in intake of non-mature
students who are most disadvantaged is associated with a decrease of £90
pounds. Therefore, it can be estimated that an increase of 10% of most
disadvantaged entering a university is associated with a decrease of
approximately £900 in median annual earnings.
Varying Effect of Proportion from POLAR3 Quintile 1
This effect was also found to significantly vary depending on the university
attended. A slope variance of 30,360.96 and slope and intercept variance of
−448794.2 were both statistically significant. Therefore, there is evidence to
suggest that the negative relationship between proportion of disadvantaged
students entering universities and graduates median annual earnings varies
between universities. We can therefore reject the null hypothesis.
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3 Your turn
Using the data provided, see whether you can run an MLM and replicate the
results using this step-by-step guide.
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