Contextual effects

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Contextual effects
In the previous sections we found that when
regressing pupil attainment on pupil prior
ability schools vary in both intercept and slope
resulting in crossing lines.
Similarly, we found that when regressing person
hedonism on person income countries vary in both
intercept and slope resulting in crossing lines.
The next question is are there any higher level
variables that can explain these patterns of
variation?
That is are there school or country level
variables that can explain some of this between
school and between country variance we have found.
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Example Research questions

• Interest lies in how the outcome of individuals
  in a cluster is affected by their social contexts
  (measures at the cluster level). Typical
  questions are
•  Does school type effect students' educational
  progress?
• Do teacher characteristics effect students'
  educational progress?
• Is a students progress effected by the ability
  of his or her peers?
• Does area deprivation effect the health status
  of individuals in the area?
• Children within families is also a multilevel
  structure. Does family background effect child
  developmental outcomes?

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An Educational example
A question of widespread interest is do children
make better academic progress in single or mixed.
This is a question about the contextual effect of
school gender on child progress.
Consider a random intercept model on our
educational data set
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Including school gender contextual effects
Make inference that girls in girl schools
progress better than girls in mixed schools.
Z0.245/0.852.9, p0.005
Do not make inference that boys in boy schools
progress better than boys n mixed schools.
Z0.097/0.1090.89, p0.390
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Comparing the models with and without contextual
effects
Between school variance reduced by 13 as a
result of fitting school gender
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Comparing random intercept contextual effects
model, with single level contextual effects model
Inference changes, under single level model we
infer boys do better in single sex schools.
Single level models which ignore clustering will
give rise to mis-estimated precision,
particularly for contextual effects. Why?
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Mis-estimated precision
In this data set we have 4059 pupils in 65
schools.
We therefore only have 65 independent bits of
information with which to identify the school
effects. The standard errors calculated for the
single level model assume we have 4059 bits of
information.
With level 1 predictors we also get mis-estimated
precision because the clustering at higher levels
induces correlations between individuals in the
same group, so we also have less than 4059 bits
of information. However, the problem is typically
far less severe. Indeed in the above analysis
there are no differences between the standard
errors for the coefficients of level 1 predictor
variable prior ability between the single level
and random intercept models.
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Why not just include dummy variables for each
school in the single level model
This would be the equivalent of fitting an
analysis of covariance model.
Unfortunately this will not work. We only have 65
pieces of information at the school level and if
we use this up by fitting 65 dummy variables we
can add no more variables at the school level. So
we can not add the school gender dummies.
Multilevel modelling provides a richer and more
accurate exploratory framework for modelling
contextual effects.
We get the right standard errors
We can see how much of the higher level variation
was explained by the contextual effects and how
much remains.
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Within group, between group and contextual
effects of for level 1 predictors
Often contextual variables are constructed by
aggregating level 1 predictors, country level
income, family level aggression etc
We will simulate a small data set with 100
individuals from four groups, for the sake of
argument lets say that our response is happiness,
we have a predictor variable that is income and
the groups are 4 different countries. Lets
simulate the mean income in each country as
2,4,6,8. We then simulate our response as
Note that this model does not include a country
level random effect. Therefore the only
differences between country happiness levels are
produced by differing incomes in the countries.
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Fitting a regression model ignoring average income
And we might conclude as income rises happiness
reduces
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Including the average income variable
Now we have a different picture. Within all
countries the relationship between income and
happiness is positive. This is the within country
slope(1.128)
However the intercept for country j is 5.772 -
2.32 ?av_incomej -2.32 is the contextual effect.
Recall the country average incomes are 2,4,6,8,
which gives intercepts of 1.13,-3.51, -8.15,
-12.8
Now within all countries we have a positive
relationship between income and happiness,
however people in richer countries tend to be
less happy than poorer ones.
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What about the between group effect?
We have identified the within group and
contextual effects of the income predictor.
The between group effect is the slope of the
regression of the country mean for happiness on
the country mean for income.
That is the slope of the regression line through
the four red points and is -1.20
within group 1.128 contextual -2.332 between
group -1.204
Notice that the between group the within group
the contextual B W
C -1.2041.128(-1.204)
Or C B W that is the contextual effect is
the the difference of the between and within
regressions
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Examples of different contextual effects
W1,B -1,C-2
W 1, B1,C0
W1,B3,C2
Contextual effect pushes up intercept as
av_income increases
Contextual effect pulls down intercept as
av_income increases
Contextual effect 0 so intercept unchanged as
av_income increases
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Why does BWC?
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Group mean centering of a level 1 predictor
If we
Something interesting happens. In our example
this corresponds to
The intercept and within slope remain unchanged ,
but ?2 now estimates the between group
coefficient not the contextual coefficient for
income
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Comparison of uncentered and group mean centered
level 1 income predictor
Note that if you grand mean centre income then ?2
remains the contextual coefficient ie behaves
in the same way as the uncentered case
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With group mean centering why is ?2 the between
group effect
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And now to some real data..
Consider the random slopes model on the hedonism
data where we regress hedonism on income
Note income centered around grand mean of 6.
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Adding contextual effect for income
The contextual effect is positive but
non-significant.
If we obtain model predictions for a low(3.5) and
high(8) average income country we obtain Low
(-0.278 0.016 ? 3.5) 0.043 ? income -0.222
0.043 ? income High (-0.278 0.016 ?8)
0.043 ? income -0.150 0.043 ? income
Thus the positive contextual effect means
countries with higher average incomes have higher
intercepts and this has reduce the intercept
variation by a small amount from 0.069 to 0.066
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Cross-level interactions
The fixed part of the previous model assumes the
within country income slopes do not change as
functions of country level variables. So the
within country lines for low and high income
countries are parallel.
It may be that the with country income slopes may
change as a function of the contextual variable
av_income.
We can test this assumption by allowing the
within country income slopes to change as a
function of av_income that is fit an interaction
between income and av_income
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Cross level interaction
The cross level interaction is negative and
statistically significant. The model predictions
for countries with low and high average incomes
are now Low (-0.465 0.049 ? 3.5) (0.123
-0.137 ? 3.5) ? income -0.29 0.076
?income High (-0.465 0.049 ? 8) (0.123
-0.137 ? 8) ? income -0.08 0.016 ? income
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Interpreting the graph
At low levels of individual income people in rich
countries are more hedonistic. As individual
income rises the rich-poor country differential
diminishes
For what incomes is the rich country-poor country
gap significant?
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Contextual effects explaining level 2 variance
Given a random slope model
Contextual variables explain level 2 variation in
the intercept and cross-level interactions of
contextual variables with x1 explain level
variability in the slope coefficient
This can be seen by re-arranging
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