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In many
studies, data arise from sample surveys in which units have
been selected with known but unequal probabilities. In these
cases, it will often be desirable to weight observations in
order to produce unbiased estimates of population parameters.
According to standard practice in such cases, the information
from each unit is weighted inversely proportional to its probability
of selection.
Suppose, for instance, that in a pre-election
poll, ethnic minority voters are over-sampled to insure that
various ethnic groups are represented in the sample. Without
weighting, the over-sampled groups would exert undue influence
on estimates of the proportion of voters in the population
favoring a specific candidate. Use of design weights can yield
unbiased estimates of the population parameters.
Design weights are also commonly used to
correct for differential non-response of sub-groups. Response
rates are estimated for relevant sub-groups, and information
from each respondent is weighted inversely proportional to
the probability of response. That way, respondents who are
over-represented in a sample as a function of non-response
are appropriately weighted down.
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Hierarchical
data can be described as arising from a multi-stage sampling
procedure. For example, schools might be sampled from a national
frame of schools and then, within each school, students might
then be sampled from a list of all students attending the
school. Probabilities at each level might be known but unequal.
For example, one might over-sample private schools and then
over-sample minority students within each school. Weights
might be constructed at each level to be inversely proportional
to the probability of selection at that level. In some cases,
weights might be available at only one level. For example,
in a two-level design with students nested within schools,
one might compute the marginal probability that a student
is selected as the product of the probability that student’s
school is selected multiplied by the conditional probability
that the student is selected given that his or her school
is selected. In another context, suppose persons are selected
with known probability and then followed longitudinally over
time. In this case, we have occasions at level 1 nested within
persons at level 2. The only weight may be a level-2 weight,
inversely proportional to the probability of selection of
that person. It is, of course, possible to include level-1
weights as well, but it is common to have weights only at
level-2 in such longitudinal studies.
HLM 6 uses a method of computation devised
by Pfefferman et al. (1998) for hierarchical data. This method,
based on weighting the information of each case in the framework
of maximum likelihood, is more appropriate than the method
of weighting in earlier versions of HLM, which used a more
conventional approach of weighting observations.
In the
two-level context, weights might be available at level 1,
at level 2 or at both levels. If weights are available at
level-1 only, the methodology used in HLM 6 assumes that these
weights are inversely proportional to  , the marginal
probability of that student i in school j is
selected into the sample. HLM 6 will then normalize
the weight to have a mean of 1.0. Thus we have
(2.1)
in which case
 (2.2)
where N is the total sample size of
level-1 units. In contrast, if weights are available only
at level 2, the methodology assumes that these weights are
inversely proportional to  the probability of selection of the level-2
unit. In this case, HLM 6 will again normalize the weight
to have a mean of 1.0, yielding
 (2.3)
in which case
 (2.4)
where J is the total number of level-2
units. If weights are available at both level-1 and level-2,
the methodology assumes that the level-1 weight is  , the conditional probability
of selection of unit i given that unit j was selected, so
that  . The level-2 weight
is assumed to be inversely proportional to . In this case, HLM will normalize the level-1
weight within level-2 units:
 (2.5)
so that the sum of these weights within a
level-2 unit will be
(2.6)
where  is the sample size of level-1 units
in level-2 unit j .
In HLM6, weights are selected at the time
of analysis, not when the MDM file is made. To select weights
for an HLM2 analysis, select the Estimation Settings
option from the Other Settings menu, and use the pull
down menus to select the weighting variables at any level.
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In the
three-level context, weights might be available at any one
of the three levels, at any pair of them, or at all three
levels. Normalization proceeds in a fashion completely analogous
to that in the case of two levels. If weights are available
only at level 1, we assume these are inversely proportional
to  , the marginal probability of selection of unit
ijk. Similarly, if weights are available only
at level 2 or only at level 3, the corresponding probabilities
are  or  , respectively.
If the weights are at levels 1 and 2 but not 3, the corresponding
probabilities are  and ; if at levels 2 and 3 (but not 1), the corresponding
probabilities are  and ; if the weights are at levels 1 and 3 (but not
2), the corresponding probabilities are  and . If weights
are present at all three levels, the probabilities are , and .
In HLM6, weights are selected at the time
of analysis, not when the MDM file is made. To select weights
for an HLM3 analysis, select the Estimation Settings
option from the Other Settings menu, and use the pull
down menus to select the weighting variables at any level.
No weighting options are available for the
multivariate and cross-classified models fitted using HMLM,
HMLM2, and HCM2.
 
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