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Add info on alpha for frequency calculation #40

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52 changes: 52 additions & 0 deletions measure_risk.rst
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Original file line number Diff line number Diff line change
Expand Up @@ -383,6 +383,58 @@ frequency.
10 Urban Female Secondary incomplete Non-LF 76 2 262 0.0074
==== ========== ======== ====================== ============== ======== =============== =============== ========

In the calculation of frequencies, missing values (‘NA’s in R [8]) are treated as if they
were any other valid value in the variable under consideration. :numref:`tab46` reproduces
the records from :numref:`tab41`. However, in record 4, the values for the variables
Education level and Labor status are recorded as missing. The sample frequencies :math:`f_{k}`
for the records 4 and 8 change as a result of the missing values. Record 4 could
have the same key as record 6 and record 8, depending on the true value of the recorded
missing values. Therefore, the sample frequency :math:`f_{k}` of record 4 increases to 3.
The key of record 8 could coincide with the key of record 4, based on the missing values. The
sample frequency :math:`f_{k}` of record 8 changes to 2.

This treatment of missing values in the calculation of sample frequencies may lead
to an overestimation of the sample frequencies. The value for the variable
Education level of record
4 is once interpreted as Secondary incomplete and once as Post-secondary.
However, record 3 has only one true (unknown) value for the variable Education level. The same
holds true for the variable Labor status.
In order to take this observation into account, it is possible to count matches of keys
that are based on one or more missing values in the keys with a value less than 1.
This reflects the probability distribution of the the true value.

The parameter alpha specifies the weight of a match based on missing values
in the sample frequencies. Alpha can be a value between 0 and 1 with the default at 1.
The same value for the parameter
alpha applies to the frequency calculation of all key variables
in the complete dataset. The last column in :numref:`tab46` shows the sample
frequencies with alpha equal 0.5. The
other risk measures, such as population frequencies and individual risk,
are calculated based on the adapted sample frequencies :math:`f_{k}`.

.. _tab46:

.. table:: Example dataset showing sample frequencies,
population frequencies and individual disclosure risk
:widths: auto
:align: center


==== ========== ======== ====================== ============== =============== ==================================
No Residence Gender Education level Labor status :math:`f_{k}` :math:`f_{k}` with alpha = 0.5
==== ========== ======== ====================== ============== =============== ==================================
1 Urban Female Secondary incomplete Employed 2 2
2 Urban Female Secondary incomplete Employed 2 2
3 Urban Female Primary incomplete Non-LF 1 1
4 Urban Male NA/missing NA/missing **3** 3
5 Rural Female Secondary complete Unemployed 1 1
6 Urban Male Secondary complete Employed 2 1.5
7 Urban Female Primary complete Non-LF 1 1
8 Urban Male Post-secondary Unemployed **2** 1.5
9 Urban Female Secondary incomplete Non-LF 2 2
10 Urban Female Secondary incomplete Non-LF 2 2
==== ========== ======== ====================== ============== =============== ==================================

In :numref:`code41`, we show how to use the *sdcMicro* package to create a
list of sample frequencies :math:`f_{k}` for each record in a dataset.
This is done by using the *sdcMicro* function freq(). A value of 2 for
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10 changes: 8 additions & 2 deletions sdcMicro.rst
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Expand Up @@ -292,6 +292,10 @@ sufficient to write the name of the object.
# sensitive variables for l-diversity computation
selectedSensibleVar = c('health')

# Options
alphaVal <- 1 # parameter alpha for frequency calculation
seedVal <- 12345 # seed

# creating the sdcMicro object with the assigned variables
sdcInitial <- createSdcObj(dat = file,
keyVars = selectedKeyVars,
Expand All @@ -300,8 +304,10 @@ sufficient to write the name of the object.
weightVar = selectedWeightVar,
pramVars = selectedPramVars,
hhId = selectedHouseholdID,
strataVar = selectedStrataVar,
sensibleVar = selectedSensibleVar)
strataVar = selectedStrataVar,
sensibleVar = selectedSensibleVar,
alpha = alphaVal,
seed = seedVal)

# Summary of object
sdcInitial
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