New Measures of Data Utility

Mi-Ja Woo

National Institute of Statistical Sciences

Question How to evaluate the characteristics of SDL methods?

Previously, data utility measures were studied in context of moments and linear regression models.- Differences in inferences obtained from the original and masked data.- Regression model and KL distance rely on the multivariate normality assumption.

Questions : - Is the assumption satisfied in the realistic situation?- What if the assumption is violated?

Example

Example: Two-dimensional original data and two masked data by synthetic and resampling methods.

Different distributions, but the same moments and estimates of regression coefficients.

New measures are needed.

1. CDF utility measure Extension of univariate case. Kolmogorov statistics

Cramer-von Mises statistics

, where are empiricaldistributions of original and masked data. Large MD and MCM indicate two data are distributed differently.

2. Cluster Data Utility A loose definition of clustering could be “the process of

organizing objects into groups whose members are similar in some way”.

A cluster is therefore a collection of objects which are “similar” between them and are “dissimilar” to the objects belonging to other clusters.

A data set is said to be randomly assigned when proportion of observations from original data for each cluster is constant (1/2 with equal number of observations for two groups) :

where is the total number of records, is the number of records from original data, and is the weight assigned to i-th cluster.

3. Propensity Score Data Utility A propensity score is generally defined as the

conditional probability of assignment to a particular treatment given a vector of observed covariates (Rosenbaum and Rubin 1983).

A data set is said to be randomly assigned when propensity score for each covariate is constant (1/2 with equal number of observations for two groups).

In the propensity score method, a propensity score is estimated for each observed covariate, and utility is measured by:

Estimation of propensity scores: Combine original and masked data sets, and create an

indicator variable Rj with the value 0 for observations from original and 1 otherwise. 1) Logistic regression model such as where

2) Tree model.3) Modified logistic regression model : Classify all data points into g groups, and fit a logistic model for each group. It combines logistic model with clustering, and it borrows strength of logistics model and clustering method.

Cluster utility is one way of propensity score utility.

4. Simulation Eight different types of two-dimensional data with n=10,000:

1) Symmetric/non-symmetric2) High/ low correlated3) Negative/ positive correlated.

Masking strategies considered: Synthetic, microaggregation, microaggregation followed by noise, rank swapping, and resampling.

Computational details:1) Cluster Utility: g=500 (5%) and g=1,000 (10%).2) Propensity score utility with logistic model:

3) Propensity score utility with tree model: Sizes of tree considered are complexity parameter cp=0.001, and 0.0001. That is, any split that does not decrease the overall lack of fit by a factor of cp is not attempted.

4) Propensity score utility with modified logistic model:The number of group is g=100 (1%), and linear and quadratic logistic functions are used to fit logistic regression models.

Results:Symmetric high negative case.

Symmetric low negative case.

Non-symmetric high negative case.

Non-symmetric low negative case.

Summary: CDF utility:

1) Do not involve parameters.2) It is favorable to rank swapping SDL method.

Cluster utility: 1) Do not measure the differences between two structures of original and masked data within a cluster, within-cluster variation. 2) Generally, it is consistent to overall results.3) For non-symmetric cases, large number of clusters tend to produce worse utility for the masked data by microaggregation method since there are three overlaps in microaggregated data.

Propensity score with logistic model: 1) The choice of degree is very crucial.2) It is hard to deal with high-dimensional data.

Propensity score with tree model: 1) Small size of tree can not distinguish utility of Rank from that of Resample.2) Large size of tree leads to bad utility for the micro-aggregation method. For some cases, large size of tree can not partition space for Rank method. 3) It is favorable to Rank SDL method.

Propensity score with modified logistic model: 1) It possesses both advantages and disadvantages of logistic model and clustering since it is the combination of cluster and propensity score utilities.2) It looks consistent to overall results for all data structures.

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