Investigation of Treatment of Influential Values

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Investigation of Treatment of Influential Values

• Mary H. Mulry
• Roxanne M. Feldpausch

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Outline

• Current practices
• Methods investigated
• Results
• Next steps

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Influential Observation

• An observation is considered influential if its
  weighted contribution has an excessive effect on
  the estimate of the total (Chambers et al 2000)

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The Data - U.S. Monthly Retail Trade Survey

• Collect sales and inventories
• Monthly survey of about 12,500 retail business
  with paid employees
• Sample selected every 5 years
• Sample is stratified based on industry and sales
• Quarterly sample of births
• Deaths are removed

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The Data

• Analysis done at published NAICS level
• Hidiroglou-Berthelot algorithm ran on the data
  before looking for influential values
• Horvitz-Thompson estimator

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Causes of Influential Units

• One time or rare event
• Erroneous measure of size
• Change in the make-up of the unit
• Seasonal Businesses

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Current Practices

• Analyst review an effect listing of micro level
  data and investigates units that may be
  influential
• When the analyst determines a correctly reporting
  unit may be influential, the case is referred to
  a statistician

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Current Practices

• One time influential value
• Imputation
• Recurring influential value
• Weight adjustment based on the principles of
  representativeness
• Moving the unit to a different industry when the
  nature of the business changes

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Goals

• To improve upon current methodology by making it
  more objective and rigorous
• To find methodology that uses the observation but
  in a manner that assures its contribution does
  not have an excessive effect on the total

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Assumptions

• Influential observations occur infrequently, but
  are problematic when they appear.
• The influential observation is true, although
  unusual. It is not the result of a reporting or
  coding error.

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Strategy

• Identify candidate methodologies and test with
  real data from one industry (about 700
  businesses) for a month that contains an
  influential value

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Evaluation Criteria

• Number of influential observations detected,
  including the number of true and false detections
  made
• Estimate of bias
• Impact on month-to-month change

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Notation

• where
• Yi is the sales for the i-th business in a
  survey sample of size n
• wi is the sample weight for the i-th unit
• Xi is the previous months sales for the ith
  business

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Methods Examined

• Weight trimming
• Reverse calibration
• Winsorization
• Generalized M-estimation

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Weight Trimming

• Does not identify influential units
• Adjusts the weight of the observation

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Weight Trimming

• Truncate the weight of the influential
  observation
• Adjust the weights of the non-influential
  observations to account for the remainder of the
  truncated weight
• Sum of the new weights is the same as the sum of
  the original weights
• (Potter 1990)

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Weight Trimming Notes

• Calculations were done within sample stratum.
• Choice of correction factor could be
  investigated. We arbitrarily chose ciwi/3.

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Reverse Calibration

• Does not identify influential units
• Adjusts the value of the observation

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Reverse Calibration

• Use a robust estimation method to estimate the
  total
• Modify the influential observations to achieve
  that total
• (Chambers and Ren 2004)

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Winsorization

• Identifies influential units
• Adjusts the value of the observation

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Winsorization

• Type I
• Type II

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Winsorization Defining K

• Define a separate Kh for each stratum in a manner
  than minimizes the mse (Kokic and Bell 1994)
• Define a separate Ki for each observation in a
  manner that minimizes the mse (Clarke 1995)

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Winsorization Defining K

• Use unweighted data to define Kh for each stratum
  where Kh mh 2sh
• Use weighted data to define Kh for each stratum
  where Kh mh 2sh where mh and sh are based on
  the weighted data

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Winsorization-Our Implementation

• Used a robust regression in SAS to estimate
  the parameters needed in the calculations

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M-estimation

• M-estimators are robust estimators that come
  from a generalization of maximum likelihood
  estimation

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M-estimation

• Identifies influential units
• Adjusts either the weight or the value of the
  influential observation

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M-estimation

• Used a weighted M-estimation technique that is
  able to modify the weights or the values of the
  influential observations (Beaumont and Alavi 2004)

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Results
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Number of Outliers Detected
Method does not detect outliers, one outlier was
specified
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Replacement Values (in Millions)
Weight trimming adjusts the other 18 weights in
the stratum Winsor wgt ?2s identified 3 other
values
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Total Sales for the Industry
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Chosen for Further Study

• Winsorization by each observation
• M-estimation by observation
• M-estimation by weight

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Contact Information

• Mary.H.Mulry_at_census.gov
• Roxanne.Feldpausch_at_census.gov