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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 Y i is the sales for the i-th business in a survey sample of size n w i is the sample weight for the i-th unit X i is the previous months sales for the i th 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 c i =w i /3.

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Reverse Calibration Does not identify influential units Adjusts the value of the observation

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Reverse Calibration 1.Use a robust estimation method to estimate the total 2.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 K h for each stratum in a manner than minimizes the mse (Kokic and Bell 1994) Define a separate K i for each observation in a manner that minimizes the mse (Clarke 1995)

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Winsorization – Defining K Use unweighted data to define K h for each stratum where K h = h +2s h Use weighted data to define K h for each stratum where K h = h +2s h where h and s h 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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