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Introduction to Emerging Methods for Imputation in Official Statistics

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Overview

1. Imputation in the quality framework of official statistics

2. Classes of imputation methods

3. Requirements for imputation in official statistics

4. Past research work

5. Statistical clustering

6. Best imputation methods

7. Processing imputation (and editing)

8. New research plans

9. Multiple imputation

10. Fractional imputation

11. Multilevel model based imputation

Pasi Piela

Overview

1. Imputation in the quality framework of official statistics

2. Classes of imputation methods

3. Requirements for imputation in official statistics

4. Past research work

5. Statistical clustering

6. Best imputation methods

7. Processing imputation (and editing)

8. New research plans

9. Multiple imputation

10. Fractional imputation

11. Multilevel model based imputation

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Imputation is defined as the process of statistical replacement of missing values

- Editing and imputation are undertaken as part of a quality improvement strategy to improve accuracy, consistency and completeness.

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Classes of imputation methods

- A1. Deterministic imputation, or
- A2. Stochastic imputation
- B1. Logical imputation,
- B2. Real donor imputation, or
- B3. Model donor imputation
- C1. Single imputation, or
- C2. Multiple imputation
- D1. Hot-deck
- D2. Cold-deck

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Five requirements for imputation in official statistics (Chambers, 2001)

- Predictive Accuracy: The imputation procedure should maximise the preservation of true values.
- Distributional Accuracy: The preservation of the distribution of the true values is also important.
- Estimation Accuracy: The imputation procedure should reproduce the lower order moments of the distributions of the true values.
- Imputation Plausibility: The imputation procedure should lead to imputed values that are plausible.
- Ranking Accuracy: The imputation procedure should maximise the preservation of order in the imputed values.

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Past research work

- traditional methods: cell-mean imputation, regression imputation, random donor, nearest neighbour, etc.
- advanced imputation techniques
- based on statistical clustering (e.g. K-means)
- homogenous imputation classes
- hierarchical clustering (e.g. classification and regression trees)
- ”modern” statistical pattern recognition methods

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Statistical clustering for imputation

- imputation classes, cells, clusters, groups

average point locating the cluster

Computational methods?

...or searching appropriate imputation cells by using categorical sorting variables?

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K-means Clustering

- The basic variance minimization clustering algorithm.
- The ”Voronoi region” (as part of the tesselation) for cluster i is given by

where || refers to the Euclidean norm (distance). At the iteration time t + 1 the weight wi is updated by

where #Vi refers to the number of units xk in Vi.

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Classification and Regression Trees

- WAID - Weighted Automatic Interaction Detection software
- EU FP4 Project AUTIMP

Target variable: categorical or continuous

Node

Y

Predictor variable

The original data

Binary splits

X1<5

X1>= 5

X2<7

X2>=7

Original data splitted into two separate parts

Only one variable in turn defines the split.

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Neural networks for clustering: Self-Organizing Maps, SOM

- The basic SOM defines a mapping from the input data space Rnonto a latent space consisted typically of a two-dimensional array of nodes or neurons.

Best methods

- Nearest neighbour (hot-deck) by Euclidean distance metrics
- The bestmethod is actually a system that includes several competitive imputation methods
- The development and evaluation of new imputation methods is closely connected to the software development.

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Processing imputation (and editing)

- The Banff system of Statistics Canada is a good example about computerized editing and imputation process. It is a collection of specialized SAS® procedures "each of which can be used independently or put together in order to satisfy the edit and imputation requirements of a survey" as stated in the Banff manual (Statistics Canada, 2006).
- successor to more well-known GEIS
- currently Statistics Finland is evaluating Banff

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Verifying edits, generating implied edits and extremal points

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Three basic outlier detection methods

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Identifying which fields require imputation and how to satisfy edits.

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Here : logical imputation (one possible value allowing to pass the edits).

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Adjusting and rounding the data so that they add to a specific totals.

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Plans for future research - intro

- Multiple imputation, MI, was not handled here because of the context of the research. But also detailed research in imputation variance and careful analysis of the datasets with hierarchical, multilevel nature (note the difference to the previously mentioned hierarchical clustering methods) containing cross-classifications and missingness were also excluded. This will lead us to the forthcoming research that will be next outlined.

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Multiple imputation

- Donald Rubin, 1987
- Bayesian framework
- very famous and popular family of the imputation methods
- rarely used in official statistics
- several imputed datasets are combined to reach an estimate for the imputation variance
- assumptions are strict and there also exist some difficulties with MI variance estimation as discussed by Rao (2005) and Kim et al. (2004)

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Fractional imputation

- a sort of mixed real donor and model donor “hot-deck” imputation method for a population divided into imputation cells
- involves using more than one donor for a recipient
- simply: three imputed values might be assigned to each missing value, with each entry allocated a weight of 1/3 of the nonrespondent's original weight
- Kim and Fuller (2004): superior to MI
- designed to reduce the imputation variance while MI gives only a simple way to estimate it

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Fractional imputation

- a sort of mixed real donor and model donor “hot-deck” imputation method for a population divided into imputation cells
- involves using more than one donor for a recipient
- simply: three imputed values might be assigned to each missing value, with each entry allocated a weight of 1/3 of the nonrespondent's original weight
- Kim and Fuller (2004): superior to MI
- designed to reduce the imputation variance while MI gives only a simple way to estimate it

Assume the within-cell uniform response model in which the responses in a cell g are equivalent to a Bernoulli sample sample selected from the elements ng.

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Fractional imputation 2

AR

sample non-respondents

AM

- Formulating the fractionally imputed

estimator:

sample respondents

imputation fraction

population quantity of interest

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Fractional imputation 3

- Kim and Fuller (2004) showed also how fractional imputation combined with the proposed replication variance estimator gives a set of replication weights that can be used to construct unbiased variance estimators for estimators based on imputed data (and for estimators based on the completely responding variables).
- fully efficient fractional imputation, FEFI
- produces zero imputation variance

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Fractional imputation 3

Approximations to FEFI by fixed number of donors.

-- e.g. systematic sampling with probability proportional to the weights to select donors for each recipient. After the donors are assigned, the initial fractions are adjusted so that the sum of the weights gives the fully efficient fractionally imputed estimator.

- Kim and Fuller (2004) showed also how fractional imputation combined with the proposed replication variance estimator gives a set of replication weights that can be used to construct unbiased variance estimators for estimators based on imputed data (and for estimators based on the completely responding variables).
- fully efficient fractional imputation, FEFI
- produces zero imputation variance

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Multilevel modeling for imputation

- Many kinds of data have a hierarchical or clustered nature.
- We refer to a hierarchy as consisting of units grouped at different levels. Thus children may be the level 1 units in a 2-level structure where the level 2 units are the families and students may be the level 1 units clustered within schools that are the level 2 units.
- More recently there has been a growing awareness that many data structures are not purely hierarchical but contain cross-classifications of higher level units and multiple membership patterns.

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Multilevel modeling for imputation 2

- Multilevel models (Goldstein, 2003) take advantage of the correlation structure between different levels of hierarchy.
- Correlation structures and connections between the study variables can be a challenge in imputation tasks as well.
- Currently in literature, there are some papers about the use of multiple imputation with multilevel models.
- Carpenter and Goldstein (2004): if a dataset is multilevel then the imputation model should be multilevel too.
- MLwiN

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Plans for future research

- new imputation methods and the analysis of imputed and missing data when data structures are complex
- use of multilevel imputation models in which clustersin thecluster sampling design can be incorporated in an imputation model as random effects
- information on the complex sampling design can be incorporated, for example, by using strata indicators as fixed covariates
- modified fractional and single imputation will be considered

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Plans for future research 2

- Another new avenue of research is the use of multilevel models and imputation in the context of small area estimation (Rao, 2003).

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References 1/2

- Aitkin, M., Anderson, D., and Hinde, J. (1981): Statistical Modelling of Data on Teaching Styles (with discussion). Journal of the Royal Statistical Society, A 144, 148-1461.
- Breiman, L., Friedman, J.H., Olsen, R.A., and Stone, C.J. (1984): Classification and Regression Trees. Wadsworth, Belmont, CA.
- Carpenter, J.R., and Goldstein, H. (2004): Multiple Imputation Using MLwiN, Multilevel Modelling Newsletter, 16, 2. **
- Charlton, J. (2003): Editing and Imputation Issues. Towards Effective Statistical Editing and Imputation Strategies - Findings of the Euredit project. *
- Chambers, R. (2001): Evaluation Criteria for Statistical Editing and Imputation. National Statistics Methodological Series, United Kingdom, 28, 1-41. *
- Eurostat (2002): Quality in the European Statistical System - the Way Forward. European Comission.
- Goldstein, H. (2003). Multilevel Statistical Models (Third Edition). Edward Arnold, London.
- Hill, P.W., and Goldstein, H. (1998): Multilevel Modelling of Educational Data with Cross Classification and Missing Indentification of Units, Journal of Educational and Behavioural Statistics, 23, 117-128.
- Kalton, G., and Kish, L. (1984): Some Efficient Random Imputation Methods. Communications in Statistics, A13, 1919-1939.
- Kim, J.K., and Fuller, W.A. (2004): Fractional hot deck imputation, Biometrika, 91, 559-578.
- Fuller, W.A., and Kim, J.K. (2005): Hot Deck Imputation for the Response Model, Survey Methodology, 31, 2, 139-149.
- Kim, J.K., Brick, J.M., Fuller, W.A. and Kalton, G. (2004): On the Bias of the Multiple Imputation Variance Estimator in Survey Sampling. Technical Report.
- Kohonen, T. (1997): Self-Organizing Maps. Springer Verlag, New York.
- Koikkalainen, P., Horppu, I., and Piela P. (2003): Evaluation of SOM based Editing and Imputation. Towards Effective Statistical Editing and Imputation Strategies - Findings of the Euredit project. *
- Lehtonen, R., Särndal, C.-E., and Veijanen, A. (2003): The Effect of Model Choice in Estimation for Domains, Including Small Domains, Survey Methodology, 29, 33-44.
- Lehtonen, R., Särndal, C.-E., and Veijanen, A. (2005): Does the Model Matter? Comparing Model-assisted and Model-dependent Estimators of Class Frequencies for Domains, Statistics in Transition, 7, 649-673.

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References 2/2

- Piela, P., and Laaksonen, S. (2001): Automatic Interaction Detection for Imputation – Tests with the WAID Software Package. In: Proceedings of Federal Committee on Statistical Methodology Research Conference,Statistical Policy Working paper, 34, 2, 49-59, Washington, DC.
- Piela, P. (2002): Introduction to Self-Organizing Maps Modelling for Imputation - Techniques and Technology. Research in Official Statistics, 2, 5-19.
- Piela, P. (2004): Neuroverkot ja virallinen tilastotoimi, Suomen Tilastoseuran Vuosikirja 2004. Helsinki.
- Piela, P. (2005): On Emerging Methods for Imputation in Official Statistics. Licentiate Thesis. University of Jyväskylä.
- Rasbash, J., Steele, F., Browne, W., and Prosser, B. (2004): A User’s Guide to MLwiN (version 2.0). London. **
- Rao, J.N.K. (2003): Small Area Estimation. Wiley, New York.
- Rao, J.N.K. (2005): Interplay Between Sample Survey Theory and Practice: An Appraisal, Survey Methodology,Statistics Canada, 31, 2, 117-138.
- Ritter, H., Martinez, T., and Schulten, K. (1992): Neural Computation and Self-Organizing Maps: An Introduction. Addison-Wesley, Reading, MA.
- Schafer, J.L., and Graham, J.W. (2002): Missing Data: Our View of the State of the Art, Psychological Methods7, 2, 147-177.
- Statistics Canada (2003): Statistics Canada Quality Guidelines. Catalogue no. 12-539-XIE.
- Statistics Canada (2006): Functional Description of the Banff System for Edit and Imputation. Statistics Canada.
- Rubin, D. (1987): Multiple Imputation in Surveys. John Wiley & Sons, New York.
- *) The Euredit project: http://www.cs.york.ac.uk/euredit/
- **) Centre for Multilevel Modelling: http://www.mlwin.com/

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