"Head-banging" is a weighted two-dimensional median-based smoothing algorithm, developed to reveal underlying geographic patterns in data where the values to be smoothed do not have equal variances. The original idea was proposed by Tukey and Tukey (1981), then studied and implemented by Hansen (1991). The staff at the National Center for Health Statistics worked with Hansen to add weights to the original algorithm (Mungiole, Pickle, Simonson, 1999).

The process is implemented as follows. We use population-weighted county values for illustration, but it can be applied to any geographic units with any user-specified weights.

For each county, up to NN (default=30) neighboring counties comprise its "smoothing window", which is of sufficient size to begin to show regional patterns while retaining patterns apparent in the raw data maps.

Three nearly collinear triples are used for smoothing at the center point of this figure. Observed values are printed at each data location, with corresponding weights given in parentheses. The low screen is the weighted median of the low values (2, 4, 5)=4. The high screen is the weighted median of the high values (7, 8, 9)=8. The observed value at the center point lies below the low screen, so the weights are used to determine whether adjustment is needed. Because the sum of endpoint weights (10) exceeds the number of triples times the center point weight (3x2=6), the smoothed value at the center point is set to the low screen.

This process is repeated ITER (default=10) times across the map. This process can be thought of as a roughly circular moving average (median) of neighboring counties applied to each county in turn across the country.

Geographic smoothing algorithms "borrow information" from neighboring areas to stabilize results from sparsely populated areas. This reduces the variability in the data, allowing patterns to emerge, but increases the bias in the estimates for each small area. Consequently, the user should not attempt to interpret the results for any single county. The variance reduction, however, allows the user to identify and compare clusters of counties with similar values. To illustrate using a recently published map of smoking habit (Pickle and Su, 2002), the mean proportion of current male smokers within a band of counties along the Mexican border is 30%, compared to 20% within an adjacent band of counties. The sample sizes of these county clusters, 1468 and 3218, respectively, would generate clearly non-overlapping 95% confidence limits, even though any pairwise comparison of single county values would probably be nonsignificant. Although these maps are to be used for exploratory, not inferential, purposes, this example illustrates the strength gained from smoothing.

Use of population weights in the smoothing process ensures that unusually high or low proportions that are reliable due to large populations are not modified, i.e., smoothed away, whereas values based on sparse populations are modified to be more like those of the surrounding counties.

## Suggested Citation

A citation for Head-Bang, indicating the software version, is recommended. See Suggested Citation on Head-Bang's help menu for the citation specific to your version of the program. The general format is:

Hansen Simonson and Statistical Research and Applications Branch, NCI. Headbang software (surveillance.cancer.gov/headbang) version <version>.

## Methods

• Katherine M Hansen. Head-banging: robust smoothing in the plane. IEEE Transactions on Geoscience and Remote Sensing 1991;29(3): 369-378.
• Michael Mungiole, Linda Williams Pickle, Katherine Hansen Simonson. Application of a weighted head-banging algorithm to mortality data maps. Stat Med 1999;18:3201-3209.[View Abstract]
• Michael Mungiole, Linda Williams Pickle. Determining the optimal degree of smoothing using the weighted head-banging algorithm on mapped mortality data. In ASC '99 - Leading Survey & Statistical Computing into the New Millennium, Proceedings of the ASC International Conference, September 1999.

## Applications

• Linda Williams Pickle, Yuchen Su. Within-state geographic patterns of health insurance coverage and health risk factors in the United States. Am J Prev Med 2002;22(2):75-83.[View Abstract]
• Linda Williams Pickle, Michael Mungiole, Gretchen K Jones, Andrew A White. Atlas of United States Mortality. Hyattsville, Maryland: National Center for Health Statistics. 1996.(https://www.cdc.gov/nchs/products/other/atlas/atlas.htm)