Balanced repeated replication

Balanced repeated replication

Balanced repeated replication is a statistical technique for estimating the sampling variability of a statistic obtained by stratified sampling.

Outline of the technique

# "Select balanced half-samples" from the full sample.
# "Calculate the statistic of interest" for each half-sample.
# "Estimate the variance of the statistic" on the basis of differences between the full-sample and half-sample values.

Selection of half-samples

Simplified version

Consider first an idealized situation, where each stratum of our sample contains only two units. Then each half-sample will contain exactly one of these, so that the half-samples share the stratification of the full sample. If there are "s" strata, we would ideally take all 2"s" ways of choosing the half-stratum; but if s is large, this may be infeasible.

If fewer half-samples must be taken, they are selected so as to be "balanced" (hence the name of the technique). Let "H" be a Hadamard matrix of size s, and choose one row per half-sample. (It doesn't matter which rows; the important fact is that all the rows of "H" are orthogonal.) Now, for each half-sample, choose which unit to take from each stratum according to the sign of the corresponding entry in "H": that is, for half-sample "h", we choose the first unit from stratum "k" if "Hhk" = −1 and the second unit if Hhk = +1. The orthogonality of rows of H ensures that our choices are uncorrelated between half-samples.

Realistic version

Unfortunately, there may not be a Hadamard matrix of size s. In this case, we choose one of size slightly larger than s. Now the submatrix of H which defines our choices need no longer have exactly orthogonal rows, but if the size of H is only slightly larger than s the rows will be approximately orthogonal.

The number of units per stratum need not be exactly 2, and typically will not be. In this case, the units in each stratum are divided into two "variance PSUs" (PSU = primary sampling unit) of equal or nearly-equal size. This may be done at random, or in such a way as to make the PSUs as similar as possible. (So, for instance, if stratification was done on the basis of some numerical parameter, the units in each stratum may be sorted in order of this parameter, and alternate ones chosen for the two PSUs.)

If the number of strata is very large, multiple strata may be combined before applying BRR. The resulting groups are known as "variance strata".

BRR formula

Let a be the value of our statistic as calculated from the full sample; let ai (i = 1,...,nh) be the corresponding statistics calculated for the half-samples. (nh is the number of half-samples.)

Then our estimate for the sampling variance of the statistic is the average of (aia)2. This is (at least in the ideal case) an unbiased estimate of the sampling variance.

Fay's method

Fay's method is a generalization of BRR. Instead of simply taking half-size samples, we use the full sample every time but with unequal weighting: k for units outside the half-sample and 2 − k for units inside it. (BRR is the case k = 0.) The variance estimate is then V/(1 − k)2, where V is the estimate given by the BRR formula above.

See also

* Resampling_(statistics)

References and external links

* [ Balanced Repeated Replication] , from the American Institutes for Research

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • Hadamard matrix — In mathematics, a Hadamard matrix is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. This means that every two different rows in a Hadamard matrix represent two perpendicular vectors. Such matrices can… …   Wikipedia

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

  • Varianzschätzer — In diesem Artikel oder Abschnitt fehlen folgende wichtige Informationen: In diesem Artikel wird als einzige Varianzschätzung die korrigierte Stichprobenvarianz erwähnt. Es fehlen konkrete Verfahren zur Schätzung der Varianz, die auf den… …   Deutsch Wikipedia

  • Medical Expenditure Panel Survey — The Medical Expenditure Panel Survey (MEPS) is a family of surveys intended to provide nationally representative estimates of health expenditure, utilization, payment sources, health status, and health insurance coverage among the… …   Wikipedia

  • MediaWiki — namespace redirects here. For help regarding the MediaWiki namespace on Wikipedia, see Help:MediaWiki namespace. For general information about Wikipedia namespaces, see Wikipedia:Namespace. Talk page and MediaWiki talk page redirect here. For… …   Wikipedia

  • evolution — evolutional, adj. evolutionally, adv. /ev euh looh sheuhn/ or, esp. Brit., /ee veuh /, n. 1. any process of formation or growth; development: the evolution of a language; the evolution of the airplane. 2. a product of such development; something… …   Universalium

  • Analysis of variance — In statistics, analysis of variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of… …   Wikipedia

  • international relations — a branch of political science dealing with the relations between nations. [1970 75] * * * Study of the relations of states with each other and with international organizations and certain subnational entities (e.g., bureaucracies and political… …   Universalium

  • genetic disease, human — Introduction       any of the diseases and disorders that are caused by mutations in one or more genes (gene).       With the increasing ability to control infectious and nutritional diseases in developed countries, there has come the realization …   Universalium