Restricted randomization: Difference between revisions

Revision as of 17:30, 26 March 2012

Latest revision as of 20:59, 20 August 2025

(59 intermediate revisions by 29 users not shown)

Line 1:

{{more footnotes|date=April 2012}}

<!-- Please do not remove or change this AfD message until the issue is settled -->

{{Article for deletion/dated|page=Restricted randomization|timestamp=20120318184004|year=2012|month=March|day=18|substed=yes|help=off}}

<!-- For administrator use only: {{Old AfD multi|page=Restricted randomization|date=18 March 2012|result='''keep'''}} -->

<!-- End of AfD message, feel free to edit beyond this point -->

{{Cleanup|date=June 2009}}

{{Lead rewrite|date=September 2009}}

In [[statistics]], '''restricted randomization''' occurs in the [[design of experiments]] and in particular in the context of [[randomized experiment]]s and [[randomized controlled trial]]s. Restricted randomization allows intuitively poor allocations of treatments to experimental units to be avoided, while retaining the theoretical benefits of randomization.<ref>{{cite book|last1=Dodge|first1=Y.|title=The Oxford Dictionary of Statistical Terms|publisher=OUP|year=2006|isbn=978-0-19-920613-1|url-access=registration|url=https://archive.org/details/oxforddictionary0000unse}}</ref><ref>{{cite journal|last1=Grundy|first1=P.M.|last2=Healy|first2=M.J.R.|author-link2=Michael Healy (statistician)|title=Restricted randomization and quasi-Latin squares|journal=Journal of the Royal Statistical Society, Series B|date=1950 |volume=12|issue=2 |pages=286–291 |doi=10.1111/j.2517-6161.1950.tb00062.x }}</ref> For example, in a [[clinical trial]] of a new proposed treatment of obesity compared to a control, an experimenter would want to avoid outcomes of the randomization in which the new treatment was allocated only to the heaviest patients.

Many [[process (engineering)|processes]] have more than one source of [[statistical dispersion|variation]] in them. In order to [[variance reduction|reduce variation]] in processes, these multiple sources must be understood, and that often leads to the concept of nested or hierarchical data structures. For example, in the semiconductor industry, a [[batch production|batch process]] may operate on several [[wafer (electronics)|wafers]] at a time (wafers are said to be '''nested''' within batch). Understanding the input variables that control variation among those wafers, as well as understanding the variation across each wafer in a run, is an important part of the strategy for minimizing the total variation in the system.

The concept was introduced by [[Frank Yates]] (1948){{full citation needed|date=November 2012}} and [[William J. Youden]] (1972){{full citation needed|date=November 2012}} "as a way of avoiding bad spatial patterns of treatments in designed experiments."<ref name="ref1">{{Cite journal |jstor = 2288775|title = Restricted Randomization: A Practical Example|last1 = Bailey|first1 = R. A.|journal = Journal of the American Statistical Association|year = 1987|volume = 82|issue = 399|pages = 712–719|doi = 10.1080/01621459.1987.10478487}}</ref>

==Example of nested data==

Consider a batch process that uses 7 monitor wafers in each run. The plan further calls for measuring a [[response variable]] on each wafer at each of 9 sites. The organization of the [[sampling plan]] has a hierarchical or nested structure: the batch run is the topmost level, the second level is an individual wafer, and the third level is the site on the wafer.

The total amount of data generated per batch run will be 7&nbsp;&middot;·&nbsp;9&nbsp;=&nbsp;63 observations. One approach to analyzing these data would be to compute the [[mean]] of all these points as well as their [[standard deviation]] and use those results as responses for each run.

Analyzing the data as suggested above is not absolutely incorrect, but doing so loses information that one might otherwise obtain. For example, site 1 on wafer 1 is physically different from site 1 on wafer 2 or on any other wafer. The same is true for any of the sites on any of the wafers. Similarly, wafer 1 in run 1 is physically different from wafer 1 in run 2, and so on. To describe this situation one says that sites are nested within wafers while wafers are nested within runs.

As a consequence of this nesting, there are restrictions on the randomization that can occur in the experiment. This kind of restricted randomization always produces nested sources of variation. Examples of nested variation or restricted randomization discussed on this page are [['''split-plot]]''' and [['''strip-plot]] [[experimental design|designs]]'''.

The objective of an experiment with this type of sampling plan is generally to reduce the variability due to sites on the wafers and wafers within runs (or batches) in the process. The sites on the wafers and the wafers within a batch become sources of unwanted variation and an investigator seeks to [[robustification|make the system robust]] to those sources—in other words, one could treat wafers and sites as noise factors in such an experiment.

Line 29 ⟶ 26:

In many industrial experiments, three situations often occur:

# some of the factors of interest may be 'hard to vary' while the remaining factors are easy to vary. As a result, the order in which the treatment combinations for the experiment are run is determined by the ordering of these 'hard-to-vary' factors

# experimental units are processed together as a batch for one or more of the factors in a particular treatment combination

# experimental units are processed individually, one right after the other, for the same treatment combination without resetting the factor settings for that treatment combination.

===Split-plot experimental examples===

Line 37 ⟶ 34:

{| class="wikitable"

|+ Orthogonally Scaledscaled Treatmenttreatment Combinationscombinations from a 2³ Fullfull Factorialfactorial

|-

! Current

Line 43 ⟶ 40:

! Concentration

|-

| −1

| &minus;1

| −1

| &minus;1

| −1

| &minus;1

|-

| −1

| &minus;1

| −1

| &minus;1

| +1

|-

| −1

| &minus;1

| +1

| −1

| &minus;1

|-

| −1

| &minus;1

| +1

| +1

|-

| +1

| −1

| &minus;1

| −1

| &minus;1

|-

| +1

| −1

| &minus;1

| +1

|-

| +1

| +1

| −1

| &minus;1

|-

| +1

Line 79 ⟶ 76:

Consider running the experiment under the first condition listed above, with the factor solution concentration of the plating agent (S) being hard to vary. Since this factor is hard to vary, the experimenter would like to randomize the treatment combinations so that the solution concentration factor has a minimal number of changes. In other words, the randomization of the treatment runs is restricted somewhat by the level of the solution concentration factor.

As a result, the treatment combinations might be randomized such that those treatment runs corresponding to one level of the concentration (&minus;1−1) are run first. Each copper strip is individually plated, meaning only one strip at a time is placed in the solution for a given treatment combination. Once the four runs at the low level of solution concentration have been completed, the solution is changed to the high level of concentration (1), and the remaining four runs of the experiment are performed (where again, each strip is individually plated).

Once one complete replicate of the experiment has been completed, a second replicate is performed with a set of four copper strips processed for a given level of solution concentration before changing the concentration and processing the remaining four strips. Note that the levels for the remaining two factors can still be randomized. In addition, the level of concentration that is run first in the replication runs can also be randomized.

Line 89 ⟶ 86:

There are 16 subplot experimental units for this experiment. Solution temperature and current are the subplot factors in this experiment. There are four whole-plot experimental units in this experiment. Solution concentration is the whole-plot factor in this experiment. Since there are two sizes of experimental units, there are two error terms in the model, one that corresponds to the whole-plot error or whole-plot experimental unit and one that corresponds to the subplot error or subplot experimental unit.

The [[ANOVA]] table for this experiment would look, in part, as follows:

{| class="wikitable"

|+ Partial ANOVA table

Line 102 ⟶ 99:

|1

|-

|Error (Wholewhole plot) = Rep* × Conc

|1

|-

Line 108 ⟶ 105:

|1

|-

|Rep* × Temp

|1

|-

Line 114 ⟶ 111:

|1

|-

|Rep* × Current

|1

|-

|Temp* × Conc

|1

|-

|Rep* × Temp* × Conc

|1

|-

|Temp* × Current

|1

|-

|Rep* × Temp* × Current

|1

|-

|Current* × Conc

|1

|-

|Rep* × Current* × Conc

|1

|-

|Temp* × Current* × Conc

|1

|-

|Error (Subplot) = Rep* × Temp* × Current* × Conc

|1

|}

Line 145 ⟶ 142:

====Example: batch process====

Consider running the experiment under the second condition listed above (i.e., a batch process) for which four copper strips are placed in the solution at one time. A specified level of current can be applied to an individual strip within the solution. The same 16 treatment combinations (a replicated 2³ factorial) are run as were run under the first scenario. However, the way in which the experiment is performed would be different. There are four treatment combinations of solution temperature and solution concentration: (&minus;1−1, &minus;1−1), (&minus;1−1, 1), (1, &minus;1−1), (1, 1). The experimenter randomly chooses one of these four treatments to set up first. Four copper strips are placed in the solution. Two of the four strips are randomly assigned to the low current level. The remaining two strips are assigned to the high current level. The plating is performed and the response is measured. A second treatment combination of temperature and concentration is chosen and the same procedure is followed. This is done for all four temperature / concentration combinations.

Running the experiment in this way also results in a split-plot design in which the whole-plot factors are now solution concentration and solution temperature, and the subplot factor is current.

Line 153 ⟶ 150:

The smaller size experimental unit is again referred to as the subplot experimental unit. There are 16 subplot experimental units for this experiment. Current is the subplot factor in this experiment.

The larger-size experimental unit is the whole-plot experimental unit. There are four whole plot experimental units in this experiment and solution concentration and solution temperature are the whole plot factors in this experiment.

There are two sizes of experimental units and there are two error terms in the model: one that corresponds to the whole-plot error or whole-plot experimental unit, and one that corresponds to the subplot error or subplot experimental unit.

Line 170 ⟶ 167:

|1

|-

|Error (Wholewhole plot) = Conc* × Temp

|1

|-

Line 176 ⟶ 173:

|1

|-

|Conc* × Current

|1

|-

|Temp* × Current

|1

|-

|Conc* × Temp* × Current

|1

|-

|Error (Subplotsubplot)

|8

|}

Line 214 ⟶ 211:

|1

|-

|Error (Wholewhole plot)

|4

|-

Line 220 ⟶ 217:

|1

|-

|Conc* × Current

|1

|-

|Temp* × Current

|1

|-

|Conc* × Temp* × Current

|1

|-

|Error (Subplotsubplot)

|4

|}

Line 245 ⟶ 242:

Consider the following example from the semiconductor industry. An experiment requires an implant step and an anneal step. At both the anneal and the implant steps there are three factors to test. The implant process accommodates 12 wafers in a batch, and implanting a single wafer under a specified set of conditions is not practical nor does doing so represent economical use of the implanter. The anneal furnace can handle up to 100 wafers.

The settings for a two-level factorial design for the three factors in the implant step are denoted (A, B, C), and a two-level factorial design for the three factors in the anneal step are denoted (D, E, F). Also present are [[interaction effects]] between the implant factors and the anneal factors. Therefore, this experiment contains three sizes of experimental units, each of which has a unique error term for estimating the significance of effects.

To put actual physical meaning to each of the experimental units in the above example, consider each combination of implant and anneal steps as an individual wafer. A batch of eight wafers goes through the implant step first. Treatment combination 3 in factors A, B, and C is the first implant treatment run. This implant treatment is applied to all eight wafers at once. Once the first implant treatment is finished, another set of eight wafers is implanted with treatment combination 5 of factors A, B, and C. This continues until the last batch of eight wafers is implanted with treatment combination 6 of factors A, B, and C. Once all of the eight treatment combinations of the implant factors have been run, the anneal step starts. The first anneal treatment combination to be run is treatment combination 5 of factors D, E, and F. This anneal treatment combination is applied to a set of eight wafers, with each of these eight wafers coming from one of the eight implant treatment combinations. After this first batch of wafers has been annealed, the second anneal treatment is applied to a second batch of eight wafers, with these eight wafers coming from one each of the eight implant treatment combinations. This is continued until the last batch of eight wafers has been implanted with a particular combination of factors D, E, and F.

Line 254 ⟶ 251:

==See also==

{{Portal|Mathematics}}

* [[Hierarchical linear modeling]]

* [[Mixed-design analysis of variance]]

Line 259 ⟶ 257:

* [[Nested case-control study]]

==External linksReferences==

{{reflist}}

*[ {{cite web | url=http://www.itl.nist.gov/div898/handbook/pri/section5/pri55.htm | title=How can I account for nested variation (restricted randomization)?] | publisher=(U.S.) National Institute of Standards and Technology: Information Technology Laboratory | access-date=March 26, 2012}}

==Further reading==

For a more detailed discussion of these designs and the appropriate analysis procedures, see:

* {{cite book

|authorauthor1 = Milliken, G. A.

|coauthorsauthor2 = Johnson, D. E.

|year = 1984

|title = Analysis of Messy Data

Line 272 ⟶ 271:

|publisher = Van Nostrand Reinhold

|___location = New York}}

* {{cite journal

|author = Miller, A.

|year = 1997

Line 279 ⟶ 278:

|volume = 39

|issue = 2

|page = 153

|doi = 10.2307/1270903

|publisher = Technometrics, Vol. 39, No. 2

|jstor = 1270903

|pages = 153–161}}

==External links==

* [https://www.southampton.ac.uk/~cpd/anovas/datasets/index.htm Examples of all ANOVA and ANCOVA models with up to three treatment factors, including randomized block, split plot, repeated measures, and Latin squares, and their analysis in R]

{{Statistics}}

Line 292:

[[Category:Analysis of variance]]

[[Category:Design of experiments]]

[[Category:Statistical models]]