Generalized randomized block design
Encyclopedia
In randomized
statistical experiments
, generalized randomized block designs (GRBDs) are used to study the interaction
between blocks
and treatments. For a GRBD, each treatment is replicated
at least two times in each block; this replication allows the estimation and testing of an interaction term in the linear model
(without making parametric assumptions about a normal distribution for the error).
(RCBD), a GRBD is randomized. Within each block, treatments are randomly assigned
to experimental units: this randomization is also independent between blocks. In a (classic) RCBD, however, there is no replication of treatments within blocks. Without replication, the (classic) RCBD's linear model
lacks a block-treatment interaction
-term that may be estimated (using the randomization distribution rather than using a normal distribution for the error). In the RCBD, the block-treatment interaction cannot be estimated using the randomization distribution; a fortiori there exists no "valid" (i.e. randomization-based) test for the block-treatment interaction in the analysis of variance
(anova) of the RCBD.
The distinction between RCBDs and GRBDs has been ignored by some authors, and the ignorance regarding the GRCBD has been criticized by statisticians like Oscar Kempthorne
and Sidney Addelman. The GRBD has the advantage that replication
allows block-treatment interaction to be studied.
or the (post-hoc) secondary analysis of data from a GRBD, statisticians may check for non-additivity using residual diagnostics.
Randomized experiment
In science, randomized experiments are the experiments that allow the greatest reliability and validity of statistical estimates of treatment effects...
statistical experiments
Design of experiments
In general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments...
, generalized randomized block designs (GRBDs) are used to study the interaction
Interaction (statistics)
In statistics, an interaction may arise when considering the relationship among three or more variables, and describes a situation in which the simultaneous influence of two variables on a third is not additive...
between blocks
Blocking (statistics)
In the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups that are similar to one another. For example, an experiment is designed to test a new drug on patients. There are two levels of the treatment, drug, and placebo, administered to male...
and treatments. For a GRBD, each treatment is replicated
Replication (statistics)
In engineering, science, and statistics, replication is the repetition of an experimental condition so that the variability associated with the phenomenon can be estimated. ASTM, in standard E1847, defines replication as "the repetition of the set of all the treatment combinations to be compared in...
at least two times in each block; this replication allows the estimation and testing of an interaction term in the linear model
Linear model
In statistics, the term linear model is used in different ways according to the context. The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However the term is also used in time series analysis with a different...
(without making parametric assumptions about a normal distribution for the error).
GRBDs versus RCBDs: Replication and interaction
Like a randomized complete block designRandomized block design
In the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups that are similar to one another. Typically, a blocking factor is a source of variability that is not of primary interest to the experimenter...
(RCBD), a GRBD is randomized. Within each block, treatments are randomly assigned
Random assignment
Random assignment or random placement is an experimental technique for assigning subjects to different treatments . The thinking behind random assignment is that by randomizing treatment assignment, then the group attributes for the different treatments will be roughly equivalent and therefore any...
to experimental units: this randomization is also independent between blocks. In a (classic) RCBD, however, there is no replication of treatments within blocks. Without replication, the (classic) RCBD's linear model
Linear model
In statistics, the term linear model is used in different ways according to the context. The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However the term is also used in time series analysis with a different...
lacks a block-treatment interaction
Interaction (statistics)
In statistics, an interaction may arise when considering the relationship among three or more variables, and describes a situation in which the simultaneous influence of two variables on a third is not additive...
-term that may be estimated (using the randomization distribution rather than using a normal distribution for the error). In the RCBD, the block-treatment interaction cannot be estimated using the randomization distribution; a fortiori there exists no "valid" (i.e. randomization-based) test for the block-treatment interaction in the analysis of variance
Analysis of variance
In statistics, analysis of variance 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 variation...
(anova) of the RCBD.
The distinction between RCBDs and GRBDs has been ignored by some authors, and the ignorance regarding the GRCBD has been criticized by statisticians like Oscar Kempthorne
Oscar Kempthorne
Oscar Kempthorne was a statistician and geneticist known for his research on randomization-analysis and the design of experiments, which had wide influence on research in agriculture, genetics, and other areas of science...
and Sidney Addelman. The GRBD has the advantage that replication
Replication (statistics)
In engineering, science, and statistics, replication is the repetition of an experimental condition so that the variability associated with the phenomenon can be estimated. ASTM, in standard E1847, defines replication as "the repetition of the set of all the treatment combinations to be compared in...
allows block-treatment interaction to be studied.
GRBDs when block-treatment interaction lacks interest
However, if block-treatment interaction is known to be negligible, then the experimental protocol may specify that the interaction terms be assumed to be zero and that their degrees of freedom be used for the error term. During exploratory data analysisExploratory data analysis
In statistics, exploratory data analysis is an approach to analysing data sets to summarize their main characteristics in easy-to-understand form, often with visual graphs, without using a statistical model or having formulated a hypothesis...
or the (post-hoc) secondary analysis of data from a GRBD, statisticians may check for non-additivity using residual diagnostics.
See also
- Block design
- Complete block design
- Incomplete block design
- Randomized block designRandomized block designIn the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups that are similar to one another. Typically, a blocking factor is a source of variability that is not of primary interest to the experimenter...
- RandomizationRandomizationRandomization is the process of making something random; this means:* Generating a random permutation of a sequence .* Selecting a random sample of a population ....
- Randomized experimentRandomized experimentIn science, randomized experiments are the experiments that allow the greatest reliability and validity of statistical estimates of treatment effects...