randomized In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rand ...

statistical experiments, generalized randomized block designs (GRBDs) are used to study the

interaction Interaction is action that occurs between two or more objects, with broad use in philosophy and the sciences. It may refer to: Science * Interaction hypothesis, a theory of second language acquisition * Interaction (statistics) * Interactions o ...

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 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 ...

(without making parametric assumptions about a

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

for the

error An error (from the Latin ''error'', meaning "wandering") is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. The etymology derives from the Latin term 'errare', meaning 'to stray'. In statistics ...

Univariate response

GRBDs versus RCBDs: Replication and interaction

Like a

randomized complete block design In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ran ...

(RCBD), a GRBD is randomized. Within each block, treatments are randomly assigned to

experimental unit In statistics, a unit is one member of a set of entities being studied. It is the main source for the mathematical abstraction of a "random variable". Common examples of a unit would be a single person, animal, plant, manufactured item, or country ...

s: this randomization is also independent between blocks. In a (classic) RCBD, however, there is no replication of treatments within blocks.

Two-way linear model: Blocks and treatments

The experimental design guides the formulation of an appropriate

. Without replication, the (classic) RCBD has a two-way linear-model with treatment- and block-effects but ''without'' a block-treatment

. Without replicates, this two-way linear-model that may be estimated and tested without making parametric assumptions (by using the randomization distribution, without using a normal distribution for the error). In the RCBD, the block-treatment interaction cannot be estimated using the randomization distribution;

a fortiori ''Argumentum a fortiori'' (literally "argument from the stronger

eason Eason is a surname. The name comes from Aythe where the first recorded spelling of the family name is that of Aythe Filius Thome which was dated circa 1630, in the "Baillie of Stratherne". Aythe ''filius'' Thome received a charter of the lands of F ...

) (, ) is a form of Argumentation theory, argumentation that draws upon existing confidence in a proposition to argue in favor of a second proposition that is held to be Logi ...

there exists no "valid" (i.e. randomization-based) test for the block-treatment interaction in the

analysis of variance Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statisticia ...

(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 (January 31, 1919 – November 15, 2000) was a British 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 ...

and Sidney Addelman. The GRBD has the advantage that replication 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. GRBD designs for models without interaction terms offer more degrees of freedom for testing treatment-effects than do RCBs with more blocks: An experimenter wanting to increase power may use a GRBD rather than RCB with additional blocks, when extra blocks-effects would lack genuine interest.

Multivariate analysis

The GRBD has a real-number response. For vector responses,

multivariate analysis Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable. Multivariate statistics concerns understanding the different aims and background of each of the dif ...

considers similar two-way models with main effects and with interactions or errors. Without replicates, error terms are confounded with interaction, and only error is estimated. With replicates, interaction can be tested with the

multivariate analysis of variance In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is often followed by significance tests ...

and coefficients in the linear model can be estimated without bias and with minimum variance (by using the least-squares method).

Functional models for block-treatment interactions: Testing known forms of interaction

Non-replicated experiments are used by knowledgeable experimentalists when replications have prohibitive

cost In production, research, retail, and accounting, a cost is the value of money that has been used up to produce something or deliver a service, and hence is not available for use anymore. In business, the cost may be one of acquisition, in which ...

s. When the block-design lacks replicates, interactions have been modeled. For example, Tukey's F-test for interaction (non-additivity) has been motivated by the multiplicative model of Mandel (1961); this model assumes that all treatment-block interactions are proportion to the product of the mean treatment-effect and the mean block-effect, where the proportionality constant is identical for all treatment-block combinations. Tukey's test is valid when Mandel's multiplicative model holds and when the errors independently follow a normal distribution. Tukey's F-statistic for testing interaction has a distribution based on the randomized assignment of treatments to experimental units. When Mandel's multiplicative model holds, the F-statistics randomization distribution is closely approximated by the distribution of the F-statistic assuming a normal distribution for the error, according to the 1975 paper of Robinson. The rejection of multiplicative interaction need not imply the rejection of non-multiplicative interaction, because there are many forms of interaction. Generalizing earlier models for Tukey's test are the “bundle-of-straight lines” model of Mandel (1959) and the functional model of Milliken and Graybill (1970), which assumes that the interaction is a known function of the block and treatment main-effects. Other methods and heuristics for block-treatment interaction in unreplicated studies are surveyed in the monograph .

Notes

References

* * * * * * * * * * {{Experimental design, state=expanded Design of experiments Analysis of variance