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I am self studying an introductory course on Designed Experiments and have come across the notion of a Completely Randomised Design (CRD) defined as follows:

Completely Randomised Design: the simplest form of a designed experiments where there is no distinction between units (ie. no blocking). Therefore, there are no partitions of the experimental area. Application of the treatments is randomised across time periods and assignment of the treatments to experimental units is also randomised.

I have come across two advantages of using a CRD experiment that I am unsure about the meaning of.

  • "Unequal replication is fine" - what does this mean? I know that replication refers to applying the same treatment to different experimental units. Does this mean it's okay to apply different treatments to different numbers of experimental units? And if so, why is this specifically allowed in this case and not others?

  • "Maximum degrees of freedom in the error term" - this is also unclear to me. What do they mean by this and why is this specifically an advantage for a CRD experiment?

Note: There were other advantages listed, however, these were the two that I didn't understand and so for clarity I have excluded the listed advantages that I already understand.

I would be grateful for any clarification here.

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You should edit your post to include a reference for the claims! By bullet points:

  • A CRD is typically analyzed by a one-way anova, and unequal replication (that is, unequal number of observations in the different treatment groups) do not cause any problems for one-way anova. The design is still orthogonal, for example. If you are able to plan for equal number of observations in each group, that is generally the best (optimizing power under assumption of equal variances, for example) but if not, no complications arise.

  • If you include blocks in the design, then some degrees of freedom is used for estimating block means, and that reduced the degrees of freedom for variance estimation. If differences between blocks are slight, that might give you a loss of power. If blocking is effective, that is, there is really important differences between blocks, the error reduction by blocking is more important than the loss of degrees of freedom. See When to use blocks?, Moving from RCB to CR design, how much bias is introduced?,

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  • $\begingroup$ Thank you for your answer. Unfortunately there are no public references (that I know of) for these claims as they came from my university lecture notes - so I can’t add in any meaningful references to the post. $\endgroup$
    – FD_bfa
    Commented Feb 16, 2023 at 14:25

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