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We designed a RCB experiment and assigned the factor levels to the experimental units randomly inside each block. Let's pretend we changed our mind and we would like to go for a Completely Randomized design. A new completely randomized assignment of factor levels to experimental units is not possible since the experiment has already begun. We have to stay with the RCB assigment (thus having 1 replicate in each former block).
The question is: how much bias would we introduce in the analysis of variance if we analysed the RCB configuration pretending it to be a CR configuration? I know that the RCB configuration can be considered as one of the $n$ equally probable random configurations, but is there a way to account for this bias in the further analysis?

Update
The reason we would like to move from RCB to CR is that, from modelling studies, our block has no significant effect on the response varible, and, as gung pointed out, it would decrease the power of the test we will perform. Since Blocks do not significantly affect the response variable of the experimental units, in case we moved to a CR design, can the replicates (previously randomized within non-significant blocks) be considered as randomized within the whole population?
This question is relevant also for post-hoc analysis, if a RCB ANOVA results in blocks not being significant, is a CR ANOVA appropriate on a configuration that was not completely randomized?

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In statistics, 'bias' is generally understood as being a property of an estimator, specifically an estimator whose expected value is not the true value of the population parameter. This is unrelated to whether you choose a randomized block design or a completely randomized design for your experiment. In either case, if you have all qualitative predictors, a continuous response variable distributed with equal variance in each level of each factor, are representing the central tendencies of your groups with their sample means, and analyzing your data with an ANOVA, there is no bias.

On the other hand, there is a legitimate question of efficiency. Whether the RB is more efficient (i.e., offers greater power) depends on how the MSE term changes. If it gets smaller, your F-value will go up resulting in greater 'significance', and vice-versa. Whether or not it gets smaller depends on whether the error term loses more sums of squares or more degrees of freedom during partitioning. Imagine SSE = 10, and dfE = 10, then MSE = 1. Now, if that is partitioned into pure error and a blocking factor, it could play out in two ways: First, if SSBlock = 2, and dfBlock = 1, then SSE = 8 and dfE = 9, and so MSE = .9. In this scenario, your F will increase and p-value will go down. On the other hand, if SSBlock = 1 and dfBlock = 2, then SSE = 9, and dfE = 8, and so MSE = 1.1. In this case, F goes down and p goes up. A simpler way to think about this is that if your block is significant, then power increases, otherwise not.

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  • $\begingroup$ Thank you for your nice example. Actually we are well aware of the differences between RCB and CR, see here for example. We chosed to go with RCB, assigned the levels to the experimental units, started the experiment but, on a successive rethinking, we reckoned that a CR analysis would have been better. Since we cannot stop the experiment, reassign levels to plots in a randomized way and start again, how much bias would we introduce analysing the current configuration pretending it is a CR one? $\endgroup$ – Charlie Jan 13 '12 at 10:26

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