Fixed Blocks in a Custom Design

[#1]

Hello

I would like to generate fixed blocks in a custom design. In practice, it is often the case that within these blocks the same experimental runs are not repeated (the factors are the same, but the factor levels differ), meaning that different factor settings occur within the blocks.

Furthermore, blocks can also exhibit correlations with other blocks and, for example, with main effects. Therefore, I see a difficulty in evaluating block effects cleanly. Is a block effect actually a real block effect, or is it merely induced, for instance, by a relatively high correlation with a significant main effect, or by different factor levels within the blocks?

Regards

Klaus

[#2]

You can treat the block as a restriction of the experimental units. It is not a factor of interest, but an additional source of variation observed in the response. You can treat the effect of the blocks as fixed or random. in the linear model. I am not sure what you mean when you say, "a difficulty in evaluating block effects cleanly."

[#3]

Adding to Mark's comments. Typically a bock is a surrogate for a set of noise variables. Noise is factors you are not willing to or able to control in the future (e.g., raw material variation, environmental conditions, operator technique, measurement error). When blocking, you are replicating identical treatment combinations in each block. The variation in the replicates must be due to noise as the factors and levels did not change between the replicates. If the noise variables have not been identified and therefore cannot be explicitly manipulated during the experiment, blocking is usually considered a random effect in a model and you will have a mixed model. If you have identified the noise factors confounded with the blocks, then treating the block as a fixed effect is recommended.  If you do treat the blocks as a fixed effect, then you will add block and all block by factor interactions to the model and you will have a fixed effects model.

Your situation is, however, different. The description you give appears to me as you want to nest treatment combinations within block. When doing this, you will confound block effects with changes in levels for the factors. This does not seem wise?

[#4 · Accepted solution]

Hi @NominalGemsbok3,

Besides the excellent comments and responses brought by Mark and Bill, here is some complementary information from my side.
The decision to have fixed or random block effect is set before the analysis, depending on how you consider this blocking factor :

• If the levels can be reproducible or if the specified levels are the only ones of interest (for example in the case of identified noise sources as mentioned by Bill), then you can treat it as a fixed block effect and it will affect response mean (bias).
• If the levels are a sample from a larger population or if you are not really interested in these two (or more) particular levels (or if noise sources have not been identified), but variation across the population, then treat it as a random effect and it will affect response variance.

To create fixed blocks in a design, you can add a blocking factor in the Factors panel and specify the number of runs per block. This way, JMP automatically creates blocks with the right factor column properties Value Order, RunsPerBlock, Design Role (Blocking) and Factor Changes (Easy). These column properties help JMP assess the role of this factor when modeling the results. When using the Fit Model script, you'll see that the fixed blocking factor is present in the Construct Model effects panel as a fixed effect like your other factors.

Blocks are created in order to avoid correlations between the block effect and the model effects you're interested in. If possible, they are created so that the block effect is orthogonal (no correlations) to the effects in your model. If not possible, the correlations will still be minimized to avoid any confusion between the influence of block effect or other factors effects. In some situations (particularly with limited experimental budget), you may end up with small correlations between the block effect and other model effects ; but this correlation is usually very low, so it is not a problem in the analysis, it is still possible to differentiate block effect from the other effects.
Finally, you might not have the same exact experiments in each block (so factor settings may differ like you mentioned), but the repartition of the treatments are done to avoid any bias regarding the factor levels used in each block : the factors levels will be (as much as possible) balanced across the blocks, so you have a similar number of runs with high and low levels for each factors in each block. This ensures that the block effect may not be correlated with other model effects. 

 Do you have any design/dataset we could look at to further answer your concerns and questions ?