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I have a design with four factors:

  • Frame (positive negative neutral)
  • Graphic ( graph no graph)
  • Color (red grey) nested in graphic
  • Position of color (nested in color, which is nested in graphic)

I am having difficulty understanding if this is an incomplete nested design or a split plot design.

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Split-plot is a type of nested design. The phrase comes from Statistics' youth, spent at the agricultural testing station of Rothhamstead, UK, where so many different experimental designs were created. Split-plots really were portions of plots. You might put a particular type of fertilizer over an entire plot, and then plant different varieties of peas on that plot. A different plot would have different fertilizer - but the same varieties of pea. So you envisage 2 or more factors, but different levels of randomization. Fertilizers are randomly assigned to plots - then for each plot, variety is randomly assigned to the split portion. All nested designs have that feature: different levels of randomization, applied to different subjects.

Back to your question:

What I need to know is how these factors are distributed in your actual design. I also need to know what the response is: what are you measuring from all of these design features?

If you have "red" on one design, do you have "grey" somewhere else on the same design? Or is it omitted from that particular design? If Red and Grey appear on one design or the other (but not together) - it seems that they are not nested.

I can't say exactly from what you have said, but it looks like a single-plot, un-nested factorial design - only unbalanced. Frame is crossed with graphic - colour and position are probably crossed also - but in that subset of the experiment that contains graphs. You can then test for those effects using the contrasts. If everything is balanced within the "graph" section, you want to look at "mean with red" - "mean with grey"; also "Position 1"-"Position 2" - or whatever seems to be of interest.

If I have understood your question correctly, the design is neither nested nor split-plot - but it is incomplete - simply because some factor combinations do not apply.

This can be analysed by obtaining an estimate of the standard deviation and using that to test for particular contrasts - i.e. comparisons of the means - to test hypotheses of interest to you.

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  • $\begingroup$ (+1) I am confused by your first sentence and in general the first paragraph. In your example, all pea varieties get combined with all fertilizers; to me it therefore sounds like "crossed" factors and not like "nested" factors. See stats.stackexchange.com/questions/228800 about crossed vs nested. At the same time, your answer is not the only place where I see claims that split plots (or repeated measures) are nested. Can you clarify? This is related to my active question stats.stackexchange.com/questions/232109 (where, by the way, I would very much appreciate your input). $\endgroup$ – amoeba Aug 29 '16 at 13:39
  • $\begingroup$ All combinations of the factors occur in the design - so it's like a crossed design in that sense. But the combinations are not assigned with equal probability to the sampling units. In fact, there are 2 types of sampling unit in my example - the large plots (where the fertilizer goes) and the subplots, where the varieties go. That's the difference. $\endgroup$ – Placidia Aug 30 '16 at 0:58
  • $\begingroup$ Thanks. Actually, now I think that whole plots are nested within fertilizers. But pea varieties are crossed with whole plots. In a biostatistics example, subjects (plots) are nested within between-subject factors (fertilizers) but they are crossed with within-subject factors (pea varieties). Nevertheless, I have seen people saying that within-subject factors "are nested" within subjects, but this is either wrong or at least a different meaning of the word "nested". Does it make sense, do you agree? Thanks a lot. $\endgroup$ – amoeba Aug 30 '16 at 1:05
  • $\begingroup$ The whole plot is an experimental unit - fertilizer is a factor. The fertilizers are randomly assigned to the whole plots, not nested. The sub-plots are also experimental units, to which varieties are randomly assigned. What makes it a split-plot design is that there are two levels of experimental unit. I think I would say that varieties are nested within fertilizer - not that varieties are nested within plots - but that's probably just semantics. The crucial point is that the denominator of the F test for comparing fertilizers is not the denominator for doing the varieties comparison. $\endgroup$ – Placidia Aug 30 '16 at 1:55
  • $\begingroup$ BTW, I took a shot are your active question stats.stackexchange.com/a/232412/14188. I don't think the other responder got it right. But it's confusing. I'll say that much. $\endgroup$ – Placidia Aug 30 '16 at 1:56

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