Split block with subsampling and no replication of blocks

[ward77]

I have a researcher who is conducting testing of area-wide treatments (done on large swaths of land, not easy to apply to small areas and not easy to replicate). There are two factors, factor A with three treatment levels (2 trt and 1 control) and factor B with two levels. The physical layout is like a split block (or strip plot), with each level of A applied to one of three vertical strips and each level of B applied to one of two horizontal strips (so each is like a whole plot factor), and the six subplots that are created by the criss-cross design will each have one level of the A*B interaction (so six combined treatment levels, each in one of six subplots). The outcome variable will be counts of insects, so likely Poisson unless the counts are large enough to be approximately normal. The counts will come from monitoring at 30 properties in each of the six subplots, so 180 observations total. I think the 30 properties are subsamples, not replicates, and the entire setup is established only once; there are no replicate blocks.

If the design were repeated in multiple blocks, I think the appropriate code would be

proc glimmix data = dat;
class block A  B;
model response = A B A*B / dist= poisson link = log;
random block block*A block*B;

But with no replicate blocks, I won't have block in the model, and I don't think I can estimate the A*B interaction without the replicates. I do, however, have the subsampling. I'm not sure how to proceed. The best I could think of was to add a random effect for the subplots to account for the subsampling and skip the interaction effect (which is essentially the same as the subplot though).

proc glimmix data = dat;
class subplot A  B;
model response = A B / dist= poisson link = log;
random subplot;

Any suggestions?

Edit: I attached a csv file, with subplot 1-6, factor A with 3 levels, factor B with 2 levels, properties 1-30 in each subplot, and simulated counts as response.

DATA dat;
    INFILE 'ExampleData.csv'
   	DSD MISSOVER PAD firstobs = 2;
	INPUT subplot $ A $ B $ property $ response;
run;

[SteveDenham]

I don't think you have any random effects in this design - you have counts (dependent variable) in six treatment combinations.  The subplot is the experimental unit.  The only thing you might do is build in some heterogeneity of variance with a RANDOM _RESIDUAL_ statement or with a GROUP= option in the LSMEANS, but without any replication of the experimental unit, those two things may not converge.

In the end, your model is response = A B; The interaction serves as the error term.  To accommodate the Poisson, you could use either GLIMMIX or GENMOD.

SteveDenham

[ward77]

Steve, thank you for your reply. Don't I need to indicate that I have subsampling from the six EU (30 properties within each subplot)?

[jiltao]

Can you send us your data set? 

[ward77]

I added a simulated dataset; I don't have actual data yet.

[jiltao]

The subplot and the A*B interaction are confounded. So you either model the A*B interaction and do not model any random effects, or model the SUBPLOT as the random effect but no A*B interaction, as what you have already done. I assume the properties are nested within the subplot, correct? If so, I cannot think of other ways to model this data.

[ward77]

Thank you for your response. Yes, the properties are nested within the subplots and the interaction is confounded with subplot.  I wasn't sure if adding the subplot as a random effect was the appropriate thing to do, but I don't know how else I would specify the subsampling. Those measurements should be correlated because they were taken from the same subplot, not 30 replications of the experiment. I'm sure the A*B interaction would be of interest but would it be appropriate to estimate it without true replication? If I estimate the interaction instead of the random effect for subplot, would that be the same as treating those subsamples as true replicates?