For more on means models, see the first volume of Analysis of Messy Data by Milliken and Johnson. It is just a different way of parameterizing a linear model that is particularly useful for unbalanced datasets. Any of the SAS procedures that allow CLASS statements or implement dummy coding can be used.   As far as a definition for "moderator",  Mr. Google offers "In statistics, a moderator variable (or simply a moderator) is  a third variable that influences the relationship between two other variables ." In this case, it appears that weed cover is a moderator in the relationship between yield and treatment. It is a variable that you don't control in the design. It really works better for continuous variables, but it appears for categorical interactions as well.   SteveDenham ... View more

What happens when you fit an interaction as in a means model, rather than trying to fit an effects model?  Something like class trt weed_cover; model yield=trt*weed_cover/solution; /* RANDOM and REPEATED statements to reflect the study design */ /* LSMESTIMATE statements to get main effects and main effect differences, or ESTIMATE statements to accomplish the same ends */ Your PROC FREQ tables should identify any empty cells. If those are present, you may need to collapse the weed_cover categories to get something that may work. Thinking of weed cover as something other than a covariate (in the usual agricultural sense), such as a moderator could help.   SteveDenham ... View more

Without going into how to fit an RCBD, I will take a swing at the large standard error that gets reported for treatment A. It comes down to the assumption of homogeneous variances that underlies GLM. GLM uses the pooled root mean squared error to calculate the standard errors, which may not be appropriate in a design where you treat BLOCK as a fixed effect, and the number of observations per block is unequal. Work through the first and third examples for PROC GLM in the SAS documentation. The first is for the analysis of an RCBD, the third is for an unbalanced ANOVA for a two-way design.   SteveDenham ... View more

One nice feature of PROC POWER is its ability to give power or sample size estimates for a variety of values of the various parameters. In this case, you may want to look at several values for Rsq, to see how sensitive sample size (or power) is to this value. If it turns out that the result is not too sensitive to changes in Rsq, then you can use the standard formula that @Ksharp gave. If it is otherwise, then you need to look at other estimators for Rsq that reduce bias.   SteveDenham ... View more

Thanks @JackieJ_SAS !   The MATLAB website (https://www.mathworks.com/help/optim/ug/equation-solving-algorithms.html ) says:  The trust-region-dogleg algorithm is efficient because it requires only one linear solve per iteration (for the computation of the Gauss-Newton step). Additionally, the algorithm can be more robust than using the Gauss-Newton method with a line search.   There is also extensive discussion about trust region methods in general being able to find solutions (e.g, convergence to a minimum) even when the initial starting point isn't near the final solution.   SteveDenham ... View more

That is what I get from the GLIMMIX code I posted as well. The standard errors of the by-genotype residual estimates are at least an order of magnitude greater than the point estimates and the chi-squared test for homogeneity is nonsignificant (pr>chisq = 0.2591). The F test for genotype is also non-significant (pr > F = 0.1851). My conclusion is that there is insufficient data to come to any frequentist conclusion about yield as a function of genotype.   So I tried a quick look at the same design, but using the Bayesian approach provided in PROC BGLIMM. the 95% HPD intervals for the means of the genotypes all overlapped, as did the variance 95% HPD intervals. Same conclusions - insufficient data to come to a conclusion about yield as a function of genotype.  My code for this was:   proc bglimm data=yield plots=all seed=12321; class genotype rep; model yield = genotype/noint; random intercept/subject=rep(genotype) group=genotype; run; No ESTIMATE statements were used to look at pairwise comparisons as the overlapping intervals covered the whole of possible comparisons. Also, no need for multiple comparison adjustment with this approach.   ... View more

I would model the heterogeneous variances, at least for yield. The QQ plot is nearly straight, with only a bit of curvature at the low end. That looks to me to be acceptable. Just my opinion. For me, the main issue in this case is the nested nature of the random effects in this model. A question, are all genotypes present in each rep, so that this is an RCB design (3 plots with each having all 7 genotypes present). If that is the case, consider this heterogeneous variance approach:   title "Consider REP as a block, model heteroscedasticity due to genotype at REP level"; proc glimmix data=yield plots=residualpanel noprofile; ods exclude diffplot linesplot; nloptions maxiter=1000; *lnyield=log(yield); /* in case you still want to try a log transformation */ class genotype rep; model yield /*lnyield*/ = genotype; random intercept/subject=rep group=genotype; lsmeans genotype/adj=simulate(seed=111) diff adjdfe=row; covtest homogeneity; ods output diffs=ppp lsmeans=mmm; run; SteveDenham     ... View more

@wateas - Have you done any sensitivity checking for the PROC MIXED runs? What kind of results do you see if you use a grid search in a PARMS statement? Does it look like the convergence is insensitive to starting values, such that the final -2 log likelihood is the same no matter where you start. I respect wanting to  reflect the experimental design in the analysis - I also believe in hierarchical interaction inclusion (e.g. a two way interaction requires the main effects are included in the model, a three-way requires the main effects and the 3 two-way interactions. etc.). In any case, the comments by @jiltao and @JackieJ_SAS both have some approaches that might help.   /* the following was added later, and I don't know if it would help with convergence issues */   One more idea - what about using a spline for the repeated measures variance-covariance structure to handle the irregular spacing? See Example 51.6 (SAS/STAT 15.2 documentation) Radial Smoothing of Repeated Measures Data, which looks at body weights of cows taken at unequally spaced time points. Time is treated as a continuous variable in this case. What I don't see in this example is a way to compare LSMEANS at various time points, but that could possibly be handled with an AT= option in LSMEANS or LSMESTIMATE statements.    SteveDenham ... View more

Hi @JackieJ_SAS ,  inquiring minds want to know where they could find the results to that simulation, please and thank you.🤔   SteveDenham ... View more

I know this has been answered, but wouldn't a PROC MEANS (or SUMMARY) give all of the levels needed for this? It would then just be a sorting task to get something to print out. I have to admit the PROC REPORT approach more quickly yields a more esthetically pleasing output, but if the dataset is really large, with a lot of nesting, and you need to use the results for additional work, it might be worth taking a look.   SteveDenham ... View more

What does running a proc freq of this sort tell you:   proc freq data= tables harvest*wks*variety*PercStemEndRot/cmh; run; The CMH option should give you a test for association of variety with PercStemEndRot, after adjusting for harvest and wks. In addition, it should let you know where the zeroes are in your data. Consolidating categories is probably the best way to handle this.    SteveDenham (I can't believe I am not offering some sort of exact approach to a generalized linear model, but I think this has two advantages - you will know where the zeroes are, and I believe you will still get some useful inferential information).   ... View more

Just for fun, consider a modeling approach that doesn't assume homogeneity of variance. Working from your GLIMMIX code, try something like:   /* Changes are in red */ proc glimmix data=one; where Season=2021; PercDMp=PercDM/100; class Harvest Variety; model PercDMp=Harvest*Variety/ dist=beta ddfm=kr2; random _residual_ / group=Variety; lsmeans Harvest*Variety/slicediff=Harvest adjust=simulate(seed=1); covtest 'common variance' homogeneity; run; If it turns out that the likelihood ratio test for variance homogeneity for Variety is not significant, try it again grouping by Harvest. I really don't know if either will affect your conclusions, but at least you have dealt with a common assumption (homogeneity) that may not be true for your data.    The other thing to look at is to try a different method than the default RSPL. Consider method=laplace, so that the error variance component is included in the optimization. That may yield standard errors that are more in line with expectations.    SteveDenham   ... View more

Look up leaf area index (LAI) and see what methods have been used for analyzing that endpoint. LAI is the proportion (or percentage/100) of the area of a given plot or transect that is covered by at least one leaf when viewed perpendicular to the ground. It is defined on the interval (0,1), bounded away from zero and one. When I last looked at the analyses that various folks used, there were a lot of options. Some have been mentioned here (beta regression, fractional logistic regression), but I am going to throw my support to some sort of resampling with replacement. Judging from the histogram you have a lot of observations, so taking samples of a relative size of (for example) total plots/20 and generating 5000 samples should not be difficult. From that, you can appeal to the central limit theorem to get means and confidence intervals. This might be more appropriate for your long right tail and non-unimodal data, which really looks like a mixture of two distributions to me.   No guarantees, no warranty implied.   SteveDenham ... View more