I am not sure I understand your reply... in any group we have only 2 replicates. Attached is the data sheet showing the experimental design.

Appreciate your input very much!

Elena

Hi,

I would simply use linear model (below is given an R code) where the explanatory variables are the "treatment" and "sub.treatment" as factors, plus their interaction, meaning that you will estimate both, the independent effects of both factors (because the sub.treatments 1 and 2 always mean the same); as well as whether the sub.treatment had a different effect in different treatments:

lm(formula = response ~ as.factor(treatment) * as.factor(sub.treatment),

data = data)

The summary (using the data from attached table) reads:

treatment "A" and sub.treatment "1" are taken as the base that others are compared against; so the Intercept is actually the mean level of Treatment A and sub.treatment 1, and the treatment D has a significant effect compared to A; there's no significant overall effect of sub.treatment 2, but it does have strong effect in treatments B and C, especially in C.

Call:

lm(formula = response ~ as.factor(treatment) * as.factor(sub.treatment),

data = data)

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 5.5000 0.9487 5.798 0.000174 ***

as.factor(treatment)B -2.0000 1.3416 -1.491 0.166890

as.factor(treatment)C 0.5000 1.3416 0.373 0.717165

as.factor(treatment)D -4.0000 1.3416 -2.981 0.013774 *

as.factor(treatment)E -2.5000 1.3416 -1.863 0.092002 .

as.factor(sub.treatment)2 2.0000 1.3416 1.491 0.166890

as.factor(treatment)B:as.factor(sub.treatment)2 -4.0000 1.8974 -2.108 0.061229 .

as.factor(treatment)C:as.factor(sub.treatment)2 -10.5000 1.8974 -5.534 0.000250 ***

as.factor(treatment)D:as.factor(sub.treatment)2 2.0000 1.8974 1.054 0.316643

as.factor(treatment)E:as.factor(sub.treatment)2 2.5000 1.8974 1.318 0.217020

Residual standard error: 1.342 on 10 degrees of freedom

Multiple R-squared: 0.9081, Adjusted R-squared: 0.8253

F-statistic: 10.98 on 9 and 10 DF, p-value: 0.0004249

Skipping the interaction would make the B and C significantly different from A, but from above you know it's because of the strong effect of sub.treatment 2 on those treatments; and the R-squared as well as AIC values of both models suggest using the interaction as well as it significantly improves the fit to the data.

Call:

lm(formula = response ~ as.factor(treatment) + as.factor(sub.treatment),

data = data)

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 6.500e+00 1.698e+00 3.829 0.00184 **

as.factor(treatment)B -4.000e+00 2.192e+00 -1.825 0.08940 .

as.factor(treatment)C -4.750e+00 2.192e+00 -2.167 0.04795 *

as.factor(treatment)D -3.000e+00 2.192e+00 -1.369 0.19263

as.factor(treatment)E -1.250e+00 2.192e+00 -0.570 0.57749

as.factor(sub.treatment)2 -5.027e-16 1.386e+00 0.000 1.00000

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.1 on 14 degrees of freedom

Multiple R-squared: 0.3131, Adjusted R-squared: 0.06774

F-statistic: 1.276 on 5 and 14 DF, p-value: 0.328

Riina Klais

Thank you, Riina. GLM was my first move too...but I don't think this is the way to go (and frankly I don't think there is any sensible way to go with this dataset). The interaction is important and having a significant interaction means that sub-treatment effect should be estimates for each treatment separately, which in turn means that 2 replicates from one treatment are compared with 2 replicates of the other treatment... this is what it all boils down to:(

The most interesting is the sub-treatment effect. The data are different bacteria strains (A, B, C...) performing a specific function in different media (1 and 2). The point is to select an optimal medium.

Dear Elena

You may use GLM but with other model.

If I understand you correctly you have to take care of the different errors for the treament and for the sub-treatment and interactions that are nested. I read your data and I assumed that the first two rows correspond to the treatment A and subtreatment 1 in units 1 and 2, and that the next two rows correspond to treatment A and subtreatment 2 in the same units 1 and 2, then I did the ANOVA for split plots that may be what you are looking for.

The plot (or unit) is where you assign the treatment, each plot (there were two for treatment) may be divided in a 2 subplots where you assigned at random the subtreatments.

I adjunt the results in an Excel table and the SAS Program that I used.

By the way, you have to verify if the assumptions behind the ANOVA are met before considering the results.

If the assumptions are verified, you may see that there is interaction between the treatment and the subtreatment. You may repeat the analysis modifying a little the model to consider the subtreatment (treatment). I think that which is the best media depends on the treatment considered.

Here I adjoint the SAS program.

Hi Guillermo, and thank you! Does this mean that units 1 and 2 are consistent amont the sub-treatments? The split-plot design would imply so, I think. In my case, they are just replicate wells on the microplate and there is no specific order in time or space for them. To my understanding, the algorithm would compare all units named 1 with all units named 2, pretty much in the same way as it does with the subtreatments. Am I wrong?

Dear Elena.

There are not a test for the units. I assumed that you have two plots (or units) for each treatment and that within each unit you assigned the two subtreatments. Then the subtreatments and the interaction is compared to the usual MSE but the MSTreatment is compared to the MSUnit(Treatment). Did I understand OK your problem?