# Test for yield and disease incidence in block design

I need to analyse a set of data coming from an agricultural experiment, with the intention of assessing the incidence of a certain disease in potatoes under three different treatments (plus blank). I'm using R. The experiment was set as follows:

• Four blocks, each with four land parcels

• In each block the treatments were assigned randomly

I have the data for each parcel, consisting of:

• number of healthy potatoes

• number of potatoes with disease

The number of potatoes in each parcel (yield) varies roughly between 130 and 200. I would like to know which tests I need to use for difference between treatments for:

• incidence (diseased/total proportion)

• yield (total)

I was thinking about ANOVA, but if so, how many degrees of freedom there would be? I'm confused because in principle the degrees would be too small, but at the same type I'm looking at many specimens. Or do I need to use another test?

Here's the output of some commands I've tried:

> summary(a<-aov(incidence~treatment,data=df))
Df Sum Sq Mean Sq F value Pr(>F)
treatment    3 0.0992 0.03306   0.837  0.499
Residuals   12 0.4739 0.03949
> summary(a<-aov(incidence~treatment+repetition,data=df))
Df Sum Sq Mean Sq F value Pr(>F)
treatment    3 0.0992 0.03306   0.656  0.599
repetition   3 0.0203 0.00677   0.134  0.937
Residuals    9 0.4536 0.05040

• Assuming you have normal data, the model is $Y_{ij} = \mu + \tau_i + \beta_j + e_{ij},$ for $i = 1, 2, 3, 4 = a$ treatments and $j = 1,2,3,4=b$ blocks, where $e_{ij}$ are independent errors from $\mathsf{Norm}(\mu=0,\sigma).$ (With one replication in each treatment-by-block cell, interaction is not supported.) Your second output looks right; it has DF(Treat) $= a-1 - 3$ and DF(Block) $=b-1=3$ and 9 DF for error (residuals). You should check residuals for normality. (It looks as if neither treatment nor block effect is significant.) – BruceET Feb 24 at 8:27