If you have udder measurements daily then I assume you have a time varying covariate. This is difficult to handle as a repeated measures ANOVA, but you can use a more general, flexible model (of which repeated measures ANOVA can be considered a special case). I would suggest using a multilevel model (also called linear mixed models or hierarchical linear models).

In R you can fit these using packages such as nlme and lme4. For example in lme4 the model be something like this:

lmer(yield ~ parity + udder.trait1 + (1|ewe_id), data=your.data.frame)

One make the model more-or-less complex to deal with interactions and complex random structures:

lmer(yield ~ parity + udder.trait1 + (parity||ewe_id), data=your.data.frame)

lmer(yield ~ parity * udder.trait1 + (parity||ewe_id), data=your.data.frame)

... and so on.

Hello Samuel, you can see it as a factorial design with repeated measures, the statistical model could be the following:

y_{ijkl} = \mu + \alpha_i + \beta_j + d_{ijl} + \gamma_k (\alpha\beta)_{ij} + (\alpha\gamma)_{ik} + (\beta\gamma)_{jk} + (\alpha\beta\gamma)_{ijk}+\epsilon_{ijkl}

where \mu is the general mean, \alpha_i is the effect of the i-th treatment (parity), \beta_j is the effect of the j-th treatment (udder), d_{ijl} is the random experimental error for the experimental units (the ewes) within treatments (parity and udder) with variance \sigma^2_d, \gamma_k is the effect of the k-th day, (\alpha\beta)_{ij} is the interaction between the parity and udder, (\alpha\gamma)_{ik} is the interaction between parity and day, (\beta\gamma)_{jk} is the interaction between udder and day, and (\alpha\beta\gamma)_{ijk} is the interaction between parity, udder and day, \epsilon_{ijkl} is the normally distributed random experimental error on repeated measures with variance \sigma^2_e.

To carry out the analysis with R, use the ez package,

library(ez)

# then use the following function:

model = ezANOVA(

data = csvfile

, dv = .(milk_yield)

, wid = .(id_experimental_unit)

, within = .(day)

, between = .(parity,udder)

, type = 3

)

print(model)

I don't have a perfect example, but the following links may help you walk through the steps using R.

They use the nmle package, from which you are able to obtain p-values, and then mean separation tests.

You'll have to change the models, obviously.  Note that the model notation for nmle is slightly different than for lmer indicated in Thom Baguley's comment above.

http://rcompanion.org/rcompanion/d_08.html

Search for "Using mixed effects model"

http://rcompanion.org/rcompanion/d_07.html

Search for "Using a mixed effects model for a nested anova"

Note that ez can also run the multilevel model (though it does so by calling lme4).

Thanks Thom, Gomez and Mangiafico for your useful and informative guidance and suggestions.I am totally new in the use of R for data analysis!

Considering the models listed (Thom and Mangiafico), could you kindly guide me further on how to present the data table to fit into the model for analysis.

Regards

It is often not easy getting started with R.  It will be worth the work in the end, but it will take some work, particularly if you are trying to start with a mixed model.

The first step is to get the data into a data frame in R.  You can import a .csv file, or use space-separated values.  See the link below for some simple examples, at the sections "Creating data frames from a text string of data" and "Reading data from a file".

     http://rcompanion.org/rcompanion/a_06.html

You want the data to be in "long format".  That is, you want one observation on each line.

After that, I would recommend working through some some statistics and analyses and plots to get used to R.  I recommend working through some analyses that you are familiar with, in my book, or in another source:

     http://rcompanion.org/rcompanion/

Then you need to decide which approach you want to take for the mixed model: ezANOVA, lmer, or nmle (Omar, Thom, or Sal above).   

For any approach, find examples on the internet you can follow.  And be sure to visit the link below, look up the package you will using for information and examples of how to use the package. (ez, nlme, or lme4)

     https://cran.r-project.org/web/packages/available_packages_by_name.html

I know that's not the most complete answer, but really it depends on what kind of output you are expecting, and which package will give you that output.

Hi Mangiafico, I am very grateful for your answers and suggestions. Indeed, the suggested links have been very useful. Thanks