# Mixed models formulation (GAMM, GLS, nlme): from R to mathematical notation

I have just finalised the analysis of a dataset fitting Generalised additive mixed models (GAMMs) with mgcv package in R v3.3.0. But I am having trouble in writing my models into their correct mathematical notations.
I am not an expert in this filed but despite that, I have “understood” how this model works and how to choose which would give the best fit. However, now I am writing a paper and I have to report the models with mathematical notation but I am not sure if I am doing it correctly.

To give you an overview of my design, I am measuring the shape variability of some individuals that has been quantified into a continuous variable (SV my response variable). For each individual I got 5 SV observations from different body parts = PC (categorical) and I got repeated individuals (30) from each of 20 samples sites = SITE (categorical). So what I have are PC and SITE as “completely crossed factors” (just to give you an idea, I know that this definition applies to random effects).
What I wanted to analyse was the variability (SV) for each of the measured body parts (PC) with temperature = TEMP and salinity=SAL (continuous response variables for which I have just ONE value for each of the 20 sampled SITEs) and length = LENGTH (continuous, 1 observation per individual). (only the individual interaction terms are of interest to me)

To sum up the varaibles involved are:

• SV (response)
• PC (indipendant categorical, 5 levels, fixed effect)
• SITE (ind. categorical, 20 levels, random effect)
• TEMP (ind. continous, ONE value for each SITE) non-linear
• SAL (ind. continous, ONE value for each SITE) linear
• LENGTH (ind. continous, one value per observation) linear

This is the model with R (I included just the core of the model):

gamm(SV ~ s(TEMP, by = PC) + SAL*PC + Length*PC,
random = list(SITE= ~ 1),
method = "REML", data = xxx)


The mathematical notation should be:

$$SV_{ijk}\ =\ \alpha_j\ +\ f_j(TEMP_i)\ +\ \beta_{1j} SAL_i\ +\ \beta_{2j} Length_{ik}\ +\ Site_i\ +\ \epsilon_{ijk}$$

SV is the observation from $i^{th}$ =1...20 Site, $j^{th}$=1...5 PC, $k^{th}$ individual, and I fit a smoother for each PC;
in a more "extended" form showing interaction terms

$$SV_{ijk}\ =\ \alpha\ +\ f(TEMP_i)\ \times PC_j\ +\ \beta_1 SAL_i\ +\ \beta_2 Length_{ik}\ +\ \beta_3 PC_j\\ +\ \beta_4 SAL_{i} \times PC_j\ +\ \beta_5 Length_{ik} \times PC_j\ +\ Site_i\ +\ \epsilon_{ijk}$$

$Site_i\ \sim\ N(0,\sigma^2_{Site})$

$\epsilon_{ijk}\ \sim\ N(0,\sigma^2)$

Are the above mathematical notations correct? Specifically what I don't get is:

• Is it correct stating $TEMP_i$ if I have just one measure of TEMP per Site, or should it be $TEMP_{ik}$?
• as a consequence is $Length_{ik}$ correct since I have a measure per individual per site?
• are the two ways I specify the interaction correct?

Thank you so much in advance for any help and andvise! (and let me know if it is not clear or more info are needed!)