[R-sig-ME] non-linear slope in random regression model

[David Villegas Ríos]

Hello,
I'm fitting some random regression models to investigate variation over
time of a response trait.
The time variable is "month", and by fitting a random regression model I
want to investigate variation in plasticity across individuals, i.e.,
differences in the "slope" between trait and time across individuals.

The relationship between the response trait and month is non-linear.
Basically, it describes a seasonal cycle.

I have considered two model candidates:

*Model 1*: fitting a polynomial of month in the fixed effects part to
describe the non-linearity and get the population mean effect of month, and
then a random slope using again the polynomial for month, to ge the
individual differences.

trait~poly(month,3),random=~poly(month,3)|ID


*Model 2*: fitting a polynomial of month in the fixed effect part to
describe the non-linearity, and then the individual-specific deviation from
the fixed-effect means is modelled as a funtion of month (linear), assuming
that the non-linearity is already accounted for with the fixed effect.

trait~poly(month,3),random=~month|ID

*My question is*:
Is it neccesary to include the non-linearity in the random part if it was
already included in the fixed-effects part?

The idea of fitting model 2 comes from the following reference (page 488):

Dingemanse, N. J., Barber, I., Wright, J., & Brommer, J. E. (2012).
Quantitative genetics of behavioural reaction norms: genetic correlations
between personality and behavioural plasticity vary across stickleback
populations.*Journal of evolutionary biology*, *25*(3), 485-496.


Thank you.

David

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