2 improved formatting
source | link

Say one wishes - and believes - to fit a mixed effects model where time ($x_{ij}$ below) is included in a linear and quadratic term in the fixed part, but only linear in the random part (hence random intercept and slope).

$$ y_{ij} = \beta_1 + \beta_2 x_{ij} + \beta_3 x_{ij}^2 + \zeta_{1j} + \zeta_{2j}x_{ij} + \epsilon_{ij} $$

Question

Is the model mis-specified? Is it the case that the quadratic term has to be included also in the random part? 

References

Looking a bit around, looks like there are mixed views:

Against: Link 1

In favor: Link 1 and also Link 3

(I'm sure there will be much more for one or the other)

Say one wishes - and believes - to fit a mixed effects model where time ($x_{ij}$ below) is included in a linear and quadratic term in the fixed part, but only linear in the random part (hence random intercept and slope).

$$ y_{ij} = \beta_1 + \beta_2 x_{ij} + \beta_3 x_{ij}^2 + \zeta_{1j} + \zeta_{2j}x_{ij} + \epsilon_{ij} $$

Is the model mis-specified? Is it the case that the quadratic term has to be included also in the random part?

References

Looking a bit around, looks like there are mixed views:

Against: Link 1

In favor: Link 1 and also Link 3

(I'm sure there will be much more for one or the other)

Say one wishes - and believes - to fit a mixed effects model where time ($x_{ij}$ below) is included in a linear and quadratic term in the fixed part, but only linear in the random part (hence random intercept and slope).

$$ y_{ij} = \beta_1 + \beta_2 x_{ij} + \beta_3 x_{ij}^2 + \zeta_{1j} + \zeta_{2j}x_{ij} + \epsilon_{ij} $$

Question

Is the model mis-specified? Is it the case that the quadratic term has to be included also in the random part? 

References

Looking a bit around, looks like there are mixed views:

Against: Link 1

In favor: Link 1 and also Link 3

(I'm sure there will be much more for one or the other)

1
source | link

Mixed Effects model with quadratic term in fixed, but not in random part

Say one wishes - and believes - to fit a mixed effects model where time ($x_{ij}$ below) is included in a linear and quadratic term in the fixed part, but only linear in the random part (hence random intercept and slope).

$$ y_{ij} = \beta_1 + \beta_2 x_{ij} + \beta_3 x_{ij}^2 + \zeta_{1j} + \zeta_{2j}x_{ij} + \epsilon_{ij} $$

Is the model mis-specified? Is it the case that the quadratic term has to be included also in the random part?

References

Looking a bit around, looks like there are mixed views:

Against: Link 1

In favor: Link 1 and also Link 3

(I'm sure there will be much more for one or the other)