Imph
☆

Algeria,
2022-10-25 11:47
(103 d 17:38 ago)

Posting: # 23349
Views: 1,077

## Intercept in mixed effects model [Software]

Hello,

To assess average bioequivalence based on a mixed effect model in Phoenix WinNonlin, do we have to include the intercept term for the fixed effects and the random intercept for the random effects? does the analysis change with the inclusion or non-inclusion of the intercept term.

Best regards.
Helmut
★★★

Vienna, Austria,
2022-10-25 13:09
(103 d 16:17 ago)

@ Imph
Posting: # 23350
Views: 887

## Intercept in mixed effects model

Hi Imph,

in any linear model – independent from the software – always include an intercept.

Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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ElMaestro
★★★

Denmark,
2022-10-31 18:46
(97 d 09:39 ago)

@ Imph
Posting: # 23352
Views: 860

## Intercept in mixed effects model

Hi Imph,

❝ To assess average bioequivalence based on a mixed effect model in Phoenix WinNonlin, do we have to include the intercept term for the fixed effects and the random intercept for the random effects? does the analysis change with the inclusion or non-inclusion of the intercept term.

The choice is yours, really. The very few textbooks or papers that I know of, which specify a
model for BE (i.e. with an equation, not with verbal mention of effects) seem to give it with an intercept term for fixed effects, but even that can be debated depending on your purpose.
If you need to extract a difference in treatment effects (and in BE you always do), then it is much more straightforward to simply fit the model without intercept and keep treatment as the first effect. You get what you want right off the bat from the first two elements of the model effects vector. It is not necessary to understand all aspects of regressors and contrast coding to make easy use of it.
The choice does not affect the model residual, so the 90% CI is the same. Or you can take the long way around the problem and extract LSMeans (e.g. package emmeans in R). This option is fine, too, and it works regardless of whether the model was fit with an intercept or not. What goes on behind the curtains is complicated, to say the least, and I shall not claim I understand it too well. Quite comforting to me was when another user on this forum tried to work out LSMeans "by hand" and got something unexpected. I didn't feel so hopeless after reading that post.
Intercepts on the random effects are a default trait with some software packages. Random terms with an intercept term is a cosmic mindf#cker, simply because I am used to directly thinking of a covariance matrix with only variance terms. Easier in my head for me.
But it is fully valid when specified with the intercept, too. It does not provide better convergence properties, so perhaps habits, taste and software defaults prevail here.

Pass or fail!
ElMaestro
PharmCat
★

Russia,
2022-11-03 16:02
(94 d 12:24 ago)

(edited by on 2022-11-03 16:34)
@ Imph
Posting: # 23356
Views: 798

## Intercept in mixed effects model

Hello!

As Helmut said - the intercept term is practically always included in the model.

upd: HaHah)) I was mistaken... corrected interpretation is:

For random effect - you can include intercept, but what will it mean?

For example, if you have repeated effect model with factor 'trt' in sequence AABB for each subject and use random effect model ~ trt (correctly: ~ 0 + trt) with covariance structure "DIAG" you will get Z1 matrix (only 2 obs for subject, but you can make 4):

A B 1 0 0 1 ...

With intercept your random effect model (~ 0 + trt) you will get Z2 matrix:

I B 1 0 1 1 ...

G matrix same:

a  b σ₁ 0 0  σ₂

and
σ₁ = 0.2 σ₂ = 0.3 V' = ZGZ'

So in the first case (no intercept), you will get variance estimate vector [a, b] where a - variance for A, b - variance for B.

Z1*G*Z1' =  0.2  0.0  0.0  0.3

So in the second case (with intercept), you will get variance estimate vector [a, b] where a - variance for intercept term (actually base level A), b - a "difference" of variance between A and B.

Z2*G*Z2' =  0.2  0.2  0.2  0.5

upd
ElMaestro
★★★

Denmark,
2022-11-10 01:34
(88 d 02:51 ago)

@ PharmCat
Posting: # 23359
Views: 604

## More obvious to discuss the model.... for me...

Hi all,

The mixed model is for our purposes
Y= Xb + Zu + e

Z is the design matrix for the random effects expressed in u, typically the betweens.
Now, in BE if we decide to fit the model with an intercept in Z then obviously we will get a column of ones followed by a column with indices for either treatment A or treatment B, but not both. Under this model specification we are thus estimating the intercept (whose interpretation is not so straightforward if you ask me, but see Pharm Cat's example above) along with the betweens for A or B (implementation-dependent), but not both.
One the other hand, if we want to have separate variance estimates for A and B extracted directly from u once optimised, then we would make sure that the random effects are fit without intercept, implying Z without a column of ones, and Bob's your uncle.

I do not see any good arguments for fitting random effects with an intercept column in Z, but I do see arguments for doing it without. If the software takes care of everything from optimisation to generation of a CI behind the curtains, then there is no preference and software defaults will suffice whatever they are since the CI will be the same.

Pass or fail!
ElMaestro
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