[R-sig-ME] LMM reduction following marginality taking out "item" before "subject:item" grouping factor

Jake Westfall j@ke@@@we@tf@ll @ending from gm@il@com
Thu Nov 29 15:54:18 CET 2018


Maarten,

Just to double-check if I get this right: the entries in each cell of the
> table are the numbers by which the variance components are divided in the
> equation of the noncentrality parameter. Is this correct?


Almost. They multiply the variance components, not divide them. Essentially
each row gives the weights of a weighted sum of variance components. Then
to translate that to what appears in the denominator of the noncentrality
parameter, the entire thing is divided by the total sample size *and we
remove the variance component for the effect in question *(I forgot to
mention that part in my last email).

For example, consider the simple design with random participants (P) nested
in fixed groups (G). So g is the number of groups, p is the number of
participants per group, and # is the number of replicates. (This is design
2 in the dropdown menu of examples.) The EMS table shows that, for the
between-group effect, the coefficients for the error, participant, and
group variance components are, respectively, 1, #, and #p. So the expected
mean square is var_error + # * var_participants + # * p * var_groups. The
total sample size is pg#, so in the noncentrality parameter expression this
becomes sqrt(var_error / pg# + var_participants / pg). Note that this only
gives most of the denominator of the of the noncentrality parameter
expression -- it ignores the variance of the contrast weights -- you can
see more in the PANGEA working paper, linked in the app.

Jake

On Thu, Nov 29, 2018 at 7:36 AM Maarten Jung <
Maarten.Jung using mailbox.tu-dresden.de> wrote:

> Hi Jake,
>
> So, regarding this issue, there is no difference between taking out
>>> variance components for main effects before interactions within the same
>>> grouping factor, e.g. reducing (1 + A*B | subject) to (1 + A:B | subject),
>>> and taking out the whole grouping factor "item" (i.e. all variance
>>> components of it) before "subject:item"?
>>
>>
>> I think that if you have strong evidence that this is the appropriate
>> random effects structure, then it makes sense to modify your model
>> accordingly, yes.
>>
>
>  This makes sense to me.
>
> Do all variances of the random slopes (for interactions and main effects)
>>> of a single grouping factor contribute to the standard errors of the fixed
>>> main effects and interactions in the same way?
>>
>>
>> No -- in general, with unbalanced datasets and continuous predictors,
>> it's hard to say much for sure other than "no." But it can be informative
>> to think of simpler, approximately balanced ANOVA-like designs where it's
>> much easier to say much more about which variance components enter which
>> standard errors and how.
>>
>> The standard error for a particular fixed effect is proportional to the
>> (square root of the) corresponding mean square divided by the total sample
>> size, that is, by the product of all the factor sample sizes. So examining
>> the mean square for an effect will tell you which variance components enter
>> its standard error and which sample sizes they are divided by in the
>> expression.
>>
>
> Your app is very useful, too. Just to double-check if I get this right:
> the entries in each cell of the table are the numbers by which the variance
> components are divided in the equation of the noncentrality parameter. Is
> this correct?
>
>
> Regards,
> Maarten
>

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