[R-sig-ME] by-item random intercepts

Chunyun Ma mcypsy at gmail.com
Thu Sep 14 04:02:50 CEST 2017

Hello all,

I am facing a dilemma of whether or not I should include by-item random
intercepts in my model. Here are the details of my problem.

I have a dataset of repeated measure in which participants solved
single-digit arithmetic problems (e.g., 4x5, 2+7, ) and their response
latencies were recorded.

The dependent variable is response latency. The independent variables
include characteristics of the stimuli (i.e., level 1) and of the
participants (i.e., level 2).

I set up the structure of random effects following recommendations from
Barr et al. (2013). For simplicity, let's say the model contains one IV.

DVti = gamma00 + gamma10IVti + u0i + u1iIVti + I0i + rti

gamma00, gamma10 are fixed effects
u0i is the random intercept
u1j is the random slope
I0i is the by-item random intercept
rti is the residual

I used lme4 to test the model
lmer(DV ~ IV + (1 + IV|sub) + (1|item), data= DT)

As I mentioned, the stimuli in my experiments are single-digit arithmetic
problems. Unlike stimuli such as English words, there are only 100
single-digit arithmetic problems for each operation and all of them were
included in my experiment. So here is my dilemma:

On one hand, a random by-item intercept would allow me to account for the
fact that there are repeated observations on each item and they are not
independent from each other.
On the other, a random by-item intercept implies there exists more items
which were not included in my experiment. However, this is not the case. I
have included all single-digit arithmetic problems in my experiment.

I could adopt a fixed-effect approach and use 100 dummy variables to
account for the item-based clustering but this would be practically

To iterate my question:
should I include a random by-item intercept given the special feature of my
A few follow-up questions:
what's the consequence of including/excluding this random effect?  How are
type-I error and power affected?
Should I use a nested structure instead of the crossed one I have mentioned
above? For example, if each participant contributed multiple observations
on each item, should I nest the by-item random intercept under subject?

Thank you very much!


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