# Random factor in linear mixed model

I need to model Reaction Time data (RT) for data collect with a classic lexical decision task. Participants have to decide whether an item is a word or not. Fixed fators are ITEMs' characteristics (such as FREQUENCY or emotional VALENCE). Each item is assessed by 60 participants.

First five data lines (only the first participant appers here):

SUBJ ITEM VALENCE FREQ RT
1 book 7.05 5,24 531
1 rope 4.67 3,70 630
1 plate 4.02 2,84 779
1 horse 2.30 0,68 958
1 pillow 7.32 0,43 893

I am not using categorical fixed factors (such as high vs low Frequency) but instead continuos factors (the numeric frequency associated to each item).

My question is: can I use ITEM as a random factor when each item has its own frequency level? When controlling for ITEM varability I am afraid being excluding the variability due to the fixed ffect FREQUENCY.

For instance, a very simple model (lme4): RT ~ FREQUENCY + VALENCE + (1|ITEM)

More generally, can we use ITEM as random factor when we use a between-ITEMs variable as a fixed factor? Does it make diffeernce when fixed fators are categorical or continuos variables?

Thanks!

• You seem to imply that you have one observation per item. Can you edit to confirm that and give us some more detail? Perhaps a snapshot of the first ten lines of your dataset might help too. Feb 3, 2021 at 17:14
• I have 60 subjects per item. Thank you
– l_f
Feb 3, 2021 at 17:27

I think that it's helpful to consider what you're telling your model is true. In the case of specifying Item as a random factor, you are essentially saying that the words used in the study are random selections from a larger body of possible words (which is probably true). You're also telling your model that, right now, each word has its own unique intercept (... + (1 | ITEM)) but that there is a some shared intercept for reaction time (not specified directly, but assuming you're using something like lme4, then you model implies the intercept unless specified otherwise). These, to me, seem like reasonable assumptions: there's nothing special about any single word outside of these word traits (i.e., frequency and valence), there is likely some general basal reaction time that people have (i.e., the fixed intercept), and each word does likely have a unique effect on that basal reaction time (i.e., the random intercept assigned to each word).
Now, the question beyond this is whether you're telling your model everything it needs to know. For example, you're observing multiple participants over repeated tests. I would hazard a guess that each participant has their own basal reaction time, so you probably need to add participants as a random effect as well (... + (1 | ITEM) + (1 | SUBJ)). You may then ask whether or not each person may have their own effect for frequency and valence (e.g., maybe some words are more frequent in a personal lexicon than in the general population, or maybe some people have special emotional associations to some words). If you have reason to believe this, then you may need to add random effects as well.
More directly to your stated question, you actually have to make ITEM have its own random effect in a model that also includes FREQ. Since each item has its own unique frequency, the model would become rank deficient if you included both variables together as fixed effects. The only way to avoid that issue would be to treat the words themselves are random selections from a population of possible words and treat specific word traits as having some fixed effect for predicting reaction time. In reality, your model is not dissimilar to an item response theory model: see here (IRT models in lme4), here (additional IRT in lme4), and here (response time IRT in R)
• Yes: the ITEM variable should be a random effect, though the exact reason for this should be theoretically informed. From a statistical perspective, to include FREQ and VALENCE, which have unique values for each individual word, you must have ITEM be a random variable or else the resulting model will be rank deficient. In my mind, the reason that ITEM is a justifiable random effect is that you've selected words that can be treated as random selections from some larger lexicon to which you want to generalize the effects of FREQ and VALENCE Feb 5, 2021 at 0:30