How to set up my partial cross classification model in lme4 [closed]

I have a dataset with 300 individuals $$i$$ that provided ratings on objects $$o$$ that are $$y_{io}$$. Each individual rated a random sample of 3 objects out of 20 possible objects so that I have 900 observations. I think this would be called a partial cross-classified mixed model.

My basic equation is $$y_{io} = 1 + u_i + v_o + \varepsilon_{io}$$ I have a fixed intercept, a random intercept for individuals and a random intercept for objects.

I understand that the lme4 command for this would be:

fit <- lmer(y ~ 1 + (1 | i) + (1 | o), data = dta)

However, I also have some individual specific variables $$X_i$$ and some object specific variables $$Z_o$$. In addition, I have some additional variables from each individual about each rated object $$W_{io}$$.

First, I want to explain the random intercepts $$u_i$$ and $$v_o$$ with my individual and object specific variables:

$$u_i = 1 + \beta X_i + \phi_i$$ $$v_o = 1 + \gamma Z_o + \zeta_o$$

How do I set this up in lme4? My intuition would be to simply add them as fixed effects. Then the $$\phi_i$$ and $$\zeta_o$$ would replace the $$u_i$$ and $$v_o$$. Is that correct?

fit <- lmer(y ~ 1 + X + Z + (1 | i) + (1 | o), data = dta)

Or do I have to add them within the brackets?

How would I assess the explained variance in the second-level equations? Is that the reduction in variance of the random effect due to the inclusion of the predictors?

And how do I deal with the $$W_{io}$$. What would be a correct place for them conceptually and within the lme4 framework?

closed as off-topic by mkt, Michael Chernick, user158565, Peter Flom♦Aug 9 at 11:11

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and you will get the estimates for their effects. In particular, the model that you will fit is $$y_{io} = \beta_0 + \beta_1 X_i + \beta_2 Z_o + \beta_3 W_{io} + u_i + v_o+ \varepsilon_{io},$$ with $$u_i \sim \mathcal N(0, \sigma_u^2)$$, $$v_o \sim \mathcal N(0, \sigma_v^2)$$, and $$\varepsilon_{io} \sim \mathcal N(0, \sigma^2)$$.
The change is the variance components $$\sigma_u^2$$, $$\sigma_v^2$$ and $$\sigma^2$$ when you include the covariates $$X_i$$, $$Z_o$$ and $$W_{io}$$ will indicate the variance explained by these predictors.