Interactions

Interactions with multilevel models can be tricky because they can be within or across levels, and the notation changes depending on whether the random effect for the lower level (in the interaction term) is specified as varying randomly within the given level. Luckily, {equatiomatic} handles all of this for you.

First, let’s look at a model with interactions only at level 1.

l1_hsb_int <- lmer(math ~ minority * female + (1 | sch.id),
  data = hsb
)
extract_eq(l1_hsb_int)

\[ \begin{aligned} \operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\ \mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{minority}) + \beta_{2}(\operatorname{female}) + \beta_{3}(\operatorname{female} \times \operatorname{minority}) \\ \alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right) \text{, for sch.id j = 1,} \dots \text{,J} \end{aligned} \]

And now an interaction at only level 2

l2_hsb_int <- lmer(math ~ meanses * sector + (1 | sch.id),
  data = hsb
)
extract_eq(l2_hsb_int)

\[ \begin{aligned} \operatorname{math}_{i} &\sim N \left(\alpha_{j[i]}, \sigma^2 \right) \\ \alpha_{j} &\sim N \left(\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{meanses}) + \gamma_{2}^{\alpha}(\operatorname{sector}) + \gamma_{3}^{\alpha}(\operatorname{meanses} \times \operatorname{sector}), \sigma^2_{\alpha_{j}} \right) \text{, for sch.id j = 1,} \dots \text{,J} \end{aligned} \]

But more complicated are cross level interactions. Here’s a quick example.

cl_long1 <- lmer(score ~ wave * treatment + (wave | sid) + (1 | school) +
  (1 | district),
data = sim_longitudinal
)
extract_eq(cl_long1)

\[ \begin{aligned} \operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i]}(\operatorname{wave}), \sigma^2 \right) \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{j} \\ &\beta_{1j} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) \\ &\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{treatment}_{\operatorname{1}}) \end{aligned} \end{array} \right) , \left( \begin{array}{cc} \sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\ \rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}} \end{array} \right) \right) \text{, for sid j = 1,} \dots \text{,J} \\ \alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right) \text{, for school k = 1,} \dots \text{,K} \\ \alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right) \text{, for district l = 1,} \dots \text{,L} \end{aligned} \]

Note that the treatement variable is shown as a predictor of the level 1 intercept and the level 1 slope. But if the slope is not randomly varying at this level, then the notation has to change.

cl_long2 <- lmer(score ~ wave * treatment + (1 | sid) + (1 | school) +
  (1 | district),
data = sim_longitudinal
)
#> boundary (singular) fit: see help('isSingular')
extract_eq(cl_long2)

\[ \begin{aligned} \operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1}(\operatorname{wave}), \sigma^2 \right) \\ \alpha_{j} &\sim N \left(\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) + \gamma_{2}^{\alpha}(\operatorname{treatment}_{\operatorname{1}} \times \operatorname{wave}), \sigma^2_{\alpha_{j}} \right) \text{, for sid j = 1,} \dots \text{,J} \\ \alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right) \text{, for school k = 1,} \dots \text{,K} \\ \alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right) \text{, for district l = 1,} \dots \text{,L} \end{aligned} \]

This works even for really complicated models, including three-way interactions that contain a cross-level interaction. For example

cl_long3 <- lmer(
  score ~ wave * group * treatment + wave * prop_low * treatment +
    (wave | sid) + (wave | school) +
    (wave + treatment | district),
  sim_longitudinal
)
#> boundary (singular) fit: see help('isSingular')
extract_eq(cl_long3)

\[ \begin{aligned} \operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{j} \\ &\beta_{1j} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{2}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{3l[i]}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) + \gamma_{4}^{\alpha}(\operatorname{group}_{\operatorname{low}} \times \operatorname{treatment}_{\operatorname{1}}) + \gamma_{5}^{\alpha}(\operatorname{group}_{\operatorname{medium}} \times \operatorname{treatment}_{\operatorname{1}}) \\ &\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{group}_{\operatorname{low}}) + \gamma^{\beta_{1}}_{2}(\operatorname{group}_{\operatorname{medium}}) + \gamma^{\beta_{1}}_{3}(\operatorname{treatment}_{\operatorname{1}}) + \gamma^{\beta_{1}}_{4}(\operatorname{group}_{\operatorname{low}} \times \operatorname{treatment}_{\operatorname{1}}) + \gamma^{\beta_{1}}_{5}(\operatorname{group}_{\operatorname{medium}} \times \operatorname{treatment}_{\operatorname{1}}) \end{aligned} \end{array} \right) , \left( \begin{array}{cc} \sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\ \rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}} \end{array} \right) \right) \text{, for sid j = 1,} \dots \text{,J} \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{k} \\ &\beta_{1k} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{prop\_low}) + \gamma_{2}^{\alpha}(\operatorname{prop\_low} \times \operatorname{treatment}_{\operatorname{1}}) \\ &\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{prop\_low}) + \gamma^{\beta_{1}}_{1}(\operatorname{prop\_low} \times \operatorname{treatment}_{\operatorname{1}}) \end{aligned} \end{array} \right) , \left( \begin{array}{cc} \sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\ \rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}} \end{array} \right) \right) \text{, for school k = 1,} \dots \text{,K} \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{l} \\ &\beta_{1l} \\ &\gamma_{3l} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &\mu_{\alpha_{l}} \\ &\mu_{\beta_{1l}} \\ &\mu_{\gamma_{3l}} \end{aligned} \end{array} \right) , \left( \begin{array}{ccc} \sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} & \rho_{\alpha_{l}\gamma_{3l}} \\ \rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}} & \rho_{\beta_{1l}\gamma_{3l}} \\ \rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\beta_{1l}} & \sigma^2_{\gamma_{3l}} \end{array} \right) \right) \text{, for district l = 1,} \dots \text{,L} \end{aligned} \]

Alternative variance-covariance specifications

Finally, there is some support for alternative variance-covariance specifications. For example, you may want to specify a model with only the variance terms estimated, and not the covariances.

hsb_varsonly <- lmer(math ~ minority * female + (minority * female || sch.id),
  data = hsb
)
#> boundary (singular) fit: see help('isSingular')
extract_eq(hsb_varsonly)

\[ \begin{aligned} \operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\ \mu &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{minority}) + \beta_{2j[i]}(\operatorname{female}) + \beta_{3j[i]}(\operatorname{female} \times \operatorname{minority}) \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{j} \\ &\beta_{1j} \\ &\beta_{2j} \\ &\beta_{3j} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &\mu_{\alpha_{j}} \\ &\mu_{\beta_{1j}} \\ &\mu_{\beta_{2j}} \\ &\mu_{\beta_{3j}} \end{aligned} \end{array} \right) , \left( \begin{array}{cccc} \sigma^2_{\alpha_{j}} & 0 & 0 & 0 \\ 0 & \sigma^2_{\beta_{1j}} & 0 & 0 \\ 0 & 0 & \sigma^2_{\beta_{2j}} & 0 \\ 0 & 0 & 0 & \sigma^2_{\beta_{3j}} \end{array} \right) \right) \text{, for sch.id j = 1,} \dots \text{,J} \end{aligned} \]

Or maybe you want to group by the same thing multiple times. In this exact model I don’t think this makes any sense, but there are cases where it can. Note that, again, this model produces a warning.

hsb_doublegroup <- lmer(math ~ minority * female +
  (minority | sch.id) + (female | sch.id),
data = hsb
)
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
#> Model failed to converge with max|grad| = 0.00319906 (tol = 0.002, component 1)
extract_eq(hsb_doublegroup)

\[ \begin{aligned} \operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\ \mu &=\alpha_{j[i],k[i]} + \beta_{1j[i]}(\operatorname{minority}) + \beta_{2k[i]}(\operatorname{female}) + \beta_{3}(\operatorname{female} \times \operatorname{minority}) \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{j} \\ &\beta_{1j} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &\mu_{\alpha_{j}} \\ &\mu_{\beta_{1j}} \end{aligned} \end{array} \right) , \left( \begin{array}{cc} \sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\ \rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}} \end{array} \right) \right) \text{, for sch.id j = 1,} \dots \text{,J} \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{k} \\ &\beta_{2k} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &\mu_{\alpha_{k}} \\ &\mu_{\beta_{2k}} \end{aligned} \end{array} \right) , \left( \begin{array}{cc} \sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{2k}} \\ \rho_{\beta_{2k}\alpha_{k}} & \sigma^2_{\beta_{2k}} \end{array} \right) \right) \text{, for sch.id.1 k = 1,} \dots \text{,K} \end{aligned} \]

And finally, you can have a mix of different things and it should generally still work.

long_mixed_ranef <- lmer(
  score ~ wave +
    (wave || sid) + (wave | school) + (1 | school) + (wave || district),
  sim_longitudinal
)
#> boundary (singular) fit: see help('isSingular')
extract_eq(long_mixed_ranef)

\[ \begin{aligned} \operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i],m[i]} + \beta_{1j[i],k[i],m[i]}(\operatorname{wave}), \sigma^2 \right) \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{j} \\ &\beta_{1j} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &\mu_{\alpha_{j}} \\ &\mu_{\beta_{1j}} \end{aligned} \end{array} \right) , \left( \begin{array}{cc} \sigma^2_{\alpha_{j}} & 0 \\ 0 & \sigma^2_{\beta_{1j}} \end{array} \right) \right) \text{, for sid j = 1,} \dots \text{,J} \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{k} \\ &\beta_{1k} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &\mu_{\alpha_{k}} \\ &\mu_{\beta_{1k}} \end{aligned} \end{array} \right) , \left( \begin{array}{cc} \sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\ \rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}} \end{array} \right) \right) \text{, for school k = 1,} \dots \text{,K} \\ \alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right) \text{, for school.1 l = 1,} \dots \text{,L} \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{m} \\ &\beta_{1m} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &\mu_{\alpha_{m}} \\ &\mu_{\beta_{1m}} \end{aligned} \end{array} \right) , \left( \begin{array}{cc} \sigma^2_{\alpha_{m}} & 0 \\ 0 & \sigma^2_{\beta_{1m}} \end{array} \right) \right) \text{, for district m = 1,} \dots \text{,M} \end{aligned} \]

With that said, this is the part of the code that I would consider the most “experimental” at this point, so please do reach out if you have issues or run into use cases that are not supported.

Using estimated coefficients

Because of the multilevel nature of the notation used by {equatiomatic}, tracking the estimated coefficients can at times be a little difficult. To help with this, we keep the Greek notation with the estimated coefficients as subscripts. For example, let’s extract the equation from our model with group-level predictors.

extract_eq(group_preds_m1, use_coef = TRUE)

\[ \begin{aligned} \operatorname{\widehat{score}}_{i} &\sim N \left(-20.6_{\alpha_{j[i],k[i],l[i]}} + 0.17_{\beta_{1j[i],k[i],l[i]}}(\operatorname{wave}), \sigma^2 \right) \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{j} \\ &\beta_{1j} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &-5.23_{\gamma_{1}^{\alpha}}(\operatorname{group}_{\operatorname{low}}) - 4.03_{\gamma_{2}^{\alpha}}(\operatorname{group}_{\operatorname{medium}}) - 1.97_{\gamma_{3}^{\alpha}}(\operatorname{treatment}_{\operatorname{1}}) \\ &0 \end{aligned} \end{array} \right) , \left( \begin{array}{cc} 9.29 & -0.2 \\ -0.2 & 0.29 \end{array} \right) \right) \text{, for sid j = 1,} \dots \text{,J} \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{k} \\ &\beta_{1k} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &12.81_{\gamma_{1}^{\alpha}}(\operatorname{prop\_low}) \\ &0 \end{aligned} \end{array} \right) , \left( \begin{array}{cc} 2.84 & 1 \\ 1 & 0 \end{array} \right) \right) \text{, for school k = 1,} \dots \text{,K} \\ \left( \begin{array}{c} \begin{aligned} &\alpha_{l} \\ &\beta_{1l} \end{aligned} \end{array} \right) &\sim N \left( \left( \begin{array}{c} \begin{aligned} &1.21_{\gamma_{1}^{\alpha}}(\operatorname{dist\_mean}) \\ &0 \end{aligned} \end{array} \right) , \left( \begin{array}{cc} 0 & -1 \\ -1 & 0 \end{array} \right) \right) \text{, for district l = 1,} \dots \text{,L} \end{aligned} \]

Future developments

There are still a few things I’m planning to implement that I haven’t quite gotten around to yet. These include

• Implementation of other features currently included in the lm extraction, including intercept, and raw_tex
• Inevitable bug fixes

Additionally, the Greek notation as subscripts may not always be desirable, particularly if the model is fairly simple. Future developments will likely make this optional.

The range of models that you can fit with lme4::lmer is huge, but I hope this will handle a wide range of models. I also recognize that some users may want a different notation for the models. At this point, I’m not planning to implement other notations, but that could change if there’s enough demand for it. I’m open to any/all suggestions for how to improve the package, and would love pull requests to help support the package, big or small.