Mixed Effects Model: Writing out and interpreting coefficients on Level 1, 2, and 3 Models

Question

Question: Have I written formulas that convey the correct mathematical representation for my three-level model? Is my written interpretation of the coefficients in the equations correct?

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I have a three-level model representing experimental time series data. I am testing if participants' ratings on three variables (A, B, and C) predict their time series values. Values are nested within stimulus number, which is nested within the participant.

• [time point] (t): each time series trial consists of 120 data points; i.e., there are 120 data points per stimulus_num
• stimulus_num (k): each experimental session consisted of 36 trials; i.e., there are 36 stimuli per participant
• ID (i): each unique participant; 77 participants total
• A, B, and C: three ratings made by the participant before s/he completed each trial
• value: the time series value provided by participant i at time point t for stimulus k

To clarify, each participant completed 36 trials. In each of the 36 trials, there are 120 data points (value) collected. Therefore, each participant should have 4,320 data points.

Here is the model:

lmer(value ~ A + B + C + (1|ID/stimulus_num), data = data)

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For the equations at each level, I have:

Level 1: $$ Y _{ikt} = \beta _{0ik} + \beta _{1ik} A_{ikt} + \beta _{2ik} B_{ikt} + \beta _{3ik} C_{ikt} + e _{ikt} $$

Level 2: $$ \beta _{0ik} = \gamma _{00i} + u _{0ik} $$ $$ \beta _{1ik} = \gamma _{10i} + u _{1ik} $$ $$ \beta _{2ik} = \gamma _{20i} + u _{2ik} $$ $$ \beta _{3ik} = \gamma _{30i} + u _{3ik} $$

Level 3: $$ \gamma _{00i} = \pi _{000} + r _{00i} $$ $$ \gamma _{10i} = \pi _{100} + r _{10i} $$ $$ \gamma _{20i} = \pi _{200} + r _{20i} $$ $$ \gamma _{30i} = \pi _{300} + r _{30i} $$

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My interpretation of the terms in the model:

At level 1, the outcome $Y _{ikt}$ represents person i's time series value for stimulus k at time point t. This outcome was modeled as a function of a random intercept $\beta _{0ik}$ and the Level 1 fixed effects of A, B, and C ($\beta _{1ik}$, $\beta _{2ik}$, and $\beta _{3ik}$).

At Level 2, $\gamma _{00i}$ is the random intercept, while $\gamma _{10i}$, $\gamma _{20i}$, and $\gamma _{30i}$ represent the time series value for stimulus k at time point t as a function of A, B, and C respectively. The effects $u _{nik}$ represent the variation each individual stimulus has relative to the grand mean for that equation.

At Level 3, $\pi _{000}$ is the random intercept, while $\pi _{100}$, $\pi _{200}$, and $\pi _{300}$ represent the time series value for stimulus k at time point t and for participant i as a function of A, B, and C respectively. Effects $r _{n0i}$ represent the variation each individual person has relative to the grand mean for that equation.

I am not sure that I have a) written the Level 1-3 formulas correctly to reflect how I built the model, and b) explained the coefficients in the model correctly, especially for levels 2 and 3. Also, have I reflected the random intercepts on "stimulus number" and "participant" correctly?

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Edit: adding output of summary(data), str(data) and head(data, 10) as requested.

    > summary(data)
           ID             value             stimulus_num        A                B                C      
     1188   :  4248   Min.   :-10.0000   Min.   : 1.00   Min.   :  0.00   Min.   :  0.00   Min.   :  0.00  
     4431   :  4248   1st Qu.: -3.0000   1st Qu.:10.00   1st Qu.: 30.00   1st Qu.: 10.00   1st Qu.: 25.00  
     f6498  :  4248   Median :  0.0000   Median :18.00   Median : 50.00   Median : 30.00   Median : 50.00  
     f7876  :  4248   Mean   : -0.1655   Mean   :18.49   Mean   : 47.81   Mean   : 35.14   Mean   : 47.36  
     f8100  :  4248   3rd Qu.:  2.0000   3rd Qu.:27.00   3rd Qu.: 60.00   3rd Qu.: 58.00   3rd Qu.: 65.00  
     f8102  :  4248   Max.   : 10.0000   Max.   :36.00   Max.   :100.00   Max.   :100.00   Max.   :100.00  
     (Other):292522 

     > str(data)
    'data.frame':   318010 obs. of  6 variables:
     $ ID          : Factor w/ 77 levels "1188","4431",..: 1 1 1 1 1 1 1 1 1 1 ...
     $ value       : int  -2 -1 -3 -1 -1 0 -1 -2 -1 -1 ...
     $ stimulus_num: int  1 1 1 1 1 1 1 1 1 1 ...
     $ A           : int  25 25 25 25 25 25 25 25 25 25 ...
     $ B           : int  75 75 75 75 75 75 75 75 75 75 ...
     $ C           : int  75 75 75 75 75 75 75 75 75 75 ...

    > head(data, 10)
         ID value stimulus_num  A       B       C
    1  1188    -2        1      25      75      75
    2  1188    -1        1      25      75      75
    3  1188    -3        1      25      75      75
    4  1188    -1        1      25      75      75
    5  1188    -1        1      25      75      75
    6  1188     0        1      25      75      75
    7  1188    -1        1      25      75      75
    8  1188    -2        1      25      75      75
    9  1188    -1        1      25      75      75
    10 1188    -1        1      25      75      75

Answer 1

Question: Have I written formulas that convey the correct mathematical representation for my three-level model? Is my written interpretation of the coefficients in the equations correct?

Unfortunately, no. The model you are fitting:

lmer(value ~ A + B + C + (1|ID/stimulus_num), data = data)

has the following features:

• A global intercept (fixed effect)
• Fixed effects for A, B and C, which vary at the participant level, $i$, so these will generate 3 fixed effects estimates
• Random intercepts for ID and the ID:stimulus_num interaction, which means that stimulus_num is nested in ID, so this will generate 2 random intercepts estimates

Therefore we expect the model to produce 7 estimates (4 fixed effects, 2 random interecps and 1 unit-level residual). When writing out the mathematics of a specific model it is always good to know how many, and what kind of, estimates are expected

It looks like your equations are on the right track, but note that, for level 2 and 3 you only need the first equation - the others would be needed only if you were fitting random slopes. So the level 2 and 3 equations are only random intercepts. Also, your index notation is not quite right because, with the usual convention, the first index should refer to the lowest level, not the highest. Perhaps you were confused because commonly $i$,$j$ and $k$ refer to levels 1, 2 and 3, whereas you are using $t$,$k$ and $i$

Also, you have the fixed effects indexed by $ikt$ which, apart from being in the wrong order is incorect because they only vary at the the individual ($i$) level.

Thus to write the mutilevel model equations we will adopt standard notation (for instance in the book by Snijders and Bosker), using subscripts ordered from level 1 to level 3. For example $Y_{tki}$ refers to time point $t$ in stimulus $k$ in participant $i$.

Thus, for level 1 we can write:

$$ Y_{tki} = \beta_{0ki} + \beta_{1}A_{i} + \beta_{2}B_{i} + \beta_{3}C_{i} + e_{tki} $$

where $\beta_{0ki}$ is the intercept in level-2 unit (stimulus) $k$ within level-3 unit (participant) $i$. For this we have the level 2 model:

$$ \beta_{0ki} = \gamma_{00i} + u_{0ki} $$

where $\gamma_{00i}$ is the average intercept in level-3 unit (participant) $i$. For this average intercept we have the level-3 model:

$$ \gamma_{00i} = \pi_{000} + r_{00i} $$

Putting it all together we have:

$$ Y_{tki} = \pi_{000} + r_{00i} + u_{0ki} + \beta_{1}A_{i} + \beta_{2}B_{i} + \beta_{3}C_{i} + e_{tki} $$

and this would result in 7 estimates from the model, as expected - 4 fixed effects: $\pi_{000}$, $\beta_1$, $\beta_2$, and $\beta_3$; and 3 random effects: $r_{00i}$, $u_{0ki}$, and $e_{tki}$

Regarding interpretation:

$\pi_{000}$ is the global intercept: it is the mean of the time series' when the fixed effects, A, B and C are all at zero.

$\beta_1$, represent the expected difference in the time series' for a 1 unit change in A, with the other fixed effects held constant. Sinmilarly for $\beta_2$ and $\beta_3$

$r_{00i}$ is the random intercept for individuals and the software will estimate a variance for this

$u_{0ki}$ is the random intercept for stimulus and the software will estimate a variance for this

$e_{tki}$ is the unit-level (time series level) residual and the software will estimate a variance for this.