[R-sig-ME] Multilevel equation

[Brian Hudson]

Thierry,

Thank you! I appreciate your help and explanation - that makes sense and I
can see where my other attempts were incorrect.

A couple questions-

1) The two lines that defined eta confused me - are they supposed to be
equal to each other? I edited the equation below such that pi has the link
function and eta has the linear equation - does that work?
2) I added epsilon for the individual error
3) There was no k index (individual level) in the equations, just i and j,
so i added some indexing in the predictors.

Does this equation make sense? Any issues with what I did? Thanks again for
your (and the community's) help.

https://quicklatex.com/cache3/9f/ql_4a4eb44285f65ea0dcaf93d551c44c9f_l3.png

$$b_i\sim \mathcal{N}(0, \sigma_s^2)$$
$$b_{ij}\sim \mathcal{N}(0, \sigma_{y}^2)$$
$$\eta_{ijk} = \beta_0 + \beta_1 \textrm{X}\textsubscript{m[i]} + \beta_2
\textrm{X}\textsubscript{y[i,j]} + \beta_3
\textrm{X}\textsubscript{s[i,j,k]} + b_i + b_{ij} + \epsilon_{ijk}$$
$$\pi_{ijk} =\frac{e_{ijk}^{\eta}}{1+e_{ijk}^{\eta}}$$
$$Y_{ijk} \sim Binom(1, \pi_{ijk})$$


On Mon, Jul 19, 2021 at 1:34 PM Thierry Onkelinx <thierry.onkelinx using inbo.be>
wrote:

> Dear Brian,
>
> I'd write it as follows. In the case of a Gaussian model, you only have to
> write $Y_{ijk} \sim \mathcal{N}(\eta_{ijk}, \sigma^2)$ and drop the link
> function. (And you could replace \eta with \mu). Basically, Y depends on a
> distribution defined by some parameters. And these parameters might need
> some further definition.
>
> $i$: state index
> $j$: year index
> $k$: observation index
> $X_m$: state_mnthyr_pred
> $X_y$: state_year_pred
> $X_s$: state_pred
> $$Y_{ijk} \sim Binom(\pi_{ijk})$$
> $$\eta_{ijk} = \frac{\pi_{ijk}}{1- \pi_{ijk}}$$
> $$\eta_{ijk} = \beta_0 + \beta_1X_m + \beta_2 X_y + \beta_3 X_s + b_i +
> b_{ij}$$
> $$b_i\sim \mathcal{N}(0, \sigma_s^2)$$
> $$b_{ij}\sim \mathcal{N}(0, \sigma_{y}^2)$$
>
> Best regards,
>
>
> ir. Thierry Onkelinx
> Statisticus / Statistician
>
> Vlaamse Overheid / Government of Flanders
> INSTITUUT VOOR NATUUR- EN BOSONDERZOEK / RESEARCH INSTITUTE FOR NATURE AND
> FOREST
> Team Biometrie & Kwaliteitszorg / Team Biometrics & Quality Assurance
> thierry.onkelinx using inbo.be
> Havenlaan 88 bus 73, 1000 Brussel
> www.inbo.be
>
>
> ///////////////////////////////////////////////////////////////////////////////////////////
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> than asking him to perform a post-mortem examination: he may be able to say
> what the experiment died of. ~ Sir Ronald Aylmer Fisher
> The plural of anecdote is not data. ~ Roger Brinner
> The combination of some data and an aching desire for an answer does not
> ensure that a reasonable answer can be extracted from a given body of data.
> ~ John Tukey
>
> ///////////////////////////////////////////////////////////////////////////////////////////
>
> <https://www.inbo.be>
>
>
> Op ma 19 jul. 2021 om 17:44 schreef Brian Hudson <bhudson.gsu using gmail.com>:
>
>> Hello,
>>
>> I am fitting a multilevel model in `lme4` and am having trouble writing
>> the
>> equation for it. I very much appreciate any help. The formula and code is
>> below, but I am not sure if the equation represents the error correctly -
>> do I need to include error terms or is that captured by the distributions?
>> I am also not sure if I am representing the logit function correctly with
>> the indexing or functional form.
>>
>> The data are comprised of US-state months nested within US-state-years and
>> US-states. I include predictors at each level and a varying intercept for
>> both state-years and states.
>>
>> The formula looks like this in R:
>>
>> ```
>> as.formula(outcome ~ state_mnthyr_pred + state_year_pred + state_pred +
>>                          (1 | state) + (1 | state_year))
>> ```
>> Where the outcome is dichotomous. The state months (e.g. jan-2010,
>> feb-2010
>> ... jan-2013) are nested with state years and within states.
>>
>> The formula I am using can be seen here:
>>
>>
>> https://quicklatex.com/cache3/e9/ql_038eeb4e4e1b0af94d3ef69fe4ff7be9_l3.png
>> And the LaTeX code:
>>
>> $$
>> \begin{aligned}
>>     \mu &=\alpha_{j[i],k[i]} +
>> \beta_{0}(\operatorname{state\_mnthyr\_pred})\ \\
>>     \alpha_{j}  &\sim N \left(\gamma_{0}^{\alpha} +
>> \gamma_{1}^{\alpha}(\operatorname{\textrm{state\_year\_pred}}),
>> \sigma^2_{\alpha_{j}} \right)
>>     \text{, for \textrm{State-Year} j = 1,} \dots \text{, J} \\
>>     \alpha_{k}  &\sim N \left(\gamma_{0}^{\alpha} +
>> \gamma_{1}^{\alpha}(\operatorname{\textrm{state\_pred}}),
>> \sigma^2_{\alpha_{k}} \right)
>>     \text{, for State k = 1,} \dots \text{, K}\\
>> \pi_{i} &=\frac{e_{i}^{\mu}}{1+e_{i}^{\mu}}\\
>> y_{i j k} \sim & \operatorname{Binom}\left(1, \pi_{i}\right)\\
>> \end{aligned}
>> $$
>>
>> I really appreciate any help. Thank you.
>>
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>>
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>

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