The little I did understand from googling on R is that the model here is that we have a binary outcome $Y_{ij}$ some sort of bounded variable $\theta_{ij}$. Where $j$ is the group within which random intercept is kept constant. So the model is:

$$ \begin{align} c_j&\sim N\left(\mu,\,\sigma^2\right) \\ \eta_{ij}&=c_j+w_1\sin\theta_{ij}+w_2\cos\theta_{ij}\\ \pi_{ij}&\sim Logistic\left(\eta_{ij},s\right)\\ Y_{ij}&\sim Bernoulli(\pi_{ij}) \end{align} $$

Where $\mu$, $\sigma$, $w_{1,2}$ and $s$ are to be fitted

With some work you could write the likelihood for your data under this model and then use something like Fisher Information to extract the standard errors.

Instead, it may be easier to supplement this model with $\theta_{ij}\sim Uniform\left(0,2\pi\right)$ generate some $Y_{ij}$, and then use your current fitting procedure to get estimates of $w_{1,2}$. Their distribution, and distribution of their sum of squares will tell you the distributions of the statistics under null hypothesis.

The little I did understand from googling on R is that the model here is that we have a binary outcome $Y_{ij}$ some sort of bounded variable $\theta_{ij}$. Where $j$ is the group within which random intercept is kept constant. So the model is:

$$ \begin{align} c_j&\sim N\left(\mu,\,\sigma^2\right) \\ \eta_{ij}&=c_j+w_1\sin\theta_{ij}+w_2\cos\theta_{ij}\\ \pi_{ij}&=logit\left(\eta_{ij}\right)\\ Y_{ij}&\sim Bernoulli(\pi_{ij}) \end{align} $$

Where $\mu$, $\sigma$, $w_{1,2}$ are to be fitted

With some work you could write the likelihood for your data under this model and then use something like Fisher Information to extract the standard errors. It also, so happens that for Bernoulli, the algebraic expressions can be relatively simple

Instead, it may be easier to supplement this model with $\theta_{ij}\sim Uniform\left(0,2\pi\right)$ generate some $Y_{ij}$, and then use your current fitting procedure to get estimates of $w_{1,2}$. Their distribution, and distribution of their sum of squares will tell you the distributions of the statistics under null hypothesis.

The little I did understand from googling on R is that the model here is that we have a binary outcome $Y_{ij}$ some sort of bounded variable $\theta_{ij}$. Where $j$ is the group within which random intercept is kept constant. So the model is:

$$ \begin{align} c_j&\sim N\left(\mu,\,\sigma^2\right) \\ \eta_{ij}&\sim N\left(c_j+w_1\sin\theta_{ij}+w_2\cos\theta_{ij},\,\epsilon^2\right)\\ \pi_{ij}&=\mbox{logit}\left(\eta_{ij}\right) \\ Y_{ij}&\sim Bernoulli\left(\pi_{ij}\right) \end{align} $$

Where $\mu$, $\sigma$, $w_{1,2}$ and $\epsilon$ are to be fitted

With some work you could write the likelihood for your data under this model and then use something like Fisher Information to extract the standard errors.

Instead, it may be easier to supplement this model with $\theta_{ij}\sim Uniform\left(0,2\pi\right)$ generate some $Y_{ij}$, and then use your current fitting procedure to get estimates of $w_{1,2}$. Their distribution, and distribution of their sum of squares will tell you the distributions of the statistics under null hypothesis.

The little I did understand from googling on R is that the model here is that we have a binary outcome $Y_{ij}$ some sort of bounded variable $\theta_{ij}$. Where $j$ is the group within which random intercept is kept constant. So the model is:

$$ \begin{align} c_j&\sim N\left(\mu,\,\sigma^2\right) \\ \eta_{ij}&=c_j+w_1\sin\theta_{ij}+w_2\cos\theta_{ij}\\ \pi_{ij}&\sim Logistic\left(\eta_{ij},s\right)\\ Y_{ij}&\sim Bernoulli(\pi_{ij}) \end{align} $$

Where $\mu$, $\sigma$, $w_{1,2}$ and $s$ are to be fitted

With some work you could write the likelihood for your data under this model and then use something like Fisher Information to extract the standard errors.

Instead, it may be easier to supplement this model with $\theta_{ij}\sim Uniform\left(0,2\pi\right)$ generate some $Y_{ij}$, and then use your current fitting procedure to get estimates of $w_{1,2}$. Their distribution, and distribution of their sum of squares will tell you the distributions of the statistics under null hypothesis.