Significance test of the amplitude of sinusoidal regression

Question

I am working on a multi-level logistic regression model with a binary outcome and a circular predictor. Thanks to this post, I know that I have to add sine and cosine of the circular variable as a predictor and that I can calculate the amplitude $a$ and the phase shift $\phi$ from the regression weights $w_1$ and $w_2$ corresponding to the sine and cosine predictors, respectively.

$a = \sqrt{w_1^2 + w_2^2}$

$\phi = \tan^{-1}\frac{w_1}{w_2}$

Say I am not particularly interested in testing the significance of sine and cosine components individually, which is what the model provides. Instead, I would like to test the overall effect of the circular variable (i. e., the amplitude). My thought was to run a model comparison (in R, anova(m1, m2)) between an intercept-only model m1 and the model including the circular variable m2 to obtain a $\chi^2$ statistic. However, I would like to know if there is a cleaner way of testing the significance of $a$ and $\phi$ - and how the corresponding standard errors are calculated. Thanks in advance for your help!

Answer 1

The relevant tests can be undertaken by converting them to tests on the underlying regression coefficients in linear form. For the test of amplitude the relevant null hypotheses convert as follows:

$$\begin{matrix} H_0: a = 0 \quad \quad \quad & & & H_A: a \neq 0 \quad \quad \quad \quad \quad \\[6pt] H_0: w_1=w_2=0 & & & H_A: w_1 \neq 0 \text{ or } w_2 \neq 0 \\[6pt] \end{matrix}$$

You can test this hypothesis with a standard F-test for the subset of the two coefficients for the sine and cosine terms. For the test of phase angle the relevant null hypotheses convert as follows:

$$\begin{matrix} H_0: \phi = \phi_0 \ \ \quad \quad \quad & & & H_A: \phi \neq \phi_0 \ \ \quad \quad \quad \\[6pt] H_0: w_1 = w_2 \tan \phi_0 & & & H_A: w_1 \neq w_2 \tan \phi_0 \\[6pt] \end{matrix}$$

This latter hypothesis test is effectively testing the ratio of the linear regression coefficients against a fixed value. This would require a custom test for a ratio of coefficients, which is non-standard but should not be too difficult to construct.

Answer 2

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.