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I have a model with $n$ participants. From 5 different angles $\theta_1, \theta_2, \dots, \theta_5$, each participant either responds to a stimulant with either a $0$ or a $1$, so $y_{i,\theta_j} \in \{0,1\}$ for $i=1 \dots n$, $j = 1, \dots, 5$.

For each angle $\theta_j$, I have created a response variable $z_j = \sum_{i=1}^n y_{i,\theta_j}$. I believe that as $\theta_j$ increases, $z_j$ also increases: i.e. that $\theta_j$ and $z_j$ are positively correlated.

How can I formulate this to be able to perform/get a test statistic to address the correlation between the two variables?

Thanks!

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If $n$ is big, by the Central Limit Theorem the conditional distribution [$Z$ given $\Theta$] is approximately Gaussian for each level $j = 1, ..., 5$. However, we cannot say much about the distribution of $\Theta$ over $j = 1, ..., 5$. For that reason the relationship between $Z$ and $\Theta$ may not be linear even if it is monotonic. To test whether a (non-)linear relationship exists you can use rank-based measures of association, like Kendall's tau or Spearman's rho. Randomization tests can be used to check their significance.

You do have to play down your expectations since the sample size of 5 offers very little statistical power. Most tests are likely to come out as statistically non-significant even if the true underlying relationship is non-zero... If there were reasonable justifications for a Gaussian nature of $\Theta$, then we could prove that any potential relationship between $Z$ and $\Theta$ must be linear. With that we could use the good-old t-test for correlation between $Z$ and $\Theta$. It would be somewhat more powerful than randomization tests for Kendall's tau and Spearman's rho.

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