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4

I suspect that a major problem is the time-series nature of the data. As the Wikipedia page says, inference with linear regression assumes that "errors of the response variables [around the values predicted by the linear model] are uncorrelated with each other." That is often not the case with time series, for which the errors around the predicted values ...

3

There is a confounder: The Earth's position on its orbit around the Sun determines on one hand the season (and, consequently, the ground temperature) and, on the other hand, the amount of light the Moon is receiving. There are subtle traces in the data. The Sun-Earth-Moon angle and the Moon brightness are tightly related, which is not surprising. The Moon ...

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tl;dr: There is no known distribution. i.e. it doesn't have a name. You use simulations instead. I found this to give some insight: Basically, Shapiro and Wilk calculated the distribution of their statistic only for $n=3$, to be a truncated $Beta(\frac{1}{2}, \frac{1}{2})$ distribution for $\frac{3}{4} \le w \le 1$ and zero elsewhere. For $3 < n \le ... 2 Why not just look it up a T Table or just punch the numbers in excel? I get the elaborate explanation above and good job at it but feels a bit overkill. In excel you can use T.INV. Just that you need degrees of freedom (which for the t distribution is n-1) for the second argument in T.INV(). And then just adjust based on what test it was upper, lower, two ... 1 The null hypothesis is that the$p_i$all have a uniform distribution on the unit interval. For more details and a discussion of the alternative hypotheses see Test for significant excess of significant p-values across multiple comparisons 1 Reading the code and vignette for FDR from astsa, it seems like it is a lift from this link and they modified it to return the number of rejects. If we use the code they lifted from, it is a benjamini-hochberg correction: fdr.basic <- function(pvals,qlevel=0.05){ n <- length(pvals) sorted.pvals <- sort(pvals) sort.index <- order(pvals) ... 1 I think I got the answer on the mechanical level by searching the scipy repo (the function performs equivalently to the R function): https://github.com/scipy/scipy/search?q=binom_test&unscoped_q=binom_test and locating binom_test. They basically take the terms on the other side of the tail which have a PMF lower than the input,$x\$. In the example given ...

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You are correct in saying that a failure to reject the null hypothesis does not mean that the null hypothesis is true. However, our default position in hypothesis testing is that that null hypothesis is assumed to be true until proven otherwise. There is no statistical test to prove that an interaction effect does not exist. You will have to design your ...

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