Non parametric test for regression slope>0? I have some data for a bernoulli variable $Y$ which I believe is a function of time $X$. I have plotted the data below, the slope is negative but I want to test the null hypothesis that the slope is greater than 0.
I know how to perform this hypothesis test for an ordinary regression model by using a t-test but I think that this Bernoulli data greatly violates the assumptions of the t test. Therefore I'm searching for a non-parametric slope test. Of course a test that is specifically for bernoulli regressors would be better but I'm willing to use a non-parametric test

 A: I don't know of a parametric test, but non-parametrically you could always do permutation- or bootstrap tests. I would avoid permutation in this case because there are actually only 9 choose 4 = 126 possible permutations as the response is only ones and zeroes.
If you can make a confidence interval you can do a test, as they are two sides of the same thing. You simply reject if the $H_0$ value falls outside the $(1-\alpha)$-interval. If you want a p-value, find the smallest $\alpha$ for which the $H_0$ value falls outside. 
Bootstrapping is a nice non-parametric way to build confidence intervals for your parameters. There are many ways to build bootstrap confidence intervals of varying intricacy. I'll use my favourite, the very straight-forward quantile interval: use the empirical quantiles of the bootstrap distribution over your parameter. 
Below is R-code with results for such a test. NB that although you called $X$ time, I have assumed that it doesn't have time series properties. If that were the case, you'd probably have to do block bootstrapping, which adds another layer of complexity.


library(plyr)

# set up data
x <- 1:9
y <- c(1,0,1,1,0,0,1,0,0)
dataset <- data.frame(x,y)

B <- 50000 # probably overkill
bootstrapped <- raply(B, function() {
  dataset_bootstrap <- dataset[sample(nrow(dataset), replace=T),]
  coef(lm(y~x, data=dataset_bootstrap))["x"]
})

hist(bootstrapped, col="grey", border="grey", nclass=60)
abline(v=0, col="red")


# your p-value
mean(bootstrapped > 0)
#> [1] 0.0922

I'd say there is weak evidence that $\beta_1 \leq 0$
