Probit analysis using 13 cases and checking normal distribution Can I use probit if my dependent variable has data that looks something like this, 1 case is equal to 0, and 12 cases are equal to 1?
Thanks
 A: You can use any statistical algorithm on any data it will successfully compute with.
The underlying questions are (1) would this be meaningful? and (2) would it have any power to detect anything?  The answers in this case are yes and no, respectively.
The analysis begins by looking at a dataset like the one described. Let's take the x-values to be evenly spaced. Because there's essentially no difference between the applicability of probit and logistic models, I will perform this analysis with the logit in R:
x <- 1:13
y <- x==2
fit <- glm(y ~ x, family=binomial(link="logit"))
summary(fit)

The output shows nothing is significant:
            Estimate Std. Error z value Pr(>|z|)
(Intercept)    0.712      2.297    0.31     0.76
x             -0.842      0.858   -0.98     0.33

I set the second $y$ value (corresponding to $x=2$) equal to $1$ because if the first (most extreme) value were $1$, the data would not enable us to determine how quickly the curve slopes down from the first to the second data point.  (The output will tell us the algorithm failed to converge.)
Here's what the fits and the p-values look like depending on where in the dataset the $1$ is located:

(Fits are shown as the red curves.)
None of the p-values gets any lower than 0.33: there's just no chance of getting a significant result.
This picture would not change much if we were to change the distribution of the $x$ values to give the $1$ as much leverage as possible.  This is the best I can do with some experimentation:

A p-value of $0.17$ would not be considered significant in most applications.  This answers the second question: such a dataset just doesn't have the power to identify a relation between the response and the independent variable(s).
As to the first question, arguably the regression can be meaningful as an exploratory tool. But the graphics displayed here suggest the result is not any more informative than merely plotting the points.
Finally, what about alternative analyses? One possible application is to determine whether there is some relationship between $x$ and $y$. If the data are a random sample of a population, for instance, and we wanted to test the hypothesis that a $y$ value of $1$ is associated with low values of $x$ (a one-sided hypothesis), then we might apply a Wilcoxon rank sum test.  Now, unlike with logistic or probit regression, there is no problem if the $1$ occurs at an extreme value of $x$:
x <- 1:13
y <- x==1
wilcox.test(x ~ y, alternative="greater")

The p-value of $0.07692 = 1/13$ is of course expressing the chance that the $1$ landed on the extreme value of $x$ at random.  But this is starting to get into the range of "suggestively significant" results.  We conclude that if the hypothesis is one-sided and if the solitary value of $y=1$ occurs for the expected extreme value of $x$, then there is some mild evidence of the association.
