Hello,
I'm currently analyzing my survey data for my master thesis and my advisor suggested to do a one-sided t-test. However, I'm not quite sure if I can really do this or if I should use a test of proportion or a binominal probability test?
I had questions with three options (only one could be chosen). Let's say A, B, C. I generated a new variable in Stata, called x. For each subject, let x=0 if he/she answered C. Furthermore, let x=1 if the subject answered A or B. Thus, x is a binary variable, with the mean x_m telling us how many percentage of subjects in the sample did not choose C.
He then said: "Now, we would like to test whether the population mean of x (say, mu_x) is larger than 0.5 (50%). By construction of x, we can employ a one-sided t-test for this (Note that x satisfy the distributional assumptions; if we could keep drawing samples from the population, then x_m would follow a normal distribution with mean mu_x). Thus, you can simply employ a t-test to test whether the mean is significantly larger than 0.5."
Can I really use a one-sided t-test here? How is this data normally distributed?
I'm not sure about the non-parametric options that I could use instead...
What I have found was a one-sample test of proportion. In stata I would then type "prtest x == .5". Another option would be the binominal probability test "bitest x == .5".
I'm quite new to these statistical tests and have never done it. Thank you so much for any help!!
The mean of a binary variable, when scored 0/1, is the proportion saying 1. There are several procedures that could be used depending what variables you want to use to predict this (e.g., logistic regression). If none, a proportion or binomial test could be used.
You did not ask whether a one- or two-tailed test is appropriate, so I will not address this. You probably will get some thoughts from commentators about this.
ps. I am not a Stata user so won't comment on the syntax.
If you use a 1-tailed or 2-tailed test depend on how you formulate your hypothesis. Obey your supervisor. "By construction of x, we can employ a one-sided t-test for this (Note that x satisfy the distributional assumptions; if we could keep drawing samples from the population, then x_m would follow a normal distribution with mean mu_x). Thus, you can simply employ a t-test to test whether the mean is significantly larger than 0.5.""
You can use a 1-sided t-test.
There are dozens of specific tests for binomial proportions and at least one will be implemented in State. Technically these are the best options. However, if you have reasonably large n (say 30+) and the mean you are looking at is not extreme (e.g., 0.2 to 0.8) the t test (or any of the common normal approximations) will be be OK as the normal approximation to the binomial is fairly good in these situations.
The video Jos Feys directed you to uses -prtest-, which carries out the usual z-test for a single proportion. The -bitest- command you mentioned carries out an exact binomial test. As Thom Baguley noted, when n is reasonably large, the z-test will give a pretty good approximation to the binomial test. But the binomial test is never going to be wrong, and you have easy access to it via -bitest-. So unless the sample is enormous to the point that is causing computational problems, why would you not use the binomial test? (Unless you want to introduce explanatory variables, as Daniel Wright suggested.)
PS- A small example using Stata is in the attached text file. (I didn't paste it here because the line-wrap would have been ugly!)
Slight digression. The main reason not to use the binomial test is if you want an interval estimate. It turns out that the interval test created from the binomial test don't have particularly good properties in small samples. This creates the additional problem that the inference from the CI and the test don't match. In these cases its possible that a confidence interval from a t test might have better coverage for example. A very simple modification of the normal approximation is known to work quite well in practice. This is to add two successes and failures before calculating the normal approximation (usually attributed to Agresti and Coull).
There is another reason not to use the binomial test - which is if you want to include prior information. It is very easy to conduct Bayesian inference with simple proportions like these. It turns out that the add two successes and failures CI I mentioned is more-or-less equivalent to Bayesian posterior probability interval with a beta(2,2) prior. This is a reasonable prior if you expect a proportion to be somewhere in the middle of the range rather than close to 0 or 1.
Excellent point about 'exact' interval estimates, Thom. Couldn't agree more. You mentioned the Agresti & Coull interval. I like the Wilson-score method. Clara can use the -ci proportion- command in Stata to get either. Here are all of the options:
exact calculate exact confidence intervals; the default
wald calculate Wald confidence intervals
wilson calculate Wilson confidence intervals
agresti calculate Agresti-Coull confidence intervals
jeffreys calculate Jeffreys confidence intervals
- Badges
- Science topic
- Similar topics
- Chemistry
- Pharmacology
- Pharmaceuticals
- Drugs