Equivalence test for one proportions test I want to apply an equivalence test on my sample to infer whether they are equivalent or not.
Since my data are binominal [0,1] I don’t know whether the TOST procedure (tost() in R) can handle my problem or not.
My data consists of one group (G) which are not equal in numbers of samples. E.g., G= 264 people. The outcome variable is binominal, such that G includes the number of Yes and NO. I want to know if I can run an equivalence test for one-portion test.
Hypothesis:
H0:p1=0.5;
H1: p1>0.5
 A: Your hypothesis is not formulated as an equivalence null hypothesis, which in the general form for TOST should be:
$H^{-}_{0}: |p-p_{0}| \ge \Delta$, with $H^{-}_{1}: |p-p_{0}| < \Delta$, 
where $\Delta$ is the smallest relevant difference between $p$ (the true proportion in the population you sampled), and $p_{0}$ (the known population proportion which you are testing against). $\Delta$ is expressed in the same scale as proportions, so $\Delta=0.025$ would define the relevance/equivalence threshold as $\pm 2.5\%$.
You have not provided quite enough information for me to work through your equivalence test, however, the TOST test is relatively straightforward, and results from translating the above general form of the equivalence test null to the two one-sided null hypotheses:
$H^{-}_{01}: p-p_{0} \ge \Delta$, with $H^{-}_{11}: p-p_{0} < \Delta$, and
$H^{-}_{02}: p-p_{0} \le -\Delta$, with $H^{-}_{12}: p-p_{0} > \Delta$.
The first one-sided test poses the null that the difference between $p$ and $p_{0}$ is greater than your equivalence/relevance threshold, and if you reject this null, you conclude that the difference must be less than $\Delta$.
The second one-sided test poses the null that the difference between $p$ and $p_{0}$ is less than $-1$ times your equivalence/relevance threshold, and if you reject this null, you conclude that the difference must be greater than $-\Delta$.
If your reject both $H^{-}_{01}$ and $H^{-}_{02}$, then you have found evidence that $- \Delta < p-p_{0} < \Delta$.
Next you need to calculate your test statistics for the two one-sided tests (both test statistics have been formulated to use upper tail probabilities, so that you do not have to keep track of which tail corresponds to which null hypothesis when obtaining p-values):
$z_{1} = \frac{\Delta - (\hat{p}-p_{0})}{\sigma}$, and
$z_{2} = \frac{(\hat{p}-p_{0})+\Delta}{\sigma}$,
where $\sigma = \sqrt{\frac{p_{0}(1-p_{0})}{n}}$.
Now you need to obtain your p-values:
$p_{1} = P(Z \ge z_{1})$, and
$p_{2} = P(Z \ge z_{2})$.
$H^{-}_{01}$ and $H^{-}_{02}$ are each rejected at the $\alpha$ level.
If, and only if, you reject both $H^{-}_{01}$ and $H^{-}_{02}$ in favor of $H^{-}_{11}$ and $H^{-}_{12}$ do you conclude that you found evidence that $p$ is equivalent to $p_{0}$ within a tolerance of $\pm \Delta$ at the $\alpha$ level of significance.
