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I've been reading on one-tailed tests. I get that the direction is determined in the hypotheses. However, in doing the test statistic itself, how do I know if I'm doing it in the right way or the other? Or is the process the same and only the process of looking up the critical values change?

NoobPers0n
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One simple example that makes this pretty clear is comparing a drug and a placebo on some continuous outcome. You can look at the null hypothesis that the means $\mu_\text{treatment}$ are the same $$H_0: \mu_\text{drug} = \mu_\text{placebo}$$ vs. the alternative hypothesis that they are different $$H_A: \mu_\text{drug} \neq \mu_\text{placebo}.$$ That's the two-sided approach.

Or, sometimes people argue that they only care about investigating whether the drug has a higher mean than placebo. That's assuming that higher is in "better", while if the means are the same or the mean is lower for the drug, you would in either case not use the drug. You could then look at the null hypothesis $$H_0: \mu_\text{drug} \leq \mu_\text{placebo}$$ vs. the alternative hypothesis $$H_A: \mu_\text{drug} > \mu_\text{placebo}.$$ If lower is better then you might want to flip the direction around.

Alternatively, perhaps your research question is about an outcome that is a potential side-effect of the drug. You could, again, use a two-sided approach or a one-sided approach. E.g. if a higher mean means more side-effects, you might look at the null hypothesis $$H_0: \mu_\text{drug} \leq \mu_\text{placebo}$$ vs. the alternative hypothesis $$H_A: \mu_\text{drug} > \mu_\text{placebo}.$$

You will mostly see two-sided tests in practice for clinical trials of drugs (some other fields may have other conventions) for several reasons:

  • You will usually be asked to (e.g. by journals and drug regulators) to use either a two-sided test at level $\alpha$ (often 0.05) or a one-sided test at level $\alpha/2$. I.e. it does not become easier to "show your drug works" by using a one-sided test.
  • In fact, a major medical journal did make me and my co-authors report two-sided p-values, even when we had a protocol that said we would do one-sided hypothesis tests.
  • Just because you only did a one-sided test, people will not ignore negative outcomes in the direction you said you were not interested in. E.g. if you said you wanted to show your drug reduces mortality and you in fact saw an increase in mortality compared to a placebo, then you cannot really ignore that just because you did a one-sided hypothesis test.
  • The most common example of one-sided tests you would see in practice in clinical trials is non-inferiority tests with a null hypothesis of $H_0: \mu_\text{drug} \leq \mu_\text{other drug}-\delta$ vs. an alternative hypothesis of $H_A: \mu_\text{drug} > \mu_\text{other drug}-\delta$, where $\delta>0$ is a non-inferiority margin. I.e. the alternative hypothesis is that the the mean for the drug is at most worse by $\delta$ compared with the mean for the other drug.
Björn
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