## What is a Two-Sided Hypothesis?

A two-sided hypothesis is an alternative hypothesis which is not **bounded from above or from below**, as opposed to a one-sided hypothesis which is always bounded from either above or below. In fact, a two-sided hypothesis is nothing more than the union of two one-sided hypotheses. For example, H_{1}: δ < 0 ∪ δ > 0 (alt.: H_{1}: θ∈(-∞,0) ∪ θ∈(0,+∞)) is a two-sided hypothesis. The corresponding null hypothesis would necessarily be a point hypothesis: H_{0}: δ = 0 (alt.:H_{0}: θ∈[0])).

A two-sided alternative hypothesis can be used when one wants to set his type I error against a very precise null hypothesis, for example that the effect is **exactly zero** (no smaller, no larger). A two-sided hypothesis will not be applicable to a superiority test which are most A/B tests performed, nor will it apply in the less-common non-inferiority test. As a statement it corresponds to the claim that the treatment will perform either better or worse than the control.

Despite its intuitive appeal, this is **rarely the claim we want to defend** via an online controlled experiment and the counter-position is rarely a strict "no effect" claim. Most of the time stakeholders will only approve a proposed change to a website, app or software if it is better than the existing state of affairs and will certainly not approve it if it is worse, therefore their position corresponds to a one-sided null hypothesis which begs the complimentary one-sided alternative. In fact, you will likely have trouble keeping your job as a conversion rate optimization specialist if you frequently allow tests to reach statistical significance for an effect in the negative direction (sequential testing with a futility boundary can help prevent that).

When a two-sided hypothesis is used the respective p-value should also be two-sided (or a two-tailed test as it is sometimes called). If a confidence interval is used to support a two-sided claim it should also be two-sided.