Alternative Hypothesis

The alternative hypothesis, denoted $H_1$ (or $H_a$), is the hypothesis a test is designed to find evidence in favor of, typically representing the presence of an effect, difference, or association that the null hypothesis denies. It may be one-sided (e.g. $H_1:\mu >{\mu }_0$, testing only for an increase) or two-sided (e.g. $H_1:\mu \ne {\mu }_0$, testing for a difference in either direction), a choice made before the data is examined, since switching sides after seeing the results invalidates the test's error-rate guarantees. Whether the alternative is simple (a single specific value) or composite (a range of values) determines what tools are available for constructing an optimal test, e.g. the Neyman-Pearson lemma applies cleanly only to simple-versus-simple hypothesis pairs.

• The choice between a one-sided and two-sided alternative hypothesis must be made based on the research question before the data is collected or examined, choosing the direction after peeking at the data (to match whichever direction happens to look significant) inflates the true Type I error rate well beyond the nominal significance level.
• A composite alternative hypothesis (spanning a range of parameter values, e.g. μ > μ₀) is far more common in practice than a simple alternative (a single specific value), and the two require different theoretical tools: the Neyman-Pearson lemma yields a uniquely most powerful test only for simple-versus-simple comparisons, composite alternatives generally require the weaker uniformly most powerful test framework, which doesn't always exist.
• The alternative hypothesis is what statistical power is defined with respect to, a test's power is always evaluated at a specific value (or range of values) under the alternative, which is why power calculations require specifying not just that an effect exists, but how large that effect is assumed to be.