Null and Alternative Hypotheses
Introduction
Every hypothesis test begins with two hypotheses: a null that captures the status quo and an alternative that captures what we would conclude if the null is rejected. Stating both clearly, before the data arrive, is the crucial first step – and a remarkably common source of error when skipped.
Prerequisites
Random variables and population parameters.
Theory
The null hypothesis \(H_0\) is typically a statement of no effect, no difference, or a specific parameter value. The alternative \(H_1\) (or \(H_A\)) is what we would conclude if we reject \(H_0\).
Two-sided alternatives are the default when the sign of the effect is unknown or of interest in both directions:
\[H_0: \mu = \mu_0 \quad \text{vs.} \quad H_1: \mu \neq \mu_0.\]
One-sided alternatives specify a direction, increasing power for that direction at the cost of giving up inference in the opposite:
\[H_0: \mu \leq \mu_0 \quad \text{vs.} \quad H_1: \mu > \mu_0.\]
Use a one-sided test only when:
- A priori, only one direction is of scientific interest.
- A result in the other direction would be treated identically to a null result.
- The directionality is pre-specified in the protocol, not chosen after seeing the data.
Simple vs. composite: \(H_0: \mu = 0\) is simple (a single value); \(H_0: \mu \leq 0\) is composite (a set). Test statistics and rejection rules differ slightly.
Equivalence/non-inferiority testing flips the roles: the null is that the effect is at least as large as some margin, and rejection establishes equivalence.
Assumptions
Hypotheses must be specified before examining the data. Post-hoc redefinition invalidates the sampling distribution of the p-value.
R Implementation
set.seed(2026)
# Two-sided test
x <- rnorm(40, mean = 2, sd = 5)
t.test(x, mu = 0, alternative = "two.sided")
# One-sided test (H1: mu > 0)
t.test(x, mu = 0, alternative = "greater")
# One-sided test (H1: mu < 0)
t.test(x, mu = 0, alternative = "less")
# TOST (equivalence: |mu| < 1)
library(TOSTER)
tsum_TOST(m1 = mean(x), sd1 = sd(x), n1 = length(x),
low_eqbound = -1, high_eqbound = 1)Output & Results
The two-sided \(p\) will be roughly twice the one-sided \(p\) when the point estimate is consistent with \(H_1\)’s direction. For equivalence testing, rejection of both one-sided components establishes equivalence.
Interpretation
Pre-register or pre-specify the hypotheses in the protocol; do not switch from two-sided to one-sided after seeing the sign of the effect. Misuse of one-sided tests is a common source of reviewer objections.
Practical Tips
- Default to two-sided tests unless a strong, pre-specified rationale for one-sidedness exists.
- Never compute both one-sided and two-sided p-values and report whichever is smaller.
- In equivalence trials, the “null” is dissimilarity and “alternative” is equivalence – reversed from classical tests.
- Composite null hypotheses are handled by evaluating the test statistic at the boundary.
- Frame hypotheses in terms of the parameter of scientific interest, not the test statistic.