Match the Alternative Hypothesis to the Question

Why the Alternative Must Match the Question

A significance test is designed to assess a claim about a population, not to describe whichever direction happens to appear in a sample. Before examining results, the research question should determine whether the test looks for an increase, a decrease, or any difference. That choice is expressed by the alternative hypothesis, \(H_a\).

For a comparison between two groups, the direction depends on the order in which the population quantities are subtracted. If the difference is defined as “Group A minus Group B,” then a positive difference means A is higher, and a negative difference means A is lower. Reversing the subtraction reverses the interpretation. The group order and the alternative must therefore be stated together.

Definition: The alternative hypothesis, \(H_a\), is the claim about a population parameter that represents the effect or difference the research question seeks evidence for. A two-sided alternative specifies a difference in either direction; a directional (one-sided) alternative specifies a difference in one direction. The null hypothesis, \(H_0\), represents no difference for the parameter being tested.

For example, let \(\Delta=\mu_A-\mu_B\), the difference between two population means. “Any difference” corresponds to \(H_a:\Delta\ne 0\); “A has a higher mean than B” corresponds to \(H_a:\Delta>0\); and “A has a lower mean than B” corresponds to \(H_a:\Delta<0\). For two population proportions, the same logic applies to \(\Delta=p_A-p_B\). A one-sided alternative is not simply a more convenient version of a two-sided question: it makes a narrower scientific claim.

In the usual notation for a test, the null is written at the no-difference boundary, such as \(H_0:\Delta=0\). For a one-sided question, the substantive null also includes differences in the opposite direction: for \(H_a:\Delta>0\), the null claim is that \(\Delta\le 0\). The test is calibrated at the boundary value \(\Delta=0\). This convention does not change the key task here: specify the direction in \(H_a\) from the research question.

Translate the Question Before Looking at Results

A reliable translation begins by identifying the outcome and the population parameter. Then fix the order of the comparison in words and symbols. Only after those choices are made should the alternative be written. A phrase such as “is there an effect?” can be ambiguous: if the scientific goal is to detect either improvement or worsening, it calls for a two-sided alternative; if the question specifically asks whether the outcome improves in a defined direction, it calls for a directional alternative.

Name the population quantity.
Decide whether the outcome is represented by a mean, proportion, or paired difference, and identify the population groups or measurements being compared.

Fix the subtraction order.
For example, define \(\Delta=\mu_A-\mu_B\). Say what a positive and a negative value mean in context.

Read the question for direction.
“Higher” or “increases” points to a positive difference in the chosen order; “lower” or “decreases” points to a negative difference; “different” without a specified direction points to either sign.

Write \(H_a\) before inspecting results.
Use \(>\), \(<\), or \(\ne\) to represent the claim. The null is written at no difference.

Worked Example: Asking Whether a Response Rate Increases

A hypothetical trial compares the proportion of patients who respond to a new treatment with the proportion who respond to a standard treatment. The research question, written before the trial results are examined, is whether the new treatment increases the response rate.

Define the parameter. Let \(p_N\) be the population response proportion for the new treatment and \(p_S\) the population response proportion for the standard treatment. Define \(\Delta=p_N-p_S\). A positive \(\Delta\) means a higher response proportion with the new treatment.

The word “increases” asks specifically whether \(\Delta>0\), so the hypotheses are \(H_0:\Delta=0\) and \(H_a:\Delta>0\). This is a directional alternative. In substantive terms, the null covers no increase, \(\Delta\le 0\), with equality as the boundary used for the test.

Suppose the hypothetical sample has 72 responses among 100 patients receiving the new treatment and 65 among 100 receiving the standard treatment. The sample proportions are

\hat{p}_N=\frac{72}{100}=0.72,\qquad \hat{p}_S=\frac{65}{100}=0.65,\qquad \hat{p}_N-\hat{p}_S=0.72-0.65=0.07.

The observed sample difference is 0.07, or 7 percentage points, in the predicted direction. But that calculation does not determine \(H_a\): the research question already did. If the observed difference had instead been negative, the question would still be whether the new treatment increases the population response rate; the data might then provide little support for that claim, but the alternative would not be rewritten.

“Any Difference” Is a Different Question

A two-sided alternative is appropriate when differences in either direction matter to the research question. It does not mean that the investigator is uncertain about which direction the sample will go. It means that the scientific claim being tested treats either an increase or a decrease as a relevant departure from no difference.

Worked Example: Asking Whether an Adverse-Event Rate Decreases

A hypothetical study compares the proportion of patients experiencing a particular adverse event under a new care protocol with the proportion under usual care. The stated research question is whether the protocol decreases the adverse-event rate.

Let \(p_P\) be the population adverse-event proportion under the protocol and \(p_U\) the proportion under usual care. Define \(\Delta=p_P-p_U\), so a decrease under the protocol corresponds to a negative difference. The hypotheses are \(H_0:\Delta=0\) and \(H_a:\Delta<0\). This is a one-sided alternative in the lower direction.

In hypothetical samples, 12 of 120 patients in the protocol group and 19 of 120 in the usual-care group experience the event. The observed proportions and difference are

\hat{p}_P=\frac{12}{120}=0.100,\qquad \hat{p}_U=\frac{19}{120}=0.1583,\qquad \hat{p}_P-\hat{p}_U=0.100-0.1583=-0.0583.

The sample difference is about \(-0.0583\), or 5.83 percentage points lower under the protocol. Its sign agrees with the proposed decrease. The direction of \(H_a\), however, comes from the pre-specified research question, not from seeing that the protocol group happened to have fewer events. Had the question instead been whether the rates differ, the alternative would be \(H_a:\Delta\ne 0\), even with these same sample counts.

The wording of a question is not always enough by itself. “Does the treatment work?” could refer to a particular improvement or to any change, including a harmful one. The outcome must also be coded so its direction is meaningful. If lower recovery time is beneficial, for example, a beneficial effect is a decrease in days, not an increase. State the outcome and the comparison explicitly rather than relying on an everyday word such as “better.”

Worked Example: Asking Whether a Paired Outcome Changes

A hypothetical group of patients has a measurement recorded before and after a program. The research question is whether the program changes the population mean measurement, in either direction. Define each paired difference as \(d=\text{after}-\text{before}\), and let \(\mu_d\) be the population mean of these differences.

With this order, a positive \(d\) means the measurement increased and a negative \(d\) means it decreased. Because the question asks about any change, whether positive or negative, the hypotheses are \(H_0:\mu_d=0\) and \(H_a:\mu_d\ne 0\). This is two-sided. If the question had specifically asked whether the program lowers the measurement, the alternative would instead be \(H_a:\mu_d<0\).

Suppose the hypothetical differences for five participants, in the stated order, are \(-4, 2, -1, 3,\) and \(-2\) units. Their sample mean is

\bar{x}_d=\frac{-4+2-1+3-2}{5} =\frac{-2}{5}=-0.4\text{ units}.

The observed mean change is \(-0.4\) units. It does not make the two-sided alternative directional: a sample mean in either direction can be relevant to the question “does it change?” The observed value is a sample summary; the alternative describes the population claim being assessed.

Why the Direction Changes the Evidence Considered

The alternative determines which sample results count as evidence against the null. To see why, consider a test statistic \(T\) whose null distribution is symmetric around zero, as in common \(z\) and \(t\) tests. A positive value indicates a sample difference in the positive direction, and a negative value indicates one in the negative direction.

For a two-sided alternative, results far from zero in either direction count as evidence. If the observed statistic is \(t_{\text{obs}}\), the two-sided p-value is the null probability of a value at least as far from zero as the observed value:

P\left(|T|\ge |t_{\text{obs}}|\mid H_0\right) =2P\left(T\ge |t_{\text{obs}}|\mid H_0\right),

where the second expression uses symmetry and applies to a continuous symmetric null distribution. For an upper-sided alternative, only results at least as large as the observed positive statistic count in that direction; for a lower-sided alternative, only results at least as small as the observed negative statistic count. Thus the selected alternative changes the relevant tail probability and can change the p-value. Choosing a tail after seeing the statistic would make the test’s evidence rule depend on the observed result.

This argument is a justification for deciding the alternative in advance, not a reason to choose whichever alternative produces a smaller p-value. The direction is part of the research claim and testing plan. A two-sided test is not interchangeable with a one-sided test just because the observed statistic points in the direction someone hoped to see.

Common Mistakes

Choosing the direction from the sample. Wrong reasoning: “The sample difference is positive, so use \(H_a:\Delta>0\).” Correction: choose \(H_a\) from the research question before examining the results. A positive sample difference can occur even when the question was two-sided or asked about a decrease.
Forgetting the subtraction order. If \(\Delta=\mu_A-\mu_B\), then \(\Delta>0\) means A has the larger population mean. If the difference is defined as \(\mu_B-\mu_A\), the same verbal claim requires the opposite inequality.
Treating “different” as directional. “Does A differ from B?” includes both \(\Delta>0\) and \(\Delta<0\), so it calls for \(\Delta\ne 0\), not just one inequality.
Assuming “improvement” always means an increase. A larger score may be desirable, but a lower symptom count or shorter recovery time may be desirable. Define the outcome and its units before deciding which sign represents the proposed benefit.
Using a one-sided alternative because it seems more powerful. A directional test addresses a narrower claim. It is justified only when the research question genuinely specifies that direction in advance, not as a way to make a result easier to declare statistically significant.

Key takeaway: Name the population parameter, state the subtraction order, and translate the research question into \(H_a\) before looking at results. Use \(\ne\) for a meaningful difference in either direction, \(>\) for an increase in the defined difference, and \(<\) for a decrease.

Check Your Understanding

For each item, decide whether the alternative is two-sided or directional, and write its direction using the stated parameter.

Let \(\Delta=\mu_A-\mu_B\). A researcher asks whether Group A’s population mean is higher than Group B’s. What is \(H_a\)?
Let \(\Delta=p_N-p_S\), where \(p_N\) and \(p_S\) are response proportions for new and standard treatments. The question is whether the response proportions differ. What is \(H_a\)?
For a paired measurement, define \(d=\text{after}-\text{before}\). The research question is whether the program lowers the population mean measurement. What is \(H_a\) in terms of \(\mu_d\)?
A sample yields \(\hat{p}_A-\hat{p}_B=-0.04\), but the pre-specified question asks whether \(p_A>p_B\). Should the alternative be changed to \(p_A<p_B\) after seeing the sample? Explain.
A lower number of symptom-days is considered beneficial. If \(\Delta=\mu_{\text{new}}-\mu_{\text{usual}}\), which sign of \(\Delta\) represents the proposed benefit, and what alternative expresses a decrease?