# When is a one-sided hypothesis required?

*Author: Georgi Z. Georgiev, Published: Aug 6, 2018*

In many cases in both applied and theoretical science the decisions, claims and conclusions made depend on the direction of the observed effect in an experiment. When one wants to estimate the error statistics that best describe the data that warrants the decision, claim or conclusion, the error probability, usually a p-value or confidence interval, should be calculated under a one-sided null and complimentary one-sided alternative hypothesis.

Put simply, if the data-based claim is directional, the
appropriate statistic to report must be based on a directional hypothesis. **If one
is not satisfied in reporting "there is a discrepancy |δ|" without caring
if it was – δ or δ, then one needs a one-sided statistical
hypothesis**. Using a two-sided calculation would mean there is a disconnect
between the research hypothesis (claim) and the reported probability, resulting
in nominal probability which overestimates the actual probability (reported
p-value is larger than actual p-value), increasing the risk of type II errors.

For confidence intervals, if one is comparing a value to the upper or lower boundary of an interval and claiming that it is above or below it, then it should be a one-sided interval. Only if one is only interested if it is between the bounds or outside the bounds (regardless in which direction) are they in need of a two-sided interval.

## Example 1: Straightforward case for one-tailed significance test and CI

If the experiment is testing the efficacy of a new drug or
treatment, the research hypothesis is often that it is **more effective** than placebo
or an existing drug or treatment. The results which would reject the
corresponding null hypothesis that the drug has no effect or is in fact harming
the patient’s recovery would come from one side of the distribution of the outcome
variable.

Constructing a statistical hypothesis is therefore predicated on the one-sidedness of the claim we would like to make, e.g.:

"drug X is decreasing the risk of cardiovascular events,
observed relative risk difference δ = -0.3 (p=0.01; H_{0} : δ
≥ 0), 95%CI_{high}: -0.06"

or

"treatment Y increases survival rate of patients with Z, observed
increase is δ = 0.25 (p=0.004; H_{0} : δ ≤ 0), 95%CI_{low}:
0.15"

I would encourage reporting p-values for more precise claims
than H_{0} : δ ≤ 0 or H_{0} : δ ≥ 0,
probably additional to these conventional ones. The claims should be informed
by practical and scientific considerations and the wider context of the
experiment. Alternatively confidence intervals at different significance levels
or SEV(claim) for different claims can be reported. The goal in doing so is using
the data in the most-informative way.

## Example 2: A less obvious case for a one-sided hypothesis

Let us say we are studying whether height is correlated with lifetime earnings. It is an observational study so causal claims would not be allowed, but we would be happy with claims that there is some connection between the two (there are 5 different possible explanations for even the strongest statistically significant correlation).

**We would be happy to detect both a positive or a negative
correlation**, even though one of the results is likely predicted and explained
to some extent by existing literature while the other would be unexpected. In
this case the overwhelming majority of the literature seems to support a small
to moderate positive correlation.

Once the data gathered and t-values are calculated using the
observed direction of the effect, if one is to state that, for example "each
one-inch increase in height results in $789 increase in yearly income", then
the corresponding p-value (p_{1}) should be **one-tailed with a null
hypothesis that increases in height either have no relation or an inverse
relation to income**. If the outcome was in the unexpected direction and we want
to speak about decrease in height being related with an increase in yearly
income (or increase in height being related with a decrease in income) then the
one-sided test in the other direction should be reported.

If the null of "no effect" is of interest, then the claim
"we observed some correlation between height and income (p = p_{1} ⋅ 2)" can be made (p_{1}
is the p-value of a simple one-sided test as described above). If the size of
the discrepancy is to be reported it has to be stated that it is an absolute
value (without a sign) to avoid confusion. As with Example #1 specifying the
null hypothesis next to the p-value is recommended.

And now here is an example in which a two-sided p-value or CI might seem like the only choice, but under a more critical examination it seems a one-sided p-value and CI should at least be considered.

## Example 3: Quality control in a production environment

This seemed to me one of the intuitive situations in which a two-sided test would be warranted and there would be no need to report a one-sided value or CI. I now think differently.

If we are producing a certain good, it usually has parameters it must satisfy before it is released to the market.

In **acceptance sampling** we would randomly inspect a subset
of the products in a batch for conformity to standard and if defects are
discovered in more than the allowed proportion, the whole batch will need to be
discarded. This is a one-sided scenario since our null hypothesis is "no more
than X% of the batch is defective" and in rejecting it we would claim that
greater than X% of the batch is defective.

In **statistical process control** we would monitor the
produce as it comes out by inspecting random samples from it to see if they
deviate from the specification by more than an allowable amount. If they do,
then production might need to be halted until the error is fixed (changing
source material, machinery settings, broken or malfunctioning parts, etc.). This
is usually done by setting boundaries for deviations from the standard in both
directions, e.g. a product is too thick or too thin, too short or too long,
weighs too much or too little, contains too much or too little of a given ingredient.

Despite the usage a two-sided test when deciding whether there is an abnormal deviation in the production process, when we make statements about the direction of those deviations it would still be incorrect to use the two-sided CIs or p-values in some cases.

For example, if we claim "such a deviation is below the lower 95% two-sided CI bound hence we would expect to see such a negative deviation in only 5% of inspections" we would be wrong. In fact, we would only see it in 2.5% of inspections if the distribution of errors is symmetrical around zero. I can see such statements being made when an argument is put forth to make a certain adjustment in a production line to compensate for an unacceptable deviation in a certain direction.

I have to admit my very limited knowledge about quality control and thus I will not speculate how often such claims are made and how important it is for business decision-makers to have exact probabilities for them, but I think the scenario above in which the wrong type of statistical hypothesis is used is certainly possible.

Starting from the clear-cut case of claims for efficacy, improvement, increases or decreases of key performance indicators and going through less obvious cases I hope they provide a good idea about what claims require the supporting statistical analysis (p-value, CI, etc.) to be based on a one-sided hypothesis.

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#### Cite this article:

If you'd like to cite this online article you can use the following citation:

Georgiev G.Z., *"When is a one-sided hypothesis required?"*, [online] Available at: https://www.onesided.org/articles/when-to-use-one-sided-hypothesis.php URL [Accessed Date: 25 Jan, 2022].