Use this advanced **sample size calculator** to calculate the sample size required for a one-sample statistic, or for differences between two proportions or means (two independent samples). More than two groups supported for binomial data. **Calculate power** given sample size, alpha, and the minimum detectable effect (MDE, minimum effect of interest).

Quick navigation:

- Parameters for sample size and power calculations
- Calculator output

- Post-hoc power (Observed power)

## Using the power & sample size calculator

This calculator allows the evaluation of different statistical designs when planning an experiment (trial, test) which utilizes a Null-Hypothesis Statistical Test to make inferences. It can be used both as a **sample size calculator** and as a **statistical power calculator**. Usually one would determine the sample size required given a particular power requirement, but in cases where there is a predetermined sample size one can instead calculate the power for a given effect size of interest.

### Parameters for sample size and power calculations

**1. Number of test groups.** The sample size calculator supports experiments in which one is gathering data on a **single sample** in order to compare it to a general population or known reference value (one-sample), as well as ones where a control group is compared to one or more treatment groups (**two-sample, k-sample**) in order to detect differences between them. For comparing more than one treatment group to a control group the sample size adjustments based on the Dunnett's correction are applied. These are only approximately accurate and subject to the assumption of about equal effect size in all k groups, and can only support equal sample sizes in all groups and the control. Power calculations are not currently supported for more than one treatment group due to their complexity.

**2. Type of outcome**. The outcome of interest can be the **absolute difference of two proportions** (binomial data, e.g. conversion rate or event rate), the **absolute difference of two means** (continuous data, e.g. height, weight, speed, time, revenue, etc.), or the **relative difference** between two proportions or two means (percent difference, percent change, etc.). See Absolute versus relative difference for additional information. One can also calculate power and sample size for the mean of just a single group. The sample size and power calculator uses the Z-distribution (normal distribution).

**3. Baseline** The baseline mean (mean under H_{0}) is the number one would expect to see if all experiment participants were assigned to the control group. It is the mean one expects to observe if the treatment has no effect whatsoever.

**4. Minimum Detectable Effect**. The minimum effect of interest, which is often called the minimum detectable effect (**MDE**, but more accurately: MRDE, minimum *reliably* detectable effect) should be a **difference one would not like to miss**, if it existed. It can be entered as a proportion (e.g. 0.10) or as percentage (e.g. 10%). It is always relative to the mean/proportion under H_{0} ± the superiority/non-inferiority or equivalence margin. For example, if the baseline mean is **10** and there is a superiority alternative hypothesis with a superiority margin of **1** and the minimum effect of interest relative to the baseline is 3, then enter an MDE of **2**, since the MDE plus the superiority margin will equal exactly 3. In this case the MDE (MRDE) is calculated relative to the baseline plus the superiority margin, as it is usually more intuitive to be interested in that value.

If entering means data, one needs to specify the mean under the null hypothesis (worst-case scenario for a composite null) and the standard deviation of the data (for a known population or estimated from a sample).

**5. Type of alternative hypothesis**. The calculator supports **superiority**, **non-inferiority** and **equivalence** alternative hypotheses. When the superiority or non-inferiority margin is zero, it becomes a classical left or right sided hypothesis, if it is larger than zero then it becomes a true superiority / non-inferiority design. The equivalence margin cannot be zero. See Types of null and alternative hypothesis below for an in-depth explanation.

**6. Acceptable error rates**. The type I error rate, **α**, should always be provided. Power, calculated as **1 - β**, where β is the type II error rate, is only required when determining sample size. For an in-depth explanation of power see What is statistical power below. The type I error rate is equivalent to the significance threshold if one is doing p-value calculations and to the confidence level if using confidence intervals.

### Calculator output

The **sample size calculator will output** the sample size of the single group or of all groups, as well as the total sample size required. If used to solve for power it will output the power as a proportion and as a percentage.

## Why is sample size determination important?

While this online software provides the means to determine the sample size of a test, it is of great importance to understand the context of the question, the "why" of it all.

Estimating the required sample size before running an experiment that will be judged by a statistical test (a test of significance, confidence interval, etc.) allows one to:

- determine the sample size needed to detect an effect of a given size with a given probability
- be aware of the magnitude of the effect that can be detected with a certain sample size and power
- calculate the power for a given sample size and effect size of interest

This is crucial information with regards to making the test cost-efficient. Having a proper sample size can even mean the difference between conducting the experiment or postponing it for when one can afford a sample of size that is large enough to ensure a high probability to detect an effect of practical significance.

For example, if a medical trial has low power, say less than 80% (β = 0.2) for a given minimum effect of interest, then it might be unethical to conduct it due to its low probability of rejecting the null hypothesis and establishing the effectiveness of the treatment. Similarly, for experiments in physics, psychology, economics, marketing, conversion rate optimization, etc. Balancing the risks and rewards and assuring the cost-effectiveness of an experiment is a task that requires juggling with the interests of many stakeholders which is well beyond the scope of this text.

## What is statistical power?

Statistical power is the **probability of rejecting a false null hypothesis with a given level of statistical significance**, against a particular alternative hypothesis. Alternatively, it can be said to be the probability to detect with a given level of significance a true effect of a certain magnitude. This is what one gets when using the tool in **"power calculator"** mode. Power is closely related with the **type II error** rate: β, and it is always equal to (1 - β). In a probability notation the type two error for a given point alternative can be expressed as ^{[1]}:

**β(T _{α}; μ_{1}) = P(d(X) ≤ c_{α}; μ = μ_{1})**

It should be understood that the type II error rate is calculated at a given point, signified by the presence of a parameter for the function of beta. Similarly, such a parameter is present in the expression for power since POW = 1 - β ^{[1]}:

**POW(T _{α}; μ_{1}) = P(d(X) > c_{α}; μ = μ_{1})**

In the equations above **c _{α}** represents the critical value for rejecting the null (significance threshold), d(X) is a statistical function of the parameter of interest - usually a transformation to a standardized score, and μ

_{1}is a specific value from the space of the alternative hypothesis.

One can also calculate and plot the whole power function, getting an estimate of the power for many different alternative hypotheses. Due to the S-shape of the function, power quickly rises to nearly 100% for larger effect sizes, while it decreases more gradually to zero for smaller effect sizes. Such a power function plot is not yet supported by our statistical software, but one can calculate the power at a few key points (e.g. 10%, 20% ... 90%, 100%) and connect them for a rough approximation.

Statistical power is directly and inversely related to the significance threshold. At the zero effect point for a simple superiority alternative hypothesis power is exactly 1 - α as can be easily demonstrated with our power calculator. At the same time power is positively related to the number of observations, so increasing the sample size will increase the power for a given effect size, assuming all other parameters remain the same.

### Post-hoc power (Observed power)

Power calculations can be useful even after a test has been completed since failing to reject the null can be used as an argument for the null and against particular alternative hypotheses to the extent to which the test had power to reject them. This is more explicitly defined in the severe testing concept proposed by Mayo & Spanos (2006).

Computing observed power is only useful if there was no rejection of the null hypothesis and one is interested in estimating **how probative the test was towards the null**. It is absolutely **useless** to compute post-hoc power for a test which resulted in a statistically significant effect being found ^{[5]}. If the effect is significant, then the test had enough power to detect it. In fact, there is a 1 to 1 inverse relationship between observed power and statistical significance, so one gains nothing from calculating post-hoc power, e.g. a test planned for α = 0.05 that passed with a p-value of just 0.0499 will have exactly 50% observed power (observed β = 0.5).

I strongly encourage using this power and sample size calculator to compute observed power in the former case, and strongly discourage it in the latter.

## Sample size formula

The formula for calculating the sample size of a test group in a one-sided test of absolute difference is:

where Z_{1-α} is the Z-score corresponding to the selected statistical significance threshold *α*, Z_{1-β} is the Z-score corresponding to the selected statistical power *1-β*, *σ* is the known or estimated standard deviation, and *δ* is the minimum effect size of interest. The standard deviation is estimated analytically in calculations for proportions, and empirically from the raw data for other types of means.

The formula applies to single sample tests as well as to tests of absolute difference between two samples. A proprietary modification is employed when calculating the required sample size in a test of relative difference. This modification has been extensively tested under a variety of scenarios through simulations.

## Types of null and alternative hypotheses in significance tests

When doing sample size calculations, it is important that the null hypothesis (H_{0}, the hypothesis being tested) and the alternative hypothesis is (H_{1}) are well thought out. The test can reject the null or it can fail to reject it. Strictly logically speaking it cannot lead to acceptance of the null or to acceptance of the alternative hypothesis. A null hypothesis can be a **point** one - hypothesizing that the true value is an exact point from the possible values, or a **composite** one: covering many possible values, usually from -∞ to some value or from some value to +∞. The alternative hypothesis can also be a point one or a composite one.

In a Neyman-Pearson framework of NHST (Null-Hypothesis Statistical Test) the alternative should exhaust all values that do not belong to the null, so it is usually composite. Below is an illustration of some possible combinations of null and alternative statistical hypotheses: superiority, non-inferiority, strong superiority (margin > 0), equivalence.

All of these are supported in our power and sample size calculator.

Careful consideration has to be made when **deciding on a non-inferiority margin, superiority margin or an equivalence margin**. Equivalence trials are sometimes used in clinical trials where a drug can be performing equally (within some bounds) to an existing drug but can still be preferred due to less or less severe side effects, cheaper manufacturing, or other benefits, however, non-inferiority designs are more common. Similar cases exist in disciplines such as conversion rate optimization ^{[2]} and other business applications where benefits not measured by the primary outcome of interest can influence the adoption of a given solution. For equivalence tests it is assumed that they will be evaluated using a two one-sided t-tests (TOST) or z-tests, or confidence intervals.

Note that our calculator does not support the schoolbook case of a point null and a point alternative, nor a point null and an alternative that covers all the remaining values. This is since such cases are non-existent in experimental practice ^{[3][4]}. The only two-sided calculation is for the equivalence alternative hypothesis, all other calculations are **one-sided (one-tailed)**.

## Absolute versus relative difference and why it matters for sample size determination

When using a sample size calculator it is important to know what kind of inference one is looking to make: about the absolute or about the relative difference, often called percent effect, percentage effect, relative change, percent lift, etc. Where the fist is **μ _{1} - μ** the second is

**μ**or

_{1}-μ / μ**μ**(%). The division by μ is what adds more variance to such an estimate, since μ is just another variable with random error, therefore a test for relative difference will require larger sample size than a test for absolute difference. Consequently, if sample size is fixed, there will be less power for the relative change equivalent to any given absolute change.

_{1}-μ / μ x 100For the above reason it is important to know and state beforehand if one is going to be interested in percentage change or if absolute change is of primary interest. Then it is just a matter of fliping a radio button.

#### References

1 Mayo D.G., Spanos A. (2010) – "Error Statistics", in P. S. Bandyopadhyay & M. R. Forster (Eds.), Philosophy of Statistics, (7, 152–198). *Handbook of the Philosophy of Science*. The Netherlands: Elsevier.

2 Georgiev G.Z. (2017) "The Case for Non-Inferiority A/B Tests", [online] https://blog.analytics-toolkit.com/2017/case-non-inferiority-designs-ab-testing/ (accessed May 7, 2018)

3 Georgiev G.Z. (2017) "One-tailed vs Two-tailed Tests of Significance in A/B Testing", [online] https://blog.analytics-toolkit.com/2017/one-tailed-two-tailed-tests-significance-ab-testing/ (accessed May 7, 2018)

4 Hyun-Chul Cho Shuzo Abe (2013) "Is two-tailed testing for directional research hypotheses tests legitimate?", *Journal of Business Research* 66:1261-1266

5 Lakens D. (2014) "Observed power, and what to do if your editor asks for post-hoc power analyses" [online] http://daniellakens.blogspot.bg/2014/12/observed-power-and-what-to-do-if-your.html (accessed May 7, 2018)

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