For the latest version of JMP Help, visit JMP.com/help.


Design of Experiments Guide > Sample Size Explorers > Power Explorers for Hypothesis Tests > Power for Two Independent Sample Equivalence of Means
Publication date: 06/21/2023

Power for Two Independent Sample Equivalence of Means

Use the Power for Two Independent Sample Equivalence of Means Explorer to determine a sample size for an equivalence test of two groups. Select DOE > Sample Size Explorers > Power > Power for Two independent Sample Equivalence. Explore the trade offs between variability assumptions, sample size, power, significance, and the equivalence range. Sample size and power are associated with the following hypothesis test:

Equation shown here or Equation shown here

versus the alternative:

Equation shown here

where μ1 and μ2 the true group means and (δm, δM) is the equivalence range. For the same significance level and power, a larger sample size is needed to detect a small difference than to detect a large difference. It is assumed that the populations of interest are normally distributed.

Power Explorer for Two Independent Sample Equivalence Settings

Set study assumptions and explore sample sizes using the radio buttons, text boxes, and menus. The profiler updates as you make changes to the settings. Alternatively, change settings by dragging the cross hairs on the profiler curves.

Test Type

Options to specify your test.

Equivalence

Specifies a test for equivalence of the mean to a reference value.

Superiority

Specifies a test for superiority of the mean to a reference value.

Non-inferiority

Specifies a test for non-inferiority of the mean to a reference value.

Upper Margin

Specifies the maximum value, above which the mean is considered different from the reference mean

Lower Margin

Specifies the minimum value, below which the mean is considered different from the reference mean.

Use symmetric bounds

Select for symmetric margins or bounds. If both bounds are negative, the upper bound is set to the positive of the lower bound. If both bounds are positive, the lower bound is set to the negative of the upper bound. If bounds are on either side of zero, the upper bound is set to the absolute value of the largest bound and the lower bound is then set to the negative of the upper bound.

Note: Typically, the equivalence margin is symmetric. However, it does not have to be symmetric.

Preliminary Information

Alpha

The probability of a type I error, which is the probability of rejecting the null hypothesis when it is true. It is commonly referred to as the significance level of the test. The default alpha level is 0.05.

Population Standard Deviation

Specifies the distribution for calculations.

Yes

Specifies known group standard deviations, calculations use the z distribution.

No

Specifies unknown group standard deviations, calculations use the t distribution.

Power Explorer for Two Independent Sample Equivalence Profiler

The profiler enables you to visualize the impact of sample size assumptions on the power calculations.

Total Sample Size

Specifies the total number of observations (runs, experimental units, or samples) needed for your experiment. Select Lock to lock the total sample size

Solve for

Enables you to solve for a sample size, difference to detect, or a group standard deviation.

Power

Specifies the probability of rejecting the null hypothesis when it is false. With all other parameters fixed, power increases as sample size increases.

Group 1 Sample Size

Specifies the number of observations (runs, experimental units, or samples) needed for Group 1 in your experiment.

Group 2 Sample Size

Specifies the number of observations (runs, experimental units, or samples) needed for Group 2 in your experiment.

Difference to Detect

Specifies the smallest difference between the group means that you want to be able to declare statistically significant.

Group 1 StdDev (σ1)

Specifies the assumed standard deviation for one of your groups, Group 1. An estimate of the error standard deviation could be the root mean square error (RMSE) from a previous model fit.

Group 2 StdDev (σ2)

Specifies the assumed standard deviation for the second group, Group 2. An estimate of the error standard deviation could be the root mean square error (RMSE) from a previous model fit.

Note: Adjusting the sample size for one group adjusts the total sample size unless the total sample size is locked. In that case, adjusting the sample size for one group adjust the sample size for the second group. Use the text boxes to specify group sample sizes.

Power Explorer for Two Independent Sample Equivalence Options

The Explorer red triangle menu and report buttons provide additional options:

Simulate Data

Opens a data table of simulated data based on the explorer settings. View the simulated response column formula for the settings used.

Make Data Collection Table

Creates a new data table that you can use for data collection. The table includes scripts to facilitate data analysis.

Save Settings

Saves the current settings to the Saved Settings table. This enables you to save a set of alternative study plans. See Saved Settings in the Sample Size Explorers.

Help

Opens JMP help.

Statistical Details for the Power Explorer for Two Independent Sample Equivalence

The power calculations for testing equivalence of two group means is based on methods described in Chow et al. (2008).

If σ1 and σ2 are unknown, the power (1-β) is computed as follows:

Equation shown here

where:

Equation shown here

α is the significance level

n1 and n2 are the group sample sizes

σ1 and σ2 are the assumed group standard deviations

δ is the difference to detect

(δm, δM) is the equivalence range

t1-α,ν,is the (1 - α)th quantile of the central t-distribution with ν degrees of freedom

T(t; ν, λ) is the cumulative distribution function of the non-central t distribution with ν degrees of freedom and non-centrality parameter λ.

If σ is known, then power (1-β) is computed as follows:

Equation shown here

Want more information? Have questions? Get answers in the JMP User Community (community.jmp.com).