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News|Articles|October 9, 2026
Power of Two One-Sided Tests with Unequal Variances
Author(s)Prasanth Sambaraju
Equivalence test using two one-sided tests are widely used for demonstrating the comparability of treatment effects in different research fields. The method described in the manuscript aims to use simulations in Microsoft Excel to compute power for 2 one-sided tests for two groups with unequal variances.
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The two one-sided test (TOST) procedure to test equivalence is the most commonly method to establish equivalence. The 1992 FDA guidance document recommends that use of statistical analysis for pharmacokinetic measures, such as area under the curve and peak concentration, be based on TOST procedure.1 The TOST procedure is used to determine whether the average values for the pharmacokinetic measures determined after administration of reference and test products are comparable.1 It was observed that during method transfers that testing for equivalence using TOST is more appropriate than using two-sample t-test. The equivalence TOST test controls the type I error at 5% irrespective of method precision. In analytical method transfer, type I error represents the risk of accepting an unsuccessful method transfer.2 The TOST was also used to evaluate the comparability of two groups containing cleanability data.3
The test statistic is calculated by using the Equations 1 and 2, and the degrees of freedom is given by Equation 3. The equations are similar to the Student’s and Welch’s t-statistic for traditional significance test. The additional term used in the equation correspond to equivalence bound, which is subtracted from the mean difference between the 2 groups.4
Where 𝛿 = difference between the two means and this is the value at which power is computed
EL and EU = Lower and Upper equivalence limits
n1 and n2 = Sample size of first and second group respectively
s1ands2 = Standard deviation of first and second group respectively
υ= degrees of freedom calculated by using Satterthwaite approximation
The TOSTs are rejected if ∆U ≤ -t(df, α) and ∆L ≥ t(df, α), where t(df, α) is the upper 100α percentile of a t-distribution.5
The Student’s t-distribution (t-distribution) has one parameter, degrees of freedom (υ), which can be obtained by subtracting sample size minus one (for one-sample t-test). This parameter defines the shape of the t-distribution.6 The pdf (probability density function) of t-distribution is given by Equation 4:
where Γ(.) = Gamma function
y = probability of observing a particular value of x from the t-distribution.7
The cumulative distribution function (cdf) of the t-distribution is given by Equation 5:
where p is the probability that a single observation from the t distribution with υ degrees of freedom falls in the interval [-∞, x].8
Newton-Raphson method. Microsoft Excel built-in function for t-distribution can handle only integer values as degrees of freedom. When fractional degrees of freedom are used, then it truncates to an integer value.9 To address this limitation, the Newton-Raphson method was used to estimate critical t-value with fractional degrees of freedom. The Newton-Raphson method is an iterative root-finding algorithm, which is based on first-order Taylor series expansion. For a function f(x) at a point xn, root at f(x)=0 can be approximated as follows (Equations 6-9):
a) The first-order Taylor expression of f(x) at xn is
b) To find the root, set f(x) = 0
c) Solving for x
d) By iterative process, the estimate of xnis updated
The value of xn converges to the root of f(x) = 0, provided the initial value of x is close to the root and f(x) is a well-behaved function.10, 11
Numerical integration. Microsoft Excel can be used to implement trapezoidal rule for numerical integration. Trapezoidal rule works by approximating the function using a piecewise linear function and evaluating the integral of each piece. If the intervals [a, b] is divided into equal sub intervals (n) with width (h), then the approximate integral is given by Equation 10.12
Simulation using Data Table function. Thepower of TOST can be estimated without any programming by usingbuilt-in Data Table function in Microsoft Excel (Home → Data → What-If Analysis → Data Table). The number of simulations performed by using Data Table function can be increased to get a more precise estimate of power.13 The introduction to the application of Data Table and Data Table exercises can be found in reference 14.
Method
Critical value of t-distribution with fractional degrees of freedom. The iterative estimate of xn as described in Equation 9 was modified to include the (1-α) term, so that the critical t-value can computed at a specified confidence level.
The term f(xn) in Equation 11 corresponds to cdf value for t-distribution. This value was computed by numerical integration using trapezoidal rule in Microsoft Excel. The term f’(xn) in Equation 11 is the pdf, which is the derivative of cdf.15 The formulas to compute cdf, pdf, and the iterative estimate value of xn in Microsoft Excel are shown in Figure 1. A custom written Visual Basic for Application (VBA) code as shown Figure 2. was used to automate this computation.
Results and Discussion
The comparison of critical t-values for different degrees of freedom at various α values are included in Table 1. The results demonstrate that the critical t-values obtained from the method described are comparable with values obtained from R, which is an open-source statistical software.16 The error percentage was calculated using Equation 12. Table 2 shows the comparison of power value obtained using the above method with the reported values. The different formulas used for power calculations are included in Figure 3.
Conclusion
The TOST power obtained is found to be comparable with reported results. The method described in manuscript provides a way to compute power with fractional degrees of freedom using Microsoft Excel, without using statistical software. The Microsoft Excel spreadsheet used for calculations can be easily modified to compute power for different combinations of sample sizes and variances.
Supporting Files: The macro enabled Microsoft Excel file used to estimate critical t-value with fractional degrees of freedom, and the Microsoft Excel file used for power calculations are available on request.
References
1. Statistical Approaches to Establishing Bioequivalence, Guidance for Industry. FDA. Accessed April 23, 2026.
2. Kaminski L, Schepers U, and Wätzig H. Analytical method transfer using equivalence tests with reasonable acceptance criteria and appropriate effort: extension of the ISPE concept.J. Pharma. Biomed. Anal. 2010 53 (5), 1124–1129
3. Chen C, Rathore N, Ji W, and Germansderfer A. Statistical equivalence testing for assessing bench-Scale Cleanability. BioPharm Int. 2010 23 (2), 40-45.
4. Lakens D. Equivalence tests: A practical primer for t tests, correlations, and meta-analyses. Soc. Psychol. Pers. Sci. 2017 8 (4), 355-362.
5. Two-Sample T-Tests for Equivalence Allowing Unequal Variance. NCSS. Accessed April 23, 2026.
6. JMP Statistical Knowledge Portal. The t-Distribution. accessed January 07, 2026.
7. MATLAB. Student's t probability density function. Accessed April 23, 2026.
8. MATLAB. Student's t cumulative distribution function. Accessed April 23, 2026.
9. Microsoft Support. TTDIST function. Accessed April 23, 2026.
10. EARS1160. Numerical Methods. Accessed April 23, 2026.
11. Gawade S, Kudtarkar A, Sawant S, and Wadekar H., The Newton-Raphson Method: A detailed analysis. Int. J. Res. Appl. Sci. Eng. Technol. 2024 12 (11), 729-734.
12. El-Gebeily M and Yushau B. Numerical methods with MS Excel. Math. Enthus. 2007 4 (1), 84-92 .
13. Microsoft Help. Calculate multiple results by using a data table. accessed April 23, 2026.
14. Introduction to Excel2007 Data Tables and Data Table Exercises. Tools for Excel Modeling. Accessed January 07, 2026.
15. ECE302. Lecture 4.3 Cumulative Distribution Function. accessed April 23, 2026.
16. R. The R Project for Statistical Computing. accessed November 26, 2025.
About the Author
Prasanth Sambaraju is an independent researcher, prashanth.kng1@gmail.com.
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