Two one-sided t-tests

For ln-transform or data already ln-transformed, the first t-test is a left-tail test of the hypotheses:

H₀: trueDifference < ln(1 – fractionToDetect) (bioinequivLeftTest)
H₁: trueDifference ³ ln(1 – fractionToDetect) (bioequivLeftTest)

The test statistic for performing this test is:

t₁=((TestLSM – RefLSM) – ln(1 – fractionToDetect))/DiffSE

The p-value is determined using the t-distribution for this t-value and the degrees of freedom. If the p-value is <0.05, then the user can reject H₀ at the 5% level, i.e. less than a 5% chance of rejecting H₀ when it was actually true.

For log10-transform or data already log10-transformed, the first test is done similarly using log10 instead of ln.

For data with no transformation, the first test is (where Ratio here refers to the true ratio of the Test mean to the Reference mean):

H₀: Ratio < 1 – fractionToDetect
H₁: Ratio ³ 1 – fractionToDetect

The test statistic for performing this test is:

t₁=[(TestLSM/RefLSM) – (1 – fractionToDetect)]/RatioSE

where the approximation RatioSE=DiffSE/RefLSM is used.

The second t-test is a right-tail test that is a symmetric test to the first. However for log₁₀ or ln-transforms, the test will be symmetric on a logarithmic scale. For example, if the percent of reference to detect is 20%, then the left-tail test is Pr(<80%), but for ln-transformed data, the right-tail test is Prob(>125%), since ln(0.8) = –ln(1.25).

For ln-transform or data already ln-transformed, the second test is a right-tail test of the hypotheses:

H₀: trueDifference > –ln(1 – fractionToDetect) (bioinequivRightTest)
H₁: trueDifference £ –ln(1 – fractionToDetect) (bioequivRightTest)

The test statistic for performing this test is:

t₂=((TestLSM – RefLSM)+ln(1 – fractionToDetect))/DiffSE

For log10-transform or data already log10-transformed, the second test is done similarly using log₁₀ instead of ln.

For data with no transformation, the second test is:

H₀: Ratio > 1+fractionToDetect
H₁: Ratio £ 1+fractionToDetect

The test statistic for performing this test is:

t₂=((TestLSM/RefLSM) – (1+fractionToDetect))/RatioSE

where the approximation RatioSE=DiffSE/RefLSM is used.

The output for the two one-sided t-tests includes the t₁ and t₂ values described above, the p-value for the first test described above, the p-value for the second test above, the maximum of these p-values, and total of these p-values. The two one-sided t-tests procedure is operationally equivalent to the classical interval approach. That is, if the classical (1 – 2 x alpha) x100% confidence interval for difference or ratio is within LowerBound and UpperBound, then both the H₀s given above for the two tests are also rejected at the alpha level by the two one-sided t-tests.