Bioequivalence and Bioavailability Forum

Dear all,

due to Detlew’s detective work (THX!) I see it clearer now. Seems that it was a bug in earlier versions.

“Equivalence Tests for the Ratio of Two Means in a Higher-Order Cross-Over Design (Log-Normal Data)” were added in PASS v14. Although nothing is stated about an improvement/update in later versions, according to the online manual (identical to the one which came with the PASS 2020 Trial) I could reproduce the examples with the internal function power.PASS() of PowerTOST. Code upon request.
All examples for ABE {0.8000|1.2500}, α 0.05.

Example 1 – Finding Power: ABB|BAA (Manual 545-9)
 CV  N reported shifted    nct  exact PT.shifted PT.nct PT.exact
0.4 10   0.0000  0.0000 0.0000 0.0299     0.0000 0.0000   0.0285
0.4 20   0.3051  0.3051 0.3111 0.3120     0.3060 0.3118   0.3126
0.4 30   0.5858  0.5858 0.5887 0.5887     0.5861 0.5889   0.5889
0.4 40   0.7483  0.7483 0.7501 0.7501     0.7484 0.7503   0.7503
0.4 60   0.9035  0.9035 0.9045 0.9045     0.9035 0.9045   0.9045
0.4 80   0.9627  0.9627 0.9633 0.9633     0.9627 0.9634   0.9634
The default in power.PASS() is the approximation by the shifted central t-distribution (method = "shifted"), although the noncentral t (method = "nct") and the exact method by Owen’s Q (method = "exact") are implemented as well. Columns starting with PT give results obtained by power.TOST().
Confirmed that PASS uses the shifted t, which I solely used in the other examples. Good agreement with power.TOST().

Example 2 – Finding Sample Size: ABB|BAA (Manual 545-11)
 CV target  Power reported N1   pwr1 N2   pwr2 PT.N.shifted PT.pwr.shifted PT.N.exact PT.pwr.exact
0.4    0.8 0.8026       45 45 0.8024 46 0.8119           46         0.8119         46       0.8134
0.4    0.9 0.9035       60 60 0.9035 60 0.9035           60         0.9035         60       0.9045
Good agreement (N1) though in practice one would round up to N2 in order to get balanced sequences like in all sample size-functions of PowerTOST.

Example 3 – Validation using Chen et al. (1997): AA|BB|AB|BA (Manual 545-12)
    SE  CV target  Power reported N1   pwr1 N2   pwr2 PT.N.shifted PT.pwr.shifted PT.N.exact PT.pwr.exact
0.1003 0.1    0.8 0.8106       16 16 0.8106 16 0.8106           16         0.8151         16       0.8239
0.1003 0.1    0.9 0.9085       20 20 0.9085 20 0.9085           20         0.9104         20       0.9192
Good agreement again. Note that in order to reproduce the results of Chen et al. – despite CV is stated in the paper – we have to work with the standard error of residuals. Here it is with 0.1002505 close to the CV of 0.1 but see also there.

Now the troublesome one of the OP.
Example 4; PASS 15.05.5: ABBA|BAAB (SE instead of CV)
    SE  CV target  Power reported N1   pwr1 N2   pwr2 PT.N.shifted PT.pwr.shifted PT.N.exact PT.pwr.exact
0.5329 0.5    0.8 0.8053       54 55 0.8053 56 0.8123           50          0.812         50       0.8128
I could not reproduce it exactly (the different design constants and dfs due to carry-over cut also in) but it explains what is going on in this earlier version of PASS and the discrepancy to sampleN.TOST().

Now what we can expect* in PASS 2000 2020 (and perhaps in a version >15):
Example 5 = 4; PASS 2000 (use CV)
 CV target N1  pwr1 N2  pwr2 PT.N.shifted PT.pwr.shifted PT.N.exact PT.pwr.exact
0.5    0.8 49 0.804 50 0.812           50          0.812         50       0.8128
Seemingly OK.

Conclusion: If you use PASS, update to v2000 v2020. If you are a sponsor receiving a sample size estimation in an earlier version, demand an update (or use PowerTOST ).

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• Expect, yes. Will you get these values? No.
  Still not corrected in PASS²⁰²⁰.