Bioequivalence and Bioavailability Forum

Hi ElMaestro,

❝ […] there must be a step for derivation of the denominator DF which happens to coincide with the DF you get via the Welch correction when just doing ordinary unpaired t-test with unequal variances.

Given the two variances and sample sizes PHX/WNL’s degrees of freedom agree with R (21.4306 for the full data set and 10.7555 for the modified one); checked to 14 significant digits.

According to Phoenix’ User’s Guide

The linear model used in linear mixed effects modeling is:

Y~N(Xb,V)

where X is a matrix of fixed known constants, b is an unknown vector of constants, and V is the variance matrix dependent on other unknown parameters. Wald (1943)¹ developed a method for testing hypotheses in a rather general class of models. To test the hypothesis H₀: Lb = 0, the Wald statistic is:

(Lβ)^T[L([X^TV^–1X)^–1L^T]^–1(Lβ)

where β and V are estimators of β and V. L must be a matrix such that each row is in the row space of X. The Wald statistic is asymptotically distributed under the null hypothesis as a χ² random variable with q = rank(L) degrees of freedom. In common variance component models with balanced data, the Wald statistic has an exact F-distribution. Using the χ² approximation results in a test procedure that is too liberal. LinMix uses a method based on analysis techniques described by Allen and Cady (1982)² for finding an F-distribution to approximate the distribution of the Wald statistic.
Giesbrecht and Burns (1985)³ suggested a procedure to determine denominator degrees of freedom when there is one numerator degree of freedom. Their methodology applies to variance component models with unbalanced data. Allen derived an extension of their technique to the general mixed model.

… followed by two pages in matrix-notation I can’t comprehend (aka hazelnut brain).

Let’s have a look at the modified data set (full precision).
s²₁ 0.129240461704103, n₁ 12
s²₂ 0.563916088422299, n₂ 9
If I trust Wikipedia
(s²₁+s²₂)²/[s⁴₁/(n₁–1)+s⁴₂/(n₂–1)] = 11.64240178 (≠ 10.75546079)

Aha!
(s²₁/n₁+s²₂/n₂)²/{s⁴₁/[n₁²(n₁–1)]+s⁴₂/[n₂²(n₂–1)]} = 10.75546079



Funny enough the radio buttons above turn out to be a gimmick without any effect. Whatever you choose – the dfs are identical.

Strange:

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1. Wald A. Tests of statistical hypotheses concerning several parameters when the number of observations is large. Transactions of the American Mathematical Society 1943;54(3):426–82.*
   
2. Allen DM, Cady FB.[/b] Analyzing Experimental Data By Regression. Belmont: Lifetime Learning Publications; 1982.
   
3. Giesbrecht FG, Burns JC. Two-Stage Analysis Based on a Mixed Model: Large-Sample Asymptotic Theory and Small-Sample Simulation Results. Biometrics. 1985;41(2):477–86.

• Wow, 57 pages! This must be serious and important stuff since in comparison Mr Einstein needed only 31 pages to describe special relativity (in German, which is more verbose than English)…
  Einstein A. Zur Elektrodynamik bewegter Körper. Annalen der Physik. 1905;4(17–5): 891–921.