Bioequivalence and Bioavailability Forum

Hi Mittyri and Antigoni,

❝ As I see the problem is in quantile function, looks like approximation is not correct for low DF

Confirmed. I checked the 64bit-versions of PHX/WNL 6.3, 6.4, 7.0, and 8.0 (Linear Mixed Effects and WNL Linear Model 502, 1–7 degrees of freedom). The critical t-value is not reported in Model 502. But we can ­calculate it from the confidence limit(s), the estimate, and its standard error as    (UnivarCI_Upper – Estimate) / StdError or (Estimate – UnivarCI_Lower) / StdError.
 Interesting  Scary that the reported critical t-value in Linear Mixed Effects is also wrong! I didn’t expect that… Results are identical (full numeric precision) in all versions of PHX/WNL. The custom function tinv(0.025, df) introduced in v6.4 is practically correct.

df   R/OO/Gnumeric    LME (rep’d)      %RE     502 (calc)       %RE  tinv(0.025, df)    %RE 
 1 12.706204736175 11.300117888951 –11.0661 16.500679825590 +29.8632 12.706204736175 ±0.0000
 2  4.302652729749  4.270551416408  –0.7461  4.327589042282  +0.5796  4.302652729750 +0.0000
 3  3.182446305284  3.178617037716  –0.1203  3.183479996442  +0.0325  3.182434925786 –0.0004
 4  2.776445105198  2.775582256369  –0.0311  2.776407538431  –0.0014  2.776445033314 –0.0000
 5  2.570581835636  2.570308254272  –0.0106  2.570544577883  –0.0014  2.570581685935 –0.0000
 6  2.446911851145  2.446804416220  –0.0044  2.446913617243  +0.0001  2.446911763977 –0.0000
 7  2.364624251593  2.364575396335  –0.0021  2.364643234749  +0.0008  2.364624210734 –0.0000

Given that, use only LME (map time as Regressor, concentration as Dependent, drag time from Regressors/Covariates to Model Specification). After execution I suggest these workarounds:

• In v6.3 (might work in earlier versions as well; untested!).
  • Data > New > Worksheet; rename to tcrit
    Two columns: Denom_DF and tinv
    To get the critical t-values:
    ♦ R-code
    for (df in 1:28) {
  cat(sprintf("%i %17.14f", df, qt(0.025, df, lower.tail=FALSE)), "\n")
}
    ♦ Open Office Calc, Gnumeric, or current version of M$-Excel
    column A: integer degrees of freedom starting with 1
    name column A as df
    column B: =TINV(0.05, df): drag down to the maximum DFs
    ♦ or enter them manually from UsableStats.
    
  • Navigate to Linear Mixed Effects > Output Data > Final Fixed Parameters > Send To > Data > Merge Worksheets
    Sort: Denom_DF
    Included Column: Effect_Level, Estimate, StdError
    Worksheet 2: tcrit
    Sort: Denom_DF
    Included Column: tinv
    Execute.
    
  • Output Data > Result > Send To > Data > Data Wizard
    Add a filter: Action: Exclude, Find: NULL, Column: Effect_Level
    Add a filter: Action: Exclude, Find: int, Column: Effect_Level
    Add > Transformation >
    Transformation Type: Custom
    New Column Name: CL_lo
    Formula: Estimate-tinv*StdError
    Add > Transformation >
    Transformation Type: Custom
    New Column Name: CL_hi
    Formula: Estimate+tinv*StdError
  
  
• In ≥6.4:
  • Results > Final Fixed Parameters > Send To > Data > Data Wizard > Add > Transformation >
    Transformation Type: Custom
    New Column Name: CL_lo
    Formula: Estimate-tinv(0.025, Denom_DF)*StdError
    Add another one named CL_hi, Formula: Estimate+tinv(0.025, Denom_DF)*StdError

In BE we don’t need the CI of fixed effects and the degrees of freedom are generally high. However, I’m dumbfounded by such defects in commercial software. Even for high degrees of freedom tinv(x, y) is three orders of magnitude “better” than what is given by LME/BE. Why is it not used?
df   R/OO/Gnumeric       LME = BE        %RE        tinv(0.025, 28)   %RE
28 2.04840714179524  2.04840711367364 –1.37·10–6  2.04840714175209 –2.11·10–9

If you don’t trust in the open source software I used, below an excerpt of R.A. Fisher’s Statistical Methods for Research Workers (1934), the Geigy tables (1980), WolframAlpha, and UsableStats:
df             t(p 0.025)
 1  12.706 12.7062 12.706 12.70620474
 2   4.303  4.3027  4.303  4.30265273
 3   3.182  3.1824  3.182  3.18244912
 4   2.776  2.7764  2.776  2.77644504
 5   2.571  2.5706  2.571  2.57058169
 6   2.447  2.4469  2.447  2.44691177
 7   2.365  2.3646  2.365  2.36462421
28   2.048  2.0484  2.048  2.04840714