Bioequivalence and Bioavailability Forum • Rounding

Helmut
★★★

Vienna, Austria,
2013-01-01 17:51
(4494 d 02:08 ago)

Posting: # 9773
Views: 14,273

Rounding [Software]

Hi Simulators!

How does your software round? Mine:

Excel 2000:       round(125.005, 2) → 125.01

OO Calc:          round(125.005, 2) → 125.01

Gnumeric 1.10.16: round(125.005, 2) → 125.01

WinNonlin 5.3:    round(125.005, 2) → 125.01

Phoenix 6.3:      round(125.005, 2) → 125.01

JavaScript:       Math.round(125.005*100)/100 → 125.01

PHP:              round(125.005, 2) → 125.01

PHP ≥5.3.0:       round(125.005, 2, PHP_ROUND_HALF_UP) → 125.01

                  round(125.005, 2, PHP_ROUND_HALF_DOWN) → 125.00

                  round(125.005, 2, PHP_ROUND_HALF_EVEN) → 125.00

                  round(125.005, 2, PHP_ROUND_HALF_ODD) → 125.01

R 2.15.2:         round(125.005, 2) → 125.00

R claims to be compliant with IEC 60559. In normal life not sooo important, but in simulations. Example (10⁶ sim’s, empiric alpha ~ T/R 1.25 in conventional balanced cross-overs).

 n   CV   alpha        alpha

         no rounding  CI rounded

42  15%  0.049842     0.050746 (sig. >0.05)

36  17%  0.050010     0.050706 (sig. >0.05)

@ElMaestro: Do you round in C?

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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d_labes
★★★

Berlin, Germany,
2013-01-02 13:51
(4493 d 06:07 ago)

(edited on 2013-01-03 09:31)
@ Helmut
Posting: # 9777
Views: 13,121

Rounding

Post reply

Hi Helmut!

❝ How does your software round?

I'm currently not in reach of my SAS but I strongly suppose that it will give 125.01 if one uses the ordinary ROUND() function. To be in accordance with the R method (or other software) one has to use the ROUNDE() function which rounds to even.

❝ Mine:

❝ Excel 2000: round(125.005, 2) → 125.01

Excel 2010: round(125.005,2) → 125.01
[edit]Just checked (03-Jan-2013)
SAS 9.2: round(125.005,0.01) → 125.01
(Note the definition of rounding to 2 decimals as rounding to multiples of 0.01)
[/edit]

❝ OO Calc: round(125.005, 2) → 125.01

❝ ...

❝ R 2.15.2: round(125.005, 2) → 125.00

Seems the R folks did it again the way different from what all others do :cool:

❝ In normal life not sooo important

A more drastically example, may be important also in normal life I think:

round(c(0.5,1.5,2.5,3.5,4.5,5.5),0)

[1] 0 2 2 4 4 6

At least in Germany one would expect:
[1] 1 2 3 4 5 6

❝ ... but in simulations. Example (10⁶ sim’s, empiric alpha ~ T/R 1.25 in conventional balanced cross-overs).

❝ n CV alpha alpha

❝ no rounding CI rounded

❝ 42 15% 0.049842 0.050746 (sig. >0.05)

❝ 36 17% 0.050010 0.050706 (sig. >0.05)

Wow! Seems rounding the CI anyhow is not such a good idea.
[edit]Also the rounding seems to be implied by regulatory definitions of the acceptance range as 80.00 - 125.00 (EMA & FDA)![/edit]

But ... IMHO this must be independent from the used rounding method for 5 except the magnitude of deviation from the not rounded results since all the other results >125.0000 - 125.0049 will be also counted as BE if rounded and as not BE if not rounded.

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2013-01-04 18:35
(4491 d 01:23 ago)

@ d_labes
Posting: # 9786
Views: 13,106

Rounding

Post reply

Dear Detlew!

❝ SAS 9.2: round(125.005,0.01) → 125.01

❝ (Note the definition of rounding to 2 decimals as rounding to multiples of 0.01)

There is a similar function in Excel (has to be activated in Add-ins: Analysis Toolpak):
mround(125.005,0.01) → 125.01
But – like R:
Maxima: round(125.005*100)/100 → 125.00

❝ A more drastically example, may be important also in normal life I think:

❝ round(c(0.5,1.5,2.5,3.5,4.5,5.5),0)

❝ [1] 0 2 2 4 4 6

❝ At least in Germany one would expect:

❝ [1] 1 2 3 4 5 6

Splendid.

❝ Wow! Seems rounding the CI anyhow is not such a good idea.

Yes. Not so important for the conventional acceptance range, but we have double rounding for NTIDs (AR 90.00–111.11%). Must not forget to state theta2=1.1111 in sampleN.TOST explicitly.

❝ [edit]Also the rounding seems to be implied by regulatory definitions of the acceptance range as 80.00 - 125.00 (EMA & FDA)![/edit]

Well, the width of FDA's/EMA's AR is not 45% but 125.004–79.995=45.009%…

</nitpick>

❝ But ... IMHO this must be independent from the used rounding method for 5 except the magnitude of deviation from the not rounded results since all the other results >125.0000 - 125.0049 will be also counted as BE if rounded and as not BE if not rounded.

Right. Maybe I will run a set and save individual results to a file. In the meantime I finished my balanced cross-over sim’s. n 12–60, CV 6–100%. With my grid I have 306·10⁶ sim’s each. With rounding I got 24 (7.84%) empiric alphas significantly >0.05; without 11 (3.59%). If I take power.TOST(CV=…, n=…, theta0=1.25) as the gold standard and calculate the %RSE of the simulated alphas I see a positive bias of the rounding again:

Q I median Q III rounded -0.110 +0.218 +0.574 unrounded -0.404 -0.036 +0.325

Jiří showed me another gem. Try

sprintf("%.2f", round(125.005, 2)) sprintf("%.2f", 100*round(1.25005, 4))

Edit: 2012-01-05
After reading a lot of stuff on the net I guess the only way to get “commercial rounding” (the one we learned in school; 1–4 down, 5–9 up) in R without requiring a library is by a function (THX to an anonymous poster):

cround <- function(x,n){ z <- abs(x)*10^n z <- z + 0.5 z <- trunc(z) z <- z/10^n z*sign(x) }

round(c(79.994, 79.995, 125.004, 125.005), 2)

79.99  80.00 125.00 125.00

cround(c(79.994, 79.995, 125.004, 125.005), 2)

79.99  80.00 125.00 125.01

But note:

round(c(0.79994, 0.79995, 1.25004, 1.25005), 4)

0.7999 0.8000 1.2500 1.2501

cround(c(0.79994, 0.79995, 1.25004, 1.25005), 4)

0.7999 0.8000 1.2500 1.2501

Sometimes R drives me nuts.

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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d_labes
★★★

Berlin, Germany,
2013-01-05 20:25
(4489 d 23:33 ago)

@ Helmut
Posting: # 9787
Views: 12,601

Rounding

Post reply

Dear Helmut!

❝ ... In the meantime I finished my balanced cross-over sim’s. n 12–60, CV 6–100%. With my grid I have 306·10⁶ sim’s each. With rounding I got 24 (7.84%) empiric alphas significantly >0.05; without 11 (3.59%).

How do you interpret the un-rounded results?
Empirical evidence of an Alpha inflation of TOST :cool:

?
Hopefully not.

❝ Sometimes R drives me nuts.

Here I have another one:

round(c(1.2500499999999999, 1.250049999999999999), 4)

[1] 1.2500 1.2501

More trouble spots can be found in "The R inferno".
Should be a must-read for all R-users.

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2013-01-05 21:13
(4489 d 22:45 ago)

@ d_labes
Posting: # 9788
Views: 12,618

Rounding

Post reply

Dear Detlew!

❝ ❝ ... In the meantime I finished my balanced cross-over sim’s. n 12–60, CV 6–100%. With my grid I have 306·10⁶ sim’s each. With rounding I got 24 (7.84%) empiric alphas significantly >0.05; without 11 (3.59%).

❝

❝ How do you interpret the un-rounded results?

The jury is out. I ran two sim’s with identical seeds to compare the results. Nice intermediate result files (33.1MB unrounded, 14MB rounded). Have to find a clever way to filter for the suspects.

❝ Empirical evidence of an Alpha inflation of TOST :cool: ?

❝ Hopefully not.

Don’t think so. On the other hand it puts occasional significant alphas I got in Methods B/C/D into perspective.

Sim’s are sim’s are sim’s.

It’s like with lab values. The more your physician requests, the more “*” you’ll get – though not being ill. ;-)

❝ Here I have another one:

❝ round(c(1.2500499999999999, 1.250049999999999999), 4)

❝ [1] 1.2500 1.2501

You are a nasty person.

❝ More trouble spots can be found in "The R inferno".

❝ Should be a must-read for all R-users.

Wow!

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
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d_labes
★★★

Berlin, Germany,
2013-01-05 21:57
(4489 d 22:01 ago)

@ Helmut
Posting: # 9789
Views: 12,566

Sim’s are sim’s are sim’s

Post reply

Dear Helmut!

❝ Sim’s are sim’s are sim’s.

The Lord may give that some regulatory bodies could realize this fundamental Rule of thumb. Remember the story that Potvin Method C was called by them "alpha inflation" because some empirical alpha's of the sims of Potvin et. al. were slight above 0.05.

❝ ❝ Here I have another one:

❝ ❝ round(c(1.2500499999999999, 1.250049999999999999), 4)

❝ ❝ [1] 1.2500 1.2501

❝

❝ You are a nasty person.

You are welcome :-D

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2013-01-06 03:21
(4489 d 16:37 ago)

@ d_labes
Posting: # 9791
Views: 12,693

Another vicious circle

Post reply

Dear Detlew!

❝

❝ How do you interpret the un-rounded results?

Had to have a break. Let’s see what I got from n 12, CV 6%, 10⁶ sim’s:
50331 studies passed when BE was defined as the unrounded upper CL ≤1.25 (α_emp n.s.). 50503 studies passed based on round(CLhi,4) ≤1.25 (α_emp sign. >0.05). In increasing order (skipping 170):

    Unrounded.lo     Unrounded.hi  Rd.lo  Rd.hi

1.14406215516423 1.25000008857710 1.1441 1.2500

…

1.16009942086582 1.25004974319996 1.1601 1.2500

and

    Unrounded.lo         Unrounded.hi         Rounded.lo         Rounded.hi

Min.   :1.11998223   Min.   :1.25000009   Min.   :1.12000000   Min.   :1.25

1st Qu.:1.14730961   1st Qu.:1.25001184   1st Qu.:1.14730000   1st Qu.:1.25

Median :1.16346327   Median :1.25002528   Median :1.16345000   Median :1.25

Mean   :1.16222013   Mean   :1.25002554   Mean   :1.16221977   Mean   :1.25

3rd Qu.:1.17608605   3rd Qu.:1.25003838   3rd Qu.:1.17610000   3rd Qu.:1.25

Max.   :1.20299871   Max.   :1.25004974   Max.   :1.20300000   Max.   :1.25

Verdict: Duno. Wasn’t there a guy having a heading “Science vs. Regulations” in one of his slides? I would say the AR is based on the maximum acceptable ∆ and BE means 1–∆ ≤ CI ≤ (1–∆)^-1. No fucking rounding here.
It’s ^so_o_oo convenient^* that 100%/0.8 gives an integer. Stupidity starts with rounding the periodic decimal 100%/0.9=111.11% to 111.11% precisely (for Canada’s lumberjacks to 112.00%).
There is no rounding in theoretical statistics at all. So I have mixed feelings.

If I don’t round I’m closer to theory and it is less likely to get significant results.
Mr X will tell me “Nice simulations but not based on our BE criterion.”
If I round I get more significant results.
Mr X will tell me “Nice simulations proving the patient’s risk is not maintained.”

Convenience [sic] was the winning argument when regulations changed from an additive (AR 80–120%) to a multiplicative model (AR 80-125%). Really. See also this slide.

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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d_labes
★★★

Berlin, Germany,
2013-01-07 16:52
(4488 d 03:06 ago)

@ Helmut
Posting: # 9796
Views: 12,488

Where all these numbers came from?

Post reply

Dear Helmut!

❝ Had to have a break. Let’s see what I got from n 12, CV 6%, 10⁶ sim’s:

❝ 50331 studies passed when BE was defined as the unrounded upper CL ≤1.25 (α_emp n.s.). 50503 studies passed based on round(CLhi,4) ≤1.25 (α_emp sign. >0.05). In increasing order (skipping 170):

❝ Unrounded.lo Unrounded.hi Rd.lo Rd.hi

❝ 1.14406215516423 1.25000008857710 1.1441 1.2500

❝ …

❝ 1.16009942086582 1.25004974319996 1.1601 1.2500

❝ and

I'm not certain if I understand your numbers given. How do the above correspond to below? Numbers from cases which were judged BE if rounded and not BE if not rounded?

❝ Unrounded.lo Unrounded.hi Rounded.lo Rounded.hi

❝ Min. :1.11998223 Min. :1.25000009 Min. :1.12000000 Min. :1.25

❝ 1st Qu.:1.14730961 1st Qu.:1.25001184 1st Qu.:1.14730000 1st Qu.:1.25

❝ Median :1.16346327 Median :1.25002528 Median :1.16345000 Median :1.25

❝ Mean :1.16222013 Mean :1.25002554 Mean :1.16221977 Mean :1.25

❝ 3rd Qu.:1.17608605 3rd Qu.:1.25003838 3rd Qu.:1.17610000 3rd Qu.:1.25

❝ Max. :1.20299871 Max. :1.25004974 Max. :1.20300000 Max. :1.25

❝ If I round I get more significant results.

❝ Mr X will tell me “Nice simulations proving the patient’s risk is not maintained.”

Augmented with the reply “If and only if one uses your (assuming Mister X to be a regulator) f*#*g rule of rounding the CI's.”

BTW: My original question was more concerned with empirical alpha>0.05 significant without rounding. I wouldn't expect such cases to be real. Otherwise the theory behind our BE statistics is wrong.

BTW2: There is a question that bothers me, every time I think about it:
assuming BE if

   0.8 ≤ lCL and uCL ≤ 1.25 (I)

or better

   0.8 < lCL and uCL < 1.25  (II)

.
At least in formulating the bioequivalence alternative hypothesis it is always written:
Θ₁< µ_T/µ_R < Θ₂
and the corresponding two one-sided t-statistics have to be t_l < -t(1-α,df) and t_u > t(1-α,df). Does this transform really to (I) for the confidence interval inclusion rule? The EMA guidance is here clear: "To be inside the acceptance interval the lower bound should be ≥ 80.00% when rounded to two decimal places and the upper bound should be ≤ 125.00% when rounded to two decimal places." But the regulatory point of view is not necessarily the scientific one as we noticed more than once.

In case of no rounding this doesn't make much difference since lCL=0.8 and uCL=1.25 (without rounding) are obtained with probability of nearly zero. But in case of rounding ...

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2013-01-07 18:29
(4488 d 01:29 ago)

@ d_labes
Posting: # 9797
Views: 12,572

Flaw in the GL?

Post reply

Dear Detlew,

sorry for the confusing post.

Summary of CIs of the 50331 sim’s passing the unrounded criterion:

     Unrd.lo              Unrd.hi               Rd.lo                Rd.hi

Min.   :1.03984217   Min.   :1.15101402   Min.   :1.03980000   Min.   :1.15100000

1st Qu.:1.13630237   1st Qu.:1.23050864   1st Qu.:1.13630000   1st Qu.:1.23050000

Median :1.15065573   Median :1.23972299   Median :1.15070000   Median :1.23970000

Mean   :1.15006990   Mean   :1.23640877   Mean   :1.15006972   Mean   :1.23640889

3rd Qu.:1.16470993   3rd Qu.:1.24556820   3rd Qu.:1.16470000   3rd Qu.:1.24560000

Max.   :1.22939131   Max.   :1.24999989   Max.   :1.22940000   Max.   :1.25000000

Summary of CIs of the 50503 sim’s passing the rounded criterion:

     Unrd.lo              Unrd.hi               Rd.lo                Rd.hi

Min.   :1.03984217   Min.   :1.15101402   Min.   :1.0398000   Min.   :1.15100000

1st Qu.:1.13631513   1st Qu.:1.23056103   1st Qu.:1.1363000   1st Qu.:1.23060000

Median :1.15069540   Median :1.23976572   Median :1.1507000   Median :1.23980000

Mean   :1.15011128   Mean   :1.23645515   Mean   :1.1501111   Mean   :1.23645518

3rd Qu.:1.16475001   3rd Qu.:1.24561372   3rd Qu.:1.1647500   3rd Qu.:1.24560000

Max.   :1.22939131   Max.   :1.25004974   Max.   :1.2294000   Max.   :1.25000000

❝ ❝ Unrounded.lo Unrounded.hi Rd.lo Rd.hi

❝ ❝ 1.14406215516423 1.25000008857710 1.1441 1.2500

❝ ❝ …

❝ ❝ 1.16009942086582 1.25004974319996 1.1601 1.2500

❝

❝ I'm not certain if I understand your numbers given. How do the above correspond to below? Numbers from cases which were judged BE if rounded and not BE if not rounded?

❝ ❝ Unrounded.lo Unrounded.hi Rounded.lo Rounded.hi

❝ ❝ Min. :1.11998223 Min. :1.25000009 Min. :1.12000000 Min. :1.25

❝ ❝ 1st Qu.:1.14730961 1st Qu.:1.25001184 1st Qu.:1.14730000 1st Qu.:1.25

❝ ❝ Median :1.16346327 Median :1.25002528 Median :1.16345000 Median :1.25

❝ ❝ Mean :1.16222013 Mean :1.25002554 Mean :1.16221977 Mean :1.25

❝ ❝ 3rd Qu.:1.17608605 3rd Qu.:1.25003838 3rd Qu.:1.17610000 3rd Qu.:1.25

❝ ❝ Max. :1.20299871 Max. :1.25004974 Max. :1.20300000 Max. :1.25

The list of the the 172 studies failing the unrounded but passing the rounded criterion above were ordered by the unrounded upper CL. So the lowest was 1.2500000885771 and the highest 1.25004974319996. This matches the summary above.

❝ ❝ Mr X will tell me “Nice simulations proving the patient’s risk is not maintained.”

❝ Augmented with the reply “If and only if one uses your (assuming Mister X to be a regulator) f*#*g rule of rounding the CI's.”

I would try to use a different wording – likely after consulting our capt’n.

❝ BTW: My original question was more concerned with empirical alpha>0.05 significant without rounding. I wouldn't expect such cases to be real. Otherwise the theory behind our BE statistics is wrong.

I wouldn’t say simulations disprove theory here. The convergence is slow (see the plot for Method B). Significant results might be pure chance.

❝ BTW2: There is a question that bothers me, every time I think about it:

❝ assuming BE if

❝ 0.8 ≤ lCL and uCL ≤ 1.25 (I)

❝ or better

❝ 0.8 < lCL and uCL < 1.25 (II).

Ouch, that hurts! Wellek (2003), Patterson & Jones (2006), Hauschke et al. (2007), Chow and Liu (2009):
$$\begin{matrix}
H_0:\mu_\textrm{T}-\mu_\textrm{R}\,{\color{Red}\leq}\,\theta_\textrm{L}\;\textrm{or}\;\mu_\textrm{T}\,{\color{Red}\geq}\,\theta_\textrm{U}\\
H_\textrm{a}:\theta_\textrm{L}\,{\color{Green}<}\,\mu_\textrm{T}-\mu_\textrm{R}\,{\color{Green}<}\theta_\textrm{U}
\end{matrix}$$Minority report^*:$$-\theta_\textrm{A}\,{\color{Red}\leq}\,\mu_\textrm{T}-\mu_\textrm{R}\,{\color{Red}\leq}\,\,\theta_\textrm{A}$$

❝ At least in formulating the bioequivalence alternative hypothesis it is always written:

❝ Θ₁< µ_T/µ_R < Θ₂

Yep, based on the above.

❝ and the corresponding two one-sided t-statistics have to be t_l < -t(1-α,df) and t_u > t(1-α,df). Does this transform really to (I) for the confidence interval inclusion rule?

No. Transforms definitely into (II).

❝ The EMA guidance is here clear: "To be inside the acceptance interval the lower bound should be ≥ 80.00% when rounded to two decimal places and the upper bound should be ≤ 125.00% when rounded to two decimal places." But the regulatory point of view is not necessarily the scientific one as we noticed more than once.

Wonderful. You discovered a flaw in the GL! According to the model BE should not be [0.8, 1.25] (borders inclusive) but ]0.8, 1.25[ (borders exclusive).

❝ In case of no rounding this doesn't make much difference since lCL=0.8 and uCL=1.25 (without rounding) are obtained with probability of nearly zero. But in case of rounding ...

¡Fantástico!

             unrounded         rnd. (commerc.)        rnd. (R)   

         [80,125]  ]80,125[  [80,125]  ]80,125[  [80,125]  ]80,125[

 79.900     –         –         –         –         –         –

 79.995     –         –         +         –         +         –

 79.999     –         –         +         –         +         –

 80.000     +         –         +         –         +         –

 80.001     +         +         +         –         +         –

 80.005     +         +         +         +         +         –

 80.010     +         +         +         +         +         +

124.900     +         +         +         +         +         +

124.995     +         +         +         –         +         –

124.999     +         +         +         –         +         –

125.000     +         –         +         –         +         –

125.001     –         –         +         –         +         –

125.005     –         –         –         –         +         –

125.010     –         –         –         –         –         –

Verbeeck R, Musuamba FT. The Revised 2010 EMA Guideline for the Investigation of Bioequivalence for Immediate Release Oral Formulations with Systemic Action. J Pharm Pharmaceut Sci. 2012; 15(3): 376–8. online

P.S.: Another goodie from the FDA (see this post; downscaling the AR for NTIDs). Have a close look at this line of code:
theta=((log(1.11111))/0.1)**2;

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
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d_labes
★★★

Berlin, Germany,
2013-01-08 12:44
(4487 d 07:14 ago)

@ Helmut
Posting: # 9798
Views: 12,493

Flaw in the GL?

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Dear Helmut,

❝ ... Transforms definitely into (II).

In contrast to you I'm not quite sure.

It is explicitly stated that way (< and not ≤) in

Hauschke, Steinijans, Pigeot
"Bioequivalence Studies in Drug Development"
Wiley, Chichester, 2007, page 90

... but I have 2 other minority reports for you (proof/evidence by authority: "Well, Lieschen Mueller says it's true, so it must be." :cool:

)

Westlake, W.J.
"Symmetrical Confidence Intervals for Bioequivalence Trials"
Biometrics, 32, p 741-744 (1976)

stating the confidence interval inclusion rule explicitly with ≤ and

Diletti et.al.
"Sample size determination for bioequivalence assessment by means of confidence intervals"
Int. J. Clin. Pharm., Ther. and Tox., Vol.30, Supl. 1, p. S51-58 (192)

stating the two one-sided tests explicitly as

t₁=(m_T-m_R-ln(Θ₁))/(s_D*sqrt(2/n)) ≥ t(1-α,df) 

t₂=(m_T-m_R-ln(Θ₂))/(s_D*sqrt(2/n)) ≤ -t(1-α,df)

Other papers state the interval inclusion rule as
I ⊂ (Θ₁,Θ₂)
subset sign: an U rotated 90° clockwise
where I is an appropriate confidence interval. I as an amateur are not able to figure out what is meant: The meaning A⊆B, when A is called a subset of B; A can be equal to B (i.e. borders included). Or A⊂B, then A is called a proper subset of B; A cannot equal B (i.e. at least one border excluded).
That time the "Theory of sets" was dealt with I have skipped school :-D

.

This is only an incomplete selection of findings which led to my uncertainness. As stated above: Using real numbers (not rounded) it will not make much a difference how we implement it, thus we can't empirical test it via simulations.
Any pro-statistician out there to enlighten this issue?

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2013-01-08 20:08
(4486 d 23:51 ago)

@ d_labes
Posting: # 9799
Views: 12,604

What a mess!

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Dear Detlew,

❝ It is explicitly stated that way (< and not ≤) in

❝

❝ Hauschke, Steinijans, Pigeot

❝ "Bioequivalence Studies in Drug Development"

❝ Wiley, Chichester, 2007, page 90

That’s what I wrote above. See also on top of page 89. The Nulls are given including the boundaries and the alternatives excluding them.

❝ ... but I have 2 other minority reports for you (proof/evidence by authority: "Well, Lieschen Mueller says it's true, so it must be." :cool: )

❝ Westlake, W.J.

❝ "Symmetrical Confidence Intervals for Bioequivalence Trials"

❝ Biometrics, 32, p 741-744 (1976)

❝

❝ stating the confidence interval inclusion rule explicitly with ≤ and

❝

❝ Diletti et.al.

❝ "Sample size determination for bioequivalence assessment by means of confidence intervals"

❝ Int. J. Clin. Pharm., Ther. and Tox., Vol.30, Supl. 1, p. S51-58 (192)

❝

❝ stating the two one-sided tests explicitly as

❝ t₁=(m_T-m_R-ln(Θ₁))/(s_D*sqrt(2/n)) ≥ t(1-α,df)

❝ t₂=(m_T-m_R-ln(Θ₂))/(s_D*sqrt(2/n)) ≤ -t(1-α,df)

Yep, but before (p. S52):

H₀: ln µ_T/µ_R≤ln θ₁ or ln µ_T/µ_R≥ln θ₂ (bioinequivalence)
H₁: ln θ₁<ln µ_T/µ_R<ln θ₂ (bioequivalence)

Drives me nuts.

What about Mr Schuirmann (1987)…

H₀₁: µ_T–µ_R≤θ₁
H₁₁: µ_T–µ_R>θ₁

and

H₀₂: µ_T–µ_R≥θ₁
H₁₂: µ_T–µ_R<θ₁

Bioequivalent if

t₁= :blahblah: ≥t_1-α(ν) and t₂=≥t_1-α(ν)

Continuing with

“The two one-sided tests procedure turns out to be operationally identical to the procedure of declaring equivalence only if the ordinary 1-2α confidence (not 1-α) confidence interval for µ_T–µ_R is completely contained in the equivalence interval [θ₁, θ₂].”

Completely contained?

Kem Phillips (1990)

H₀: µ_T–µ_R<θ_L or µ_T–µ_R>θ_U
H₁: θ_L≤µ_T–µ_R≤θ_U

continuing with

“H₀ is rejected in favor of bioequivalence if T_L and -T_U equal or exceed t_1-α,ν […]”

And so on and so forth in many papers…

❝ Other papers state the interval inclusion rule as

❝ I ⊂ (Θ₁,Θ₂)

Oh yes. I use it sometimes myself as well.

❝ That time the "Theory of sets" was dealt with I have skipped school :-D .

When I was in school from one year to the next everything was given as sets. Was fashionable for a while. Didn’t bother me too much because I’ve spent many schooldays in one of the many Viennese coffee houses anyhow.

❝ This is only an incomplete selection of findings which led to my uncertainness. As stated above: Using real numbers (not rounded) it will not make much a difference how we implement it,…

Agree. In my home-brew BE software I didn’t round at all, but tested for θ_L ≤ 90% CI ≤ θ_U. Ever since I’m using commercial software I’m in limbo. I validated Phoenix/WinNonlin with data sets from the literature (and even a very small one manually). But none of them “scratched at the edge”. The manual isn’t helpful:

If the interval (CI_Lower, CI_Upper) is contained within LowerBound and UpperBound, average bioequivalence has been shown.

Contained? Meaning ⊂ or ⊆?

❝ … thus we can't empirical test it via simulations.

Not sure what you mean here. :confused:

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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d_labes
★★★

Berlin, Germany,
2013-01-09 11:33
(4486 d 08:26 ago)

@ Helmut
Posting: # 9801
Views: 12,382

What a mess!

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Dear Helmut,

❝ In my home-brew BE software I didn’t round at all, but tested for θ_L ≤ 90% CI ≤ θ_U.

Me too. From our sponsors perspective it's the better choice and if rounding comes into play it looks much better ;-)

. In that sense the R rounding is also a good choice.

❝ ❝ … thus we can't empirical test it via simulations.

❝ Not sure what you mean here. :confused:

I meant that simulations with θ_L ≤ 90% CI ≤ θ_U do not give substantial different results compared to θ_L < 90% CI < θ_U. Thus we can't choose that alternative which is closer to the analytical power calculation results (of course via PowerTOST :cool:

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2013-01-09 16:18
(4486 d 03:41 ago)

@ d_labes
Posting: # 9802
Views: 12,353

What a mess!

Post reply

Dear Detlew,

❝ ❝ ❝ … thus we can't empirical test it via simulations.

❝ ❝ Not sure what you mean here. :confused:

❝

❝ I meant that simulations with θ_L ≤ 90% CI ≤ θ_U do not give substantial different results compared to θ_L < 90% CI < θ_U. Thus we can't choose that alternative which is closer to the analytical power calculation results…

Ah, now I understand! Theoretically (!) it would be possible if one goes with the R-package Rmpfr allowing for high numeric precision or Maxima and run a zillion of sim’s. Would need a supercomputer in the backyard.

❝ … (of course via PowerTOST :cool: ).

What else? :-D

In my future sim’s of alpha (rounded CI) I will go for a ratio 0.80 instead of 1.25 since all three algos we have discussed give the same results there. Nevertheless, more studies than the 5% expected will pass making a significant result difficult to interpret.

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
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ElMaestro
★★★

Denmark,
2013-01-02 17:12
(4493 d 02:47 ago)

@ Helmut
Posting: # 9778
Views: 12,833

Rounding

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Hi HS,

❝ @ElMaestro: Do you round in C?

Yes, I gave it some thought in C. Perhaps a little oddly, but in the C standard C will round down for integers (i.e. if you convert 125.8 to an integer the standard will be to tell you the result is 125). For recent sims I therefore need to make sure that the compiler does not convert to integers before the comparison with 125 or 80 is done. This is done e.g. by comparison with a true double precision figure like 80.00 rather than 80.

—
Pass or fail!
ElMaestro

Helmut
★★★

Vienna, Austria,
2013-01-02 18:04
(4493 d 01:55 ago)

@ ElMaestro
Posting: # 9779
Views: 12,912

Abandon rounding

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Hi ElMaestro,

❝ Perhaps a little oddly, but in the C standard C will round down for integers

I see. Another hint that C is a child of Fortran (also rounding down).

❝ […] This is done e.g. by comparison with a true double precision figure like 80.00 rather than 80.

Yep. ~~In my future sim’s I will abandon rounding at all.~~ Edit: See above. In my sim’s I used the ratios (not percentages) rounded to 4 decimals.
round(x.xxxxx,4) works – the problem comes with round(xxx.xxx,2).

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

yjlee168
★★★

Kaohsiung, Taiwan,
2013-01-06 00:15
(4489 d 19:44 ago)

@ Helmut
Posting: # 9790
Views: 12,711

Rounding

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Dear Helmut,

❝ How does your software round? Mine:

❝ Excel 2000: round(125.005, 2) → 125.01

❝ OO Calc: round(125.005, 2) → 125.01

❝ Gnumeric 1.10.16: round(125.005, 2) → 125.01

❝ WinNonlin 5.3: round(125.005, 2) → 125.01

❝ Phoenix 6.3: round(125.005, 2) → 125.01

❝ JavaScript: Math.round(125.005*100)/100 → 125.01

❝ PHP: round(125.005, 2) → 125.01

❝ R 2.15.2: round(125.005, 2) → 125.00

Based on ICE 60559's rounding off a '5' standard (go to the even digit), "R 2.15.2: round(125.005, 2) ⇒ 125.00" is correct. Zero is considered as an even digit.

—
All the best,
-- Yung-jin Lee
bear v2.9.2:- created by Hsin-ya Lee & Yung-jin Lee
Kaohsiung, Taiwan https://www.pkpd168.com/bear
Download link (updated) -> here

Helmut
★★★

Vienna, Austria,
2013-01-06 04:29
(4489 d 15:30 ago)

@ yjlee168
Posting: # 9792
Views: 12,565

Rounding

Post reply

Hi Yung-jin,

❝ Based on ICE 60559's rounding off a '5' standard (go to the even digit), "R 2.15.2: round(125.005, 2) ⇒ 125.00" is correct. Zero is considered as an even digit.

I know – and the heated discussions on the R-Help list as well. :-D

As Detlew mentioned above I don’t expect regulators to be familiar with

round(c(0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5), 0)

[1]  0  2  2  4  4  6  6  8  8 10

^*

More (and an answer to your PM) tomorrow.

I call that bias towards even.

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

Helmut
★★★

Vienna, Austria,
2013-01-06 16:34
(4489 d 03:24 ago)

@ yjlee168
Posting: # 9793
Views: 12,612

Rounding

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Hi Yung-jin,

continuing our personal communication. THX for reminding me on the rule your professor in physical chemistry taught you. How could I forget this goodie?

if the number next to the required digit ≤4, then drop it: 125.003 ⇒ 125.00
if the number next to the required digit ≥6, then add 1 to the required digit: 125.008 ⇒ 125.01
if the number next to the required digit =5, then
1. if the required digit is odd, then add 1 to the required digit: 125.015 ⇒ 125.02
2. if the required digit is even, then drop it: 125.005 ⇒ 125.00

x = 124.990–125.010, step 0.0005; y = rounded to two decimal figures. Dashed line: y = 125.005

(1) Commercial rounding:
[image]

(2) R & Maxima:
[image]

(3) Chem. rounding:
[image]

Based on the regression I would prefer (3). But the down-rounding close to 125.005 extends even further up than in R & Maxima:

            (1)    (2)    (3)

124.9950  125.00 125.00 125.00

← rule 3.1
125.0045 125.00 125.00 125.00 ← rule 1
125.0050 125.01 125.00 125.00 ← rule 3.2
125.0055 125.01 125.01 125.00 ← rule 3.2
125.0059 125.01 125.01 125.00 ← rule 3.2
125.0060 125.01 125.01 125.01 ← rule 2

Edit: It’s clear that we are loosing information in rounding. No big deal if we don’t introduce bias. Nice that R & Maxima use ‘rounding half to even’ which is overall (i.e., [–∞, 0, +∞]) less biased than ‘rounding half away from zero’. However bias exists – just a different one.
In our case we have positive numbers only. ‘Best’ algo?

BTW, is disturbed me somewhat that most replies at R-Help were like “R complies with the standard. Full stop.”

As soon as we abandon our own reason, and are content
to rely upon authority, there is no end to our troubles. Bertrand Russell

A central lesson of science is that to understand complex issues
(or even simple ones), we must try to free our minds of dogma and
to guarantee the freedom to publish, to contradict, and to experiment.
Arguments from authority are unacceptable. Carl Sagan

Anyone who conducts an argument by appealing to authority
is not using his intelligence;
he is just using his memory. Leonardo da Vinci

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

yjlee168
★★★

Kaohsiung, Taiwan,
2013-01-07 00:38
(4488 d 19:21 ago)

@ Helmut
Posting: # 9794
Views: 12,636

Rounding

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Dear Helmut,

Firstly, I like to thank you for giving me the chance to look at rounding issue, although I don't think that I am a mathematician to deal with such a tough question. For me, rounding rule is just like a guideline to follow with. I will not take it as scientific issue, but just a rule or a standard. Otherwise, why do people make such a standard (IEC 60559)? Just like BE criteria is set as 90%CI of 80.00% ~ 120.00%. Fortunately, after discussion I find that the rounding rule I use is the same as IEC 60559 standard. So I will put IEC 60559 standard into my SOP of data analysis later.
So far, I don't think that R completely complies IEC 60559 standard in its rounding function (as your demonstration) under Windows (my computer OS). In R online help, it says that the rounding function may vary with different computer OS. Fortunately, in your case (round(125.005, digits=2), it complies.
I agree with that rounding will loose information. But sometimes it may also introduce bias too. Arbitrarily, we call it 'rounding error.' Because it can be difficult to estimate the rounding error, so we live with it. It will not introduce bias only when we take the final number to round off and present it. If this is the case, we don't have to worry about the bias.
Finally, I absolutely agree with you that sometimes R drives me nuts too.

—
All the best,
-- Yung-jin Lee
bear v2.9.2:- created by Hsin-ya Lee & Yung-jin Lee
Kaohsiung, Taiwan https://www.pkpd168.com/bear
Download link (updated) -> here