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Correspondence  |   February 2005
Improved Statistical Methods for Quantal Assay
Author Affiliations & Notes
  • Yehuda Ginosar, B.Sc., M.B.B.S.
    *
  • * Hadassah Hebrew University Medical Center, Jerusalem, Israel. ,
Article Information
Correspondence
Correspondence   |   February 2005
Improved Statistical Methods for Quantal Assay
Anesthesiology 2 2005, Vol.102, 477-478. doi:
Anesthesiology 2 2005, Vol.102, 477-478. doi:
In Reply:—
We appreciate Dr. Pace’s taking the trouble to examine our data on spinal bupivacaine for cesarian delivery.1 Dr. Pace obtains results nearly identical to ours for successful operation but different results for successful induction, particularly the estimate of ED95. Dr. Pace graciously attributes these differences to our use of NONMEM (NONMEM Project Group, University of California, San Francisco, CA), and raises several important questions about using NONMEM for binary data.
Unfortunately, the explanation for the difference is more prosaic. On reexamining our analysis, we discovered an error in our data files analyzed by NONMEM. Dr. Pace correctly inferred our success/failure data from figure 1 of the article. We refit the data using both NONMEM and Excel (Microsoft, Redmond, WA), obtaining results identical to those of Dr. Pace (table 1).
Table 1. Successful Anesthesia at Different Doses of Intrathecal Bupivacaine: ED50and ED95for Successinductionand Successoperation. 
Image not available
Table 1. Successful Anesthesia at Different Doses of Intrathecal Bupivacaine: ED50and ED95for Successinductionand Successoperation. 
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Believing that the differences in results were related to the use of NONMEM, Dr. Pace raised four objections to our modeling approach. His objections raise several important considerations about logistic regression analysis to which we wish to respond.
First, Dr. Pace observes that the steepness parameter γ has no mechanistic interpretation. It simply describes the shape of the distribution.
Second, Dr. Pace points out that estimating interindividual variability with only a single observation in each individual is highly suspect. We agree, which is why we did not estimate interindividual variability in our analysis. As pointed out by Dr. Pace, this has already been addressed in Anesthesiology. “Naive pooled data” is the correct description of the approach taken when data from multiple patients is fit while ignoring intersubject variability, as was done in this article.
Third, NONMEM and Rplus are performing identical calculations, and obtain nearly identical results. We also repeated the analysis with Excel, with assistance from Steven L. Shafer, M.D. (Professor, Anesthesiology Service, Palo Alto Veterans Affair System, Palo Alto, California) and obtained identical estimates to those from NONMEM. We are puzzled at Dr. Pace’s concerns about “extremely complex [calculations], involving many assumptions.” The objective function for logistic regression is simply the calculation of a probability. The probability of multiple observations is the product of the probability of each individual observation, a readily computed number. Finding the parameters of the model that maximize probability of the observations with a naive pooled data approach requires virtually no assumptions at all, which is why a simple Excel spreadsheet and NONMEM generate identical results. We do observe, though, that it is not appropriate to use the Laplacian transformation when estimating single-subject data (i.e.  , when using a naive pooled data approach). Although we reported in the manuscript that the Laplacian option was used, it had no effect because interindividual variability was not estimated.
Dr. Pace suggests fitting ED95rather than ED50. We agree and wish to elaborate on this observation. With a little algebra, one can express ED50, the dose associated with 50% probability of success, as a function of ED95, the dose associated with 95% probability of success. To be specific, ED50= 0.05263261/λED95This can be readily substituted into the relation between dose and probability:
yielding the formula to estimate the ED95from the data, rather than estimate the ED50from the data,
More generally, if one wishes to estimate EDx, the effective dose associated with probability x, the formula is simply
These formulas can be used in NONMEM, Excel, or any other estimation tool. The point estimates from such an estimation should be identical with those derived by calculating ED95from ED50and γ, but if the program can generate SEs, confidence intervals can be constructed from the SE estimates. This is the only real advantage of estimating ED95directly, but confidence bounds about the ED95estimate may be clinically important. Again, whether one uses NONMEM or another tool is irrelevant; all should return nearly identical answers, provided they maximize likelihood.
Fourth, we agree that the standard quantal sigmoid Emax model enforces a logarithmic dose transformation. Fortunately, drugs tend to work on a more or less logarithmic scale, and we think about drugs on a log scale (e.g.  , a typical clinical guideline is “the correct dose is half to twice the nominal dose,” which reflects log spacing of dose). It is nonsensical to say “the standard model enforces X, Y, or Z” because, by definition, the standard is what it is. The users of the model are free to change it any way they desire, but of course, the changed model is then no longer the “standard” model. In this case, the user is free to substitute log dose, Exp(dose), or any desired transformation of dose into the “standard quantal sigmoid Emax” model if the user believes that is a better reflection of either the data or the underlying biology. Of course, the result is no longer the standard model but a nonstandard model, and the use of a nonstandard model would have to be justified based on pharmacology, biology, or goodness of fit to the data. However, that is only a modest hurdle for the modeler. Dr. Pace notes that the observations were evenly spaced doses, suggesting a logarithmic transformation seemed unnecessary. The spacing of the doses has little to do with the preferred model, because it is the response to the dose, not the spacing of the doses, that determines the shape of the curve.
We agree that, in general, if ED95is of clinical interest, an ED95calculated from all the data (i.e.  , reestimating the model parameterized in terms of ED95and γ), rather than calculated from the ED50and γ, is probably the optimal method of analysis. The point estimate will be the same, but the SEs will be estimated about the parameter of clinical interest. However, using NONMEM or any other estimation tool should give essentially identical results for this simple problem. The standard quantal sigmoid Emax model, mathematically modified as shown above, is perfectly appropriate to calculate ED95or any other clinically relevant dosage.
Even though the results changed slightly with the corrected data analysis, the conclusion remains unchanged. The data strongly support using doses of 11–12 mg bupivacaine for single-shot spinal anesthesia for cesarean delivery. The use of smaller doses increases the likelihood of inadequate neuraxial anesthesia.
* Hadassah Hebrew University Medical Center, Jerusalem, Israel. ,
Reference
Reference
Ginosar Y, Mirikatani E, Drover DR, Cohen SE, Riley ET: ED50and ED95of intrathecal hyperbaric bupivacaine coadministered with opioids for cesarean delivery. Anesthesiology 2004; 100:676–82Ginosar, Y Mirikatani, E Drover, DR Cohen, SE Riley, ET
Table 1. Successful Anesthesia at Different Doses of Intrathecal Bupivacaine: ED50and ED95for Successinductionand Successoperation. 
Image not available
Table 1. Successful Anesthesia at Different Doses of Intrathecal Bupivacaine: ED50and ED95for Successinductionand Successoperation. 
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