IN this issue of the Journal, Lu et al. explore the arcane subject of modeling binary data using population analysis, a technique that determines the response of the typical individual, as well as inter- and intraindividual variability. 1They demonstrate that when there are small numbers of observations per individual, the population approach to data analysis results in a hugely biased estimate of the Hill coefficient in logistic regression. The article mentions minimum alveolar concentration (MAC) only in passing, but their findings raise the question, Is MAC fundamentally flawed?
MAC is among the most useful concepts in anesthetic pharmacology. MAC establishes a common measure of potency for inhaled anesthetic drugs: the partial pressure at steady state associated with 50% probability of movement to noxious stimulation (e.g. , incision). We use the concept of MAC to provide uniformity to our dosage of inhaled anesthetic drugs, establish the relative amounts of drug for different endpoints (e.g. , MACawake, MACBAR, MACthe knife), characterize drug interactions (e.g. , MAC-reduction), and guide our search for mechanisms of anesthetic action (the concentration responsible for biologic effects must be similar to MAC).
One of the great mysteries of anesthetic action is that MAC is so consistent. The inhaled anesthetic drugs are unique in pharmacology in their incredibly small amount of pharmacodynamic variability. Within a population, MAC varies by not more than 10–15% among individuals. 2,3MAC varies from species to species by approximately the same amount as it does from individual to individual. 4Someday, when we understand the mechanism of inhaled anesthetic action, we will look back on this low variability in MAC and think “it was so obvious that the mechanism had to be X, because only that could have accounted for the low variability.”
Lu et al. demonstrate that the type of study used to determine MAC in humans might produce highly biased underestimates of variability. By definition, MAC in humans is the concentration associated with 50% probability of response to initial incision. It is limited to initial incision to provide a uniform experimental design. However, because there is only one initial incision in a patient, you only get one lousy bit of information per patient: response or no response. There is no room for partial responses—either the patient responded or didn't. It takes eight patients to make a single byte of data.
This Editorial View accompanies the following article: Lu W, Ramsay JG, Bailey JM: Reliability of pharmacodynamic analysis by logistic regression: Mixed-effects modeling. Anesthesiology 2003; 99:1255-62.
A consequence of the minimal data in each observation is that estimates of MAC and its variability are vulnerable to bias. Paul and Fisher observed that the classic “up–down” experimental design to determine MAC could be expected to produce errors in MAC of 10%, and that variability in MAC was systematically underestimated. 5In a previous manuscript, Lu and Bailey demonstrated that when patient-to-patient differences are ignored, and the data are treated as arising from one giant rat (called the naïve pooled data approach), the steepness of the concentration versus response curve is grossly under estimated. 6 Figure 1shows the probability versus response relationship in many individuals (thin lines ) and the apparent curve that would result from treating the data as though arising from one individual (thick line ).
In the current article, the authors ask the question, Could population analysis describe representative individuals (fig. 1, thin lines ) and thus correct the “error” of the thick line in figure 1? Their results are quite disconcerting. They demonstrate that it takes at least 10 observations per subject to get an unbiased estimate of the Hill coefficient with the population approach. To understand the reason for this, consider a study with only two observations per patient. With two observations, there are four possibilities for the concentration versus response relationship as shown in figure 2. The thin curve has a very large Hill coefficient. This curve perfectly predicts the observations in 2A–C but provides a perfectly terrible fit of the observations in 2D. The thick curve provides OK fits of all the data points (similar to the thick curve in fig. 1). However, if A, B, C, and D were all individuals in the same study, a population approach would average the nearly infinite Hill coefficients of A, B, and C (thin lines ), with something more modest to fit D. The average of three near-infinities and something less than infinity still yields an enormous value for the Hill coefficient. Because the Hill coefficient is directly related to the SD of MAC, 7could the low variability in MAC be an artifact of the data analysis?
Fortunately, the early MAC studies preceded modern population analysis techniques and simply used the giant rat analysis technique. 8–10More recent studies continue to use the giant rat analyses technique. 11–13As a result, virtually all MAC studies estimate the response shown by the thick line in figure 1and do not attempt to estimate the response in individuals (fig. 1, thin lines ). This is a good thing. First, clinicians want to set their doses at concentrations at which the majority of individuals are anesthetized, which is the dose determined using the giant rat analysis technique. Second, the probability of response versus concentration curve is by definition steeper in each individual than in the population as a whole. Because the population as a whole shows variability of just 10% or less, in each individual the curve must be almost vertical, with individuals moving from 100% chance of responding to zero chance of responding, with very small increments in concentration. This agrees with clinical practice.
Although the observations of Lu et al. thus do not invalidate the conclusions of MAC studies to date, they convincingly demonstrate that studies with only a single observation per subject will never establish the concentration versus response curve in individuals, at least not by using population analysis techniques. More important, the article by Lu et al. reinforces the previous message of Paul and Fisher: In human MAC studies, each individual literally contributes one bit of data. As a result, modest differences in MAC values between two groups in a study, or when compared with historical controls, may be an artifact unless very careful statistical measures are used to compare the groups.