Given a random variable (think medical risk factor) X whose possible values x i ( i = 1 , ⋯ , n ) are realized with respective probabilities p i , the statistical entropy is defined by the expression

in which p ≡ ( p 1 , … , p n ) stands for the complete set of probabilities. For simplicity, the outcomes and probabilities in this section are treated as discrete quantities. In cases for which the variable is continuous, the formalism replaces a discrete probability function p x with a probability density function (PDF) p ( x ) and the summation over outcomes in Equation (1) with an integral as follows

Actually, extension of the theory to continuous variables entails redefining information entropy by incorporating a statistical measure function m ( x ) to preserve invariance under a transformation of coordinates [13]. This modification does not affect the outcomes of the analyses in this paper and therefore is not explicitly dealt with in the cases to follow. A statistical problem requiring the measure function that is worked out in detail can be found in Ref. [12].

The principle of maximum entropy (PME) asserts that maximizing the information entropy of a random variable subject to constraints posed by known information, usually in the form of expectation values, leads to the least biased statistical distribution of that variable. The term “least biased” means that the maximum entropy (MaxEnt) distribution makes use only of known prior information. Other input data that can bias the distribution, such as might be drawn from model assumptions, is not employed. As a consequence, the MaxEnt distribution does not generate properties of variables or relations among variables not consistent with the given information. For example, if the given information says nothing about the correlation of two variables, then the resulting MaxEnt distribution will lead to no correlation of those variables. This is not necessarily the case in other methods of statistical inference.

The MaxEnt distribution is also the most probable of any distribution constrained by the same given information. This characteristic follows from the Boltzmann-Planck relation [14],

expressing an exponential dependence of the multiplicity W on the statistical entropy S and number of trials N. Maximization of entropy S therefore leads to a maximum multiplicity W. The more ways an observable system can be realized, the higher is its probability of occurrence. In cases of application with large number of trials, the resulting MaxEnt distribution can be many orders of magnitude more probable than any other statistical distribution constrained by the same prior information.

Mathematically, the PME leads to a variational equation for the set of probabilities p . For example, in the absence of all prior information except for the probability completeness relation

maximization of expression (1) subject to constraint (4) leads to a system of equal probabilities

in accordance with the Principle of Insufficient Reason [15] (also referred to as the Principle of Indifference).

More generally, when additional information is given in the form of known expectation values F j ( j = 1 , ⋯ , m ) of a set of functions f j ( x i )

the variational function to be maximized takes the form

in which the set of functions λ k ( k = 0 , 1 , ⋯ , m ) are Lagrange multipliers to be determined from the given information. The symbol H, widely used in communication theory, is actually meant to be an upper case Greek eta, which stands for “Entropy”.

Variation of Equation (7)

with respect to each p i leads to the solution

in which the partition function Z ( λ )

has replaced the multiplier λ 0 after utilization of the completeness relation (4).

Knowledge of the partition function Z ( λ ) of a system permits one to calculate all the statistical information that can be known or inferred about that system. In particular,

Z ( λ ) is a normalization factor ensuring that the completeness relation Equation (4) is satisfied.First derivatives of Z ( λ ) with respect to each multiplier λ j yield expressions

for the given expectation values from which the set of Lagrange multipliers λ ≡ ( λ 1 , ⋯ , λ m ) can be determined.

Homogeneous second derivatives of Z ( λ ) give rise to expressions

for the (initially not given) variances of the functions f j ( x i ) .

Mixed second derivatives of Z ( λ ) yield 1^st order correlation functions

which reduce to Equation (12) for j = k .

Clearly, higher-order mixed derivatives of Z ( λ ) can generate higher correlation functions, such as worked out in Ref. [9] for a pair of correlated variables (human height and weight).

Further details regarding the foregoing methodology, including derivations of relations, can be found in Reference [9] and the collected papers of E. T. Jaynes [13].

In summary, implementation of the PME to a particular system of random variables yields a mathematical expression, similar to that of Equation (9), for the least biased, most probable statistical distribution of that system consistent with known prior information. The expression permits one to predict initially unknown uncertainties and correlations of functions of the variables in terms of the given information, including the simplest case (relevant to this paper) where f ( x i ) ≡ x i .

The preceding characteristics notwithstanding, one might inquire as to the reliability of the predictions drawn from the MaxEnt distribution in any given case. Perhaps the most spectacularly successful case is that of equilibrium statistical mechanics (ESM), the domain of physics for which the PME was first proposed. Given only the mean system energy E and number N of particles (which plays the same role as the number of trials Nin this paper), ESM can accurately account for the thermal properties of a vast array of systems ranging from the scale of elementary particles to that of stars, galaxies, and beyond. What makes this possible is the astronomical number of particles in the physical system. One mol (i.e. gram-molecular-mass) of a gas, for example, contains an Avogadro’s number (~6 × 10²³) of particles. Since the variance Δ 2 E of mean energy (or Δ 2 N of mean particle number) is inversely proportional to the number of particles, the ratio Δ 2 E / E is so small (generally on the order of ~10⁻¹²) that no further improvement in predictability is achieved by explicitly including the variances of energy and particle number in the analysis.

The MaxEnt distribution, however, does not require an astronomical number of samples for its success. A recent example of its spectacular reliability in a matter relating to health and medicine was demonstrated in the case of human weight and height [9]. As part of an investigation of the application of the body mass index (BMI) to identify and classify obesity [10], the author has shown that human height and weight are distributed empirically according to a joint lognormal distribution to such a degree of exactness, that one might suspect there is an underlying biophysical mechanism. Although there may well be such an intrinsic mechanism (that question is yet unanswered), use of the PME led to a MaxEnt distribution identical to the empirical one. Theoretical predictions agreed within statistical uncertainties with all moments and correlation functions testable with a large anthropometric data base [16] comprising several thousands of individuals in the male and female cohorts.

With regard to the incomplete reporting of medical risk factors by clinical laboratories, the MaxEnt distribution could in principle enable physicians to predict the variability of test results and thereby give more informed guidance to patients. However, the number of test measurements N of a specified risk factor is far fewer than the sample size in the two preceding examples. Ordinarily one presumes N = 1 because this information is not routinely included in the medical report.

In the following sections of this paper the MaxEnt distribution will be derived for several practical cases distinguished by prior information regarding (a) the number of trials (i.e. tests per variable) and (b) the independence of the variables. The MaxEnt distribution itself will reveal how sharp are its predictions in any particular case. And if the predicted statistics should turn out to be highly uncertain, that is not a failure of the PME. It is an indication that the paucity of information provided by the clinical laboratory is insufficient to permit a reliable estimate of a patient’s risk. More generally, as emphasized by Jaynes: “...the principle is most useful in just those cases where the empirical distribution fails to agree with the one predicted by maximum entropy.” [17].

A final point to note: The examples to follow use test data from an unidentified healthy patient. The predictive distributions to be derived from the principle of maximum entropy are general in that they depend solely on the kind of prior information provided; e.g. a single expectation value, or a pair of correlated expectation values, etc. Given a certain set of information, the MaxEnt procedure leads to a predictive distribution of fixed functional form containing unknown Lagrange multipliers. When the specific numerical values of the prior information are taken into account to solve for the Lagrange multipliers, the predictive distribution becomes applicable to an individual, rather than a statistical population.

Implementation of the procedure in Section 2.1 with function f ( x ) ≡ x and expectation value F ≡ 〈 X 〉 = x 0 leads to an exponential PDF

with partition function

and expectation

Thus, from Equations (14)-(16) the MaxEnt probability density takes the forms

in which the first is for a general exponential, and the second presents the PDF explicitly in terms of the prior information x 0 .

Following the procedure of Section 2.1, it is now possible to predict the variance of X

from which follows the standard deviation of a single trial

It is a well-known characteristic of the exponential distribution that the standard deviation of the variable equals the mean [18]. This property has profound implications for the diagnostic utility of risk factor test results reported by clinical laboratories.

As determined in Section 3.1, the least biased, most probable predictive distribution is exponential in form leading to a standard deviation also of magnitude x 0 . The reference range [ s L ,   s U ] is ordinarily thought of as the region within which a patient’s test result is “normal”. This inference can be misleading in several ways. The term “normal” evokes an image of the widely encountered Gaussian distribution, referred to as the “normal” distribution with implication that the test result suggests “healthy” or “low risk” because it falls within 1 or 2 standard deviations about the mean where most healthy people’s test result would fall. But this is not necessarily the case.

In a Gaussian distribution, which is symmetric about the mean, the probability of a result falling within a range of ± σ X about the mean x 0 is 68.3%. However, in the case of an exponential distribution, which is a monotonically decreasing function, the corresponding range is [ 0 ,   2 x 0 ] with integrated probability 86.5%. This suggests that the outcome of a repeat trial could fall almost anywhere within the full range, and therefore well outside the reference range.

To examine this property quantitatively, one can calculate from PDF (17) the probability P out that a point estimate falls outside the bounds of the reference range

Table 1 summarizes the values of P out for selected risk factors from two frequently requisitioned screening panels, Metabolic and Lipid, of a patient’s medical record. For each risk factor, the point estimate is within the reference range. However, the probability P out , calculated from Equation (20), is well over 50% that a repeated measurement from the same sample would fall outside the reference range in all cases except one. For the one exception no lower limit was posted for the reference range.

In regard to the content of Table 1 it is to be noted that the reference range of a risk factor is not an absolute quantity. Clinical labs, health service providers, and medical organizations may cite different ranges for the same risk factor [1]. The Lipid Panel provides an important timely example. Popular health and medical media have long conveyed to the public that high density lipoprotein cholesterol (HDL-C) is the “good” cholesterol and low density lipoprotein cholesterol (LDL-C) is the “bad” cholesterol. While this belief is largely valid, the implication that there is no upper threshold of harm for HDL-C or lower threshold of harm for LDL-C is not. Although still controversial issues, research has established a more nuanced situation whereby a too high HDL-C is associated with a variety of conditions, including elevated risk of dementia [19]-[22], and a too low LDL-C (hypocholesterolemia) is likewise associated with a variety of adverse conditions including cancer, pancreatitis, hyperthyroidism, and kidney failure [23][24]. Therefore, HDL-C and LDL-C values in Table 1 include reference ranges with upper and lower limits.

Table 1. Probability that a trial falls outside the reference range.

Another point to emphasize is that the test results in Table 1 would generally be regarded as indicative of a nominally healthy patient. Few primary care physicians or even specialists would be concerned (if they thought about it at all) that the results of a repeat trial from the same sample had a high probability of falling outside the reference range. This possibility would be regarded as a purely hypothetical, rather than practical, matter, since there is no second measurement provided by the clinical lab. Indeed, the fact that Equations (17) and (20) predict such results might be construed as a failure of the MaxEnt distribution, and one would wonder what the “true” probability distribution is.

It is important to understand that there is no “true” distribution. Probability is a measure of the knowledge one has of a system, not a characterization of the system, itself. The form taken by the MaxEnt probability function depends on what prior information is available. And the more information one has, the more reliably one can predict a system’s statistical behavior. This apparent subjectivity does not detract from the concept of probability as an objectively rigorous part of mathematics. When the PME is implemented correctly, two analysts with the same prior information about a system should be able to arrive at equivalent predictions of the system’s statistical properties.

In summary, the MaxEnt distribution does not guarantee agreement with an empirical distribution. What it has been proven to provide is the least biased, most probable predictive probability distribution consistent with the information available. Accordingly, as revealed by the content of Table 1, more information needs to be provided than a single point estimate of each risk factor if a clinical lab report is to be diagnostically useful to a physician.

In Case 1, only a point estimate x 0 was reported. In the absence of knowledge of the number of trials, the MaxEnt distribution was found to be of exponential form with mean and standard deviation both equal to x 0 and a Lagrange multiplier λ = x 0 − 1 . Now, given the number of trials N, how does the point estimate x 0 of indeterminate trial number relate to the mean x ¯ ?

Expressed as a relation connecting the expectation values of random variables ( X ¯ ,   X n ) , Equation (21) becomes

in which the expectation 〈 X n 〉 of each variable X n ( n = 1 , 2 , ⋯ , N ) is calculated by the integral

where p ( x ) is the MaxEnt PDF Equation (17). One finds from Equations (22) and (23) that the expectation of the sample mean equals the expectation of a single trial, or x ¯     =     x 0 .

When deriving the MaxEnt distribution, one ordinarily takes a given expectation value to be an exact quantity. Or, if this value is uncertain, then one supplements the information in the functional equation (7) with a known variance (or the equivalent). In the present case, the uncertainty in x 0 (and therefore x ¯ ) is not provided by the clinical lab. However, having been informed from Equation (21) that x ¯ is a sample mean of N trials, it is now possible to derive the PDF that governs its distribution. An expedient way to do this is to make use of the characteristic function (CF) [25] of the variable X

Equation (24) shows that the CF and PDF are Fourier transforms of one another.

From Equation (24) one obtains the CF of the variable X ¯ by applying the expectation operation to the function exp ( i   X ¯   t ) as follows [25]

The PDF of X ¯ is then the inverse Fourier transform of h X ¯ ( t )

where Γ ( N ) is the gamma function

equal to ( N − 1 )     ! for integer N. Evaluation of the integral in Equation (26) is accomplished by contour integration around the N t h -order pole at − i   N   λ in the lower half complex plane.

One sees from Equation (26) that the variable X ¯ representing the mean of Nindependent identically distributed variables is governed by a gamma distribution [18] with parameter N λ . Applying PDF Equation (26), one can calculate the first and second moments of X ¯

which validate the mean obtained in a different way in Equation (22) and now provide the best predictive estimate of the variance of X ¯

From Equation (30), it follows that the predicted standard deviation of the reported point estimate x 0

can be made arbitrarily small by increasing the number of independent measurements of the associated risk factor.

The number of measurements performed by a diagnostic lab, however, is at best likely to be relatively few, if not just one. Nevertheless, use of machine automation of multiple trials simultaneously from the same sample is a future possibility not to be excluded. To assess the impact of knowing the number of trials on the reduction in uncertainty, and therefore enhancement of reliability, of the MaxEnt predictions, two functions are of particular utility: the coefficient of variation CV

and the conditional probability P in ( x ¯ 0 | N ) that the mean x ¯ 0 of a repeat set of N trials

lies within the reference range. The two-parameter function of the form Γ ( a , z ) in Equation (33) is the incomplete gamma function

As an example of the information provided by knowledge of the number of trials, we consider again measurement of serum glucose—which is a risk factor for diabetes, among other conditions—as summarized in Table 1 for an indeterminate number of trials. Pertinent data in mg/dL are reproduced below for convenience:

Use of the preceding data in Equations (32) and (33) lead respectively to the blue traces in Figure 1 and Figure 2. The figures actually display results for Case 1, Case 2, and Case 3 (to be taken up in Section 5). In viewing the figures, note that the phrase “mean number” in the abscissa label is to be interpreted as “no given number” for Case 1, “exact number” for Case 2, and “mean number” for Case 3.

Figure 1. Coefficient of variation as a function of number of trials per sample for Case 1 (green), Case 2 (blue), and Case 3 (red). The cases are distinguished by (1) no number reported, (2) exact number reported, (3) mean number reported.

The blue trace in Figure 1 shows the variation of CV with N for Case 2. Theory predicts that about 10 trials suffice to reduce the CV from an initial value of 1 in Case 1 (a presumably single trial governed by an exponential distribution) to 0.3. However, to achieve a CV of 0.1 would require about 100 trials because of the slow asymptotic decrease of Equation (32). The CV for Case 1 is shown in the Figure as a horizontal solid green trace without discrete markers spanning all values of N since no prior information regarding the number of trials was provided. Thus, while it is natural to presume only 1 trial was made, the PME makes no use of speculative assumptions.

The blue trace in Figure 2 shows the variation of P in with N, a perspective complementary to the probability values given in Table 1. From Figure 2, one infers that there is a 50% probability that 10 trials suffice to produce a mean outcome that falls within the reference limits. If only a single trial is actually (rather than presumably) performed, then this probability is about 15%, consistent with the complementary probability given in Table 1. However, to achieve a 90% probability that a set of measurements yield a mean within the reference limits would require about 100 trials. The horizontal solid green trace in the figure represents the outcome of Case 1, as discussed in the preceding paragraph for Figure 1.

Figure 2.Probability, as a function of number of trials per sample, that the mean outcome of a repeated set of trial falls within the reference interval for Case 1 (green), Case 2 (blue), and Case 3 (red). Case description is the same as for Figure 1.

It is to be borne in mind that the point estimates of the risk factor in both Cases 1 and 2 are exactly the same number x 0 . In Case 1, the physician was totally uninformed as to how the number was arrived at. In Case 2, the physician learned additionally that the number was the mean of N trials—hence the initial notation x ¯ for that number. As a consequence of that extra bit of information, the most probable prediction of risk dramatically changed from a simple exponential distribution to the more informative gamma distribution. This transition is again illustrative of the fact that probability is a mathematical quantity reflecting one’s knowledge of a system and not intrinsic properties of the system itself.

Since N is an independent random variable, the solution to finding p N ( n | n ¯ ) is to apply the PME again, given only the probability completeness relation

and the expectation of trial number

As in Case 1, the solution is an exponential function of the variable n and a Lagrange multiplier here designated α

Unlike Case 1, the variable n is discrete, and the partition function evaluates to

From Equation (11), which relates the Lagrange multiplier to the expectation of the variable, one finds

from which follows the sought-for conditional probability

explicitly expressed in terms of the given information n ¯ , instead of the Lagrange multiplier α . To a reader familiar with physics, Equation (42) is recognized as the distribution of photons in thermal equilibrium [26]. In the case of photons, however, n can take the lowest value 0, whereas in the case of testing a risk factor, the lowest meaningful value is n = 1 .

Upon substitution of Equation (42) into Equation (36), with replacement of the dimensioned coordinate x ¯ by a dimensionless coordinate u

one can re-express the distribution p X ¯ ( x ¯ | n ¯ ) more simply in the form

where the differential d u = λ d x ¯ is explicitly included.

The above notational change facilitates the calculation of the lowest moments as dimensionless quantities

from which follow the predicted uncertainty relations

and

The integrations over u in Equation (44) can be performed generally, leading to any desired k t h moment ( k = 0 , 1 , 2 , ⋯ )

where k = 0 yields the correct normalization constant 1.

The conditional probability P in ( x ¯ 0 | n ¯ ) that a mean of n ¯ trials yields an outcome x ¯ 0 that falls between the lower ( s L ) and upper ( s U ) boundaries of the reference range is given by

Before considering an application to a specific example, it is necessary to clarify what may appear to be an ambiguity in Equation (50).

Being of greater generality than Equation (33), Equation (50) must therefore reduce correctly in the previous case where the number of trials is constant, including the value n ¯ = 1 . Substitution of n ¯ = 1 , however, leads to P in =     0 , which is not correct. A mean value of 1 can only occur if there is only 1 trial. What is required is to evaluate the right side of Equation (50) in the limit n → 1 , which leads to

which correctly agrees with Equation (33) for N = 1 .

The red traces in Figure 1 and Figure 2 respectively show the variation with n ¯ of the functions CV and P in ( x ¯ 0 | n ¯ ) . By comparison with the blue traces for Case 2, one sees immediately the effects of having less precise information regarding the number of trials. For example:

A mean of 10 trials leads to a CV of 0.5, compared with 0.3 for Case 2. And 100 trials reduce the CV to 0.2, rather than 0.1 for Case 2.Similarly, to achieve a 50% probability that the mean outcome of the trials falls within the reference range would require a mean of about 15 trials, rather than exactly 10 as for Case 2.A mean of 100 trials yields a probability P in of 80%, in contrast to Case 2 for which 100 trials yields a probability P in of 90%.And last, although beyond the range displayed in Figure 2, to achieve a probability P in of 90% according to Equation (50) would require close to 300 trials.

In this section we examine the most probable distribution of two variables that are correlated with one another, but independent of any other variables. This is not exactly the system of HDL-C and LDL-C, which are part of a lipid panel comprising four constituents, but it provides a simple tractable system to address the seminal question of interest here: Does knowledge of a measure of correlation, in addition to just the point estimates of the two mean values, significantly reduce predictive uncertainty? A model more closely representative of the variables HDL-C and LDL-C will be taken up afterward.

Consider, therefore, two risk factors, X and Y, continuous over the non-negative real axis, for which there is a non-zero covariance

Expectations of products such as X a Y b for real powers a, bare calculated with a bivariate PDF p X , Y ( x , y ) as follows

As in the previous cases, p X   Y ( x , y ) is the sought-for maximum entropy (MaxEnt) distribution obtained by solving a variational equation based on prior known information. In the present case we take this information to be the probability completeness relation

the means of X and Y

and the covariance of X and Y, more conveniently reported as a dimensionless covariance coefficient

from which follows the mean of the product X Y

If κ = 0 , then the integrand in Equation (58) simply factors into a product of two independent univariate distributions, each being of exponential form as treated in Section 3 for Case 1. If κ ≠ 0 , then following the procedure of Section 2.1 leads to the MaxEnt solution

with partition function

in which, for convenience, Λ has been defined as

The unknown Lagrange multipliers can be determined from Equation (11) as applied below,

provided the prior information is consistent.

The operations in Equation (62) lead to relations

Manipulation of Equations (63)-(65) results in relations

which express λ 1 and λ 2 in terms of λ 3 . Λ is then a function only of λ 3 , which can be found by solving the equation

Equation (67) is highly nonlinear and generally requires numerical solution.

Upon solution of the three Lagrange multipliers, the joint PDF p X , Y ( x , y ) is uniquely determined. One can then calculate the marginal distributions of X and Y by integration

From Equations (59), (68), and (69) then follow the conditional distributions of interest that encode the predictive statistics of one variable in light of knowledge of a corresponding value of the other variable

One sees from Equations (70) and (71) that the conditional distributions are of exponential form, just as in Case 1 for uncorrelated variables, except that now the conditional mean values

are correlated through the multiplier λ 3 . Moreover, as shown in the analysis of Case 1, the standard deviation of an exponentially distributed variable is equal to the mean. It therefore follows that the coefficients of variation predicted by the conditional probability densities are

The expressions in Equations (74) and (75) are identical by virtue of Equation (66).

Regarding the forms of CV X | Y and CV Y | X , it is to be recalled that the information actually reported by diagnostic labs includes the point estimates ( x 0 , y 0 ) , not the conditional means. If λ 3 = 0 in Equations (74) and (75), then λ 1 and λ 2 respectively equal the reciprocals of the reported point estimates, whereupon the coefficients of variation reduce to 1.

Knowing the conditional distributions of X and Y, one can then predict the probability that a repeat measurement of one of them, given knowledge of the other, would fall within the reference interval ( x 1 ,   x 2 ) or ( y 1 ,   y 2 ) :

The preceding relations are to be compared with the relations for uncorrelated exponential variables:

For the purpose of illustration, the system of HDL-C and LDL-C are here treated as an isolated pair ( X ,     Y ) of correlated variables. Since the given prior information took no account of other lipid panel variables, the PME placed no upper limits on the ranges of X and Y. Reproduced below for convenience are the pertinent patient lipid panel data from Table 1expressed in units of mg/dL except for the dimensionless covariance coefficient κ :

Since no empirical value of the correlation of HDL-C and LDL-C is known to the author, the coefficient κ in Equation (80) was set to correspond to the empirical covariance coefficient of Case 4B, which is taken up in the following subsection.

Solution of Equation (67) with subsequent use of Equation (66) leads to the following Lagrange multipliers

truncated to 4 significant figures. (Calculations were made with the Maple Computer Algebra System set to 20-digit precision.)

Although there are numerous statistics that can be predicted by means of the joint, marginal, and conditional PDFs, the focus in this section (and in the following section) is on the predictive statistics of LDL-C. The reason for this emphasis is that numerous studies have concluded that lowering a hypercholesterolemic patient’s serum LDL-C should be a primary objective of preventative and therapeutic measures to lower the risk of ASCVD and other cardiovascular disease [34]-[36].

As examined in the previous cases, the coefficient of variation (CV) and probability P in of a point estimate falling within the reference range are two statistics indicative of the predictive uncertainty (and therefore reliability) of the numbers furnished by clinical labs. The conditional distributions based on the data in relation (80) and the solutions (81) lead to

from Equations (75) and (77) respectively, instead of CV Y = 1.000 and P Y ( in ) ( 40 , 100 ) = 31.9 % for uncorrelated LDL-C. The differences are small, but do indicate a lower uncertainty when information regarding correlation is provided. The more information that goes into the MaxEnt analysis, the greater is the expected reliability of a diagnostic inference.

Methods to measure LDL-C directly are in general complex, time-consuming, and expensive. Consequently, clinical labs in the U.S. and elsewhere routinely prefer to estimate LDL-C concentrations by means of an empirical formula such as the Friedewald equation [37], Martin equation [38], Sampson equation [39], and various others [32].

The Friedewald equation, which is linear in the variables, is the simplest of the empirical equations and has been in use the longest. It was proposed in 1972 as a means of estimating LDL-C without the need for ultracentrifugation, and has provided satisfactory results for patients with plasma lipid levels that are not too high or low. Subsequent elaborations to extend the formula to include patients with extreme lipid levels entail inclusion of numerous empirical parameters and/or powers of variables greater than 1. For the analysis in this section with lipid data within reference levels, the use of the Friedewald equation will suffice.

The Friedewald equation takes the simple form

in which

and

X m is the maximum value that X can take, since concentrations of the substances in Equation (83) must be non-negative quantities. The term involving triglycerides is a proxy for VLDL-C, i.e. the cholesterol carried by very low density lipoproteins. Thus T is representative of the total cholesterol concentration. It is apparent from the second equality in Equation (83), that LDL-C anti-correlates with HDL-C in a given sample.

The symbols and numerical values (in units of mg/dL) to the right of the variables in Equations (84) and (85) are the reported point estimates of a patient’s Lipid Panel taken from Table 1 with one minor exception. A reader may notice that the point estimate y 0 is here taken to be 83, rather than 81 as in Table 1 and as used in Equation (80). The reason for the small change is that the numbers in Equation (84) satisfy the Friedewald equation, whereas the actual LDL-C point estimate reported by the clinical lab was obtained by the Martin equation. Nevertheless, the closeness of the two values is indicative of how well the Friedewald equation works for a non-hyperlipidemic patient.

Although a clinical lab is not likely to provide a correlation coefficient, knowledge of the point estimates of each of the four constituents in the Lipid Panel makes it possible, by means of the PME, to derive the most probable distribution function of the variables Y and X. Rigorous application of the PME in the present context would lead to a trivariant MaxEnt probability distribution for X, Y, G with constraint on their sum T. One could then integrate over unwanted variables to obtain the marginal distributions of Xand Y.However, a much simpler and more transparent approximate approach is to take X m constant for the given sample and derive univariate probability densities for X and Y directly.

In what follows, Yis treated as a function of the independent random variable X. Figure 3, based on the Friedewald Equation (83) and data in Equation (84), shows the linear variation of Y over the physically allowed range [ 0 ,   x m ] . From Equation (83) and the basic principles of calculus, it follows that the probability densities of X and Y are related by the following transformation

Thus, one need only find p X ( x ) to obtain p Y ( y ) .

Implementation of the MaxEnt procedure, making use only of the available

Figure 3.Variation of a patient’s low-density lipoprotein cholesterol as a function of the concentration of high-density lipoprotein cholesterol as calculated from the Friedewald equation. Concentrations are in units of mg/dL.

information.

1) probability completeness,

2) point estimates x 0 and y 0 of the variables,

3) Friedewald value of cholesterol maximum x m

Leads to the exponential PDFs

in which the partition function

differs from that of Case 1 because the upper limit of the integral is no longer infinity, but constrained to x m .

The Lagrange multiplier λ is obtained by applying Equation (11) in the form

which leads to the nonlinear equation

in the dimensionless variable

From Equations (87) and (88) can be derived the pertinent statistics of interest, including the population means x ¯ , y ¯

variances

covariance

and the integrated probabilities

corresponding to Equations (76) and (77), that a repeat trial of each variable will fall within its respective reference interval.

It is to be noted from Equations (93) and (94) that the variance is the same for variables Xand Y, and that the covariance is the negative of this variance. Thus, the Pearson correlation coefficient [40]

which is a widely used measure of linear correlation of two variables, takes the maximum negative value of the range ( 1 ≥ r X , Y ≥ − 1 ) .

Solution of Equation (90) for the variable u with x m = 153 yields the Lagrange multiplier

Substitution of λ into relations (92)-(95) provides the statistics for judging the reliability of reported lipid panel results employing the Friedewald equation.

First, the results verify that the population means ( x ¯ ,   y ¯ ) in Equation (92) do indeed reproduce the point estimates ( x 0 ,     y 0 ) in the panel (84). Second, from the means (92) and variances (93) follow the coefficients of variation

which are significantly lower than the value 1 for uncorrelated exponential variables. Third, evaluation of the integrated probabilities (95) yield

Although the value for LDL-C in Equation (99) exceeds that of Equation (82) for directly measured LDL-C, it still represents a large uncertainty.

Finally, for purposes of comparison with κ in relation (80), one can calculate the covariance coefficient

Note that the value of κ 0 in relation (100) is an actual outcome of the statistical analysis of Case 4B, whereas the value κ assigned as prior knowledge in Case 4A, relation (80), was intentionally chosen to be close to κ 0 .

The reason for use of a “covariance coefficient” in Case 4A, rather than the standard Pearson correlation coefficient, is that the latter contains the variances (or standard deviations) of the two variables whose values are ordinarily not provided by the clinical laboratories. As pointed out previously, the purpose of examining Case 4A was to see what improvement in predictability would ensue from inclusion of the most minimal correlation information. Had that minimal information comprised the Pearson coefficient without also including the variances, there would have been insufficient information to derive a maximum entropy distribution. In such a case, one would have to resort to a specific model, which, in essence, is precisely what the Friedewald equation provides. In this case, minimal prior knowledge required to carry out a maximum entropy analysis did not have to include variances and/or covariances, since this information is retrievable from the empirical equation and full Lipid Panel.