Info

The Fisher information for these sums of means assuming that |0 = 0 estimated from an OU process is compared in Figure 1 to the information when the time points are independent. To reduce the complexity in graphing the relationships, the OU information is plotted against X = a • At. If the approximate value of a is known from previous experiments, At can be chosen to achieve improved efficiency in detecting the desired time relationship of the experimental response. All efficiency appears to be lost for X greater than 4, while the information appears to increase exponentially when it is less then 1. The maximum relative efficiencies are 1, 1.5, 2.5, 9, and 69 for the constant, linear, quadratic, cubic, and quartic effects, respectively. Estimating a constant response is not at all efficient using dynamic measures. It is not immediately Figure 1 Fisher information gain from dynamic measurements relative to independent measurements as a function of X. (See insertfor color reproduction ofthefigure.)

clear why a linear relationship would be less efficient than the others, although it is well known that the best design in this case is to put the observations at beginning and end, with none in the middle. Since it was assumed that the beginning was zero, only one point is needed to estimate a linear response. For in vivo biomarkers, it would be difficult to imagine a constant or linear response. For those not inclined mathematically, it should be noted that all continuous functions of time can be approximated with a polynomial in time if the degree is sufficiently large. This means that additional contrasts, extensions of those above, can be used to get a better estimate of the time relationship as long as the degree is less than the number of time points. If the change with time is not well modeled by low-degree polynomials, a regression using the specific time-dependent function of interest should be used to save error degrees of freedom. It seems apparent from Figure 1 that the Fisher information of dynamic measurements is increased as the degree of polynomial or autocorrelation increases.

Besides obtaining additional information about the curvature of the mean time effect, the autocorrelation parameter a contains information that is seldom estimated or used. For the OU process, this parameter combined with the variance provides a measure of the biological variation. It is proportional to the homeostatic force needed to maintain the dynamic equilibrium. The Fisher information for a is mathematically independent of the mean and vari- Figure 2 Fisher information for a as a function of time between measurements (Af).

Figure 2 Fisher information for a as a function of time between measurements (Af).

ance. The relationship between the base 10 logarithm of a-information as a function of Af for various values of a is shown in Figure 2. The closed circles are the autocorrelation half-life for each a and the open circles represent a loose upper bound for Af at X = 3, which is an autocorrelation of 0.05. Those serious about capturing information about a should seriously consider X < 1. For a given a, the information approaches 2/a2 as Af goes to zero. Obviously, the larger a is, the less information about the autocorrelation is available in the data, necessitating larger m or smaller Af to get equivalent information. The time units are arbitrary in the figure but must match the time unit under which a is estimated.

Shannon Information

Shannon information is about how well a signal is communicated through some medium or channel; in this case it is the biological medium. The measurement of variables to estimate parameters that indicate which signal was transmitted is the signal detection process. Pathological states and the body represent a discrete communication system where the disease is the signal transmitted, which may affect any number of biological subsystems that act as the communication channels, and the biomarker is the signal detected in one or more biomarker measurements. The disease is then diagnosed by partitioning the biomarker space into discrete, mutually exclusive regions (R) in a way that minimizes the signal misclassification. Information, or communication, theory is usually applied to electronic systems that can be designed to take advantage of the optimal properties of the theory. In biology, it is mostly a reverse-engineering task. The signal is the health or disease state of the individual, which is transmitted through various liquid and solid tissues with interacting and redundant pathways. In this paradigm, biomarkers are the signal detection instruments and algorithms. Rules, usually oversimplified, are then constructed to determine which signal was sent based on the biomarker information. An elementary background in information theory is given by Reza .

Shannon information is really a measure of uncertainty or entropy and describes how well signals can be transmitted through a noisy environment. Probability models are the basis of this type of information which complements Fisher information rather than competing with it. The general framework for Shannon information follows.

As a reminder, Bayes ' theorem, where P[ • ] is a probability measure and P[A\B] = P[event A given event B], states that P[A\B] = P[A and B]/P[B]. In this particular representation, some elemental probabilities will be used: the probability that the signal was sent: for example, the prevalence or prior probability, n = P [S ]

the probability that the signal was received, q = P [Dj ]

and the probability that a particular diagnosis was made given that the signal was sent,

The last expression is rather ominous, but in words it is the probability that the multivariate biomarker Y, a list (vector, array) of tests, is in the region Rj given that signal S. was sent through a noisy channel, properly modeled by the probability function P. This P can be either continuous or discrete or both and is generally a function of unknown parameters and time. It reflects both the biological variability and the analytical variability. Since this function contains the parameters, it is where the Fisher information applies. In its crudest form, q. is just a proportion of counts. Although the latter is simple and tractable for a biologist, it is subject to losing the most information about the underlying signal and does not lend itself readily to the incorporation of time used in pathodynamics. Generally, any categorization of continuous data will result in information loss.

Table 1 shows the correspondence between signals (S) and diagnoses, or decisions (D), in terms of the probability structure. If the number of decision

INFORMATION FROM DATA 655 TABLE 1 Noisy Discrete-Signal Multichannel Communication Probability Table

Decision

INFORMATION FROM DATA 655 TABLE 1 Noisy Discrete-Signal Multichannel Communication Probability Table

Decision

 Signal Di 