Copyright 1984, Journal of Parapsychology. Reproduced with permission. To purchase a complete issue write Journal of Parapsychology, 402 N. Buchanan Blvd., Durham, NC 27701, USA ( For copies of complete articles, write The Genuine Article, ISI, 3501 Market St., Philadelphia, PA 19104, USA.



(Originally published in Journal of Parapsychology, Vol. 48 No. 4, Dec. 1984)

ABSTRACT: This article compares two previously described mathematical psi models. The teleological model emphasizes the space-time independence of psi and the close relationship between ESP and PK. But this model leads to a divergence problem in the sense that the role of future observers seems exaggerated. The quantum collapse model avoids this problem. But this model may appear less attractive insofar as some of the space-time independence is lost and the relationship between ESP and PK is weakened. Experiments that might show the inadequacy of on or the other model are discussed.
At an early stage of psi research, one could hope to understand psi by a straightforward extension of physics. It seemed reasonable to interpret telepathy in terms of "mental radio waves" and psychokinesis (PK) in terms of some "mental force" But this changed when precognition was discovered and, more generally, when experiments showed the surprising insensitivity of psi to physical parameters suc as space, time, and the complexity of the task. Psi appeared increasingly implausible, suggesting a need for major changes in our thinking.

But, even with psi and current physics incompatible, there seems no need yet to abandon the general mathematical physical approacl to the problem. On the contrary, physics is rather flexible and open to changes. And the abstract mathematical method of modern physics has been very successful in helping us to understand phenomena that seemed at first intuitively implausible.

Several mathematical psi models have been developed by different researchers (see Millar, 1978). None of these presently available models can be considered as a satisfactory psi theory. Nevertheless these models are stimulating for the design of new experiments and the formation of new concepts.

In the following, I will review two of these models, compare their merits and their shortcomings, and discuss experiments to possibly show their inadequacies.

Since the first model has already been discussed in the parapsychological literature (Schmidt, 1975; 1978), a brief review will be sufficient. But the second model is published only in a rather technical version (Schmidt, 1982), so that some more detailed explanations are appropriate.


The teleological model was originally developed to show by a specific example that a world with precognition could "make sense," that is, that the existence of precognition need not lead to logical, conceptual difficulties and that the "intervention paradox" presents no problem.

The model begins with an ensemble of all the possible world histories with their corresponding probabilities as they would be predicted by conventional physics. And, to introduce an element of noncausality, the model postulates a psi law that modifies the probabilities for the different world histories. This law is teleological in the sense that, for example, the outcome of a coin flip depends on its implications for the future history of the world. This is certainly implausible, but so is precognition. What matters is rather that this teleological law could be formulated in a mathematically simple form (without too many ad hoc assumptions) and that the resulting model is self-consistent. In our case, this law is very simple, and provides for a space-time independence of psi. At the same time, the psi law is Lorentz-covariant, that is, it fits into the space-time frame provided by Einstein's relativity theory.

The model does not claim to "explain" psi. It does not even try to discuss what happens inside a subject's head. It rather assumes subjects (the psi sources) with given abilities and then studies how a world with such odd elements as psychics can still be reasonable as a whole. This restricted scope of the model is in accordance with the experimental work aimed at the physical characteristics of psi (like its dependence on space, time, and task complexity) rather than at its psychological or spiritual aspects.

Let me list here the most important features of the teleological model:

1. The universality of the psi mechanism. PK, precognition, and other forms are integral parts of one common psi principle (given by the teleological law). This has the practical implication that, in a proper test arrangement, a prophet can perform PK tasks and a successful PK subject can predict future events.

2. The "weak violation" of conventional physics. The model changes only the probabilities of world histories that were already possible in the frame of conventional physics. Therefore, psi effects do not violate the established conservation laws of physics (like symmetry laws and the laws for energy and momentum conservation). Only statistical laws are affected.

3. Space-time independence of psi. The probability for a particular history is affected by the activation of psi sources. But it is irrelevant when and where this activation occurs in the course of a world history. For a PK test, this irrelevance implies that the PK effect is independent of the distance between subject and random generator. It further implies that the subject's effort does not even have to coincide in time with the activation of the random generator.

4. The complexity independence of psi. In this model, psi appears "goal oriented." In a PK experiment, for example, the subject succeeds by aiming at the desired end result rather than by working on the intermediate steps that lead to the final goal. Accordingly, the formalism implies independence of the PK success from the internal structure or complexity of the random generator.

5.The vital role of feedback. To affect the probability of a history, the psi source must be activated, that is, the subject must receive feedback on his effort.

An interesting implication is that, in cases of delayed feedback, the relevant feature is the subject's mental state at the time of the feedback rather than at the time of the "test."

6. The divergence problem. The present course of the world history can be affected by the future activation of psi sources. The existence of such retroactive effects is emphasized by the results of PK experiments with prerecorded targets (Schmidt, 1976). Nevertheless, the model seems to exaggerate the effect of the future on the present, leading to a severe problem. I call this the divergence problem in analogy to the famous divergence problem in quantum electrodynamics. There, the problem was finally overcome by "renormalization" procedures. But for the divergence in our psi model, no remedy has been found yet that would not destroy the internal simplicity and symmetry of the model.

The listed features seem closely linked, insofar as each one is an integral part of our model. But if we do not want to commit ourselves to any theoretical model, the first five features still appear as interesting hypotheses to be studied independently.


The Observer Problem in Quantum Theory

Quantum theory describes a physical system in terms of a state vector (or wave function). This vector can be considered as a set of parameters specifying the state of the system. There seems to be no question of how to use these vectors to make successful calculations and predictions. But there is still controversy about the interpretation of the state vectors.

Consider, for example, a binary random generator that randomly selects a red or green lamp to be lighted, with probabilities p and q, respectively. When the generator has been activated, but before an observer has looked at the outcome, the state vector of the system appears in the form (1)

|STATE> = sqrt(p)|RED> + sqrt(q)|GREEN>

In this equation, the state vectors |RED> and |GREEN> correspond to physically reasonable, macroscopically well-defined states, with the red or the green lamp lighted. The form of the total state vector, |STATE>, as a superposition of two different possibilities, however, makes one wonder whether Nature has already decided for one outcome, or whether physical reality at this stage is actually some intuitively implausible "ghost state" suspended between two possibilities.

When an observer looks at the outcome, he sees either the red or the green lamp lighted. At this stage, one feels, Nature must have definitely decided for one outcome. In the language of quantum mechanics, the original state vector, |STATE>, has made a quantum jump, or has been reduced, into either the state |RED> or the state |GREEN>. But the Schrodinger equation, which describes the change of the state vector with time, has no provision for such a sudden quantum jump. Then, how is this transition brought about? Is it a physically real process or some artifact resulting from an improper interpretation? Let me mention four different approaches:

Approach 1. The most simple, conventional explanation, accepted by most physicists, is that the state vector describes not "physical reality," but rather the observer's knowledge about this reality. And obviously, this knowledge changes suddenly with an observation.

The provocative idea that the equations of physics deal not with Nature per se, but with our knowledge about Nature, makes quantum theory self-consistent. At the same time, this seems to imply a very special role for the human observer, making him basically different from, say, an observing TV camera. The situation appears particularly puzzling when an observer tries to describe a system that includes another observer, as in Schrodinger's cat paradox, or in the paradox of Wigner's friend (d'Espagnat, 1976).

Approach 2. One widely explored possibility is the existence of "hidden parameters," which would give quantum theory a (possibly deterministic) substructure. The hidden parameters would supplement the conventional parameters that make up the state vector. Allowing this possibility, one could imagine new laws of motion for the hidden parameters, such as laws describing the collapse of a state vector in detail. Unfortunately, there are a large number of such possible theories, all unpleasantly complicated and based on arbitrary assumptions. And with the original quantum formalism so elegant and practically successful, one feels hesitant to introduce much more complicated theories.

For parapsychologists, hidden variables are tempting, because they show some noncausal and space-independent features. If I could change a hidden variable at my location, this might imply a simultaneous change at some distant location. That need not disturb physicists, because in their model hidden variables cannot be observed and measured. But if the mind had some nonphysical way to change and measure hidden variables, psi effects could result. Walker and others have explored some of the possibilities for constructing psi models based on hidden parameters (Mattuck & Walker, 1979; Walker, 1975).

Approach 3. With the conventional interpretation of quantum theory so attractively simple, Everett in his "Many World Interpretation" (1957) suggests a somewhat new interpretation that assigns "physical reality" to the state vector and relieves the human observer from his role as a distinguished outsider. But there is a heavy price to pay. Now the macroscopically ambiguous "ghost states" become physical reality. When the random generator makes a decision and the observer looks at the lighted lamp, the real world has to split into two equally real branches, one branch where the observer has seen "red" and one branch where the observer has seen "green". Everett argues that the observer would not notice his splitting into two branches and that, therefore, the model is self-consistent.

Approach 4. Whereas the first approach opens the possibility that the human observer might play a singular role in Nature, the new formalism suggested by Eugene Wigner (1962) would explicitly spell out the new role of the observer. Wigner's approach agrees with Everett's insofar as the state vector is physically real (not merely a measure of the observer's information), providing maximal information about the system (no hidden parameters). Then, the ghost state |STATE> of Eq. (1) represents a physical reality with two macroscopically different but equally real branches.

But as soon as an observer becomes consciously aware of the outcome, Wigner proposes, the observer's mind induces a reduction from the ambiguous ghost state into one of the "physically reasonable" states |RED> or |GREEN>. Thus, by interfering with the Schrodinger equation, the mind helps in maintaining one single reality.

Wigner had already wondered if the human mind, playing such an active role in shaping physical reality, might not contribute some PK effect in the process. The quantum collapse model to be discussed pursues this idea.

The detailed mathematical description of this model has been presented elsewhere (Schmidt, 1982). The model introduces, at the phenomenological level, a mathematical formalism that would provide the reduction of the state vector under an observation. The main requirements that led to this particular formalism were that the formalism should be mathematically as simple as possible but at the same time quite general, applicable to all situations.

To save the reader from having to go into the details of the quantum formalism of the original paper, I have summarized in the Appendix some of the general ideas and mathematical results for the simple case of a binary random decision.

The Reduction Process

In the model, the act of observation induces a gradual reduction from the original ghost state (|GHOST> = |STATE> of Eq. 1) into the well-defined states (|RED> or |GREEN>).

To formulate this transition mathematically, I shall introduce the three time-dependent functions (2):

RED (t) = probability that Nature has decided for |RED>

GREEN (t) = probability that Nature has decided for |GREEN>

GHOST (t) = probability that at time t Nature is still in the ambiguous, undecided state

At the beginning of an observation, at the time t = 0, Nature is still completely undecided, that is to say (3):

GHOST (0) = 1

RED (0) = 0

GREEN (0) = 0

The change of these parameters with time is given by the following equations (see also Appendix) (4):

GHOST (t) = R

RED (t) = p(1 + qf)(1 - R)

GREEN (t) = q(1 - pf)(1 - R)

where R = exp(-kt).

This shows an exponential decay of the ghost state, with the final result, after a sufficiently long time (t > > 1/k) (5):


RED(END) = p(1 + qf) = p'

GREEN(END) = q(1 - pf) = q'

In this model, the observer (and his momentary mental state) is determined by two parameters, the alertness parameter k and the PK coefficient f.

The value of the positive alertness parameter determines the speed of the state vector reduction. The choice of the name for this parameter should suggest that a highly alert observer might produce a faster reduction than a sleepy one. But even though state vector reduction is necessary for PK to operate, the speed of this reduction does not determine the size of the PK effect (perhaps, you don't have to be in an alert state to produce PK effects). The PK effect is given by another parameter, f.

A nonvanishing value of the PK coefficient f makes p' and q' different from the original probabilities p and q; that is, we have a PK effect. The Appendix shows that the absolute value of f is limited by the condition |f| > 1. This implies an upper limit for PK success. In the case of a symmetric coin flipper (p = q = 1/2), this maximal success rate is 75%.

The Eqs. (4) and the more detailed Eqs. (A6) in the Appendix show that PK may affect the way in which the disappearing ghost state is redistributed among the final states |RED> and |GREEN>. As the ghost state declines, the efficiency of the PK effort declines. And when the reduction is completed, there is nothing left for PK to operate on.

This mechanism implies that simultaneous PK efforts by two subjects are rather inefficient, because each subject contributes to the attrition of the ghost state, thus leaving less for the other subject to work on. The Appendix shows explicitly that the PK score produced by two subjects, working simultaneously or consecutively, cannot be higher than the score obtained by the better of the two subjects working alone. This is very different from the situation in the teleological model and explains why the quantum collapse model has no diveregence problem.

Operational Definition of Consciousness

A most interesting suggestion of the quantum collapse model is that some aspects of consciousness could be operationally defined by its ability to collapse the state vector. Conventional physics is not able to measure this collapse. But with the PK mechanism serving as a measuring probe, a successful PK subject could tell the difference between the collapsed and the noncollapsed state, because only the noncollapsed state would respond to PK efforts.

We might even look for consciousness effects from animals. To test whether, for example, a dog can collapse a state vector, we would compare the outcome of two types of tests in which the decisions made by our binary generator would or would not be preinspected by the dog before the human subject applied his PK effort. If the preinspection makes a difference, this would be our evidence for dog consciousness.

The model does not specify clearly what constitutes a conscious observation that should reduce the state vector. In the case of the dog, it might be necessary to activate the animal's attention by enforcing each red signal by a simultaneous food reward. And with a human subject, a passive observation where the observer immediately forgets the outcome, or a subliminal perception that does not fully enter consciousness, might produce only incomplete reduction. All these questions could be answered experimentally.

In this context, the results of two previous experiments might be interesting. In a PK experiment with prerecorded targets (Schmidt, 1976), the subjects listened to sequences of 256 clicks that were randomly channeled to the right or left ear. The PK target was to obtain more clicks on one specified side. The clicks were generated at a rate of 10 per second so that the subject could clearly notice the individual decisions but could not spend much time "digesting" the information. Half of the clicks were generated momentarily and presented once. The other half were prerecorded and presented four times in succession. The scoring rate on the repeatedly presented clicks was found to be higher (at a moderate level of significance) than the rate on the only-once-presented events.

In the frame of our quantum collapse model, this effect might be understood in the sense that, at the first presentation of the clicks, there was not enough time for the subject to absorb the whole information and thus to reduce the state completely. Therefore, subsequent PK efforts could lead to a strengthening of the observed PK effect.

Another possibly relevant result comes from a PK experiment with prerecorded and preinspected seed numbers (Schmidt, 1981). This experiment was performed in the following steps: (a) With the help of radioactive decays as source of randomness, a six-digit random number was generated and recorded. (b) This number was carefully inspected by the experimenter. (c) The seed number was fed into a computer "randomness" program to produce a binary quasirandom number sequence. (d) The binary sequence was displayed to the PK subject as a sequence of red and green signals (or in some other manner) while the subject tried to enforce the generation of many "red" signals.

In part of the experiment, step b was omitted (the experimenter's observation of the six-digit random number). In this case, the PK subject was the first person to observe the random result that originated from the radioactive decay. But the outcome of the experiment showed a PK effect also in the part where the experimenter had preinspected the seed numbers. Thus, there appeared no significant collapse, even though the experimenter had enough information to derive from the seed numbers, in principle, the finally displayed binary sequence.

This result might help us to a better understanding of what constitutes a "conscious observation" that collapses the state vector. Note that the seed numbers did not convey "meaningful information" to the first observer, or information that he could remember (a large block of seed numbers used for the experiment were inspected in one sitting). Refer to Schmidt (1982) for more information.


I have listed six typical features of the teleological model. Let us now look at the corresponding features of the quantum collapse model.

1. In the teleological model, all forms of psi seem intricately linked, and there is a high degree of symmetry between PK and precognition: A PK subject can be set up to act to predict future events, and a prophet can be made to acomplish PK tasks. In the quantum collapse model, however, PK seems to play a more dominant role, and the model might not be sufficient to account for all forms of precognition. Take as example the case where the subject tries to predict the outcome of a future random event. If this subject is the first to receive feedback on the results, then the subject can still succeed (by mentally enforcing the generation of the predicted event). But if somebody else looks at the results first, collapsing the state vector in the process, then the subject's efforts should be useless.

2. Both models agree insofar as only a "weak violation" of physics occurs; that is, the effects appear only in connection with random processes.

3. The space-time independence of psi is somewhat restricted by the quantum collapse model. The outcome of a PK experiment is still independent of the distance in space and time between the subject's effort and the random event. Experiments with prerecorded targets still work. But if two subjects make consecutive PK efforts on a random event, it matters which subject tries first. This feature may be an advantage of the quantum collapse model because it helps to eliminate the divergence problem.

4. The complexity independence is common to both models because the formalism makes no reference to the internal structure of the random generators.

5. Feedback is equally vital for both models.

6. There is no divergence problem in the quantum collapse model because a complete observation reduces the state vector so that PK effects from later observers are excluded.

For a most straightforward experimental comparison between the two models, consider a PK experiment where two subjects, A and B, make subsequent efforts at a total of N prerecorded binary events. Let the arrangement be such that (a) in half of the trials, subject A makes the first effort, and in the other half, subject B is first; (b) in half of the trials both subjects make an effort in the same direction, and in the other half, in opposite directions; (c) these test situations are randomly mixed, so that a subject never knows the test condition (i.e., whether the other subject has already made his effort and whether the efforts are in the same or in opposite directions).

To evaluate the results, determine the deviations of the four scoring rates from chance under the four different conditions given by Table 1. Note that for opposite target directions in a trial, the subjects have opposite scores. Therefore, the definitions of the score deviations in the table have to specify (last column) to which of the subjects this score applies.

From the viewpoint of the teleological model, it should not matter whether A or B made the first effort; that is, apart from statistical fluctuations,

S1 = S3; S2 =-S4 (Teleological Model)



Deviation of scoring rates (S)Subject orderTarget directionSubject who scored

Considering next the quantum collapse model, let us first assume that the feedback from each trial provides a complete observation with complete collapse of the state vector. Then the second subject cannot exert an effect; that is,

S1 = S2; S3 = S4 (Quantum Collapse Model)

The predictions of the two models are compatible only if Sl = S2 = S3 = S = 0, that is, in the absence of PK. Therefore, a PK experiment could easily distinguish between the two models.

In the quantum collapse model, the possibility of an incomplete reduction of the states can be covered with the help of Equation A16. But because a PK effect is necessarily accompanied by some reduction of the state, the difference between the two models remains observable.


The two models discussed in this paper are modest in their claims insofar as they do not purport to "explain" psi phenomena. Rather, the models try to provide a conceptually clean framework for mathematically describing psi effects at a phenomenological level, This approach gives a close link to laboratory experiments concerned with the "physical" aspects of psi - like its relationship to space-time and causality, and its incompatibility with currently accepted laws of physics.

For the physicist, the ultimate goal of psi research would be the discovery of some novel microscopic law of Nature of great mathematical simplicity and beauty, from which all psi effects could, in principle, be derived. That law would qualify, from the physicist's viewpoint, as an "explanation" of psi.

But the phenomenological, macroscopic approach appears as a reasonable, and perhaps necessary, first step, as a basis for a later, more complete understanding.

The two particular psi models.were selected for the discussion because these models are relatively simple mathematically and because their easily testable predictions sharply disagree on a vital question of parapsychology: the degree to which the future may affect the present, that is, the extent of theo noncausality of psi.

We have indications of such noncausality from precognition tests, from PK tests with prerecorded targets, and from experiments that suggest an effect of a later checker on previously collected test results. Furthermore, this noncausality might be the source of the uncontrollability of psi, in the sense that the future, beyond our control, affects the results of a present experiment.

The teleological model, with its space-time-independent structure, provides for all these noncausal effects and derives PK, precognition, and the other forms of psi from one basic mechanism. But this attractive high degree of symmetry leads to a divergence problem in the sense that future observers obtain an unreasonably high influence on the present.

The quantum collapse model drastically reduces this PK effect from future observers. After a complete observation of a random event, there is no more opportunity left for future observers to affect the outcome. The model retains much of the space-time independence of psi. And precognition and PK effects on prerecorded targets can still occur. But some forms of precognition seem not to work as well as they should.

Thus, the two models might be too extreme, in opposite directions. And this makes the suggested Experiments particularly interesting.

Another difference between the two models might be emphasized. The teleological model can be formulated completely within the framework of classical physics. And the model makes no reference to any concept of consciousness. Thus, there is no logical necessity that the psi problem be related to the consciousness problem, or to quantum theory.

The quantum collapse model, on the other hand, assumes a close link between psi, consciousness, and quantum theory. The most provocative implication of this model is that the effect of consciousness in collapsing the state vector should be measurable, with the PK effect serving as a measuring probe.


The Reduction Equation

Consider the simple case of a binary random generator that makes a decision on the lighting of a red or green lamp, with the associated probabilities p and q, respectively.

Before an observer has looked at the result, the status of the system is given by a state vector (A1)

|GHOST> = sqrt(p)|RED> + sqrt(q)|GREEN>

This vector describes the quantum mechanical superposition of two macroscopically different states |RED> and |GREEN> with the red or the green lamp lighted, respectively.

Our model considers this "ghost state" as a physically real state but one in which Nature has not yet decided for one or the other possibility. Physical reality, at this stage, consists of a coexistence of two branches of reality, one with the red lamp lighted and one with the green lamp lighted.

After an observer has looked at the outcome of the random decision, there is no more ambiguity because the observer clearly sees either red or green.

The model assumes that it is the act of observation that gradually reduces the initial |GHOST> state (Eq. (A1)) into either the |RED> or the |GREEN> state. To describe this reduction in a mathematical, statistical manner, let us introduce the following parameters (A2):

A(t) = RED(t)

B(t) = GREEN(t)

C(t) = GHOST(t)

with (A3)

1 = A(t) + B(t) + C(t)

Here A(t) and B(t) are the probabilities that at time t Nature has decided for |RED> or |GREEN>, respectively, and C(t) is the probability that Nature is still in the undecided |GHOST> state.

Starting from the fully undecided state at time 0, we have (A4)

A(0) = 0, B(0) = 0, C(0) = 1

For the change of the parameters A, B, and C under an observation, the model A gives the reduction equations:(A6)

(d/dt)C(t) = -kC(t)

(d/dt)A(t) = pk(1 + fq)C(t)

(d/dt)B(t) = qk(1 - fp)C(t)

In these equations, the parameters k and f depend on the observer and his mental state, but are independent of the values p and q that specify the random generator.

If k and f don't change with time, integration of the Eqs. (A6) gives (A7)

C(t) = R

A(t) = p(1 + qf)(1 - R)

B(t) = q(l - pf)(1 - R)

with (A8)

R = exp(-kt)

If the observation time was long enough (kt > > 1), then the reduction is complete (A9):

C(END) = 0

A(END) = p(1 + qf) = p'

B(END) = q(l - pf) = q'

The values p' and q' give the probabilities for the observer to see the red or the green lamp lighted, respectively. Only for f = 0 are these probabilities equal to p and q. Therefore, we call the parameter f the PK coefficient.

The parameter k from Eq. (A8) does not appear in the result of Eq. (A9). This parameter measures the speed of the reduction under the observation. And since a very alert observer might be expected to produce a faster reduction than a sleepy one, we call k the alertness parameter.

The PK coefficient f is subject to the restriction (A10):

|f| < 1

This is easily seen from Eqs. (A9). If, for example, f were larger than 1, then a p value sufficiently close to 1 would lead to a negative value of B(END). But this is not admissible because B(END) represents a probability.

The restriction of Eq.(A10) puts an upper limit to the size of the PK effect. For a symmetric (p = q = 1/2) random generator, Eqs.(A9) with f = 1 give the maximal success rate (A11):

p'(max) = 3/4 = 75%

PK Addition Effects

Imagine two subjects with, parameters (k,f) and (k',f'), respectively, who observe the same event simultaneously, then the momentary changes of A, B, and C, given by the right side of Eqs. (A6), receive an additional contribution from the second observer. The resulting final probability for red is (A12):

B(END) = p(1 + qf'')

with (A13)

f'' = (fk + f'k')/(k + k')

It is easily seen that the absolute value of f'' cannot exceed the larger one of the absolute values of f and f'; that is, the two subjects together cannot score higher than the better subject alone.

It might be difficult to have two observers acting precisely at the same time. If one starts observing only slightly earlier, then he may already have reduced the state so that there is nothing left for the second observer to do.

But consecutive action of two observers can be interesting if the first observation is not complete. If, for example, the time was so short as to give the first observer only a subliminal glimpse at the outcome, the status after this observation would be given by (A14):

C = R

A = p(1 + qf)(1 - R)

B = q(l - pf)(l - R)

with the reduction factor R somewhere in the range (A15)

0 < = R < = 1

As an example with some potential experimental interest, consider the case where the reduction factor and PK coefficients of the subsequent subjects are (R,f) and (R',f'), and where any remaining ghost state is reduced by a neutral (vanishing f value) final observer.

Simple calculation gives for the total PK effect in this case (A16)

p' = A(END) = p + pq[f(1 - R) + f'(1 - R')R]

This equation shows explicitly how the two subsequent PK efforts are unsymmetric in the sense that a complete observation by the first subject (R = 0) cuts out any effect from the second subject.

For more details with a more general discussion of the reduction mechanism, see Schmidt (1982).


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