The analysis of neurologic studies using an extended exponential model
Section snippets
Background and motivation
Before a patient undergoes surgery, it is customary for the anesthesiologist to sedate the patient who then loses either partial or complete consciousness for a random amount of time which is to exceed the surgery time. One of the post-surgery health procedures is to cause the patient to recover consciousness completely. For some reasons, if this recovery does not occur on its own before a medically suggested time, the anesthesiologist intervenes by medicating the patient with a dose of
The intervened exponential distribution
Let Y denote the random amount of time a patient might be unconscious due to an anesthetic drug. The surgery team would prefer to have Y exceed an amount of time τ>0 which may either be known when it is decided by the medical team or an unknown when it is to be disease complication dependent. At any rate, we assume that Y follows a truncated exponential modelIt is easy to recognize that τ+θ=E[Y], the expected time for a patient to be in an unconscious state. At
Parameter estimation
Consider a random sample X1,X2,…,Xn of size n≥2 from an IED population in , . Then, its log likelihood function is non-zero if and only if the minimum of the observations, X(1)>τ. Hence, the maximum likelihood estimate (MLE), of the threshold parameter, τ, is X(1). To estimate the other parameters, θ and ρ, we consider the transformation of the data, Ui=Xi−X(1) for i=1,2,…,n. It is easy to see that the log likelihood function ψ(u1,u2,…,un) of the transformed data for ρ≠1 is
Testing whether effective intervention took place
A zero value for ρ is indicative of a completely successful medical intervention, whereas ρ=1 is to be interpreted as the status quo in a conscious recovery rate of a patient. Of interest to the medical team is whether effective intervention took place. That is of interest to test H0: ρ=1 versus H1: ρ<1 based on a random sample X1, X2,…,Xn of size n from IED as in , . Since the incidence rate, θ, is unknown in a real life situation, the testing of ρ is not straight forward. For this purpose we
Illustrative example
Two small sets of data were used to illustrate this technique. Each had about eight data points.These were the transformed times, Ui, i=1,…,n for n=8. Admittedly, these are small sample sizes. However, they demonstrate the IED technique quite nicely. The MLEs of θ and ρ were derived using , , , . We then computed the chi square values with 95% confidence limits on the parameter, ρ, from , . Table 1 gives the values of . Table 2 gives the χ2 value with the 95% confidence limits on ρ.
In
Conclusions
One can see that from this newly derived intervened exponential distribution that it certainly has application to the intervention we have proposed. The shortcoming of this technique is the derivations required to isolate the intervention parameter, ρ. Also some large sample approximations are required to derive a number of the results. However, the maximum likelihood procedure works very nicely for this model as does the chi square distribution. We have not had the opportunity to apply this to
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