We consider the nature and utility of marginal decompression sickness (DCS) events in fitting probabilistic decompression models to experimental dive trial data. Previous works have assigned various fractional weights to marginal DCS events, so that they contributed to probabilistic model parameter optimization, but less so than did full DCS events. Inclusion of fractional weight for marginal DCS events resulted in more conservative model predictions. We explore whether marginal DCS events are correlated with exposure to decompression or are randomly occurring events. Three null models are developed and compared with a known decompression model that is tuned on dive trial data containing only marginal DCS and non-DCS events. We further investigate the technique by which marginal DCS events were previously included in parameter optimization, explore the effects of fractional weighting of marginal DCS events on model optimization, and explore the rigor of combining data containing full and marginal DCS events for probabilistic DCS model optimization. We find that although marginal DCS events are related to exposure to decompression, empirical dive data containing marginal and full DCS events cannot be combined under a single DCS model. Furthermore, we find analytically that the optimal weight for a marginal DCS event is 0. Thus marginal DCS should be counted as no-DCS events when probabilistic DCS models are optimized with binomial likelihood functions. Specifically, our study finds that inclusion of marginal DCS events in model optimization to make the dive profiles more conservative is counterproductive and worsens the model's fit to the full DCS data.
in distinguishing between decompression sickness (DCS, hereafter referred to as “full DCS”) and marginal symptoms (hereafter referred to as “marginal DCS”), dive medical professionals typically treat full DCS with recompression therapy and do not treat marginal DCS with recompression therapy (4). In the present study, we use the criteria as defined in US Navy studies as operational definitions for marginal and full DCS (7, 8). Marginal DCS is described as minor aches and pains, such as pain in a single joint that lasts <60 min or pain in multiple joints that lasts <30 min. In the absence of other signs and symptoms, single joint pain lasting >60 min or multiple joint pain lasting >30 min would be classified as full DCS. The outcome from empirical dive trials is typically grouped into three categories: full DCS, marginal DCS, and no DCS. However, for the development and optimization of probabilistic DCS models, the dive data are grouped into two categories: full DCS and no DCS.
The method of likelihood maximization (1, 2) is used in probabilistic DCS modeling to adjust the various model parameters until the best fit is achieved between the model predictions and the binary (DCS/no DCS) outcome from empirical dive data. The method was first applied to probabilistic DCS models by Weathersby et al. (13) for fitting two versions of a two-parameter Hill response equation and two versions of a two-parameter risk function to a data set containing 310 He-O2 man-dives to depths between 33 and 1,400 feet of seawater, which resulted in 7 marginal DCS outcomes and 18 full DCS outcomes. Since the maximum-likelihood method assumes a binomial outcome process, Weathersby et al. had to make a decision about how the marginal DCS cases were to be treated. They considered three alternatives: 1) the marginal DCS events were treated as nonevents, 2) the marginal DCS events were counted as 0.5 of a full DCS event, or 3) the marginal DCS events were treated as full DCS events. There was no clear conclusion by Weathersby et al. on how the marginal events should be scored during model optimization, nor was there an analysis of whether marginal DCS was related to decompression.
In the literature following the introduction of maximum-likelihood methods into probabilistic DCS modeling by Weathersby et al. (13), a variety of methods for incorporating marginal DCS into probabilistic models have been used. Later, Weathersby et al. (14) counted marginal DCS events as 0.5 of a full DCS event for model optimization. This choice of 0.5 was purely a judgment call as to how marginal DCS events should be counted in relation to full DCS events. The dive data they considered contained a total of 800 exposures resulting in 21 cases of full DCS and 6 cases of marginal DCS. In investigating the role of O2 in decompression sickness, Weathersby et al. (12) used a data set containing 477 exposures resulting in 11 full DCS cases and 18 marginal DCS cases. As with earlier work, Weathersby et al. used likelihood maximization of a tissue supersaturation dose-response model and weighted marginal DCS events as 0.0, 0.5, and 1.0 of a full DCS event. Again, there was no clear “best” choice of weight that should be given to marginal DCS when the DCS model is optimized. Using chamber air dives including 800 exposures with 21 full DCS cases and 6 marginal DCS cases, Tikuisis et al. (10) optimized two DCS models via likelihood maximization: a four-compartment nonlinear gas exchange model and a two-compartment monoexponential model. Tikuisis et al. assigned a weight of 0.5 for the marginal DCS cases when calculating the likelihood, again a judgment call. In studying the decompression risk of dry vs. wet chamber dives, Weathersby et al. (16) used a fractional weight of 0.5 in the likelihood function. The time of the onset of DCS symptoms was incorporated into probabilistic DCS model optimization by Weathersby et al. (15), who, again, used a weight of 0.5 for marginal DCS events. In developing statistically based decompression tables using models with exponential and exponential-linear gas exchange kinetics, Parker et al. (6) elected to use the weight of 0.1 for marginal DCS events. Their 2,383-exposure data set contained 131 DCS outcomes and 75 marginal DCS outcomes. In changing from a weight of 0.5 to 0.1, Parker et al. provided the following explanation: “In previous modelling …, we assigned marginal cases a numerical outcome of 0.5. This assignment is functionally an implementation of the assumption that two marginal cases are as important as a single DCS case. Discussions with senior medical officers indicate a much lower level of concern for marginal outcomes.” Since the publication of the report of Parker et al., many works on optimizing probabilistic DCS models used a weight of 0.1 for marginal DCS events. A notable exemplar is the comparison of exponential with linear-exponential gas kinetics by Thalmann et al. (9). Yet, we are not aware of any rigorous and conclusive investigation concerning the appropriate weight that should be assigned to marginal DCS events.
The focus of the present study is a rigorous investigation of the optimal fractional weight for marginal DCS events during probabilistic DCS model optimization using a binomial likelihood function. We will address four main points. 1) We investigate whether marginal DCS events are related to decompression exposure. 2) We examine occurrence density functions and cumulative density functions for marginal and full DCS events to determine whether the onset dynamics are similar. 3) We perform data combinability tests for full and marginal DCS cases to determine whether these two categories should be combined into a single decompression model. 4) We examine the weight that should be given to marginal DCS cases in optimizing probabilistic models. We used the study decision tree shown in Fig. 1 to investigate these four points, as described in detail in methods.
To clarify the role of fractional weight for marginal DCS events, it is useful to examine the likelihood function and demonstrate its behavior for a simple example problem. The maximum likelihood for a given data set and model maximizes the likelihood that the candidate parameter set is the correct parameter set for that model (1). Thus, given two different values of likelihood for the same data and model, the parameter set producing the greater of the two likelihoods is the better parameter set. The ith component of the likelihood (L) may be defined as where δ1 = 1 if the ith dive, with the probability of a DCS event P(E)i, resulted in a DCS event or δ1 = 0 if a DCS event was not observed. For the entire data set, the likelihood function is defined as where N is the total number of exposures in the entire data set. A given model (decompression or any other) is optimized by adjustment of the model parameters until the likelihood function is maximized. Note that the likelihood is defined for a system with a binomial outcome (DCS/no DCS). In modifying the binomial likelihood to give a partial significance for marginal DCS events, prior researchers, as we have previously reviewed, assigned a fractional weight of δ1 = 0.1 or δ1 = 0.5 when the dive trial resulted in marginal DCS. To demonstrate this process, we follow the example given by Weathersby et al. (13), who presented a particularly useful toy demonstration of the use of the likelihood function in optimizing the parameters of DCS models. In the toy model, they considered events, each with a uniform probability of c, for a trial with 10 observations resulting in 3 occurrences of the event. The resulting likelihood was By maximizing L, Weathersby et al. found c = 0.3, which was the empirically expected probability of a single event. We can extend this example to include fractional weighting. Suppose we have a second 10-observation trial with 2 occurrences of the event and 3 occurrences of the pseudoevent. Furthermore, suppose we wish to assign a weight of 1/3 to the pseudoevents, so that the three pseudoevents, taken together, have the same importance as a single event. We now have a trial with three equivalent full events. In this case, the likelihood is So we find that the same value of c maximizes the likelihood and we further find that the maximized likelihood has the same value as the previous example.
Now consider the log likelihood (LL) = ln(L). Either the likelihood or the log likelihood may be used to optimize a model, since LL is a monotonic function of L. However, trials with a large number of samples suffer from numerical difficulties when the likelihood is used, so the log likelihood is commonly used (11). For the first of the above-described examples, the log likelihood is If we take the derivative of Eq. 5 with respect to c and equate the result to zero, we find the optimal probability of a single event to be, again, c = 0.3. For the second of the above-described examples with two full events and three pseudoevents, the log likelihood function is As we found with the likelihoods, the log likelihoods are equivalent for the two trials. Therefore, the fractional weighting holds when the model is optimized under the likelihood or the log likelihood. The weight that should be given to marginal DCS events is considered below.
The dive trial data used in this work, known as “BIG 292,” comprise a standard US Navy DCS model calibration data set of air and O2-N2 dives containing 3,322 exposures and resulting in 190 cases of full DCS and 110 cases of marginal DCS (7, 8). The data were collected by US, UK, and Canadian military laboratories from 1944 to 1997 and contain detailed time-depth histories, details of gas(es) breathed, gas switches, and case reports for the divers experiencing full DCS or marginal DCS. In classifying the outcomes of the dives, “[c]ategorization of cases as DCS, marginal DCS, or not DCS was done by consensus [of 4 to 8 diving medical officers assembled into a panel to review the case reports] and if consensus could not be reached, by simple majority. Marginal cases … involve symptoms or signs that are thought to be decompression related, but not serious enough to require recompression therapy.”
For all 190 DCS cases and 68 of 110 of the marginal DCS cases, the symptom onset times were reported as the last time the diver was definitely asymptomatic and the first time the diver definitely was symptomatic. Although the onset times have been shown to be useful in calibrating probabilistic DCS models (15), we did not use onset times for model calibration in the present study, because there were no analog onset times for our third null model (null3; see below). The use or lack of use of onset times is unlikely to change the conclusion as to whether marginal DCS is related to decompression exposure or whether there is an optimal fractional weight for marginal DCS events.
For our investigation of whether marginal DCS is related to decompression exposure, we removed the 190 full DCS dives from the BIG 292 data set and assigned the weight of δ = 0 for nonevents and δ = 1 for marginal DCS events when calculating the log likelihood. To test data combinability, all profiles (no DCS, marginal DCS, and full DCS) were used with a weight of δ = 1 for marginal and full DCS.
We used the three-compartment linear-exponential (LE1) decompression model with one crossover pressure and one pressure threshold, resulting in a total of eight parameters to be fitted (9). The maximum likelihood of this model was compared with the maximum likelihood of three null models (null1, null2, and null3; see below). We rejected a null model, if, when properly accounting for the added parameters of the LE1 model, the log likelihood difference test passed with a confidence of 95%.
Null model 1.
Null1 is a model for which any exposure to decompression has a uniform probability of producing a marginal DCS event. Full DCS is not considered with null1. For this simple model, we do not allow a longer decompression exposure to have any greater probability of producing a marginal DCS event than a shorter exposure, and the probability of a marginal DCS event is where the subscript “1” denotes null1 and the positive parameter c1 is the same for any dive. If this model is calibrated on a dive data set containing D observations of marginal DCS and N observations of no DCS, such that the total number of exposures to decompression in the dive data set is D + N, then the log likelihood of null1 can be expressed as Upon taking the first derivative of Eq. 8 with respect to c1, we obtain c1 = ln(1 + D/N), so that the probability, under null1, of a diver developing marginal DCS when exposed to decompression becomes Note that this unremarkable result is what would be empirically expected. The second derivative (Sylvester) test, conducted on Eq. 8 indicates that the solution, Eq. 9, maximizes the log likelihood and is the global maximum.
Null model 2.
Null2 is a model for which there is a uniform probability of marginal DCS per minute of dive time. Null2 assigns a greater probability of the event to a longer dive. Null2, which was used as a null model in other work (9), can be written as where c2 is a hazard constant with units of inverse time, t2 is the right-censored time of the dive trial, and t1 is the time of the first decompression. Following previous work (9), we assign t2 to be 24 h after the time of final surfacing for nonsaturation dives and 48 h after the surface time for saturation dives. We do not use symptom onset times (15) when optimizing any of the null models considered in the present study, because null1 and null3 (see below) prevent their use. Upon substituting Eq. 10 into the log likelihood function and equating to zero the derivative of the log likelihood with respect to c2, we obtain the expression for the optimal hazard constant The stationary point of Eq. 11 is a maximum, and the solution of Eq. 11 for c2 is both necessary and sufficient for optimality of null2.
Null model 3.
For null3, we test the hypothesis that marginal DCS is related to decompression exposure against the alternative hypothesis that marginal DCS is a random event unrelated to decompression. For this model, as with all other models considered here, the full DCS events are eliminated from the dive data and we use only the nonevents and the marginal events. In contrast to other works, however, we grade the marginal DCS events as full events for a binomial outcome. More specifically, we assign δ = 1 for observations of marginal DCS when calculating the likelihood, Eq. 1, or the log likelihood.
For null3, the full DCS dives are eliminated from the dive data set, the marginal DCS events are graded as nonevents and returned to the dive data set, and then an equivalent number of dives corresponding to the original number of marginal events (110 for the BIG 292 data set) are selected at random from the dive data set population consisting of the union of the nonevents and non-event-graded marginal events and are graded as marginal DCS. The eight-parameter LE1 model is then optimized on this sham data set using several sets of random initial parameter values. The best outcome (optimized parameter set producing the greatest log likelihood) is selected as the optimal parameter set for that realization of sham data. For this test, it is reasonable to expect that the particular randomly selected dives that are graded as marginal events could influence the optimized parameter values and the resulting log likelihood. Therefore, we repeated the generation of the randomly selected sham data set to produce a total of 30 sham data sets and sham optimal log likelihoods. In comparing null3 with the LE1 model optimized on the real data set, we use the mean value of the optimized sham log likelihood, rather than the greatest log likelihood from the sham data sets, because, with enough generations of sham data sets, we would eventually reproduce the real data set.
To optimize null3 and the LE1 model, we used a decompression model optimization software package that we previously developed. This system makes use of exact hazard functions (a function that quantifies the potential for an event, DCS in this case, to occur; compare with Ref. 5), exact gas kinetics crossover solutions for the LE1 model, parallel algorithm execution, and multiple optimization techniques. For the present work, we added a substantial improvement to our parameter optimization system by deriving and programming an exact, albeit nonlinear and coupled, solution for the optimal gain values. This exact solution is appropriate for any DCS model with a probability function where g→ is a nonnegative gain vector and R→ is a nonnegative hazard function vector. Equation 12 shows that a large value of the hazard function leads to a large probability of DCS and a small value leads to a small probability of DCS. The optimal gain components, gi (i = 1, 2,…, C), of the gain vector are given by the simultaneous solution of C equations of the form where the index c counts over the C tissue compartments, the index d counts over the D dive profiles resulting in DCS, and the index n counts over the N no-DCS dive profiles. In Eq. 13, the notation Rab refers to the integrated hazard function (before multiplication by the gain) for the ath tissue compartment and the bth dive profile. A proof of gain optimality is given in the appendix. Equation 13 is valid for incidence-only formulations of the log likelihood, and not for calculations using symptom onset times, nor is Eq. 13 valid for a binomial log likelihood that uses a fractional weight for any outcome. The use of Eq. 13 effectively removes the gain vector from the parameters over which the DCS model must be optimized, significantly improving solution time. Furthermore, the solution of Eq. 13 gives the exact gain vector. For the LE1 model used in this work, the use of Eq. 13 reduces the number of model parameters to be fitted from 8 to 5.
To test the combinability of dive trial data containing marginal DCS events with dive data reporting full DCS events, the log likelihood difference test for data combinability (1, 11) was used. In constructing this test, we randomly split the no-DCS dives into two groups, A and B, each containing one-half of the total no-DCS dive profiles. The full DCS dive profiles were added to group A and the marginal DCS dive profiles were added to group B. In the control group C, we combined all no-DCS, marginal DCS, and full-DCS dives into one data set. For each of the three groups, any DCS outcome was graded as an “event” (δ = 1), so there were three binary-outcome data sets. That is, no fractional weighting was used for marginal DCS events in group B or group C. Because we were concerned that the particular choice of the nonevent dive profiles that were used to generate groups A and B might bias the fitted model, we generated 30 realizations of groups A and B; we used only 1 realization of group C, inasmuch as it contained all the available dives.
For each of the 30 group A data sets, the 30 group B data sets, and the 1 group C data set, we optimized the 3-compartment LE1 model containing 1 crossover pressure and 1 pressure threshold, resulting in a total of 8 parameters (5 true parameters and 3 exact gain values) to be fitted. Further details of the model and the parameters are available elsewhere (9). Five optimizations were run on each data set, with each optimization using a different randomized initial parameter set. The parameter set producing the greatest log likelihood was selected as best, and the log likelihood from that parameter set was used in the likelihood test for combining the data sets. For the combinability test, we completed a total of 305 optimizations using our previously developed software system.
Optimal fractional weight for marginal DCS events.
To determine whether there exists a fractional weight for marginal DCS cases, 0 ≤ δ ≤ 1, that maximizes the log likelihood, thus improving the fit between a model and empirical data, we begin with any DCS model that has a probability function where R is an arbitrary hazard function and may contain any number of tissue compartments. For this derivation, we include the gain terms in R. To find the optimal fractional weight, δ, we write the log likelihood using Eq. 14. Furthermore, set δ = 0 for the profiles that are DCS-free and set δ = 1 for the profiles that result in full DCS. Then we retain the variable δ for profiles that result in marginal DCS. This gives us a parameter to vary to maximize log likelihood. Under these conditions and using the properties of logarithms, the log likelihood becomes where δ only appears for profiles associated with marginal DCS outcomes. In Eq. 15, the indexes d, m, and n count over the respective D profiles that result in full DCS, the M profiles that result in marginal DCS, and the N profiles that result in no DCS. Next, we assume that R changes negligibly with δ and take the derivative of LL with respect to δ to obtain The two terms in the summation of monotonic Eq. 16 have opposite signs, so we cannot state that in all cases the slope is positive or negative. The central limit theorem offers guidance if we consider the average hazard function for the marginal dives in a given data set. In this case, for a sufficiently large trial set, we can replace Eq. 16 with where the overbar denotes the mean value. In results, we use Eq. 17 to find the optimal fractional weight for marginal DCS events. A more detailed derivation Eq. 17 and a brief discussion of its interpretation are given in the appendix.
For the data set combinability tests (which include comparison of the performance of nested models), we used the likelihood ratio test as outlined elsewhere (11). The likelihood ratio test was also used for comparing the performance of nested models when judging the fitness of a known decompression model (fitted to marginal DCS and no-DCS dive trials) to the three null models discussed above. To assign 95% confidence limits and 95% prediction limits to a fit of model-predicted DCS events and DCS observations, we used SigmaPlot 11.0 (Systat Software, San Jose, CA). To assign 95% confidence limits to model parameters, we used a software system we had previously developed. In comparing the occurrence density function (ODF) with the cumulative density function (CDF) for marginal and full DCS onset, we used the Mann-Whitney rank sum test and did not assume the data to be normally distributed. For all tests, P ≤ 0.05 was considered significant.
The results begin with comparisons between the null models and the LE1 model optimized on data containing only nonevents and marginal DCS events. In addition, the predicted probability of marginal DCS is compared with the observed marginal DCS incidence for the BIG 292 data set. This is followed by a combinability analysis on data containing marginal and full DCS events. ODF and CDF for marginal and full DCS are presented, and differences in the functions are explored. Finally, we derive the optimal fractional weight for marginal events when combined with full DCS events in a single model optimized with a binomial log likelihood function.
Are marginal DCS events related to decompression exposure?
Table 1 shows the log likelihood for the null models, the log likelihood of the LE1 model optimized on the actual data, the prediction error for each model (defined as the difference between the number of predicted DCS cases and the number of actual DCS cases), the hazard parameters for null1 and null2, and the number of parameters used by each model. Null2 outperforms null1 by 28.8 log likelihood units, indicating that longer dives are more likely than uniform random chance to result in marginal DCS (both models use 1 parameter). In comparing LE1 with null2, we must account for seven additional degrees of freedom (dof) of LE1. Using the log likelihood difference test (11), we need to have twice the difference in log likelihood exceed the critical χ2 value for 7 dof of 13.9 for P ≤ 0.05. This is, indeed, the case, so we can reject null2 in favor of the LE1 model. The LE1 model optimized on the actual marginal DCS data outperformed the mean of the 30 null3 sham marginal data sets by 79 log likelihood units (both models used 8 parameters). This leads us to conclude that the marginal DCS events in these data were not random but, rather, are related to exposure to decompression.
The performance of the null models is compared with that of the LE1 model in Fig. 2. The prediction error is defined as the number of marginal DCS cases predicted for the target data set minus the actual number of DCS events in the same data set. Thus a positive (negative) prediction error indicates overprediction and a negative prediction error indicates underprediction of the number of marginal events.
It is worth exploring the predicted vs. observed results for the LE1 model tuned on the marginal data. For this fit, as we previously noted, we used the BIG 292 data set, discarded the full DCS events, and did not use failure times in calculating the probability of DCS. This comparison, shown in Fig. 3, was made by ranking each dive profile in the data set by its LE1-predicted probability of marginal DCS. We then grouped the dives into 10 bins containing 312 or 313 dive trials, which are represented by the means of the predicted probabilities, and reported the total number of observed marginal DCS cases and the total number of predicted marginal DCS cases for each of the 10 bins. In Fig. 3, we also show the identity line, the 95% confidence limits, and the 95% prediction limits. In comparing the model predictions with the dive trial outcome, as shown in Fig. 3, we can reasonably conclude that marginal DCS events are correlated with exposure to decompression as predicted by the LE1 model tuned on this data set and are not random occurrences of minor unrelated symptoms.
ODF and CDF.
The ODF and CDF are useful methods to visualize the time course of DCS onset in a dive data set (3, 11, 15). The ODF is created by selection of a reference time for all dives in the data set, typically the time at which decompression ends. Using the available symptom onset times T1, which represents the last time the diver was definitely asymptomatic, and T2, which represents the first time the diver was definitely symptomatic, translated to the reference time, and assuming that a uniform probability of DCS occurred during the interval T1 → T2, we can construct the ODF where the ordinate is the number of DCS cases per unit time and the abscissa is the time since the end of final decompression (i.e., final surfacing time). In constructing our ODFs for the marginal and full DCS events (Fig. 4), we normalized the number of cases per unit time (hour) by the total number of occurrences of the corresponding DCS event type, so that the area under each ODF curve was 1. From the ODF, we see that the time at which full DCS was most likely to occur in this data set was 1 h after the dive. For marginal DCS, the time of most likely DCS onset was 2 h after the dive. A Mann-Whitney rank sum test indicated that the median values of the marginal DCS and full DCS ODFs were significantly different (P = 0.002).
The CDF, shown in Fig. 5, was obtained by integrating the ODF. Figure 5 also demonstrates the difference in the onset dynamics between marginal and full DCS. For example, full DCS occurred sooner in the time interval from ∼8 h before to ∼4 h after the dive. Similar to the ODFs, the marginal and full DCS CDFs were also significantly different (P = 0.005). Although we did not further explore the differences in the onset dynamics between marginal and full DCS, it was reasonable to expect that the difference in dynamics between these two event types would change the dynamics of DCS model predictions when the marginal DCS events are given more or less weight during model optimization. This fact was noted earlier by a study showing a shift in model parameters as the weight of marginal DCS was increased (13).
Combinability of marginal and full DCS dive data.
For testing the combinability of marginal and full DCS outcome dive trials into a single model using a binomial likelihood function, we created three data groupings. Group A contained the full DCS dive profiles and one-half, randomly selected, of the no-DCS profiles. Group B contained the marginal DCS dive profiles and the other half (that not used in group A) of the no-DCS dive profiles. Group C (control) included all dive profiles. For all three groups, we optimized the LE1 model with all DCS events, marginal or full, assigned a weight of δ = 1 in the binomial log likelihood function. We generated 30 realizations of groups A and B with different randomized splittings of the no-DCS dive profiles. For comparing the separately optimized groups A and B with the aggregated group C, we used the likelihood ratio test (see methods). The data and outcomes are shown in Table 2. In selecting the group A–B splitting to compare with group C, we chose the group A–B pair with the best combined log likelihood. It is important that the group A–B pairings be maintained for this test, so that every dive profile in group C is also present in the group A–B pair selected for comparison. Stated differently, we cannot choose the best log likelihood from group A and the best log likelihood from group B. However, the splitting had only a weak influence on the combined group A–B log likelihood. The mean combined group A–B log likelihood was −886.24 ± 1.51 (mean ± standard deviation), whereas the best combined group A–B log likelihood was −883.20.
In the log likelihood ratio test for data combinability, the difference in log likelihood is calculated as where the absolute value of log likelihood from Table 1 is used in Eq. 18. From the results shown in Table 1, we see that ΔLL = 52.3. In the version of the LE1 model used for our combinability test, there are eight adjustable parameters (although there are only 5 parameters after gains are removed, 8 dof are retained for χ2 calculations). Therefore, for groups A and B taken together, we have 16 parameters, so there are an additional 8 dof for the combination of groups A and B compared with group C. For P < 0.05 for 8 dof, the critical χ2 is 15.5. In comparing ΔLL with χ2, we can easily see that the data sets A and B are different and, therefore, are not combinable under this decompression model.
Optimal fractional weight for marginal DCS events.
In methods, we presented an expression for the optimal fractional weight for marginal DCS events. In the appendix, we give a more detailed derivation of this equation. Under the assumption that the mean hazard function, R̄, changes negligibly with variations in δ, we find that the slope of the log likelihood vs. δ function is Now, consider the root of Eq. 19, which is found at R̄ = 0.693. This value has special significance, inasmuch as this gives P(DCS) = 0.5. So, we conclude that if the mean predicted probability of DCS is P < 0.5, then the slope of the log likelihood given by Eq. 19 is negative and the optimal fractional weight for marginal DCS events is δ = 0 (marginal DCS should be treated as no DCS). Conversely, if the mean predicted probability of DCS is P > 0.5, then the slope of the log likelihood is positive and the optimal fractional weight for marginal DCS events is δ = 1 (marginal DCS should be treated as full DCS). For the BIG 292 data set and most other dive data sets, the mean probability of DCS is well below 0.5. In fact, for the BIG 292 data set, the mean probability of DCS has the upper limit P̄ = (190 + 110)/3,322 = 0.0903, so that the upper limit of R̄ is 0.0946, which is well below 0.693. This result is depicted graphically in Fig. 6, where the optimal weight is shown to occur for δ = 0. Therefore, as long as binomial likelihood functions are used to optimize probabilistic DCS models, marginal DCS events are best included in the no-DCS category. This gives the best possible fit of the DCS model to the full DCS cases in the dive data set.
An objective of this work was to investigate the hypothesis that marginal DCS events were random noise and were not related to decompression exposure. Our approach for accomplishing this objective was to fit a standard decompression model (LE1) to marginal DCS and nonevent data while omitting full DCS cases. The maximum likelihood for this model and data (LLtrue) was compared with the maximum likelihoods (LL1, LL2, and LL3) of three null models that were based on different assumptions concerning the nature of the dives. Null1 assumed that the probability of a marginal DCS event was the same for every dive in the data set (excluding full DCS cases), with a probability equal to the average incidence of marginal DCS events. Null2 assumed that the probability of marginal DCS was uniform per minute of dive time. Null3 assumed that marginal DCS events occurred randomly throughout the data set, so there would be no significant difference in maximum likelihood between the model fitted to marginal DCS events identified during the dive trials (i.e., “true” marginal DCS events) and sham marginal DCS events chosen at random from the same dive trials. In Fig. 2, we compared LLtrue with LL1, LL2, and LL3 and found that LLtrue provided a significantly better representation of the marginal data than all null models. Null2 was the best of the null models, and null1 was next best. Null3, which assumed that marginal DCS events were random noise, unrelated to decompression, was the poorest model of the data. These results constituted a strong argument that symptoms defined as marginal (7, 8) were truly related to the decompression exposure. On the basis of this finding alone, the use of marginal DCS events in fitting the parameters of DCS models seemed justified.
From our test of the combinability of data containing marginal DCS events with data containing full DCS events, we found that the data were not combinable under a single DCS model using a binomial likelihood function. Furthermore, in comparing the ODFs and CDFs for marginal and full DCS in the BIG 292 data set, we found significant differences in the median values of the ODF and CDF between marginal and full DCS events.
When we attempted to derive an expression for the optimal fractional weighting for marginal DCS events under the assumption of a binomial likelihood function, we found that no nontrivial optimal fractional weight exists. Specifically, we found that if the log likelihood is the objective function to be maximized and if the mean probability of marginal DCS in the training dive data set is <0.5, then the best choice of fractional weight is δ = 0. That is, marginal DCS events should be considered no-DCS events.
An examination of the marginal DCS cases in relation to the dive types in the BIG 292 data set gives an indication as to why marginal events are problematic. In the BIG 292 data set, saturation dives account for 14.4% of the total exposures. Marginal DCS events account for 3.3% of the total exposures. However, 55% of the marginal DCS events occurred on saturation dives. Thus, including any nonzero fractional weighting for marginal DCS events disproportionately increases the influence of the saturation dives.
In assigning a fractional weight to the marginal DCS events, the number of “effective” DCS cases for the entire data set increases. The optimization process, by maximization of the log likelihood, will adjust the model parameters until the model predictions give the best possible match to the training data set. For a reasonable DCS model, the number of predicted DCS cases will be close to the number of observed effective cases. As the fractional weight assigned to the marginal DCS events increases, the mean DCS probability of the dives in the training data set must, therefore, increase. This is not necessarily undesirable, inasmuch as dive plans using this model will be more conservative if a greater fractional weight is assigned to marginal DCS, thus reducing the probability of full DCS. However, the model, particularly in light of our findings of problems with marginal DCS/full DCS data combinability and significant differences in the ODF/CDF between marginal and full DCS events, might be less accurate in predicting the probability of full DCS. The fact that models optimized by including fractional weight for marginal DCS might less accurately predict full DCS is an undesirable outcome. A reasonable approach to explore, at least for the present binomial likelihood functions, would be to assign a zero fractional weight to marginal DCS events, thus treating them as no-DCS events, optimize DCS models using binary outcomes (no-DCS, full DCS), and address the issue of dive profile conservatism by reconsidering the risk that is acceptable for a given dive profile. In so doing, we obtain the best predictions for full DCS without compromising diver safety.
This work was supported by Naval Sea Systems Command (NAVSEA) Contract N61331-06-C-0014. P. W. Weber was supported by the National Defense Science and Engineering Graduate Fellowship.
- Copyright © 2009 the American Physiological Society
Let a compartmental DCS model consist of C compartments (counted by index c) and let a set of dive trials consist of D trials in which DCS occurred (counted by index d) and N trials in which no DCS occurred (counted by index n). For an incidence-only formulation of the log likelihood, the optimal gain, gi, for the ith tissue compartment is given by the simultaneous solution of the C coupled, nonlinear equations where Rab represents the integrated hazard function (before multiplication by the gain) for the ath tissue compartment and the bth dive profile.
Take the probability that a diver experiences DCS as and the probability that the diver reaches the right censoring time without experiencing DCS as Write the log likelihood of the D + N dive trials as which can be written as The derivative of log likelihood with respect to the ith gain (gi) is Upon equating Eq. A6 to zero, we arrive at the necessary condition for gain stationarity Now, the second derivative test needs to be conducted on Eq. A7 to determine whether the stationary points are extreme and, if so, whether the extrema are maxima or minima. The second derivative of Eq. A5 with respect to gi is
By inspection, Eq. A8 is found to be negative for any nontrivial data set, inasmuch as all terms are positive or squared, so that the sign of Eq. A8 is determined by the negative sign in the numerator. Therefore, the simultaneous solution of the C equations (Eq. A7) maximizes the compartmental gains. In light of this finding, the solution of Eq. A7 is necessary and sufficient for gain optimality.
Calculation of the log likelihood using the optimal gain set follows the pseudoalgorithm: 1) select a parameter set β→ (excluding the gain values), 2) calculate and store the integrated hazard functions R(β→), 3) solve the C simultaneous, nonlinear equations (Eq. A1) for the optimal gains g→, and 4) evaluate the log likelihood, Eq. A4 or A5, using R(β→) and g→.
Of course, a change of the parameter set requires a new optimal gain set. Additionally, this derivation of the optimal gain set applies to an incidence-only formulation of the binomial log likelihood. A formulation of the log likelihood that includes symptom onset times is straightforward but is not presented here. Use of this method to find the optimal gain set offers significant computational advantages. For example, the gain values are removed from the parameter space over which a model must be optimized, reducing the dimension of the parameter space by the length of the gain vector. Furthermore, the gain and tissue constant parameters are often closely collinear, resulting in an ill-conditioned Hessian matrix. The closely aligned eigenvectors of the Hessian matrix, resulting in slow convergence to the optimal parameter set. With exact gain values, the Hessian matrix is smaller and has a smaller condition number, so that convergence to the optimal parameter set β→ is greatly improved.
Optimal Fractional Weight for Marginal DCS Events
We now give a more detailed derivation of the optimal fractional weight for marginal DCS events presented earlier. This derivation begins with any DCS model that has a probability function where, as noted earlier, the arbitrary hazard function R includes the gain terms. From Eq. A9, we can easily see that the probability of not suffering DCS is Now write LL using Eqs. A9 and A10. For the profiles that do not result in DCS of any type, set δ = 0; and for the profiles that result in full DCS, substitute δ = 1. If the profile results in marginal DCS, retain δ as a variable. This gives us a parameter to vary to maximize log likelihood. The log likelihood becomes In Eq. A11, the N, M, and D profiles resulting in no DCS, marginal DCS, and full DCS, respectively, are counted by the respective indexes n, m, and d. The properties of logarithms allow Eq. A11 to be expressed in the simpler form where only the profiles associated with marginal DCS outcomes contain the variable δ. To derive a simple expression to let us understand how the log likelihood changes with variations in δ, we will assume that R changes negligibly with δ. Although this assumption introduces some level of approximation into this derivation, it does result in an equation that can be easily interpreted without the need for numeric computation. We can now take the derivative of log likelihood with respect to δ to obtain From Eq. A13, we can see that we have lost the independent variable, δ, so we cannot select a value of δ to eliminate the slope in Eq. A13 and maximize the log likelihood. We additionally see, from Eq. A13, that the second derivative of log likelihood with respect to δ is From these results, we conclude that there exists no optimal δ for 0 < δ < 1. In fact, Eq. A13 states that the slope of the log likelihood with respect to δ is constant, and Eq. A14 further reinforces the fact that the function log likelihood vs. δ is a straight line. If the slope is nonzero and there are no extrema on the interior of the domain, then the extrema must exist on the boundary. It follows, then, that we should explore at which boundary point, δ = 0 or δ = 1, the optimal fractional weight exists. The two terms in the summation of Eq. A13 have opposite signs, so we cannot state that in all cases the slope is positive or negative. The central limit theorem offers guidance if we consider the average hazard function for the marginal dives in a given data set. In this case, for a sufficiently large trial set, we can replace Eq. A13 with where the overbar denotes the mean value. As we discussed in results, the slope of the log likelihood vs. δ function is constant. Therefore, there exists no optimal value of δ between 0 and 1. For data sets with the mean probability of DCS <0.5, the optimal fractional weight for marginal DCS events is 0; for data sets with the mean probability of DCS >0.5, the optimal fractional weight for marginal DCS events is >1.
As a somewhat more sophisticated derivation of the optimal fractional weight, we can allow the mean R to vary with δ, i.e. It may now be shown that where, this time, the curvature is nonzero and positive and is given by By the odd symmetrical properties of the hyperbolic arctangent function and the model requirement that 0 < δ < 1, the optimal δ is 0 if N > D (which is true for a vast majority of data sets), 1 if D > N, and may be chosen as 0 or 1 if N = D. This duplicates our earlier finding that the optimal fractional weight for marginal DCS is δ = 0 for most dive data sets.