Operations Research Games: a Survey
Summary (4 min read)
1 Introduction
- In model (1), one might ask why a smooth term for x2 is required whereas a simple linear term seems to be sufficient to capture the impact of x1.
- These and similar multi-step procedures perform a series of locally optimal fit and selection procedures, however, the quality of the global model fit can only be investigated empirically.
- The rest of the paper is organized as follows: Structured survival models and penalized likelihood estimation schemes that will be utilized as building blocks in the boosting algorithm are introduced in Section 2.
2 Structured Survival Models
- To overcome the restrictions of Cox models, as discussed above, the authors allow the inclusion of both, timevarying and smooth effects.
- In the survival context, P-splines are frequently used to model smooth functions [18].
- The latter determines the smoothness of the resulting function estimate, where bigger values of κj correspond to smoother functions and κj = 0 corresponds to an unpenalized estimation of the j-th term.
3 Boosting in Survival Models with Time-Varying Effects
- The authors device an estimation procedure for Cox-type models with additive structure and possibly time-varying effects.
- Variable selection and model choice play another major role in this setting.
- To combine all tasks, component-wise boosting methods are applied in the following.
3.1 Basic Considerations
- Estimation of models can be done with respect to many different criteria.
- For more details on base-learners in general and on how to choose them the authors refer to Bühlmann and Hothorn [11].
- Thus, the authors have a variable selection and model choice approach based on component-wise boosting.
- As classically the authors are using B-splines of degree 3 or higher, the degrees of freedom for difference penalties of order 2 or higher remain always greater than one.
3.2 Likelihood-Based Boosting for Survival Data (CoxFlexBoost)
- The boosting algorithm, which the authors will present in the following section, is essentially based on the likelihood-based boosting approach as proposed by Tutz and Binder [20].
- As the authors specially focus on the inclusion of flexible and time-varying terms in Cox-type additive models, they call the new algorithm CoxFlexBoost.
- In the following, the authors denote the j-th base-learner by gj(x(t);βj), j = 1, . . . , J , where J is the number of base-learners (possibly after decomposing nonlinear effects into several separate base-learners as described in the previous section).
- The covariates x(t) include classical covariates and possible time-varying effects expressed as artificial time-dependent covariates or the time t itself.
- Thus, gj(x(t);βj) can correspond to a linear function of x̃, where x̃ is a covariate from x(t), or of time t, with t being the time covariate from x(t), or, more flexible, a smooth function of x̃ or t.
3.2.1 CoxFlexBoost Algorithm
- Hence, the integrand in (11) is a function depending on the coefficient βj , which the authors try to estimate, and on time t̃.
- As CV does not involve estimation of the degrees of freedom (which tend to underestimate the true degrees of freedom [24]) this is a more sensible solution.
- An appropriate stopping iteration is determined as the number of boosting iterations m̂stop,opt that maximizes the log-likelihood on the validation data.
3.2.2 Remarks on Computational Considerations
- The authors have to integrate over time t̃ for each base-learner, in each boosting iteration and in each step of the optimization method (in their implementation the Broyden-Fletcher-Goldfarb-Shanno [BFGS] method, see, e.g., [25]) used to determine β̂j .
- The authors believe that specifying the degrees of freedom df to determine the amount of smoothness of each base-learner is much more intuitive.
- To specify the smoothing parameters via the corresponding degrees of freedom the authors exploit the relation that the latter depend on κj and thus they can solve df(κj)− d̃f j != 0 (13) for κj with a pre-specified value of d̃f j .
4.1.1 Outline of Simulations
- To gain deeper insights in the properties of the proposed CoxFlexBoost procedure, two simulation studies were performed.
- Furthermore, the authors investigated the properties of variable selection and wanted to check if other effects, as linear and smooth effects, are detected and modeled “correctly”.
- The authors will utilize linear or P-spline base-learners in the following.
- (18) Like in the previous simulation scheme the authors simulated the censoring times Ci i.i.d.∼ Expo(1/t).
4.1.2 Simulation Results 1: Model Choice and Variable Selection
- In the first scheme the authors simulated 200 randomly drawn replicates of the data set.
- In the following section, the authors explore the accuracy of model choice (and thus of variable selection) given by the relative frequency of selected base-learners.
- When the authors look at the noneffective covariates they see that the frequencies of selection are much smaller than those of the effective covariates.
- The time-varying effect of x6 is always discovered and the (log) baseline hazard is almost always selected.
- Only in roughly half of the models a flexible time-varying effect is chosen.
4.1.3 Simulation Results 2: Estimated Effects
- Now, the authors look at the estimated effects and compare them with the real, specified effects.
- In Figure 1, the authors see a selection of the estimated effects for the six effective covariates.
- Caused by the sparse data at the boundaries (note that the authors used a standard normal distribution to simulate x3) the boundaries of the sine form of x3 are estimated quite poorly, whereas the middle part is estimated quite sensible (not depicted here).
- The estimation in the center region is hardly effected and only slight deviations in the areas with less observations can be identified.
- The time-varying effect of x6 (not depicted here) suffers from the same problem as the baseline hazard, i.e., the estimated function is very unstable in the sparse, right tail.
4.2 Application: Model for Surgical Patients
- In the following section, the authors aim to build a model for patients with severe sepsis.
- Baseline characteristics and detailed outcomes of that population were published recently [30, 31, 32].
- The authors obtained relevant covariates reflecting the state of the patient on admission day, and the 90-day survival time for 462 patients with severe sepsis.
- To build the model, the authors applied the proposed CoxFlexBoost algorithm to the data.
4.2.1 Application of CoxFlexBoost
- To asses the stability of the variable selection and model choice process of component-wise boosting, as implemented in CoxFlexBoost [28], the authors used 5 random subsamples, each with 362 observations, of the severe sepsis data from Großhadern.
- In contrast to the two-stage stepwise procedure, CoxFlexBoost is not able to handle preset covariates.
- In contrast, in the application of the TSS procedure [3] six mandatory covariates were used.
- Defining an inclusion rate of 2 or less negligible, only 10 out of 20 covariates can be regarded as influential covariates in the boosting model.
- For some of the covariates, time-varying effects were also selected.
4.2.2 Comparison of Model Selection Strategies
- Comparing the results of the application of the two-stage stepwise procedure (see [3]) and CoxFlexBoost to the Großhadern data set of patients with severe sepsis, the authors can conclude that both approaches have advantages with regard to different aspects:.
- The two-stage stepwise procedure is easily extended in such a way, as showed in Hofner et al. [3].
- In their application CoxFlexBoost tended to a sparser solution but this could be due to the starting model with mandatory covariates in the two-stage stepwise model.
- Altogether, the authors see that none of the approaches is superior to the other.
- Especially in high-dimensional settings with many possible predictors, boosting with its robustness against overfitting and the built in regularization is clearly the preferred method.
5 Summary and Outlook
- The authors derived boosting methods for flexible survival models with time-varying effects.
- It is hard to decide if a covariate should enter the model as a linear term, smooth term or as time-varying effect or if the covariate is not required at all.
- Boosting offers the possibility to estimate the model with inherent model choice and variable selection.
- One possible alternative to the proposed model choice scheme in CoxFlexBoost could be to fit the model in a similar fashion like that proposed in the MFPT approach by Sauerbrei et al. [8].
- Including time-varying effects for smooth effects would result in modeling an interaction of two functions:.
Acknowledgments
- The authors thank W. H. Hartl from the Department of Surgery, Klinikum Großhadern for the data set and stimulating problems and D. Inthorn and H. Schneeberger for initiation and maintenance of the database of the surgical intensive care unit.
- B. Hofner and T. Hothorn were supported by Deutsche Forschungsgemeinschaft, grant HO 3242/1-3.
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Cites background from "Operations Research Games: a Survey..."
...(44) However, these limiting assumptions have not prevented game theory from contributing to analysis in numerous related areas.(45) One response that is often offered in defense of the assumption above, that adversary objectives are strictly consequence maximizers, is that it defaults to the worst case and is therefore “erring” in the right direction....
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Cites background from "Operations Research Games: a Survey..."
...We refer the reader to Young (1985) that gives a thorough review of basic cost allocation methods and to Borm et al. (2001) for a survey of cooperative games associated with operations research problems....
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...Kruskal (1956) and Prim (1957) provide two greedy algorithms for solving this kind of problem....
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Q2. What future works have the authors mentioned in the paper "Operations research games: a survey" ?
Issues to be considered in the future involve: ² dynamics: changes in the player set and other time-related aspects, ² strategic incentives ( coopetition ), ² minimising private information exchange, ² consistency, monotonicity and continuity arguments for allocation rules, ² stochastic uncertainty, ² asymmetric information between the players with respect to the data of the underlying operations research problem.
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Q4. What is the recent contribution to the theory of °ow situations?
An interesting recent contribution to the theory of °ow situations is the characterisation of the MC solution (as a set valued solution) for so-called simple °ow situations, i.e., situations where each player dictatorially controls exactly one arc, other arcs are publicly available (with control games w with w(S) = 1 for all coalitions S) and all arcs have a capacity of 1.
Q5. What are some examples of assignments that can be used to prove the balance of permutation games?
Applications that can be analysed using assignment games are, e.g., private markets in used cars, real estate markets and auctions.
Q6. What is the common way to create a context speci c allocation rule?
Such a rule can be based either on desirable properties in this speci¯c context or on a kind of decentralised mechanism that prescribes an allocation on the basis of the algorithmic process along which a jointly optimal combinatorial structure is established (e.g., following an algorithm to create a minimal cost spanning tree).