Articles

Book: Matrix Analysis (Cox) - Mathematics


Equilibria and the solution of linear and linear least squares problems. Dynamical systems and the eigenvalue problem with the Jordan form and Laplace transform via complex integration.


A Tale of Two Procurement Books: Cox's Sourcing Portfolio Analysis (Cox) & Strategic Sourcing in the New Economy (Keith et al.)

A new face of procurement is emerging which is recognizing that a new set of value drivers must be developed in the face of massive environmental changes. Some of the emerging views on the “future of procurement” focus on the importance of being a strong internal consultant, the importance of building relationships, coaching suppliers, post-award contract management, and relational contracting. These topics are emerging in the context of the new view that procurement is about building value for the enterprise.

It is against this context that the two books in this review represent two anchors that exist in the procurement environment, both the old, traditional model of power-based, “red meat eating” buyers, and that of the procurement professional focused on driving relationship contracting and vested value to create mutual benefit for both parties. Each book represents a different perspective, and thus provide a representation of the current debate going on in procurement strategy meetings around the globe.

Andrew Cox’s new book “Sourcing Portfolio Analysis” represents the more traditional “power-based” view of procurement. The book relies on a series of prescriptive two by two matrices, that are further segmented into eight by eight matrices, that establish different typologies for supplier segmentation,. The application of these frameworks to the universe of buyer-seller relationships proposed (ostensibly) to lead to improved sourcing outcomes…for the buyer only that is! The book clearly views the supplier as a party that must be “beaten down” through the application of power-based negotiation tactics, the assumption being that buyers have been “exploited” by suppliers for years, and the tables have now turned.

This work builds on much of the prior strategic positions concepts introduced by Michael Porter’s Five Forces analysis in his book Competitive Strategy (1980), Peter Kraljic’s “Category Segmentation Matrix” published in the Harvard Business Review (1983), as well as Professor Cox’s own articles, working papers, and book chapters (which consume 3 pages of the reference section of the book). Unfortunately, the most up-to-date reference (apart from his own work) that are cited are from 2008, and it is clear that there is a gap in Cox’s use of more modern research paradigms that are related to relational contracting and negotiation. In fact, the subtitle for the book, “Power Positioning Tools for Category Management and Strategic Sourcing” gives it all away. This book essentially depicts how buyers can leverage their market positions to gain more power over suppliers in negotiating contracts. But in doing so, Cox has taken the tools of Porter and Kraljic, and put them on steroids, without adding any significant insights that help the reader better understand how to deliver value in supply chain relationships. The author spends the entire book focused on building even more detailed segmentation matrices within segmentation matrices to define different types of sourcing environments. This results in several new terms for buyer-seller relationships that he is able to form into a typology. Unfortunately, the resulting typologies are not only completely confusing, but indeed without meaning! Take, for example, this one:

Reciprocal – Supplier Development + Supply Chain Sourcing: Full lean / agile / agilean supplier collaboration at the first tier + arm’s-length sourcing from within the supply chain, in which neither party maximizes their share of value.

Buyer Dominant – Supplier Development + Partial Supply Chain Management: full lean/agile/agilean supplier collaboration at the first tier + information-based collaboration only within the supply chain, in which the buyer maximizes their share of value from all suppliers in the chain.

A good example of this type of relationship would be……what exactly? These buzz-words might be heard from a supply chain consultant who has had too much coffee at breakfast! But without a strong set of definitions, a practical set of examples, and a reference to more recent academic research, these descriptions are both confusing and lack tangible exemplars to demonstrate them. I expected more from Professor Cox, who has a lengthy set of publications in refereed journals.

What bothers me about this book is that it is entirely focused on power, and how a buyer can reduce their dependence on suppliers, and therefore “receive better value for money deals”. The notion that dependence may not be a bad thing, so long as it is governed through effective contractual mechanisms, performance measurements, open exchange of information, and mutual benefits is entirely lacking. While I am in agreement that driving competition in any type of supply chain relationship is a good thing, it can involve increasing dependence through long-term mutually beneficial relationships. For example, my experience with Honda is that they often seek to build “supplier for life” relationships, that are focused on robust should-cost models, target costing, supplier development, and continuous improvement strategies in collaboration with suppliers. In another industry, oil and gas, I have witnessed supplier relationships that span ten years or more that are based on complete open books cost plus strategies that seek to drive demand management and improvement strategies. These types of relationships do not exist in the world of Cox’s Sourcing Portfolio Analysis. But then again, not everything can be simply labeled in a two by two (or eight by eight) matrix.

Unfortunately, the typologies identified here fail to provide practical examples or cases that define exactly what is meant. And that is a telling problem with this book – these are theoretical models that have worked well, but which cannot be readily identified in practice. Perhaps this may be due to the fact that Professor Cox has had engagements primarily with organizations that are already in a position of power. His work was used for a long time in the 1990’s within IBM, a large buying company, which was known primarily for implementing Cox’s ideas around leveraging spend and driving to touch-less procurement technologies that minimized face to face contact with suppliers. Cox’s work has historically focused on an environment in which procurement is embedded in large and powerful buying organization, procuring an uncomplicated product or service from an acquiescent supply base.

Interestingly enough, it seems even large behemoths like IBM have also turned the corner on this antiquated view of procurement. I remember years ago at an IACCM conference seeing the senior director of procurement answer a question “how do you define strategic supplier relationships?” with the response “if they do a lot of business with IBM.” However, the most recent 2013 IBM Chief Procurement Officer study published by the IBM Institute for Business Value found quite a different result. They found for instance that companies with high performing procurement organizations have higher profit margins and had the three following common attributes: 1) They focus on improving enterprise success, not just procurement performance, 2) They engage with stakeholders to understand and anticipate their needs and values, and 3) They embrace progressive procurement practices and tools to drive results. These role models are also more likely to collaborate with suppliers to develop new technologies, that emphasize brokering new relationships with suppliers to introduce new ideas and innovative thinking.

These components are entirely missing from this book, as the ecosystem depicted in the Cox book is far too simplistic to capture the realities of the complex world that procurement professionals are faced with today. It assumes that buyers control their environment, and that price is the primary driver of value. And this simply is no longer the case, as IBM itself points out in their study.

If Cox’s book is anchored in the past, then the new book by Keith, Vitasek, Manrodt and Kling is completely focused on the future. This book is a remarkable contrast, in that it adopts a diametrically opposite view to the notion that power is the basis for buyer-seller relationships. In fact, the introduction of the book begins by acknowledging the contribution that Porter, Kraljic, and the firm A.T Kearney (who espoused the term “strategic sourcing” to focus on leveraging volume to reduce price). But then the authors go on to note that their time has passed, and a new approach is needed for proactive procurement strategies to emerge:

No one would debate that these pioneers have led an evolution in procurement that made a lasting impact. But times have changed. Today’s environment is more dynamic and is filled with greater uncertainty. The tried and true tools and tactics adopted over the last 30 years as the gold standard are not as effective as they once were. Organizations that historically have won by leveraging their power or by strategically maneuvering to shift power in their favor find those strategies are losing effectiveness.

Specifically, the authors adopt the acronym VUCA to describe the current business environment, standing for volatility, uncertainty, complexity, and ambiguity. Procurement must approach sourcing decisions in this environment, with the knowledge that incremental changes will not do. In a VUCA environment, traditional static models of leveraging power no longer work. Procurement must earn the trust of the business to fully succeed, and in doing so, it must embrace a vision that goes far beyond the thinking of today, leveraging creativity and know-how to change not just the function from within, but the overall business value procurement can deliver.

The authors of the book set out to achieve this, in seeking to deliver on two key objectives: 1) to build awareness that reliance on conventional power and leverage approaches in sourcing relationships is limited, and 2) to invite procurement professionals to better understand the benefits of using relational contracting approaches through the use of Sourcing Business Models. Much of this work is grounded in the work by Kate Vitasek, the developer of the “Vested” model. Vested is a Sourcing Business Model in

which buyers and suppliers carefully craft highly collaborative relationships supported by true win-win economics. A win for buyers is a win for suppliers. Buyers and suppliers are vested in each other success.

The work is also highly grounded in the work of Oliver Williamson, the Nobel Prize winning author of transaction cost economics. This paradigm is one of the most highly cited models used in the sourcing research literature, and posits that buyer-seller relationships range from highly competitive marketplaces to establishing corporate hierarchies through insourcing. This work is used as the foundation to construct the seven Sourcing Busienss Models identified in the book. The authors also provide a Business Model Mapping Toolkit that provides step-by-step instructions for determining which Sourcing Business Model is most appropriate for your situation. Interestingly, the authors provide a link for readers to download all of the models through their vested outsourcing website, believing that use of the tools will propagate their use. In the remainder of the book the authors discuss other approaches that are required to fundamentally change the nature of the dialogue between buyers and seller, and the importance of building trust in relationships.

An important differentiator of their approach is the prominent discussion of the term relational contract, defined as a combination of written contract(s), interface protocols, and distinct social norms that enable a continuously efficient and effective commercial relationship. The relationship is efficient when the parties cooperate to minimize friction toward the commercial goals (i.e.,

when the transaction costs before and after the contract award are optimized). As the authors note, “the secret to make this happen is continual alignment of interests.”

The book is written in a format that is an easy read, without the use of buzz-words and plenty of examples and illustrations for the concepts. A number of case studies are written using unnamed individuals from the procurement world who have gone through different experiences and situations. I like this approach, as it provides a real-world feel and also emphasizes the importance of individual personality and capabilities and mindset as a critical ingredient of buyer-seller relationships. These cases also clearly explain the aftermath that occurs after a “meat eating buyer” has gone through and beat up a supplier. This often involves serious lapses in supplier performance, or post-contract markups and engineering change notices that wind up costing even more money than the supposed cost savings that occurred in the contracting stage. The alternative involves Trust, choosing to be open, transparent and credible. The benefits of opening up relationships that could not occur otherwise are illustrated again and again throughout the book.

I also liked how the authors draw on a historical perspective of the sourcing function, including a description of the now-famous destruction of relational capital achieved by the “procurement pitbull” Dr. Ignacio Lopez. There are also references to other visionaries in the procurement world such as Gene Richter and Thomas Stallkamp.

In contrasting the two approaches, a comment from a former IBM procurement senior executive who helped me with my review is useful to consider:

Overall procurement strategy development does need to continue to evolve. While I understand the notion behind VUCA, and there are certainly ever-present variables, I do not see these conditions necessarily being present 24/7 in the majority of commodities/relationships in which I have participated. Similarly, I do not see these elements as necessarily providing guidance as to how to build a supply chain team nor a supplier strategy. I tend to focus and tailor the solution to the operational variables of our business, the inherent flexibility of the products and processes to react to change, and the level of dependence on suppliers for technology, innovation, and risk mitigation. “Power” and “Leverage” are the results of the strategy, not the starting points. Conversely, the idea that ‘spend’ is not an inherent lever in driving the best financial results is equally mis-guided (IMHO).

In developing and deploying procurements true value proposition to the business, everything in these two books have something to add. However procurement transformation, lessons from the past and classic maturity model thinking must be put in context. Modest changes in capability and thinking can help but will not bring the value modern business demands. There is also an emerging need for technologies that drive sourcing analytics, risk management, supplier life cycle management, and market intelligence will become critical in building effective procurement transformation components – yet the linkage to these approaches is not well described or understood in the procurement community. Procurement must progress and as George Bernard Shaw put it so eloquently “Progress is impossible without change, and those who cannot change their minds cannot change anything”.


Book: Matrix Analysis (Cox) - Mathematics

· To join zoom lecture MWF at noon please use the link provided in Canvas Announcements DO NOT click on zoom in Canvas .

Overview: Survival outcome is often the ‘ultimate’ outcome, in many critical areas of disease research such as cancer, as well as recently emerging medical AI. This course discusses the concepts, theories, and applications associated with censored and truncated survival data. The topics include likelihood for right censored and left truncated data, nonparametric estimation of survival distributions, comparing survival distributions, proportional hazards regression, semiparametric theory and other extended topics on complex survival data including competing risks etc. as time permitting.

Important Note: You are strongly encouraged to attend lectures and take notes. You are also strongly encouraged to take advantage of the office hours to discuss any questions/problems that you have - Note that you can make appointments for office hours!

Lecture: MWF 12:00-12:50pm

Teaching Assistant: Denise Rava

1. Cox and Oakes, Analysis of Survival Data, Chapman & Hall, 1984

2. Fleming and Harrington, Counting Processes and Survival Analysis, Wiley, 1991

4. Kalbfleisch and Prentice, The Statistical Analysis of Failure Time Data, Wiley, 1 st or 2 nd ed.

5. Bickel, Klaassen , Ritov and Wellner , Efficient and Adaptive Estimation for Semiparametric Models. Springer, 1998.

Not reference but read for fun: Gladwell “David and Goliath” which has the story of the Freireich (1963) leukemia survival data that D. R.Cox used and we also use.

Topics: (future topics are subject to update)

Week 1: Introduction to survival analysis right-censored and left truncated data Kaplan-Meier estimate of survival.

Week 2: Log-rank test of two-sample survival weighted log-rank tests and efficiency counting processes.

Week 3: Parametric survival distributions likelihood for right-censored and left truncated data Cox proportional hazards regression model.

Week 4: Partial likelihood inference predict survival under the Cox model time-dependent covariates.

Week 5: Martingale theory stratified Cox model goodness-of-fit methods.

Week 6: Case study model selection - stepwise, explained variation, information criteria, penalized log-likelihood.

Week 7: Design of a survival study other survival models.

Week 8: Additive hazards model semiparametric efficiency competing risks.

Week 9: Multivariate survival causal inference.

1. [ Introduction] Efron , B. and Hinkley, D.V. (1978) Assessing the accuracy of the maximum likelihood estimator: observed versus expected Fisher information. Biometrika , 65, 457-487.

2. Tsiatis A A . A nonidentifiability aspect of the problem of competing risks. Proceedings of the National Academy of Science USA, 1975 72: 20-22.

3. Cox DR. (1969) Some sampling problems in technology. In: New Development in Survey Sampling, Ed. Johnson and Smith. Wiley.

4. Vardi Y. Multiplicative censoring, renewal processes, deconvolution and decreasing density: Nonparametric estimation. Biometrika , 1989 76: 751-61.

5. Tsai, Jewell and Wang, A note on the product-limit estimator under right censoring and left truncation. Biometrika , 1987 74: 883-6.

6. Wang M-C. Nonparametric estimation of cross-sectional survival data. JASA, 1991 86: 130-143.

7. Wang M-C. A semiparametric model for randomly truncated data. JASA, 1989 84: 742-748.

8. Struthers and Farewell. A mixture model for time to AIDS data with left truncation and an uncertain origin. Biometrika , 1989 76: 814-7.

9. Asgharian M, M’Lan CE, Walfson DB. Length-biased sampling with right censoring: an unconditional approach. J Amer Stat Assoc (JASA) 2002, 97: 201-209.

10. Harrington DP, Fleming TR. A class of rank test procedures for censored survival data. Biometrika , 1982 69(3): 553-566.

11. Reid N. A conversation with Sir David Cox. Statistical Science, 1994 9: p449-450 (about the Cox model).

12. Thomsen and Keiding . A note on the calculation of expected survival. Statistics in Medicine, 1991 vol. 10, p. 733-738.

13. Xu R and O’Quigley J. Proportional hazards estimate of the conditional survival function. Journal of the Royal Statistical Society, Series B, 2000 vol.62, p. 667-680.

14. Xu R, Luo Y, Chambers, CD. Assessing the effect of vaccine on spontaneous abortion using time-dependent covariates Cox models. Pharmacoepidemiology and Drug Safety, 2012 21(8): 844-50 doi : 10.1002/pds.3301.

15. O’Quigley J and Pessione F. The problem of a covariate-time qualitative interaction in a survival study. Biometrics, 1991 47: 101-115.

16. Xu R, Adak S. Survival analysis with time-varying regression effects using a tree-based approach. Biometrics, 2002 58: 305-315.

17. Gill R. Understanding Cox’s regression model: a martingale approach. J Amer Stat Assoc (JASA). 1984 79: 441-447.

18. Andersen PK and Gill RD. Cox’s regression model for counting processes: a large sample theory. The Annals of Statistics, 1982 10: 1100-1120.

19. Lin et al. Checking the Cox model with cumulative sums of martingale-based residuals. Biometrika , 1993 vol. 80, p. 557-572.

20. Xu R, O’Quigley J. Estimating average regression effect under non-proportional hazards. Biostatistics, 2000 1: 423-439.

21. Xu R, Harrington DP. A semiparametric estimate of treatment effects with censored data. Biometrics, 2001 57:875-885.

22. Loftus JR and Taylor JE. A significance test for forward stepwise model selection. http://arxiv.org/pdf/1405.3920.pdf

23. Akaika H (1973). Information theory and an extension of the maximum likelihood principle. In: Breakthroughs in Statistics, 1992, vol.1, p.610-24. Springer, New York.

24. Xu, Vaida and Harrington. Using profile likelihood for semiparametric model selection with application to proportional hazards mixed models. Statistica Sinica , 2009 19: 819-842.

25. Volinsky , CT and Raftery, AE. Bayesian information criterion for censored survival models. Biometrics, 2000 56: 256-262.

26. Harezlak et al. Variable selection in regression – estimation, prediction, sparsity, inference. In Li and Xu ( ed ) ‘High-Dimensional Data Analysis in Cancer Research’. Springer, 2009. (available via elink )

27. Tibshirani , R. The lasso method for variable selection in the Cox model. Statistics in medicine, 1997 16(4): 385-395.

28. Huang J and Harrington D. Penalized partial likelihood regression for right-censored data with bootstrap selection of the penalty parameter. Biometrics, 2002 58: 781-791.

29. Fan J, Li R. Variable selection for Cox’s proportional hazards model and frailty model. Annals of Statistics, 2002 30(1): 74-99.

30. Bradic J, Fan J, Jiang J. Regularization for Cox's Proportional Hazards Model with NP-Dimensionality. Annals of Statistics, 2011 39(6): 3092-3120.

31. Kent J. Information gain and a general measure of correlation. Biometrika , 1983 70: 163-173.

32. O’Quigley J, Xu R, Stare J. Explained randomness in proportional hazards models. Statistics in Medicine, 2005 24: 479-489.

33. Xu R, Chambers C. A sample size calculation for spontaneous abortion in observational studies. Reproductive Toxicology, 2011 32: 490-493.

34. Gray RJ. Flexible methods for analyzing survival data using splines, with application to breast cancer prognosis. JASA, 1992: 87: 942-951.

35. Chan P, Xu R, Chambers C. A study of R-squared measure under the accelerated failure time models. Communications in Statistics – Simulation and Computation, 2018, 47(2): 380-391.

36. Struthers CA, Kalbfleisch JD. Misspecified proportional hazards models. Biometrika , 1986 73: 363-369.

37. Lagakos SW, Schoenfeld DA. Properties of proportional-hazards score tests under misspecified regression models. Biometrics, 1984 40: 1037-1048.

38. Chastang C, Byar D, Piantadosi S. A quantitative study of the bias in estimating the treatment effect caused by omitting a balanced covariate in survival model. Statistics in Medicine, 1988 7: 1243-1255.

39. Murphy SA, van der Vaart AW. On profile likelihood (with discussion). JASA. 2000 95: 449-485.

40. Maples JJ, Murphy SA, Axinn WG. Two-level proportional hazards models. Biometrics, 2002 58: 754-763.

41. Newey WK. Semiparametric efficiency bounds. J Applied Econometrics, 1990 5(2): 99-135.

42. Li X, Xu R. Empirical and kernel estimation of covariate distribution conditional on survival time. Computational Statistics and Data Analysis. 2006 50(12): 3629-3643.

43. Strandberg E, Lin X, Xu R. Estimation of main effect when covariates have non-proportional hazards. Communications in Statistics – Simulation and Computation, 2014, 43(7): 1760-1770.

44. Prentice RL. On non-parametric maximum likelihood estimation of the bivariate survivor function. Statistics in Medicine, 1999 18: 2517-2527.

45. Wei LJ, Lin DY, Weissfeld L. Failure time data by modeling marginal distributions. JASA 1989 84: 1065-1073.

46. Morris CN. Parametric empirical Bayes inference: theory and applications (with discussion). JASA, 1983 78: 47-65.

47. Vaida F, Xu R. Proprotional hazards model with random effects. Statistics in Medicine, 2000 19: 3309-3324.

48. Gamst A, Donohue M, Xu R. Asymptotic properties and empirical evaluation of the NPMLE in the proportional hazards mixed-effects model. Statistica Sinica , 2009 19: 997-1011.

49. Louis TA. Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society B, 1982 44(2): 226-233.

50. Ripatti S, Palmgren J. Estimation of multivariate frailty models using penalized partial likelihood. Biometrics, 2000 56: 1016-1022.

51. Murphy SA. Consistency in a proportional hazards model incorporating a random effect. Annals of Statistics, 1994 22(2): 712-731.

52. Hou J et al. High-dimensional variable selection and prediction under competing risks with application to SEER-Medicare linked data. Statistics in Medicine, 2018 37(24): 3486-3502.

53. Hernan MA, Brumback B, Robins JM. Marginal structural models to estimate the joint causal effect of nonrandomized treatments. JASA, 2001 96(454): 440-448.

Homework: You may discuss, but please write them independently. Write your solutions, answers and results in your own words (and in complete sentences, and clearly lay out your setup, background etc.) in the main part, and append program codes in the back all needs to be turned in. Any two students turning in exactly the same solutions may be considered plagiarism.

HW1 (35%, due on Gradescope by 11:59pm on Sunday 4/26):

1) Answer the questions in the study points file, up to and including Topic 4 (10%, see Canvas)

2) Exercise on page 18 of Topic 2 lecture notes

3) a) Simulate a clinical trial data set by taking sample size n=100, T from the standard Exponential (1) distribution, and C from Uniform (0, c) distribution. Choose a value for c so that about 30% of the data are censored. Plot the KM curve for the survival function S(t) of T and its pointwise 95% confidence intervals (CI). What is the estimated median follow-up time?

b) Now focus on estimating S( 0.3) from the above distribution. Repeat the simulation of part a) 1000 times, summarize in a table the bias, standard error (SE), standard deviation (SD) of the estimates from the 1000 repeats, and coverage probability (CP) of the 95% confidence intervals.

4) Exercise on page 38 of Topic 3 lecture notes (latest version)

5) Exercise on page 18 of Topic 4 lecture notes.

HW2 (35%, due on Gradescope by 11:59pm on Sunday 5/17):

1) Answer the questions in the study points file, up to and including design of a survival study (10%, see Canvas)

2) Consider the data set ‘ lymphoma.prognosis ’ at

Find the 5 binary covariates used in Xu and Adak (2002, which were identified in the original Shipp et al. paper).

a) Carry out log-rank tests to compare the survival of the two groups formed according to each covariate ( eg. Age > 60 vs. otherwise, etc.), plot the corresponding KM curves plot also the log-log of the KM curves to check the PH assumption.

b) Fit a Cox model with all 5 covariates, and continue with parts c) and d) below

c) Use the cumulative martingale-based residuals to check the proportional hazards assumption for each covariate in the model

d) Compute one of the R-squares measures that we talked about.

e) Fit a non-PH model with piecewise constant regression effects for all 5 covariates, where the ‘pieces’ are 3 intervals with approximately equal number of events each.

3) Do the exercise on page 6 of martingale theory notes, as well as the verification on page 10.

Papers for final presentation:

# 20[group 2], 52[group 3], 27[group 4], 34[group 5], 28[group 6], 53[group 1]