Microsoft Excel Statistics

In my last blog entry I introduced the notion that optimal decision making wasn’t ‘on the radar’ for many clients in Australasia, and laid out a couple of ideas why. I too once focussed on Monte Carlo simulation rather than decision evaluation, but last year the most obscure event changed that.

Call me a nerd of you will, but I like modelling problems in Excel. There is skill involved in setting up a problem such that the model assumptions aren’t too gross, and an art to making the model elegant. This elegance can be very important to optimisation problems, but more on that later. My first homemade optimisation problem was generated by motorcycle racing! MotoGP, to be precise. A friendly tipping competition with friends was formed at the start of the 2009 season with the following structure:

• Entrants played the role of Team Manager.
• Team Managers had a fixed budget to spend on riders.
• Either a few good riders could be purchased, or many lesser riders, or something in between.
• The team that had accumulated the most points at the end of the season was the winner and received kudos!

Although the future results could not be known of course so I set up and ran the optimisation with Evolver after the event to see what the optimal team selection would have been. Historical data could have been used to discover the type of rider mix that tended to be optimal and thus make an informed decision for this competition. The risk in having only a few riders was that any misfortune would have a big negative impact on the points won, whereas a team consisting of many (cheaper) riders was less likely to suffer such a fate. This downside scenario will be modelled into the 2010 MotoGP Team Manager predictive, optimised model (currently in production)!

What has this to do with the corporate world? Replace “team” with portfolio and “riders” with “assets”, “shares” or “projects” and you have a classic portfolio optimisation model. I hadn’t created this model with business applications in mind but I realised that was precisely what I was doing. An instant later I realised just how useful Evolver would be in many decision scenarios even though it doesn’t incorporate uncertainty (RISKOptimizer does).

In the next instalment I will further explore some practical applications for Evolver and you’ll see just how universally appropriate it can be.

» Making Optimal Choices, Part 1

Rishi Prabhakar
Trainer/Consultant

Something has troubled me for some time regarding the choices being made in risk land. I train and work with many clients whom have adopted Monte Carlo simulation techniques (via @RISK for Excel) into the day-to-day running of their businesses. By doing so they (hopefully) now have a good understanding of the exposure they are facing be it in project cost estimation, discounted cash flow analysis or, well, anything really. But this is only one facet of risk and decision assessment, specifically dealing with the descriptive statistical output from a simulation. What of the decision evaluation component? Why aren’t more of my customers analysing the decisions they make, or better yet actually optimising them? I have a few ideas why.

If you’re in business you have to make decisions. Big ones, little ones, yes/no, multiple state and continuous value decisions. Decisions that impact other decisions in simple or complex dependency structures. But are you making the best decisions possible? I’m sure important decisions aren’t being made completely randomly (I hope!) but I see many companies who rely completely upon qualitative techniques for their decision making (experience, gut feel, etc.) which of course means optimality is no more than a hoped for outcome rather than something that is actually being worked towards.
Firstly the decision model must be identified and then quantified, and this can be a difficult task. There is a level of modelling aptitude necessary for effective modelling that goes beyond merely knowing Excel and its functions, and into the construction of logical mathematical descriptions of possibly complicated processes. Relevant decisions need to be identified and the impact of those decisions combined into a formula that can be mathematically optimised. A critical component to all this is the knowledge that spreadsheet models can actually be optimised, and that in cases where Excel’s Solver fails there are Palisade products (Evolver and RISKOptimzer) that can perform optimisations under virtually any circumstance.

I too used to focus on Monte Carlo simulation rather than decision evaluation, and this was mainly a product of the clients I was dealing with almost exclusively when I first worked for Palisade. In my next blog I’ll tell you why that changed and also get a little more into the nuts and bolts of optimisation.

Rishi Prabhakar
Trainer/Consultant

New DecisionTools Suite 5.5.1 is a maintenance update that has been fully translated into Spanish, German, French, Portuguese and Japanese. It features simulation of password-protected worksheets in @RISK as well as an integrated RISKOptimizer toolbar. In addition, you can now also launch any DecisionTools program from within any other program already running. If you still have @RISK 5.0 or DecisionTools Suite 5.0, version 5.5.1 offers @RISK simulations that run 2 to 20 times faster than before, new scatter plots from scenario analysis, a freehand distribution artist, an Excel-style Insert Function dialog with graphs, and much more.
 
@RISK 5.5.1 and DecisionTools Suite 5.5.1 are free for current maintenance holders. If you don't have maintenance, contact Palisade to get up to date:

US/Canada
607-277-8000, sales@palisade.com

Europe
+44 1895 425050, sales@palisade-europe.com

Latin America
607-277-8000 x318, ventas@palisade.com

Brasil
607-277-8000 x318, vendas@palisade.com

Asia-Pacific
+61 2 9929 9799, sales@palisade.com.au

» Get your update
» Read What's New in @RISK 5.5.1 and DecisionTools Suite 5.5.1

Why attend?

This one-day forum is a great way to find out how others in the Healthcare Industry are using our software, as well as to learn new approaches to the problems Healthcare professionals face every day. We will have six software training sessions, and six real-world case studies presented by industry experts covering risk and decision analysis from all angles specific to the Healthcare sector.

You will also see how new versions of @RISK, PrecisionTree, RISKOptimizer, TopRank, NeuralTools, StatTools, and other Palisade software tools work together to give you the most complete picture possible in your situation.

Who should attend?

Professionals in risk and financial analysis in: Care Equipment & Services, Pharmaceuticals, Biotechnology & Life Sciences, Hospital Care & Management, or related services

How much?

For a limited time, the cost for attending the Health Risk Analysis Forum is has been discounted $100.

$295 covers all sessions, continental breakfast, lunch and a cocktail networking reception. Attendees will also receive a welcome package that includes a 15% discount on their next software purchase.

Please contact Jameson Romeo-Hall at jromeo-hall@palisade.com if you are interested in attending.

Location
The Westin Gaslamp Quarter
910 Broadway Circle
San Diego, CA 92101
(619) 239-2200

Book your room at a discounted rate (subject to availability.)

In the last two blogs I have challenged the idea that simulation results can be boiled down to a single statistic with any positive benefit. The context of a statistic is incredibly important, which is another reason why many statistics and charts/tables should be reported on, not simply one figure. And here’s a compelling reason why.

Consider two competing, similarly-sized projects, of which a company can only pursue one. Now let’s say this company would like to take on the project that has the “least risk”. If they are only familiar with generating the P90 for the total project cost they will be forced to select the project with the lowest P90. But what if the key drivers for exceeding the P90 are easier to mitigate in one project compared to the other? Perhaps the project with the lower P90 also has a higher P95 or P99 – this means the catastrophic failure is actually greater despite a lower P90 and is the mathematical equivalent of “when things go bad, they go really bad”. Not all P90s are created equally! Such an adverse outcome might sink a smaller company where a larger one could wear the loss. The context of the company running the analysis also impacts the context of the analysis itself.

So you can see not only do simulations generate results with which informed decisions can only be made if approached holistically, but if the language used is restrictive this outcome will never be achieved. Risk analyses are a necessary part of business because most of us wish to minimise the chance that something bad will happen, quite simply. Even if a manager tells you they “want the P90” what they are really asking is “tell me about the risk we’re facing”. The answer to this fundamental question is not found in a single figure taken from a simulation, but in a range of charts and tables which require correct interpretation.

More so, Monte Carlo simulation itself is only one piece of the risk and decision assessment pie. Decision modelling and optimisation, predictive modelling and statistical analyses should also form part of the quantitative approach to uncertainty. There is life beyond just risk simulation software, and I intend on exploring that in future blogs.

» Part 1
» Part 2

Rishi Prabhakar
Trainer/Consultant

Where I left off last time was lamenting the use of Monte Carlo simulation to create a single value (statistic etc.) from a model. It might still not be clear why this is anathema to me, so here goes.

A simulation is not a number. It’s not one possible (future) outcome – that’s a scenario. Monte Carlo simulation is a methodology for understanding one’s exposure to outcomes not situated close to the central tendency of the process/project in question. Note the plural “outcomes”. Risk analysis, when done properly, should let you know essentially all possible outcomes and how likely they are for your model. Output from a simulation can include a plot of means (over time), or P5s, or P95s, or the mean ± one standard deviation or any number of statistics. But that’s not plotting a simulation! Let’s not give a minimalist graph too much credit.

Such statements also perpetuate the idea that simulation is only used for creating means (or other centrally tending statistics) and ignores the wealth of information available. Risk simulation software exists to help you do risk analysis which must include not only several statistics but also sensitivity information. It is all too easy to turn a risk assessment into a hunt for a regularly asked for percentile (such as the P90) and there ends the task. I see this a lot, especially in project cost estimation where the pressure both from management and regulatory bodies is to accurately estimate some large percentile. Once found there is usually scant further risk analysis.

Nothing good ensues. When risk analyses are run “to get ‘the’ number” they become simply another box to tick in a process and ultimately any benefits (perceived or actual) will be forgotten and lost to the ages. The notion of context is also lost. No single number by itself really means anything, or at least shouldn’t mean anything to a decision maker. I have often heard phrases like “the model returned/gave $1.2m” followed by an audience nodding in agreement. Huh? Which statistic are you talking about there, and how about reporting a few other numbers around it to place that $1.2m somewhere meaningful?

In the next installment I will look further into this issue of context and hopefully prove the necessity of an holistic approach to understanding and reporting simulation results.

» Part 1

Rishi Prabhakar
Trainer/Consultant

Both the Conservative Party and the Labour Government have indicated that they will raise the state pensions age of men and women to help reduce the UK’s national debt.  In addition, more and more employers in the private sector are closing good pension schemes. The Association of Consulting Actuaries’ (ACA) recent survey on pension trends has revealed that 59% of employers are set to review pensions ahead of 2012 and 24% of employers will consider pension benefit reductions when they have to auto-enroll all employees into a scheme.

With taxes on business and individuals likely to rise over the next few years, it is difficult to see anything other than a deteriorating climate for pension savings unless there is a radical change of approach, says the ACA. It has proposed a standing Pension Commission that will challenge the legal and regulatory hurdles standing in the way of sensible long-term pension designs.

Perhaps, a more in-depth risk analysis may help the ACA make a stronger case to the government. As a related example, in the US, the Society of Actuaries and the Casualty Actuary Society, sponsored a research project with the Illinois State University to develop a model for projecting economic indices such as interest rates, equity price levels, inflation rates, unemployment rates, and real estate price levels. The model was created using Palisade’s @RISK and Microsoft Excel. In fact, @RISK’s built-in probability distribution functions, correlation matrices, and simulation results were essential to the study.

The UK ‘pensions’ landscape is set to undergo tremendous change, which will impact each and every one of us. Using scientific, risk analysis techniques, actuarial industry bodies can develop a strong argument and lobby the government so that informed policy decisions are made that are right for both the financial health of the nation and its citizens.

Craig Ferri
EMEA Managing Director of Risk & Decision Analysis

I read an interesting article on the causes of the Global Financial Crisis by John B. Taylor. Although the topic is interesting enough already, especially for a member of a risk analysis-specialising company, something else caught my eye. I have observed in training workshops, onsite consulting and now academic papers a phenomenon regarding probabilistic modelling. Many of those using the methods don’t understand what they should actually be getting from the methodology. There is an intellectual leap from the deterministic to the probabilistic that sometimes does not get made. This limits the usefulness of Monte Carlo simulation, and the value of performing such statistical analyses.

Back to the article which spurred me to write this blog in the first place. Or rather, the graph. Yes a single graph of housing starts vs. time (and its brief description) leapt out at me. One of the lines on the graph was claimed to show model simulations of housing starts using the actual interest rate, compared to the interest rate ‘predicted’ by the Taylor Rule and a third line showing actual data.

So what’s the problem?

The problem is that simulation techniques should not be used to create a single value. The single ‘simulation’ line implies a single modelled/returned value for each time period. This is deterministic modelling. There may be a particular scenario that has been modelled, but it certainly isn’t a simulation that is being represented by that single line. Simulations produce thousands of data, observed values and their associated percentiles as well central moments (mean, variance etc.). Not just one value (sorry Value at Risk – that includes you too) that can be plotted as a single line. I would guess that if a simulation were run as I understand the term then the line in the chart was probably constructed using the simulated means. But I shouldn’t be guessing.

This is far from the only time I’ve seen simulation results reduced to a single entity. I have heard from clients in the past “the simulation gave $X” with little to no context around it, and this is supposed to both mean something to me and to their customers and help to make better decisions under uncertainty…

In the next blog I will explore this idea further and discuss the sorts of results that should be gleaned from a simulation. In particular, why narrowing simulation results down to a single number is counterproductive to healthy business practices.


Rishi Prabhakar
Trainer/Consultant

Two weeks ago, I attended IQPC’s (International Quality & Productivity Center) Lean Six Sigma and Process Improvement Summit in Orlando, Florida. During the past 4 years, I have watched the conference, the attendees, and their projects evolve. The IQPC did an excellent job keeping the quality of the conference at an A+ level despite wrangling with the effects of a down market and near zero travel budgets for many companies. This conference has earned it place as one of the premier Six Sigma events of the year.

With attendance numbers on par with last year (which are only slightly down from a few years ago), the major difference that I noticed was the attendees' passion. As the economy has worsened and media’s perception of Six Sigma waned, practitioners and champions are more passionate and committed now than ever. Perhaps it’s because they still have jobs and their companies understand the value of cost reduction in both their processes and product/ process development programs. They - and the companies who employ them - have every right to be excited and passionate because they are making positive changes to their organizations that will hopefully lead them to recovery and stability faster than others.
 

Many companies, large and small, represented practically every industry. Farmers Insurance and Capital One were two representatives from the insurance and banking industries. Technology and pharmaceuticals were well represented by Seagate, Motorola, Merck and Johnson & Johnson. In addition, the energy sector was well represented, as were the military, aerospace and services sectors. (If you want a complet list of companies attending, it may be available at www.sixsigmaiq.com)

The overriding message heard over and over again, was, “We need to make your Six Sigma deployments stick.” Initially, I found this to be an interesting message since it came from a group of many highly intelligent and motivated individuals who were obviously very successful in doing just that: “Making it Stick”. This message serves as a clarion call for all of us. We need to look for new tools, philosophies and approaches to make our improvement initiative better and “stickier” so that they can pass the test of time.

The highlight of every year is the awards ceremony. There were many great projects honored this year, and congratulations to the winners and everyone who submitted their projects! At the awards ceremony I had the pleasure to meet a great group from the Bahamas Telecommunications Company. They are the pioneers for Lean Six Sigma for their company. (I tried to get them to need an onsite training session in some of the Palisade tools, but have been thus far unsuccessful!) Good luck on your Six Sigma Journey, I hope to see you accepting an award next year!

In Part 2 of this series I finished by asking what should be done with historical data, now that we have decided that storing it is probably a good idea. I won’t keep you waiting any longer.

Auditing and calibration of the model at both the micro and macro level. It’s as important as any other element of risk or statistical analysis, or indeed the model building itself. At the distribution level historical data helps to both parameterise the distributions and in fact select them in the first place. As a minimum a few data points will help you to understand possible central tendencies and variability for your risks, and also generate a list of feasible distributions to choose from. With a reasonable number of observations @RISK for Excel can be used to fit distributions to the data taking care of both distribution selection and parameterisation simultaneously. Only five data points are technically needed, but a reasonable fit will require either more than that or other holistic information to achieve validity.

At the macro level total project cost estimates are often ignored from the portfolio perspective. Commonly high percentiles are reported from such models to use in a ‘contingency’ calculation, such as the P90 or P95. Whilst a high percentile, the P90 (say) should still be exceeded 10% of the time! If your projects never go over this percentile then either there are some major mitigating factors not included in the model or the volatility is being consistently overstated. Likewise, the P10 for total cost (these ‘good’ percentiles are rarely if ever reported or considered in project cost estimation work) should be bettered in roughly 10% of projects. If this is not the case then the upside risk has been overstated. This may be due to misconceptions about the positive skewing present in most cost/delay risks or mistakes made in the parameterisation of the risks where the estimate (“most likely” etc.) is actually the “best case” or close to it, rather than a central tendency of the process over time. There could also be other possibilities.

No matter how you look at it, the collection and intelligent use of historical data is integral to effective and useful risk analysis and management, and critical to achieving valid Monte Carlo simulation results. If you aren’t currently recording everything you can get your hands on start right now!

 

» Part 1
» Part 2

Rishi Prabhakar
Trainer/Consultant

An anonymous comment in the Vail (Colorado) Daily News about the dangers of overanalyzing a decision reminded me that, while the benefits of risk analysis have been much vaunted, the costs of decision evaluation have not been clearly defined.  Sure, it's pretty easy to come up with a figure for a DFSS training effort or a budget for an entire risk management department. But what about the statistical analysis process itself?  

Well, there's staff time or your own time (which is worth something), Monte Carlo software, some portion of your computing costs,data acquisition, and on and on. Many variables. But the kind of costs I'm thinking of are the kind you rack up while you're analyzing, say, option valuation, and not doing something else.  These are opportunity costs.  They are what really limit how thoroughgoing your risk analysis becomes, which layer you drill down to--and they are very difficult to quantify.

How do you calculate whether the time you're spending in risk assessment is cost-effective? It's a problem of operations risk.  So I suppose you could enumerate all the other activities that would consume the same amount of time and model their paybacks.  But that would cost you more time in statistical analysis. . . . and you would be left in a positive feedback loop.

In the days ahead I'll be talking to risk management and operations research folks to find out how they decide how much analysis is just the right amount--not too much, and not too little.   I'll be surprised if I turn up any computational approaches--but who knows?  

In a recent public training workshop (for @RISK for Excel) I was reminded of an unusual fact regarding data.

Commonly @RISK for Excel is used to fit distributions to historical data for use in risk modelling, and it sure beats wildly guessing obscure parameters. However there are (naturally) a litany of woe-inducing problems with all historical data sets: non-stationary data series, extreme values/outliers, data recording errors, seasonality and heteroskedasticity to name a few. Excessive ‘cleansing’ of the data set is commonly prescribed, but the statistician in me cringes to even type those words! Quality control and transforming the data will help to eliminate most of those problems, but what about outliers?

In the early Naughties I was working for a large Australian bank, forecasting their daily call centre volumes for the purpose of planning staff levels and predicting service levels. A particular call centre averaged 30,000 calls per weekday. Yet on September 12th, 2001, calls dropped to less than 10,000. Along with the rest of the world, Australians were watching the terrorist attacks on television and the internet rather than calling to fix spelling mistakes in their contact details or transfer small sums of money between accounts. But what to do with that data point? Presuming the forecasting model is not intended to include such extreme events as terrorist attacks then the point could simply be filtered out of the data set and not thought of again.

But now consider a process that should include rarer events, such as flood damage or operational risk, as one of the risks in a model. If you have 10 years of good data (say), but the set includes an event that should only occur every 100 years. This level of impact is thus drastically overrepresented in the data and any fitted distribution will be biased toward such extremes. Yet the data point can not be completely ignored as such values can occur and the simulation models must have the capacity to sample such values (though with a reasonable likelihood). In this case the artistry that is fitting distributions to data comes to the fore. The data point could be removed from the set but not from our decision making process.

From the range of distributions that can be selected, the optimal choice should not only represent the remaining data well but also have a tail that samples events in the vicinity of those that have been excluded from the analysis with reasonable probability. No, that’s not always easy to do. But as with many elements of probabilistic modelling it simply must be done in order to provide useful information to decision makers.

Thus the context of the modelling can go a long way to determine the most appropriate steps to take with your data set. If that sounds like a subjective guideline then you read it correctly. Not enough people realise just how important experience and intuition can be in the seemingly prescriptive fields of mathematics and statistics. Fitting distributions to data is no different.

And yet that isn’t the unusual fact I was reminded of in the workshop! But I’ll leave that for Part 2 of my Data Issues blog.

Rishi Prabhakar
Trainer/Consultant

Today there is a heightened appetite for risk management due to global economic circumstances. But risk management has always been an intrinsic aspect of business to a higher or lesser degree. However, in the current technology-led business environment, the use of software to effectively manage risk makes logical sense. It provides a level of sophistication that the traditional processes simply cannot offer. Let me explain why.

Risk management essentially involves three stages – identification, quantification, and the on-going management of risks. In reality, these stages are not completely distinct from each other, with each stage influencing and informing the others. For example, an initial quantification of risks may lead to the conclusion that some of the identified risks are in fact not serious enough to warrant further consideration, or that the original description of the risk was not sufficiently precise for meaningful risk management measures to be put in place.

Each of these stages can benefit from the use of supporting risk modeling software. For instance, Microsoft Excel can be used to create a risk register, i.e. a database that records the risks identified, the assessment of the likelihood and impact of each of these risks, the mitigating actions that have been planned, and the assignment of responsibilities for these actions. However, there are many other software tools available, each designed for a specific purpose and focus. To illustrate, enterprise-wide risk management software focuses on the creation of integrated and holistic risk management systems, whereas Monte Carlo simulation and decision tree software place their emphasis on enhancing the quantitative analysis of risks.

The selection of the appropriate risk analysis software should involve very careful thought. The right decision can lead to a very effective implementation, whereas the wrong decision may result in a large amount of wasted investment.

There are some key considerations to bear in mind when selecting the risk modeling software. Choosing software based on how many staff will genuinely be required for the day-to-day risk management process is crucial. It is easy to select software based on the ideal situation that there will be a wide staff involvement in the risk management process. In reality, this may not be possible, potentially resulting in a cumbersome and inflexible solution being chosen over a more stand-alone and flexible application.

Similarly, knowing the level of risk quantification required is important. In fact, best practice risk management now involves the use of quantitative techniques, often using Monte Carlo simulation. When correctly conducted, the process of quantifying risks is rigorous and structured, can expose hidden or biased assumptions, as well as provide a more solid rationale upon which to base the major decisions.

Finally, determining the extent of on-going risk management needed for your business can assist with software selection. 

Needless to say, any software application will be most successful when used by appropriately trained and motivated staff, and when used as a supporting tool within an overall risk management process. Software is not a replacement for process.

Craig Ferri
EMEA Managing Director of Risk & Decision Analysis

Resolved for 2010:

Leavitt said his statement about lower global temperatures was based on "eyeballing" temperature data from recent years.  Now I ask you--my friends whose business is statistical analysis, environmental risk analysis (or any kind of risk assessment)--how cool is that?   

It’s clear that the financial crisis has exposed a number of failings in the practice of risk management. In my last post I talked about the relevance risk analysis and the disciplines of ‘quantitative risk management’ (QRM) and ‘decision making under uncertainty’ (DMU) are to all sizes of organisations, be it large or small. 

However, how accessible are these disciplines to the average size business across the globe today?

With the need to make more informed decisions more pressing by the day, thankfully QRM and DMU and now far more accessible than ever before.  Traditionally systems tended to be expensive, enterprise-based applications targeted at large companies who were prepared to spend considerable time, money and human resources.  The result was an all-singing, all-dancing product which often ended up underused due to confusion on the part of the very employees who were supposed to make it work.

Steady increases in computer processing have given the desktops of today as much power as the high-end servers of a few years ago, meaning that risk analysis and management is now an achievable goal for businesses of all sizes.  Palisades @RISK and Decision Tools Suite software are such desktop risk and decision analysis tools – working within Microsoft Excel and therefore being accessible to a large number of users.

‘Monte Carlo Simulation’, a technique originally conceived by scientists working to develop the atomic bomb as part of the Manhattan Project, is an inherent part of @RISK, a cornerstone of the Suite.  It enables users to introduce uncertainty into their previously static spreadsheets, which lets them look at things in a probabilistic, rather than a deterministic way.  In layman’s terms, this means that rather than companies and individuals making decisions based on estimates or best guesses, they can see all the potential outcomes to a venture – and how likely these scenarios are to occur.

For many companies this significantly improves the decision-making process.  Firstly it requires a change in the methodology of employees responsible for assessing risks and opportunities and secondly for the first time employees have a tool which allows them to communicate their recommendations to management or colleagues in a transparent and standardised way.  Equally, being able to look at scheduling risk in a probabilistic and quantitative sense allows for the allocation of labour and resources in a way which minimises slack and wastage whilst maximizing ROI.

So, it would seem that the new ‘risk management’ language that is starting to develop in the workplace and being taught to a new generation of managers on MBA courses should be welcomed.  With the accessibility of the technology available to assist them, we need to make sure that organisations do more than just pay lip service to QRM and DMU if they are to reap the rewards.

In my next blog I’ll be giving you the my top ten tips to adopting a health approach to risk, that will help businesses of all sizes maximize their risk management programmes.

Craig Ferri
EMEA Managing Director of Risk & Decision Analysis

Using this technique the researchers estimated that for every laboratory-confirmed case of H1N1, there were actually 79 cases.  In other words, predictions of a pandemic were not hysteria.  And while public health officials are unlikely to get hard data that will allow them to measure the actual extent and severity of H1N1 infections, no doubt without efforts to prepare for the virus, their rearview mirror methods would eventually tell a much grimmer story.    

I'll defer to Mr. Miller, whose blog is loaded with interesting details.