Archive for the ‘Risk analysis’ Category
On the accuracy of group estimates
Introduction
The essential idea behind group estimation is that an estimate made by a group is likely to be more accurate than one made by an individual in the group. This notion is the basis for the Delphi method and its variants. In this post, I use arguments involving probabilities to gain some insight into the conditions under which group estimates are more accurate than individual ones.
An insight from conditional probability
Let’s begin with a simple group estimation scenario.
Assume we have two individuals of similar skill who have been asked to provide independent estimates of some quantity, say a project task duration. Further, let us assume that each individual has a probability 
Based on the above, the probability that they both make a correct estimate, 
This is a consequence of our assumption that the individual estimates are independent of each other.
Similarly, the probability that they both get it wrong, 
Now we can ask the following question:
What is the probability that both individuals make the correct estimate if we know that they have both made the same estimate?
This can be figured out using Bayes’ Theorem, which in the context of the question can be stated as follows:
In the above equation, 
Similarly, 
Question: Why?
Answer: If both estimators are correct then they must have made the same estimate (i.e. they must both within be an acceptable range of the right answer).
Finally, 
Now lets apply Bayes’ theorem to the following two cases:
- Both individuals are good estimators – i.e. they have a high probability of making a correct estimate. We’ll assume they both have a 90% chance of getting it right (
).
- Both individuals are poor estimators – i.e. they have a low probability of making a correct estimate. We’ll assume they both have a 30% chance of getting it right (
)
Consider the first case. The probability that both estimators get it right given that they make the same estimate is:
Thus we see that the group estimate has a significantly better chance of being right than the individual ones: a probability of 0.9878 as opposed to 0.9.
In the second case, the probability that both get it right is:
The situation is completely reversed: the group estimate has a much smaller chance of being right than an individual estimate!
In summary: estimates provided by a group consisting of individuals of similar ability working independently are more likely to be right (compared to individual estimates) if the group consists of competent estimators and more likely to be wrong (compared to individual estimates) if the group consists of poor estimators.
Assumptions and complications
I have made a number of simplifying assumptions in the above argument. I discuss these below with some commentary.
- The main assumption is that individuals work independently. This assumption is not valid for many situations. For example, project estimates are often made by a group of people working together. Although one can’t work out what will happen in such situations using the arguments of the previous section, it is reasonable to assume that given the right conditions, estimators will use their collective knowledge to work collaboratively. Other things being equal, such collaboration would lead a group of skilled estimators to reinforce each others’ estimates (which are likely to be quite similar) whereas less skilled ones may spend time arguing over their (possibly different and incorrect) guesses. Based on this, it seems reasonable to conjecture that groups consisting of good estimators will tend to make even better estimates than they would individually whereas those consisting of poor estimators have a significant chance of making worse ones.
- Another assumption is that an estimate is either good or bad. In reality there is a range that is neither good nor bad, but may be acceptable.
- Yet another assumption is that an estimator’s ability can be accurately quantified using a single numerical probability. This is fine providing the number actually represents the person’s estimation ability for the situation at hand. However, typically such probabilities are evaluated on the basis of past estimates. The problem is, every situation is unique and history may not be a good guide to the situation at hand. The best way to address this is to involve people with diverse experience in the estimation exercise. This will almost often lead to a significant spread of estimates which may then have to be refined by debate and negotiation.
Real-life estimation situations have a number of other complications. To begin with, the influence that specific individuals have on the estimation process may vary – a manager who is a poor estimator may, by virtue of his position, have a greater influence than others in a group. This will skew the group estimate by a factor that cannot be estimated. Moreover, strategic behaviour may influence estimates in a myriad other ways. Then there is the groupthink factor as well.
…and I’m sure there are many others.
Finally I should mention that group estimates can depend on the details of the estimation process. For example, research suggests that under certain conditions competition can lead to better estimates than cooperation.
Conclusion
In this post I have attempted to make some general inferences regarding the validity of group estimates based on arguments involving conditional probabilities. The arguments suggest that, all other things being equal, a collective estimate from a bunch of skilled estimators will generally be better than their individual estimates whereas an estimate from a group of less skilled estimators will tend to be worse than their individual estimates. Of course, in real life, there are a host of other factors that can come into play: power, politics and biases being just a few. Though these are often hidden, they can influence group estimates in inestimable ways.
Acknowledgement
Thanks go out to George Gkotsis and Craig Brown for their comments which inspired this post.
Uncertainty about uncertainty
Introduction
More often than not, managerial decisions are made on the basis of uncertain information. To lend some rigour to the process of decision making, it is sometimes assumed that uncertainties of interest can be quantified accurately using probabilities. As it turns out, this assumption can be incorrect in many situations because the probabilities themselves can be uncertain. In this post I discuss a couple of ways in which such uncertainty about uncertainty can manifest itself.
The problem of vagueness
In a paper entitled, “Is Probability the Only Coherent Approach to Uncertainty?”, Mark Colyvan made a distinction between two types of uncertainty:
- Uncertainty about some underlying fact. For example, we might be uncertain about the cost of a project – that there will be a cost is a fact, but we are uncertain about what exactly it will be.
- Uncertainty about situations where there is no underlying fact. For example, we might be uncertain about whether customers will be satisfied with the outcome of a project. The problem here is the definition of customer satisfaction. How do we measure it? What about customers who are neither satisfied nor dissatisfied? There is no clear-cut definition of what customer satisfaction actually is.
The first type of uncertainty refers to the lack of knowledge about something that we know exists. This is sometimes referred to as epistemic uncertainty – i.e. uncertainty pertaining to knowledge. Such uncertainty arises from imprecise measurements, changes in the object of interest etc. The key point is that we know for certain that the item of interest has well-defined properties, but we don’t know what they are and hence the uncertainty. Such uncertainty can be quantified accurately using probability.
Vagueness, on the other hand, arises from an imprecise use of language. Specifically, the term refers to the use of criteria that cannot distinguish between borderline cases. Let’s clarify this using the example discussed earlier. A popular way to measure customer satisfaction is through surveys. Such surveys may be able to tell us that customer A is more satisfied than customer B. However, they cannot distinguish between borderline cases because any boundary between satisfied and not satisfied customers is arbitrary. This problem becomes apparent when considering an indifferent customer. How should such a customer be classified – satisfied or not satisfied? Further, what about customers who choose not to respond? It is therefore clear that any numerical probability computed from such data cannot be considered accurate. In other words, the probability itself is uncertain.
Ambiguity in classification
Although the distinction made by Colyvan is important, there is a deeper issue that can afflict uncertainties that appear to be quantifiable at first sight. To understand how this happens, we’ll first need to take a brief look at how probabilities are usually computed.
An operational definition of probability is that it is the ratio of the number of times the event of interest occurs to the total number of events observed. For example, if my manager notes my arrival times at work over 100 days and finds that I arrive before 8:00 am on 62 days then he could infer that the probability my arriving before 8:00 am is 0.62. Since the probability is assumed to equal the frequency of occurrence of the event of interest, this is sometimes called the frequentist interpretation of probability.
The above seems straightforward enough, so you might be asking: where’s the problem?
The problem is that events can generally be classified in several different ways and the computed probability of an event occurring can depend on the way that it is classified. This is called the reference class problem. In a paper entitled, “The Reference Class Problem is Your Problem Too”, Alan Hajek described the reference class problem as follows:
“The reference class problem arises when we want to assign a probability to a proposition (or sentence, or event) X, which may be classified in various ways, yet its probability can change depending on how it is classified.”
Consider the situation I mentioned earlier. My manager’s approach seems reasonable, but there is a problem with it: all days are not the same as far as my arrival times are concerned. For example, it is quite possible that my arrival time is affected by the weather: I may arrive later on rainy days than on sunny ones. So, to get a better estimate my manager should also factor in the weather. He would then end up with two probabilities, one for fine weather and the other for foul. However, that is not all: there are a number of other criteria that could affect my arrival times – for example, my state of health (I may call in sick and not come in to work at all), whether I worked late the previous day etc.
What seemed like a straightforward problem is no longer so because of the uncertainty regarding which reference class is the right one to use.
Before closing this section, I should mention that the reference class problem has implications for many professional disciplines. I have discussed its relevance to project management in my post entitled, “The reference class problem and its implications for project management”.
To conclude
In this post we have looked at a couple of forms of uncertainty about uncertainty that have practical implications for decision makers. In particular, we have seen that probabilities used in managerial decision making can be uncertain because of vague definitions of events and/or ambiguities in their classification. The bottom line for those who use probabilities to support decision-making is to ensure that the criteria used to determine events of interest refer to unambiguous facts that are appropriate to the situation at hand. To sum up: decisions made on the basis of probabilities are only as good as the assumptions that go into them, and the assumptions themselves may be prone to uncertainties such as the ones described in this article.
Six common pitfalls in project risk analysis
The discussion of risk in presented in most textbooks and project management courses follows the well-trodden path of risk identification, analysis, response planning and monitoring (see the PMBOK guide, for example). All good stuff, no doubt. However, much of the guidance offered is at a very high level. Among other things, there is little practical advice on what not to do. In this post I address this issue by outlining some of the common pitfalls in project risk analysis.
1. Reliance on subjective judgement: People see things differently: one person’s risk may even be another person’s opportunity. For example, using a new technology in a project can be seen as a risk (when focusing on the increased chance of failure) or opportunity (when focusing on the opportunities afforded by being an early adopter). This is a somewhat extreme example, but the fact remains that individual perceptions influence the way risks are evaluated. Another problem with subjective judgement is that it is subject to cognitive biases – errors in perception. Many high profile project failures can be attributed to such biases: see my post on cognitive bias and project failure for more on this. Given these points, potential risks should be discussed from different perspectives with the aim of reaching a common understanding of what they are and how they should be dealt with.
2. Using inappropriate historical data: Purveyors of risk analysis tools and methodologies exhort project managers to determine probabilities using relevant historical data. The word relevant is important: it emphasises that the data used to calculate probabilities (or distributions) should be from situations that are similar to the one at hand. Consider, for example, the probability of a particular risk – say, that a particular developer will not be able to deliver a module by a specified date. One might have historical data for the developer, but the question remains as to which data points should be used. Clearly, only those data points that are from projects that are similar to the one at hand should be used. But how is similarity defined? Although this is not an easy question to answer, it is critical as far as the relevance of the estimate is concerned. See my post on the reference class problem for more on this point.
3. Focusing on numerical measures exclusively: There is a widespread perception that quantitative measures of risk are better than qualitative ones. However, even where reliable and relevant data is available, the measures still need to based on sound methodologies. Unfortunately, ad-hoc techniques abound in risk analysis: see my posts on Cox’s risk matrix theorem and limitations of risk scoring methods for more on these. Risk metrics based on such techniques can be misleading. As Glen Alleman points out in this comment, in many situations qualitative measures may be more appropriate and accurate than quantitative ones.
4. Ignoring known risks: It is surprising how often known risks are ignored. The reasons for this have to do with politics and mismanagement. I won’t dwell on this as I have dealt with it at length in an earlier post.
5. Overlooking the fact that risks are distributions, not point values: Risks are inherently uncertain, and any uncertain quantity is represented by a range of values, (each with an associated probability) rather than a single number (see this post for more on this point). Because of the scarcity or unreliability of historical data, distributions are often assumed a priori: that is, analysts will assume that the risk distribution has a particular form (say, normal or lognormal) and then evaluate distribution parameters using historical data. Further, analysts often choose simple distributions that that are easy to work with mathematically. These distributions often do not reflect reality. For example, they may be vulnerable to “black swan” occurences because they do not account for outliers.
6. Failing to update risks in real time: Risks are rarely static – they evolve in time, influenced by circumstances and events both in and outside the project. For example, the acquisition of a key vendor by a mega-corporation is likely to affect the delivery of that module you are waiting on –and quite likely in an adverse way. Such a change in risk is obvious; there may be many that aren’t. Consequently, project managers need to reevaluate and update risks periodically. To be fair, this is a point that most textbooks make – but it is advice that is not followed as often as it should be.
This brings me to the end of my (subjective) list of risk analysis pitfalls. Regular readers of this blog will have noticed that some of the points made in this post are similar to the ones I made in my post on estimation errors. This is no surprise: risk analysis and project estimation are activities that deal with an uncertain future, so it is to be expected that they have common problems and pitfalls. One could generalize this point: any activity that involves gazing into a murky crystal ball will be plagued by similar problems.
Eight to Late 