Risk Curve: Modelling the “Ideal” Hazard

In this episode of Risk! Engineers Talk Governance, due diligence engineers Richard Robinson and Gaye Francis discuss the mathematics behind risk modelling and why relying on heat maps for decision-making can have limitations.

Richard and Gaye explore the concept of the "ideal" hazard risk curve, unpacking why every hazard carries its own unique risk profile rather than a neat line of constant risk. Drawing on Heinrich's accident triangle and the hyperbolic relationship between consequence and likelihood, Richard walks through the calculus of integrating under a risk curve, and why simply "spotting the dot" on a five-by-five risk matrix can underestimate high-consequence, low-likelihood events by an order of magnitude or more.

They discuss the limitations of the standard risk matrix for large organisations dealing with vastly different scales of risk, and why New Zealand's updated WHS legislation is shifting focus toward identifying critical hazards and credible controls first, rather than getting bogged down debating likelihood.

Key takeaways:

• Heat maps are useful for communication, but dangerous as standalone decision-making tools
• The area under the risk curve matters – it's far larger than a single dot suggests
• Safety risk assessment should prioritise critical hazards and reasonable controls over likelihood arguments

Note: This episode references slides — for the full visual experience, check out the YouTube channel https://www.youtube.com/c/R2aAu.

If you’d like us to cover a specific topic or have any feedback we’d love to hear from you. Email admin@r2a.com.au. 

For further information on Richard and Gaye’s consulting work with R2A, head to https://www.r2a.com.au, where you’ll also find their booklets (store) and a sign-up for their quarterly newsletter to keep informed of their latest news and events. 

Gaye is also founder of Australian women’s safety workwear company Apto PPE https://www.aptoppe.com.au.

Transcript

Megan (Producer) (00:00):

Welcome to Risk! Engineers Talk Governance. In this episode, due diligence engineers, Richard Robinson and Gaye Francis discuss the Risk Curve: Modeling the "Ideal" Hazard.

(00:14):

We hope you enjoy the chat. If you do, please support our work by giving us a rating and subscribing on your favourite podcast platform. And if you'd like more information on R2A, our newsletter and resources or have any feedback or topic ideas, please head to the website www.r2a.com.au.

Gaye Francis (00:33):

Hi Richard. Welcome to another podcast session.

Richard Robinson (00:36):

Good morning, Gaye.

Gaye Francis (00:38):

Today we're going to talk about modeling the ideal hazard if that really exists.

Richard Robinson (00:42):

As we mentioned earlier, small children can be in this category.

Gaye Francis (00:46):

Ideal hazards, maybe. So what we're going to do just as an introduction, I guess, we're going to use some slides in this particular podcast. So if you usually just listen to us and it might be worth transferring over to the YouTube channel (see description), but we will explain the slides as best we can as we're holding them up as well. So not critical to go to the YouTube channel, but just be aware that we will be holding some slides up.

Richard Robinson (01:13):

Now, the reason why we're doing this particular podcast, because it's come to our attention that a lot of people have decided to use heat maps again with regards to downside risk.

Gaye Francis (01:21):

For decision making purposes.

Richard Robinson (01:22):

Rather than just reporting purposes. And we've talked about this in the past, but I thought it was worth just mathematically explaining a little more robustly why using it as a decision making tool will put you in the wrong place. Now I'll put up the slide here, but normally R2A deals with sort of three core areas of risk. Downside risk, the risk from which there's no prospective gain, that's safety risk, which is all negative.

Gaye Francis (01:47):

Yep. And that's your graph on your left hand side.

Richard Robinson (01:51):

Upside, downside risk, which is what popped up in the 1980s, particularly with the commercial people. So you can buy shares in the stock market, the price can go up or the price can go down. Or downside risk from a promised upside risk position, which is what project risk is about. Those three things are actually quite different for a start, but today we're just talking about downside safety risk in the first instance. Now, if you just look at that slide, it is downside. It's going negative. But when you're actually doing this sort of stuff, most people use the heat map. They've got the heat map on the positive side of things, which is sort of on one level, a little bit peculiar.

Gaye Francis (02:28):

So when Richard says the heat map, it's typically the five by five matrix like we see in the Australian Standard.

Richard Robinson (02:34):

Yeah, red, amber, green sort of thing. Where green's low risk and red usually don't go there. Now one of the interesting things about these things, they're normally done on a log log scale. And the reason why that's the case, there's actually some technical competent reasons for that is that back in the 1930s, a fellow called Heinrich in the US was doing studies of all US accidents and that was built upon by a guy called Bird and you probably read about it, the accident triangle where one very serious incident was accompanied by 10 not so serious events and then about 30 or 40 or 50 other events. And by the time you get down to the near misses, you're at sort of the hundreds, close to a thousand, that sort of thing.

(03:10):

The numbers were always a bit flaky because how many people actually report near misses when it comes to the crunch. But the general recognition was, and I've never seen anybody disagree with this, is that generally a tenfold increase in consequences accompanied by a tenfold decrease in risk. And that's a hyperbolic curve, Y equals one on X, Gaye.

Gaye Francis (03:33):

Excellent mass lesson, Richard. <laughs>

Richard Robinson (03:34):

Well, we haven't got there yet. Now what people do on a heat map ordinarily, they just spot the dot, one point. And if you assume the risk follows a line of constant risk, on a log log scale, that will be a 45 degree slope, which is the way most people normally sort of treat these things. And a tenfold increase in consequence, accompanied by tenfold decrease in likelihood. Then if you pick one point in that line at one level, you have characterised the full possible range of outcomes of that risk because a very small one happens a lot and a very rare one, a very big one is very rare. The problem, of course, is that no hazard has a line of constant risk associated with it. Every hazard has its own unique risk curve associated with it.

Gaye Francis (04:24):

And it's often wobbly and different shaped.

Richard Robinson (04:26):

Yeah, mine is pretty wobbling. I mean, I've sort of just sort of prepared a prior argument.

Gaye Francis (04:31):

A non-ideal.

Richard Robinson (04:32):

A non-ideal <hazard>. And what that's basically showing is that there's probably wobbles. I mean, if you just think about a high voltage hazard or something like that, if the arc flashes, it's either pretty much going to hit you or it's going to miss you. Or if there's two of you walking along...

Gaye Francis (04:45):

One goes and one doesn't.

Richard Robinson (04:47):

Or maybe both of you go. But it'll have jumps and bumps in it. But nevertheless, if you were actually trying to work out the actual risk associated with this thing, what you really need to know is the area under the risk curve.

Gaye Francis (05:00):

Okay.

Richard Robinson (05:01):

And that means you've got to really integrate the area under the risk curve. Now if you take the ideal risk curve, which is Y equals one on X, which is the hyperbolic, and then you plot that on an ordinary graph paper, it looks something like that <shows slide>, which is, if you think of a hyperbolic risk curve, it's obviously a tenfold increase. Well, you get a shape like that (high then drops sharply). Now in practical terms, this always made a lot of sense to me because the one place you do not want to be is high consequence, high likelihood.

Gaye Francis (05:36):

Correct.

Richard Robinson (05:37):

And that means you're in the top left hand corner of this diagram, the bright spot, the hotspot on the chart. And if you stay equally distance away from a point up there, what shape curve must you get?

Gaye Francis (05:48):

Hyperb...

Richard Robinson (05:50):

This is really getting <difficult>.

Gaye Francis (05:51):

I know it is. <laughs>

Richard Robinson (05:56):

Now, and if you want to integrate the area under the risk curve and it's a function Y equals one over X, Gaye, what's the integral of one over X? It's the natural log.

Gaye Francis (06:07):

It's the natural log. There you go.

Richard Robinson (06:09):

Now if you start integrating the natural log, one of the difficulties you have with a hyperbola is it goes asymptopic in both directions. So if you integrate from zero to infinity, you get an infinite number, which makes it mathematically very inconvenient.

Gaye Francis (06:24):

Almost impossible, you would think.

Richard Robinson (06:25):

So therefore you have to integrate over a range. Now obviously if it's such a small matter, then you don't worry about the really small events because nobody's being hurt and maybe somebody gets a bruise or something like that - it happens an awful lot, but who cares? It's a bit like stubbing your toe in the morning when you get up. It's very annoying, but you still go to work.

Gaye Francis (06:42):

That's true.

Richard Robinson (06:46):

And then obviously you've got to put a limit on the other end because if you keep saying planet destroying meteorites constitute a hazard that we have to deal with, then it's a one by 10 to the minus 12 and minus 14. How far do you integrate the other direction? So you have to integrate under the line over a region and what change does it get. Now the plot that I just put up was just basically one over X, so it's going through the one point. So therefore if you just sort of start integrating from say one to 10, one to 100, one to a thousand, one to 10,000, something like that, what's the actual increase you get over the actual line, which at that point is just one. It's a line of constant risk going through one. And the answer is quite significant. The handy thing about integrating that particular line is that you have to... the log integral is the LM of the upper limit minus the LM of the lower limit. The lower limit has to be one that makes the natural log zero.

Gaye Francis (07:56):

OK.

Richard Robinson (07:56):

Or if anybody's interested in this sort of stuff and you want to do it yourself, that's easy. But just to give you the rough idea, if you integrate from one to 10 of one over X and the natural log, it goes up by a factor of two point something. If you integrate from one to a hundred, it goes up by a factor of 4.6, from one to a thousand it goes up 6.9, from one to 100,000 goes up to 11 point times the value of risk. And if you went to one in a million, it goes up by a factor of something like 13, an order of magnitude. That's significant, particularly when you're dealing with high consequence, low likelihood events that happen very rarely. And as I pointed out to you before, before we started this podcast, if you want to be statistically significant about failures, you've got to fail at least three times doing the same thing, which from a risk point of view is not a particularly good thing to do.

Gaye Francis (08:51):

No, and I don't think that's particularly a lot of times have to fail either.

Richard Robinson (08:56):

No, what's the line? If it first you don't succeed, try, try again. Whether it's statistically significant. Which must be useful. I don't know. Now the point about all this is everybody spots the dot and thinks, I don't know what they sometimes think. I think they think they've done a risk assessment. But what you've done is a fearfully complicated thing and you want to be clear we have only at this point considered downside, pure risk, safety risk. We're not talking about upside normal distribution risk that the commercial people like doing. That's why people do PhDs and these things. And we certainly haven't been considering downside risk from the point of view of a critical project. Because remember with the project is downside risk from the promised upside risk position.

Gaye Francis (09:37):

So where do you start your curve?

Richard Robinson (09:39):

Well, remember the whole point of our project due diligence process is you start off with the critical things because you cannot let a showstopper happen. It's not possible with a big project.

Gaye Francis (09:50):

But that means we're still ... I think what this is saying is typically you know what the potential range of consequences are. Characterising the likelihood associated with the consequences is always the difficult part.

Richard Robinson (10:03):

And if you're talking about high consequences, likehood, like for example, an explosive air fuel mixture in the chamber and you just spot the dot and say one in a million is here, you have probably underestimated by an order of magnitude.

Gaye Francis (10:17):

And I think that's, well, primarily that was one of the reasons why the WHS legislation came in and the due diligence approach was to eliminate all of those games that people played with the likelihood things, with the likelihood arguments.

Richard Robinson (10:32):

And what are the Kiwis done with their new amendment bill? They're changing their legislation to say that you must focus on the critical ones first.

Gaye Francis (10:41):

Critical ones, but you're also still focusing on the controls that you can put in place. So there's no discussion around, you're only testing the likelihood element when you're looking at the control that you can put in place as a balanced argument.

Richard Robinson (10:57):

It's a reasonableness test.

Gaye Francis (10:59):

And I think what we've seen in a lot of these sort of stuff when people are using the risk characterisation five by five matrix and characterising risk, they get bogged down in the detail because it's always an argument about what the risk is before you even start to look at the controls you can put in place.

Richard Robinson (11:18):

But the other problem they've got is the scope of their risk curve. I mean, if you're talking about a one in a million event, awful lot of people say that 99.9, well, 0.001, rare is one in a thousand. No. They don't just cover the range.

Gaye Francis (11:36):

No. And we've found that when we have been doing some of these, the risk matrix curves are, oh, sorry, risk matrices are really good at the communication aspect and we've done it for enterprise risk frameworks and things like that. So you're communicating things and characterising things in the system of the <inaudible>.

Richard Robinson (11:58):

The value system of the decision makers.

Gaye Francis (12:01):

But often you're right, the five by five scale is not big enough to characterise all of the issues of a water company, for example, because they're dealing with dam failures, but they're dealing with burst pipes as well.

Richard Robinson (12:16):

Same thing with hospitals, but any large organisation, the state governments, remember we did a couple of jobs for hospitals. You got the hospital, what's critical for individual hospital isn't necessarily critical for Department of Health and what's critical for the Department of Health isn't necessarily critical for the state as a whole. So that's another two orders of magnitude. And BHP Bilton used to do it that way too.

Gaye Francis (12:34):

So we're not saying don't use any of these things, but what we're saying is understand what they can be useful, their limitations and have that understanding when you're doing it. But safety risk is still asking you to say, what are all the credible, critical issues, what are the controls you can put in place and what is reasonable in the circumstances?

Richard Robinson (12:58):

And that's where the Kiwis are going. I hope the Australians catch up quickly.

Gaye Francis (13:02):

Here's hoping, but as you said, New Zealand's definitely leading the way. That was a very technical podcast for us.

Richard Robinson (13:10):

Well, for some.

Gaye Francis (13:11):

For some. Yes, other people enjoy this sort of stuff. I understand. But thanks, Richard, for explaining that and we will see you next time.

Richard Robinson (13:19):

Thanks, Gaye.