**Episode 1: presumption laws**

[IN PROGRESS]

OK so I’m gonna try something new. I have started listening to the SF Chronicle’s podcast and I am loving it. But also theoretically I should probably not be listening to podcasts quite so much. But listen, sometimes code is slow.

In any case, they bring up a lot of interesting topics about my beloved city and so I’m thinking I’ll pick up on some of these and go wild on the data side. We’ll see what happens.

The first post I wanna dive into is this idea of *presumption laws*. This came up in the August 13 episode of City Insider, which interviewed (bamf) Fire Chief Jeanine Nicholson. Chief Nicholson had previously battled aggressive breast cancer, and mentioned in the interview that the laws around workers compensation for firefighters are unusual.

In most cases, a worker has to prove that something about their job *caused* their injury or illness. With a presumption law, the employer has to prove that the job was not the cause of the injury/illness. Now,

*{Job caused illness} = {Job didn’t cause illness}^C*

meaning the “event” that the job caused the illness is just the converse of the “event” that the job didn’t cause the illness. This implies that the probability that the job caused the illness is just,

*P(Job caused illness) = 1-P(Job didn’t cause illness)*

This sort of makes it look like there isn’t a huge difference between what the employer has to prove under presumption laws and what the employee has to prove in the usual case, and so the number of cases that go for the employee might be about the same.

That is, for example, a trial without presumption laws might find that the probability the job caused the illness was .4 and the employee would not get worker’s comp. The same case tried without the presumption laws would find the probability the job didn’t cause the illness was .6, and the outcome would be the same.

Not so. Let’s do this.

### Firefighting and cancer

First, the numbers. Firefighters get a lot of cancer. In Boston, the rates are estimated to be about twice the overall average [1]. A CDC report [2] found increased risk of cancer, especially mesothelioma.

But is the cancer *caused by* the firefighting? On that point, we get a lot more hemming and hawing. See,

If you are a fire fighter and have cancer this study does not mean that your service caused your cancer. This study cannot determine if an individual’s specific cancer is service-related. In addition to exposures that you may have encountered as a fire fighter there are other factors that may influence whether or not you developed a particular cancer, and this study was not able to address many of these factors. [2]

And the classic, “may be/association/suggests” language, which gives authors a good way to avoid making statements about causes,

We report that firefighting may be associated with increased risk of solid cancers. Furthermore, we report a new finding of excess malignant mesothelioma among firefighters, suggesting the presence of an occupational disease from asbestos hazards in the workplace. [3]

### Proving Causation

The classic and most trusted way to prove one thing causes another is to run an experiment. Experiments should have the exposure randomly assigned and compare the exposed people to a control group who weren’t exposed, which is why we call these sorts of experiments “Randomized Controlled Trials” (RCT’s). In this case, an ideal experiment would involve randomly making some people firefighters. Or randomly sending some firefighters out to inhale/roll around in more toxins than others. In other words, not a workable solution. But in the absence of an experiment, proving causation gets way, way harder. Some even think it’s impossible [4]. Some will trot out the old “correlation is not causation” apple. Sigh. I know.

In these cases, add the fact that not only does the worker have to prove firefighting can cause cancer in general, they have to prove firefighting caused *their own* cancer. Statistics is generally much better at looking at large samples than small samples, and a sample of one is about as small as it gets.

So hopefully we’re starting to get a sense of why proving cancer is caused by the job is way harder than keeping your employer from proving the cancer wasn’t caused by the job. That is, suing your employer when the presumption is that the cancer was caused by the job makes it the employers job to prove causation and not yours. And that makes all the difference.

### Impact of the law

Unfortunately, we don’t have a lot of data readily available about how many more cases are won in states with presumption laws than without.

In Pennsylvania, the number of claims has increased since their presumption law was passed.

Before Pennsylvania enacted its presumption law, in July 2011, Kachline said, Philadelphia firefighters with cancer filed only about two or three cases a year. By December 2013, 62 had filed claims. [5]

In Texas, the law seems to be giving firefighters very little of a leg up,

Of 117 workers comp cancer claims filed by firefighters in the state since 2012, 91 percent have been denied, according to the Texas Department of Insurance. [6]

### Probability of Causation

Alright so we’ve sort of seen that proving that something caused your cancer is hard, but I want to talk about why that’s true mathematically. Turns out, it’s a pretty interesting question. Here, I’m going to lean heavily on the work of my brilliant, generous heroine Prof. Maria Cuellar of Penn.

Anyone who has had the (mis)fortune to take a stats 101 course has heard of hypothesis testing. We set our “null” hypothesis as sort of what conventional knowledge says about the world. We call the null hypothesis *H0 *(“H naught”). The alternative *Ha *(…”H A”) is the new theory we’re trying out. With a presumption law, our null is exactly what we “presume” to be true–that the job caused the cancer:

*H0: job caused cancer; Ha: job did not cause cancer*

Without the presumption law, this is flipped:

*H0: job did not cause cancer; Ha: job caused cancer*

As many of us have heard (and some of us have repeated) a hundred times, we can’t prove the null, we can only reject it in favor of the alternative. So in the presumption law setting, we might reject that the job didn’t cause the cancer in favor of the idea that the job did cause the cancer, based on the evidence presented. In the usual setting, we reject the employer’s claim that the job didn’t cause the cancer in favor of the claim that the job did cause the cancer, and we give the employee some money. Whew.

The reason these two settings are so different is that we need sufficient evidence in order to reject the null. In order to fail to reject the null, there just has to be not enough evidence to reject it. The problem is not symmetric. It’s much easier to not have enough evidence than it is to have enough evidence.

Now that we’ve established our null and alternative hypotheses, we have to figure out what we mean by “enough evidence.” Often in hypothesis testing we’ll have a question like, “is the mean of group 1 bigger than the mean of group 2?” This looks like:

*H0: E(X1) = E(X2) ; Ha: E(X1) > E(X2)*

We can estimate the true means *E(X1), E(X2) *using the average of the data we have from each group. We reject the null hypothesis if the sample average of group 1 is big enough compared to the sample average from group 2 to make us suspicious that group 1 and group 2 have the same means.

In our case, we aren’t interested in the means of groups. We want to know the probability of cancer for an individual if they hadn’t been exposed. We can’t use sample averages anymore, instead we have to look at something a bit more delicate.

Under the usual laws, you have to prove that the cancer you got was from the exposure. Meaning, if you had not been exposed, but everything about you was the same, you would not have gotten cancer. In causal inference these types of “had X, would have Y” statements are called potential outcomes. So for example,

*Y(A=1) = 1 and **Y(A=0) = 0*

would mean that if I’m exposed *(A=1)* I get cancer *(Y=1) *and if I’m not exposed *(A=0)*, I don’t get cancer *(Y=0).* The employee has to show that the probability of causation (*PC*),

*PC = P( Y(A=0) = 0| Y=1, A=1, X ) *

is high. That is, they need to show that the chances are good that the firefighter would not have gotten cancer if they hadn’t been exposed *(Y(A=0) = 0) *knowing that this firefighter was exposed *(A=1), *did get cancer *(Y=1) *and has some other characteristics (*X)* like their age, their smoking habits, gender, etc. So our new hypothesis test looks like:

(i) H0:* PC < t1 ; Ha: PC > t1*

Where *t1 * is some legally determined threshold for “likely enough.” For example, if *t1=.5*, the employee wins their case if it seems any more likely that the exposure caused the cancer than that something else did. If *t1=.75, *it means we rule for the employee if there is at least a 75% chance that the cancer is due to the exposure and not something else.

The reason estimating this is so difficult is that for a person who was exposed *(A=1), *we can’t see what would have happened if they hadn’t been. Aforementioned heroine Prof. Cuellar has some cool estimands for this problem [7].

In the presumption law case, the employer wants to show that the employee would probably have gotten cancer anyway. Meaning, they want to show

*P( Y(A=0) = 1| Y=1, A=1, X ) = 1-PC*

is high. That’s what they want to prove, which means we start with the assumption of the opposite: the probability they would have gotten sick anyway is small. This gets translated into a hypothesis test of,

*(ii) H0: 1-PC < t2 ; Ha: 1-PC > t2*

Which we can just move around terms to make:

*H0: PC > 1-t2; Ha: PC < 1-t2*

Let’s say t1 = 0.75 and t2 = 0.25. That means case (i)–no presumption law–is:

H0:* PC < 0.75 ; Ha: PC > 0.75*

And case (ii)–with a presumption law–becomes:

*H0: PC > 0.75; Ha: PC < 0.75*

Just to reiterate (because I’m confusing even myself here, there’s a reason we’re told to stay away from double negatives) case (i) now means: “I assume the probability that this person would have stayed healthy if they hadn’t been exposed is less than 75%.” To win the case, the employee has to show enough evidence that the chances they would have stayed healthy if they hadn’t been exposed are high.

Case (ii) says, “I assume the probability this person would have stayed healthy if they hadn’t been exposed is at least 75%.” To win the case, the employer has to show that the chances they would have stayed healthy without exposure are relatively low. In some sense these seem symmetric, but statistics are always measured with uncertainty.

So let’s say we **Do Math** and come up with a best guess of the probability of causation that’s 0.7, and we’re pretty sure the truth somewhere between 0.6 and 0.9. We have this range because 0.7 is just a guess, an estimate. We used modeling and sampled data and whatever else to get to it, so we can’t be totally sure it’s exact. But based on probability distributions, we can get a good idea of what a likely range is. So we’re not sure it’s exactly 0.7, but we’re pretty sure it’s not anything higher than 0.9 or lower than 0.6.

In case (i), this means I don’t have enough evidence to reject the null that the probability of causation is less than 75%, because I said there’s a pretty healthy chance it goes all the way down to 0.6. The employer wins. In case (ii), it means I don’t have enough evidence to reject the null that the probability is greater than 0.75%–it could be as high as 0.9 and I wouldn’t be surprised. The EMPLOYEE wins! As long as the range we estimate isn’t totally outside the null hypothesis, the null wins.

It’s harder to have enough evidence than it is to not have enough evidence.

Hopefully this made sense. The long and short of it is, things like presumption laws should theoretically make it way, way easier for employees to win these kinds of cases. That’s because proving you wouldn’t have gotten sick in the alternate universe where you weren’t exposed is much harder than sitting back and letting the other guy try to prove there’s an alternate universe out there where you got sick anyway.

Thanks for listening, all. Hopefully now there’s an army out there rooting for me to keep listening to podcasts and so I’m obliged to continue.

Sources (I’m being lazy with citations, I’m sorry):

[1] https://www.nbcnews.com/health/cancer/cancer-biggest-killer-america-s-firefighters-n813411

[2] https://www.cdc.gov/niosh/pgms/worknotify/pdfs/ff-cancer-factsheet-final-508.pdf

[3] https://oem.bmj.com/content/71/6/388.full

[4] My mother

[5] Daniels RD, Kubale TL, Yiin JH, et al Mortality and cancer incidence in a pooled cohort of US firefighters from San Francisco, Chicago and Philadelphia (1950–2009) Occupational and Environmental Medicine 2014;71:388-397.

[6] https://www.pewtrusts.org/en/research-and-analysis/blogs/stateline/2015/12/07/special-treatment-for-firefighters-with-cancer-some-states-say-yes

[7] https://www.houstonchronicle.com/news/houston-texas/houston/article/Despite-Texas-law-nine-in-10-firefighters-with-13144635.php

[8] https://arxiv.org/pdf/1810.00767.pdf

Some other useful info:

https://www.safetynational.com/conferencechronicles/presumption-laws/

Presumptive Legislation for Firefighter Cancer