Note: This is a working paper, so comments and suggestions for revisions are welcome. It is a bit rougher than I normally release, but I believe it produces a valid estimate and is quite adequate. What appears here is the analysis itself; a summary and further discussion appears at the antiTHRlies blog.

**Introduction**

It is recognized by any (honest and knowledgable) observer that every smoker who switches to e-cigarettes (or any other very low-risk alternative) represents, in expected value terms, some fraction of an eventually-averted premature death due to smoking. It is possible to estimate that fraction. A related interesting question is how many people have *already* avoided premature death thanks to switching to e-cigarettes. Even though the availability of e-cigarettes is a relatively new phenomenon, and thus most people destined to avoid premature death thanks to e-cigarettes would still be alive even if they were still smoking, it turns out that this is still a rather impressive number.

The following analysis first estimates the first of these statistics, the expected value of premature deaths averted for each smoker who switches to e-cigarettes. It then calculates how many Americans have *already* avoided premature death thanks to e-cigarettes. That is, how many are alive today who would have already died from smoking had they not switched. (The calculation is based on the USA for convenience and can, of course, be extrapolated to the rest of the world.) The calculation method follows the approach I used in my 2009 paper that debunked the claim that tobacco abstinence is generally lower risk than pursing harm reduction (switching to a low-risk alternative). The results are necessarily rough — with precision that is less than implied by a single significant figure — but this back-of-the-envelope still provides a substantial improvement over the unquantified discussion that are typical.

**Analysis**

Estimated premature deaths averted for each smoker switching to e-cigarettes

The fraction of a premature death averted for a specific individual depends on her probability of dying prematurely from smoking if she does not quit as well as her probability of still dying prematurely from smoking even if she quits now (i.e., she has already smoked the cigarette that will lead to her death, and it is too late to avoid that by quitting). This could be estimated as a function of age, sex, smoking history, and other variables. Estimating the total smoking-caused premature deaths averted in the population could be similarly estimated using the distribution of these variables among those who have switched to e-cigarettes.

While such a calculation is theoretically possible, it is not possible in reality. Despite billions of dollars spent on research on smoking and health, relatively little of it has any value. In particular, while it should be possible to map the aforementioned variables into one’s probability of avoiding a smoking-caused death by quitting now, and predicting when that death would have occurred, such estimates have never been produced. Moreover, it is obviously possible to know the distribution of those variables for smokers who have quit thanks to e-cigarettes, but to date those who keep statistics on tobacco use have avoided collecting that data. Indeed, in the USA there is not even very good data on how many people have quit smoking using e-cigarettes.

The best estimate available is based on the estimate of the probability of a smoker dying prematurely from smoking, and roughly reducing it to account for the possibility that she will still die from smoking even if she quits now. For the US population, the US CDC puts the former estimate at 50%, and this is accepted as conventional wisdom in the tobacco control discourse.

This figure is fairly slippery, and the basis for it has never been produced, but it is what we have to work with. In particular, exactly who it refers to is highly ambiguous: It obviously cannot refer to every individual who once smoked a few cigarettes (i.e., almost everyone) but presumably is not limited to just those who smoke until their dying day.

The context of the use of the 50% figure suggests it refers to anyone who is currently a smoker. Assuming this interpretation, someone switching to e-cigarettes represents a reduction of almost .5 of a smoking-caused premature death. It is “almost” because some of those quitters are already doomed from their smoking to date. In addition, there is an “immortal person-time” problem: Those smokers who are still alive at any given time are less likely to be those that die from smoking, because some of those who will die prematurely from smoking already did so.

The number of smokers doomed already can be calculated to be that just under 5% of all current smokers based on the lag-time until smoking-caused deaths, as presented in the analysis below. The bias from immortal person-time can be estimated to contribute almost as much. Thus we can estimate that each individual quitting smoking thanks to e-cigarettes represents about .4 of a smoking-caused premature death averted.

[Note: The phrasing used here — “premature death” and “smoking-caused” — though perhaps seeming a bit tortured, is the proper phrasing. The phrasing common in the political rhetoric (e.g., “lives saved”) implies that ever-death is contingent on smoking. The phrase “smoking related” is worse still, since “related” is basically meaningless. It is clear from context that “smoking related deaths” really means premature deaths caused by smoking, and “lives saved” are averted premature smoking-caused deaths. This said, some of the language that follows is streamlined toward the more typical phrasing.]

This estimate, and the more detailed calculation that follow is obviously dependent on the conventional wisdom statistic, that 50% of smokers die prematurely from smoking, as well as the ambiguity about what constitutes a smoker for that statistic. It seems safe to interpret the statistic as recognizing the fact that many smokers will not smoke until their dying day, and thus the reduction in total risk resulting from the (low) background rate of smoking cessation is already included in it. If those switching to e-cigarettes were shown to be more or less likely than average to be one of those who would otherwise quit then these estimates would have to be adjusted accordingly. However, lacking any basis for such conclusions, one way or the other, there is no apparent choice but to treat switchers as average smokers.

Note that it would be possible, in this estimate and the calculation that follows, to add back the estimated premature mortality that results from using e-cigarettes, which are generally believed to not be risk-free (though there is no affirmative evidence of risk). However, this would be a waste of time. It is clear that any such risk no more than a few percent of the risk from smoking. Thus any impact would be far below the uncertainties inherent in these estimates. For example, if the probability of dying from smoking were really 49% rather than 50%, it would have a bigger impact on the estimate than adding in the risk from e-cigarettes. Our knowledge is not adequate to estimate the risks from smoking at that level of precision, and thus it would merely be a silly distraction to even try to add back in the risk from e-cigarettes.

Calculating the number of premature deaths *already* averted by e-cigarettes

The indented material is the calculation itself. The material in between provides background, explanations, caveats, and other related observations. The arithmetic itself can be viewed or downloaded from this spreadsheet. Interested readers who prefer other input values can use that to change them to revise the estimate.

The calculation requires estimating the chance that someone switching at a given time in the past would have died from smoking already, and then applying this calculation to the number who had actually switched at the given time. It turns out that the first part (technically more complicated) is actually a rather more solid estimate than the latter (which is a simple matter of survey data — unfortunately the surveys were never done). Taking the latter first:

Without loss of generality, divide time into quarter-years. Estimate that 1,000,000 smokers have quit by switching to e-cigarettes as of now (end of 2014Q4), 500,000 as of 2012Q1, and no one as of 2008Q1. Interpolate linearly within the two intervals.

The 0 figure for 2008 is not quite true, of course, but consideration of the few earliest adopters is trivial in the overall bottom line. The estimate of one million currently quantifies the conventional wisdom based on sales data. The middle number is an estimate based on some (highly non-ideal, but not entirely information-free) surveys that were done.

Note that these estimates are the second-greatest contributors of uncertainty in the calculation, after the aforementioned 50% figure. Refining them would substantially improve the precision of the estimate. The remainder of the calculation is comparatively precise, except for its reliance on the 50% estimate.

For simplicity, treat smoking cessation as dichotomous. That is, someone who quits using e-cigarettes stays quit, so temporary smoking cessation is treated as non-cessation. Also, cutting down on quantity smoked thanks to e-cigarettes is not counted.

These make this a very conservative estimate. It is only the deaths averted thanks to *complete* smoking cessation caused by e-cigarettes, rather than all deaths averted thanks to e-cigarettes. The latter would be much greater, but calculating it would require estimating the reduction in deaths due to temporary or partial cessation, which are certainly positive but have never been effectively estimated.

Following the methodology introduced in my 2009 paper, the unit of analysis is the average quarter of smoking, averaged across all smokers. As I showed in that analysis, for each smoker who dies prematurely from smoking, there must be a particular period (day, quarter, year, or whatever) in which he smoked the cigarette that doomed her to die from smoking. While estimating the probability that a *particular* period of smoking will cause the premature death of a *particular* smoker requires far better epidemiology than currently exists, the average across all periods for all smokers can easily be calculated.

Start with the conventional wisdom 50% and estimate the total number of years of smoking per applicable smoker to be 45. Then any given person-quarter of smoking has a .0028 chance of causing premature mortality.

See the 2009 paper for more details about this method. Note that the estimate of 45 years is probably a bit conservative. That would be the appropriate figure if every smoker to whom the 50% applied had started at 15 and continued until 60, but it appears that most of those who the CDC counts as smokers for purposes of their statistics do not actually smoke that many years.

Premature mortality due to smoking can be divided into deaths from cardiovascular disease (CVD) and deaths from other causes (cancer, non-cancer lung disease). Estimates of the number of deaths attributed to smoking put about the same number in each of these categories so let half the deaths be in each category.

Excess deaths from CVD among former smokers return to close to baseline after two years. This means that for someone dying from smoking-caused CVD, the period that doomed her was within the previous two years, and if she had quit before that, she would have avoided the fatal CVD event. This means that the deaths from smoking occur during the 8 quarters following the period causing the fatal CVD. Assume that they are evenly divided across those periods.

It is a simplification to say that *all* smoking-caused CVD deaths occur within two years of the dooming period, but it is close enough to all do to make this a reasonable estimate. Dividing the deaths evenly across the eight periods is very conservative based on the typical tobacco control claims (e.g., the claims of huge reductions in heart attacks within a few months of reductions in smoking exposure), but is probably more realistic than those claims.

Deaths from other causes take much longer to return to baseline following smoking cessation, meaning that the caused death comes much later compared to the fatal period of smoking. Specifically, the return to near baseline takes about 15 years. Assume that the deaths are evenly divided across the 60 quarters following the dooming quarter.

There is data available that could improve somewhat on dividing the deaths evenly across the 2 and 15 year periods, but the impact of the resulting imprecision is minor compared to the two sources of uncertainty already introduced so is not really worth doing. If the adjustment were made, it would tend to increase the estimated number of deaths already averted since more of the deaths would be estimated to occur closer to the time of the dooming period.

Based on these inputs, we can calculate the number of premature deaths that have already been averted thanks to e-cigarettes. The number of former smokers who avoided being doomed in a given quarter is calculated based on the above (the number who quit smoking and the probability that the period would have doomed them had they continued). We then need to calculate what quarter those deaths would have occurred in.

To calculate the averted deaths from CVD that would have occurred in a given quarter, we need to look back at the recently avoided dooming periods. Someone who avoided being doomed to die from CVD during that quarter has a 1/8 chance of having avoided death in this particular period, as does someone who avoided the dooming quarter of smoking during each of the previous 7 quarters. Summing these gives the number of avoided CVD deaths during this period. A similar calculation applies to non-CVD deaths, substituting 60 quarters for 8.

Adding these for each quarter gives us the number of premature deaths from smoking that would have occurred during that period, but did not, thanks to e-cigarettes.

This is all straightforward calculation. The spreadsheet shows the arithmetic.

Adding up the resulting avoided deaths through 2014Q4 gives approximately 16,000 premature deaths already avoided, 13,000 from CVD and 3,000 from other causes. About half of these would have occurred in the last 20 months.

Note that the deaths that would have already occurred from non-CVD causes is much lower because most of the non-CVD deaths that have already been avoided would not have occurred yet, given the much longer lead time from being doomed until actually dying.

In addition, another 19,000 smoking-caused premature deaths have already been averted but would not have occurred yet.

That is, because of the lead times, most people who have been *saved* from non-CVD death (e.g., lung cancer) would not have died *yet* even if they had not been saved. In addition, about one-fourth of those already saved from CVD would not have died yet because most of those quitting smoking thanks to e-cigarettes has increased sharply over time.

It is important to not confuse this number with the number who quit smoking thanks to e-cigarettes and thus will *sometime* be saved from smoking-caused death due to their continuing abstinence from smoking. That number is the estimate of .4 of them as calculated in the previous section. The 19,000 figure represents those who have already avoided the smoking that would have doomed them but would not have died yet. Put another way, they are the ones who would still be doomed to die from smoking if they had not switched to e-cigarettes already, even if they quit smoking today. That emphasizes the thesis of my 2009 paper: Quitting smoking *eventually* is not a safe alternative to switching to a low-risk alternative now!

This calculation ignores deaths from other causes that occurred subsequent to the averted death from smoking. That is, if someone were saved from a smoking-caused death in 2010 thanks to switching to e-cigarettes in 2009, and would have died in 2011, but instead lives long enough to get hit by a bus in 2012, then she would still be dead now even though she was saved from smoking. So she is not actually “alive today thanks to e-cigarettes” even though her death from smoking was averted. Still, given the limited time period we are talking about, there would be relatively few such cases and so representing them as “alive today thanks to e-cigarettes” is close enough to accurate.

This estimate can be extrapolated to other populations with similar health and longevity. For example, the number of smokers who have quit thanks to e-cigarettes is similar in the UK (a large portion of a smaller total), so the numbers already saved are similar.

**Conclusions**

This back-of-the-envelope estimate is clearly rough, but it is obviously better than any estimate made without the aid of actual calculations. Even allowing for the inherent uncertainty in the calculation, if we accept as approximately correct the “50% of smokers will die from it” starting point, it is safe to conclude that more than 10,000 Americans are alive today thanks to e-cigarettes, in addition to similar numbers in the UK and in the rest of Europe. A greater number still have already been saved (they would still have died from smoking if they had quit today) even though they would not have died yet.

In the long run, many more of those who have switched will avoid dying from smoking. We can estimate the total to be in the order of 40% of them. But there is something rather more compelling about the number already saved: It is *theoretically* possible that a miracle will occur and most current smokers will quit without e-cigarettes or other low-risk substitutes. In that case, the number eventually saved thanks to switching would be lower than the 40% because many would eventually be saved by quitting by other means. But even if that unlikely even occurs, it will be too late for many who could have been saved by switching. Thus, waiting for that miracle, rather than embracing the immediate promise of tobacco harm reduction, would be a deadly mistake. In particular, we can conservatively estimate that between 10,000 and 20,000 Americans who get to ring in New Years Day 2015 would not have lived to see that day were it not for e-cigarettes.

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I believe that the 50% figure came from Hill and Doll’s ‘Doctors Study’. I quote from the final report:

“Applicability of findings.

Our 1994 report of the 40 year findings concluded that “Results

from the first 20 years of this study, and of other studies at that

time [the 1970s], substantially underestimated the hazards of

long term use of tobacco.17 It now [in the 1990s] seems that about

half of all regular cigarette smokers will eventually be killed by

their habit.” Ten years later, the 50 year findings show that for the

continuing cigarette smokers in this study who were born in the

first, second, and third decades of the 20th century, the eventual

risks vary from about one half to about two thirds (fig 2),

although the mortality rates among them from lung cancer were,

if anything, somewhat lower than among their contemporaries

in the general UK population who smoked.

It is a new finding that the risk of being killed by tobacco can

be as great as two thirds, but this applies directly only to the hazards

suffered by this particular group of professional British men

during the past half century. If these results are to be used indirectly

to help predict the risks that male, female, rich, poor, British,

and non-British populations of persistent cigarette smokers

are likely to face over the next half century, then it may be more

appropriate to retain the previous semi-quantitative conclusion

that “smoking kills about one half.” For, although we have found

a population in which persistent cigarette smoking killed somewhat

more than half (fig 2), there may be other circumstances in

which it kills somewhat less than half.

CDC seems to base their claim on a couple of secret old US datasets (though not that old) and a black-box analysis (and thus highly suspect, given who it is coming from). If their numbers were defensible, they would share their data and methods. Thus, it is safe to conclude they are cooking them. Nevertheless, it is their claim, so if they want to stick with it, then they have a rather larger number of deaths on their hands from opposing THR.

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