Probable connection between approval voting and candidate-honesty

Warren D. Smith, Sept. 2016


It appears that the most-approved candidates in approval-style polling coincide with the most-truthful candidates according to fact-checking organizations. Previously, it had been unclear why real-world approval voters approved or disapproved candidates or what approval "meant" to them. This observation may go a long way toward clarifying that, although further data, and possibly then some refined rewording of the hypothesis, are desirable. But our present data already is good enough to yield one to five "nines" worth of confidence. We explain various confidence calculations and criticisms thereof, eventually claiming confidence≥99.5% for our final model versus the null hypothesis, which bound ought to be immune to all the criticisms.

Table of contents

  1. 2016 USA presidential election
  2. Note about Gary Johnson
  3. The preceding (USA 2012) Presidential election
  4. French 2012 Presidential election
  5. First try at a Conclusion
  6. Conclusion revisited in light of "p-hacking" worry
  7. USA 2008 Presidential election
  8. A maximum likelihood model, and yet another criticism (by Gary Smith)
  9. Final Conclusion
  10. Implications for Democracy

2016 USA presidential election

I am writing this in September 2016 in the midst of the 2016 US presidential race. I noticed an interesting possible connection, see the data table below.

CandidateH0 H1 H2 H3 H4 H5FCApprovalRCPApprovalHPAppNum
Hillary Clinton(DN)2 11 15 22 28 2224942.3/52.341.6/56.20.775
Bernie Sanders(D)*0 11 17 20 39 1310650.0/36.855.8/35.61.463*
Elizabeth Warren(Dx)*0 0 11 11 33 45 928.0/26.325.2/27.10.997*
Joe Biden (Dx) 5 13 15 28 21 17 7545.7/37.751.3/36.61.307*
Martin O'Malley(D)0 6 17 56 22 0 1817.5/27.0 -- 0.648
Donald Trump(RN) 18 35 18 15 11 425038.8/58.6 38.3/59.00.656
Ted Cruz(R) 7 27 31 13 16 711429.7/54.8 23.8/54.80.488
John Kasich(R)* 5 13 14 16 28 25 5539.1/30.037.4/37.71.148*
Marco Rubio(R) 3 15 23 22 24 1314236.3/40.3 34.9/46.00.830
Jeb Bush(R) 3 6 22 22 30 14 7930.8/52.4 30.6/53.50.580
Rand Paul(R) 6 16 14 20 24 22 5125.7/41.0 27.7/44.00.628
Chris Christie(R)8 17 10 28 19 2110229.8/44.5 30.4/48.20.650
Rick Santorum(R) 8 27 20 22 12 10 5924.0/42.0 19.9/46.30.510
Rick Perry(R) 11 18 18 25 14 1516921.0/39.3 23.6/43.50.538
Scott Walker(R) 6 24 15 19 23 1317223.5/29.3 33.7/37.40.852
Bobby Jindal(R) 0 10 0 40 40 10 1018.7/30.3 17.1/35.90.547
Ben Carson(R) 14 43 25 11 7 0 28 -- 36.3/45.40.830
Lindsey Graham(R)0 17 17 33 17 17 12 -- 16.3/42.10.387
George Pataki(R) 20 80 0 0 0 0 5 -- 14.8/37.90.391
Mike Huckabee(R) 10 12 29 22 7 20 41 -- 28.5/45.00.633
Carly Fiorina(R) 9 23 23 18 14 14 22 -- 24.6/39.00.631
Jill Stein(G) 0 0 100 0 0 0 1 -- 15.4/28.90.532
Gary Johnson(L) 0 33 33 0 33 0 3 -- 24.1/28.90.834

FC=Number of "facts" claimed by that candidate and checked by ", winner of pulitzer prize" as of 13 Sept. 2016. Note: for the following 4 people politifact did not check enough facts; therefore I have added all the facts checked by for that candidate, along with numerical ratings provided by me, to their totals: George Pataki, Jill Stein, Gary Johnson, Elizabeth Warren. Even then, still below 10 facts for each of them.
H0=Percentage of those facts given the worst "pants on fire" rating.
H1=Percentage given 2nd worst "false" rating.
H2=Percentage "mostly false."
H3=Percentage "half true."
H4=Percentage "mostly true."
H5=Percentage "true."
In parentheses: (D/R)=Democratic/Republican party candidate. N=eventual nominee of that party. x=never actually ran.
ApprovalRCP=Favorable/Unfavorable rating from USA-wide poll averages compiled by, as of 13 Sept. 2016, mostly from
ApprovalHP=same thing, but instead compiled by "HuffPost Pollster" as of 15 Sept 2016.
AppNum=numerical computation of the Fav/Unfav ratio (averaged if have two).
Asterisks: Candidate-name awarded a star (*) if he/she manages to attain ≥51% "true" or "mostly true" rating from AppNum awarded asterisk if ≥1.00 (after rounding to two decimals) i.e. if candidate manages to get at least as many "favorable" as "unfavorable" ratings.

Amazingly enough, the two kinds of asterisks happened to coincide. I.e, the only two candidates (Sanders & Kasich) who got 51%-or-above honesty ratings from politifact, also happened to be the only two who were more approved than disapproved by the US public.

Is this just a fluke? Well, there are 21 candidates listed here, hence 21×20/2=210 possible candidate-pairs. If honesty and approval were completely unrelated, the chance the two pairs would coincide would be 1/210. (Actually, less, since it did not have to be a "pair.") Therefore, we conclude with confidence>99.52% versus the null hypothesis that this effect is real.

If we include also the two non-candidates (but the press acted as though they were likely to be candidates for quite a while) Warren and Biden, then Warren also would have gotten a double-*, albeit based on only 9 checked facts. (She has the greatest True+MostlyTrue rating among those listed.) But Biden would have only a single *. So, all in all, if we consider adding Warren and Biden we then would have 3 honesty stars and 4 approval stars among 23 candidates and close-to-candidates, with the honesty stars forming a strict subset of the approval stars. Is this a mere coincidence? If the two kinds of stars were wholy unrelated then considering that the number of possible 4-element-subsets of 23 pseudo-candidates is 23×22×21×20/24=8855 and the number of such subsets that include one particular triple is 20, this would again be a low-probability (specifically chance=20/8855) event, yielding 1-20/8855≈99.77% confidence versus the null hypothesis that this is a real effect.

Conclusion: Depending which of 2 ways we calculate it, we get 99.5 or 99.8% confidence based on this data that "candidate truthfulness" and "approval rating" are related. This could be an important insight into approval voter behavior.

Meanwhile, the actual voting system used by the Dem/Repub parties to select their nominees – which was not approval voting – produced results (e.g, their nominees Clinton and Trump) far less correlated with honesty! Trump was the single most-dishonest person tabulated among those with at least 30 checked facts, with only 15% True+MostlyTrue (lowest by far) and 53% False+PantsOnFire (highest by far) and an incredible 71% MostlyFalse+False+PantsOnFire. Meanwhile, Clinton was about the 4th most honest among the 21 listed, but could not quite manage to reach 51% True+MostlyTrue.

Also, if one examines the scores of candidates in plurality-style polls and the official primary votes... then again, one sees far less correlation with candidate truthfulness than one would find with approval-style voting – indeed I doubt that any statistically significant connection is discernible.

Might the cause of all this merely be some sort of bias from that nasty evil organization ""? Certainly their ratings are somewhat subjective (and I do not always agree with them, and instances exist where different fact-checking organizations disagree about the same "fact"), and their choice of which "facts" to check may be somewhat manipulable.

To try to protect ourselves from that worry, we alternatively could consult other independent fact-checking organizations. The book Deciding What's True: The Rise of Political Fact-Checking in American Journalism by Lucas Graves (Columbia University Press, Sept. 2016) is helpful for learning about that. Apparently the fact-checking entity is a fairly recent invention and it may have been an American invention: – founded 1995
  Spinsanity – founded 2001, over 400 facts checked during 2004 US election, but ended  – founded 2003 Annenberg public policy center at Univ. Pennsylvania;
            produces about 5 fact checks per week, supported by foundation philanthropy. – founded by St Petersburg Times / Congressional Quarterly in 2007,
     sold in 2009 to The Economist Group; checks about 20 facts per week making it
     the most productive fact checker (at least among USA-based ones if not worldwide).
  Washington Post fact checker – started September 2007, ended after 2008 campaign season,
     but revived in January 2011. Checks about 1 fact per day.'s "reality check" – not sure of its years of operation, but it has
    been checking Clinton & Trump during the latter part of the 2016 presidential campaign.

Internationally, we have

  "Pagella politica" and "Il politicometro" (both Italy)
  The Guardian "reality check" (UK)  (India)
  MorsiMeter (Egypt)
  "El poligrafo" a fact-checking feature in "El Mercurio" (Chile) founded 2013 (Africa)
  French fact checkers:
    Decodeurs:        Founded by Le Monde in 2009
    Les Pinocchios:
    OWNI veritometre:
                           Founded by Cedric Mathiot for in 2008
  Der Spiegel (Germany) also has perhaps the largest fact-checking team in the world.

Unfortunately for our purposes, "" does not provide numerical truth-ratings for checked facts nor summary ratings for candidates; they just provide lists of individual facts, usually claimed by or about candidates. (Anybody willing to work could tally them up and numerically rate them, however; and they do index their facts by candidate-name and provide "annual reviews.") The same problems pertain to "Fact checks of the 2016 election" by New York Times and most or all of the international and other groups. (Further, e.g, MorsiMeter focused solely on Morsi's campaign promises and not on fact-checking any of his rivals, making it nearly useless for our purposes.)

"Fact checker at the Washington Post" does provide numerical "Pinocchio ratings" with occasional summary tallies. Here is the summary they published on 15 July 2016.

   Candidate        4P  3P  2P   1P   0P   FC
  Donald Trump      63  21  10    2    2   52
  Hillary Clinton   13  36  30.5  5.5 14   36


  4P = percentage of facts regarded as "4 Pinocchios" – outright whopper lies.
  3P = "3 Pinocchios" – "mostly false."
  2P = "2 Pinocchios" – "half true."
  1P = "1 Pinocchio" – "mostly true."
  0P = the coveted "Gepetto checkmark" – completely true.
  FC = total number of facts checked for that candidate.

As you can see the Washington Post agrees with Politifact that Clinton is far more truthful than Trump, but appears to have a tougher standard in that both Clinton and Trump score worse at Washington Post than they score at Politifact. (I also think the Washington Post checks fewer facts than Politifact, but the ones it checks, it checks more thoroughly.)

The Washington Post went further by claiming that Trump was not merely the least-honest candidate in the 2016 race (that conclusion agrees with Politifact), but actually among all major politicians in the entire history of the Washington Post's fact-checking feature. Also amazing is this. Most politicians show a unimodal distribution of honesty-scores, with, for example, Graham's peak being at honesty level "H3" in our first data table. But in the case of Trump, his frequency actually keeps increasing (at least in the view of Washington Post) with more Pinocchios, up to the highest Pinocchio level the Post could conceive of! also found Trump to be "The king of whoppers" on 21 Dec 2015 and noting "in the 12 years of's existence, we've never seen his match." CNN's "reality check" one day after the first Clinton-Trump presidential debate on 26 September checked claims made by both during that debate and my tally of their checks (version of noon 27 Sept) was

   Candidate       IC  FA  TB  MT TR   FC
  Donald Trump      0  14   2   0  3   19
  Hillary Clinton   0   0   1   5  6   12

KEY: TR=true; MT="mostly true"; TB="true but misleading"; FA=false; or IC="it's complicated."

CNN concluded "From misrepresentations to half-truths and flat-out lies, Trump has talked around and away from the truth more brazenly than any major party presidential nominee in modern political history" agreeing with all the other fact-checking bodies. This debate also was fact-checked by Washington Post,, and PolitiFact.

Note about Gary Johnson

The data tabulated above shows that (at least as of mid-September 2016) the most-approved presidential candidate still running (based on approval/disapproval ratio) was not Hillary R. Clinton, and not Donald J. Trump. It was the Libertarian party's nominee (and former governor of New Mexico) Gary Johnson.

Johnson was not permitted to participate in the Presidential Debates and got very little media attention. He will undoubtably get only a tiny percentage of the official vote on election day. Nevertheless it appears the USA currently wants him as president more than any rival.

Despite that, I have no doubt that Trump will get more votes in the official election on 8 Nov. 2016, than all his rivals (besides Clinton) combined. This illustrates the massively distortionary nature of the USA's plurality voting system. A further confirmation of that is the following pairwise poll conducted by the Johnson/Weld campaign (nationwide sample, 1006, polled by telephone on 8 & 9 October 2016). Question 4: "For whom would you vote for president if the choice was between Republican Donald Trump and Libertarian Gary Johnson?" Results:

Johnson 41.6%, Trump 37.3%, Don't know 21.1%.

This margin is 43 more Johnson than Trump preferrers among the 1006 polled, which naively would be at least 2.7σ, where σ≤(√1006)/2=15.86 voters. For those worried this poll should not be trusted – because it was released by the Johnson/Weld campaign, not conducted by some independent entity – some reassurance is provided by Question 1 ("Do you have a favorable or unfavorable opinion of Gary Johnson?") with result

Fav=15.2%, Unfav=30.0%, NoOpinion=38.1%, Never heard of GJ=16.7%.

This result is worse for Johnson than the ApprovalHP numbers from independent polls, which is evidence against the hypothesis that the Johnson/Weld campaign released an atypically Johnson-favoring poll. If anything, the opposite was true.

While it is true that Gary Johnson currently has the highest approval/disapproval ratio, it is unclear how seriously that should be taken because the US public is poorly informed about GJ compared to HRC and DJT. (Indeed, only 53% of those polled in the HuffPost poll set were able to express an opinion about him; the remaining 47% said "don't know.") If our elections were using approval, then the media and his rivals would scrutinize GJ a lot more, everybody would be better informed, and then his approval might change a lot in one or the other direction, perhaps invalidating his frontrunner status.

Nevertheless it is totally clear GJ deserves far more attention than he got, and deserves far more votes than he will get – the USA's present voting system plainly is an absurd parody of democracy.

The preceding (USA 2012) Presidential election

This election was largely but not entirely independent of the USA 2016 race. "Not entirely" because Perry and Santorum ran in both races. "Largely" because all the major candidates in 2016 were a wholy-disjoint set from those in 2012, i.e. Perry and Santorum both were minor.

2012 CandidateH0 H1 H2 H3 H4 H5FCApprv
Mitt Romney(RN) 9 18 17 28 18 15 206 475
Rick Santorum(R6) 8 27 20 22 12 10 59 430
Ron Paul(R) 8 20 13 20 20 20 40 610
Newt Gingrich(R) 15 20 20 24 13 8 75 325
Herman Cain(R) 12 42 15 19 12 0 26 430
Tim Pawlenty(R) 6 18 12 18 35 12 17 420
Jon Huntsman Jr.(R) 6 6 28 28 22 11 18 490
Michele Bachmann(R) 26 36 13 10 7 8 61 420
Rick Perry(R6) 11 18 18 25 14 15 169 345
Barack Obama(DN6) 2 12 12 27 28 21 572 525

NOTES: 6 inside the parentheses warns that candidate's politifact report card honesty stats may include many statements they made not just in the 2012 race, but also during the 2016 and/or 2008 races.
Approval gives favorable/unfavorable percentages expressed as 1000F/(F+U) at latest date shown for that candidate in this multicolored approval-vs-time graph which I'd created about 4 years ago from a large number of approval-style polls.
Multicolored approval-vs-time graph
None of the 10 candidates listed managed to get True+MostlyTrue percentages ≥51% according to Politifact; the highest three were Obama with 49%, Pawlenty 47% (but that was based on only 17 facts checked), and Ron Paul with 40%. Only two candidates got more approval than disapproval:

Obama and Paul.

Again, by an amazing coincidence, the two most-approved candidates coincided with the two most-truthful candidates – at least, provided the low-data candidate Pawlenty is ignored. Pawlenty dropped out very early in the race, on 14 August 2011, many months before the first vote in the first US state primary was cast and hence in my view should not even be counted as a "candidate" at all. All the others stayed in the race until at least two states had voted.

Also, considering Pawlenty's August 2011 drop-out date, arguably his June 2011 Approval poll should be used instead of his August 2012 poll. If so, the problem disappears: Pawlenty then becomes both the 2nd-most-approved candidate and 2nd-most honest, causing the 3 most-approved and 3 most-honest (out of the 10 total) candidates to coincide, an amazing probability=1/120 event.

This amazing coincidence corresponds to a confidence level of 88 to 99% depending on which of those three methods I use to compute it.

French 2012 Presidential election

The following table is based on fact checking by OWNI's "Veritometre." There are two classes of facts checked: quantitative claims, and other.

For the 6 most important candidates in the race (e.g. they had the six highest approval ratings) I am going to tabulate each candidate's correct/incorrect ratio on quantitative claims, ignoring claims that Veritometre called "en cours" (i.e. still unchecked; apparently they planned to check those later but I do not know if that ever happened) and for the present purpose I am going to count what Veritometre calls "declarations imprecises" (meaning numbers within ±10% but not within 5%) as "correct." Our six candidates were exactly the six with over 100 quantitative claims checked by Veritometre as of 20 April 2012.

For the nonquantitative claims, Veritometre awarded "Pinocchios" (the more Pinocchios, the more untruthful the candidate); and we also tabulate the average Pinocchio-count for each candidate's other-facts collection.

CandidateFCVeritometre ratio
Jean-Luc Melenchon1642.82 1.83 (23) 39.07% 45/46=0.978
Francois Hollande 2152.71 1.97 (35) 49.44% 61/32=1.906won
Eva Joly 1071.86 2.20 (20) 26.69% 30/61=0.492
Nicholas Sarkozy 3921.11 2.35 (34) 40.47% 49/48=1.021also in runoff
Francois Bayrou 2711.08 2.43 (23) 39.20% 48/42=1.143
Marine Le Pen 1631.25 2.71 (21) 27.43% 33/62=0.532

KEY: FC = #quantitative claims made by that candidate that they checked; correct/incorrect ratio as described above
OWNI "Veritometre" fact-checking data: "OWNI data journalists have checked nearly 1300 quantitative claims by the presidential candidates – on employment, security, trade ... Of all the statements we checked, 40.2% are correct, 41.6% are incorrect and 18.6% turn out to be too vague."
Average Pinocchios: a different set of facts checked by Veritometre each were awarded "Pinocchio counts" (more Pinocchios⇒more dishonest). For example Melenchon's "1.83 (23)" means of the 23 facts claimed by Melenchon he got an average of 1.83 Pinocchios.
Approval: in a pseudo-election exit poll study of 2340 voters using approval-style voting. carried out by Antoinette Baujard, Frederic Gavril, Herrade Igersheim, Jean-Francois Laslier, and Isabelle Lebon and corrected by them for geographic biases.
Fav/Unfav: IPSOS professional approval-voting-style France-wide poll 19 April 2012. (Apparently by telephone with about 956 respondents.) I have also computed their F/U ratio numerically.

Hollande not only won the official election (both first round, and runoff), he also would have won with approval voting (either poll) and also apparently would have been the "Condorcet beats-all winner" according to pairwise polls.

And Hollande was the second-most-truthful candidate, slightly behind Melenchon, and was actually the most truthful within the statistical margin of error in the truthfulness measurement [i.e. given that only 164, a finite number, of facts were checked for Melenchon, if those facts were randomly chosen from a large pool, then his truthfulness is only measurable to within error at least about ±1 part in √164].

Warning: Concerning such "statistical errors in candidate-truthfulness estimates," keep in mind that fact-checkers actually choose the claims they check for each candidate in a nonrandom manner, concentrating on more-interesting, more-emphasized, more-requested, and less-obvious facts. And even just a single big lie by a candidate can cause many voters to despise him – and probably rightfully so – even though if that were the only fact checked for him, that would mean that candidate's truthfulness percentage was very poorly measured indeed.

After France's first voting round, the runoff was to be between Hollande and (the incumbent seeking re-election) Sarkozy. Hence there was a Hollande-Sarkozy debate, and further fact-checking of statements by Hollande & Sarkozy:

"Data journalists from OWNI checked 132 statements by Nicolas Sarkozy or Francois Hollande made during recent debates or speeches." Veritometre gave a score of 56% correct to Hollande versus 44% for the incumbent Sarkozy.

So our truthfulness↔approval connection arguably is supported by Hollande's victory in France 2012, in the sense that he was the second-most-honest candidate (and considering the statistical margins of error, given the finite number of checked facts we have, may actually have been the most honest). This support is far less clear if we go to the next most approved candidate – either Melenchon, Bayrou, or Sarkozy. All three were comparably approved, within statistical margin of error; but their truthfulnesses differed considerably according to Veritometre. So certainly our whole hypothesized connection is only partially supported by France 2012 (as opposed to USA 2012 or USA 2016 which pretty much fully supported it), but nevertheless I would vaguely contend France 2012, on overall balance, supports the hypothesis more than it opposes it.

Unfortunately, Veritometre for our table's column about correct/incorrect ratio only checked numerical claims, and their ±10% cutoff, while objective, probably does not have greatly relate to what voters care about. For example a candidate who said "three million" when the correct number was 2.6 million would be rated "incorrect" by Veritometre – but speaking as a voter, I probably would have been satisfied with that candidate's approximation. Meanwhile, such blatantly false (according to several fact-checking entities unanimously) claims by Donald Trump in the USA 2016 campaign, such as "I opposed the Iraq war from the beginning," "Hillary Clinton started the 'birther' movement, but I finished it" and "Hillary Clinton has been fighting ISIS her entire adult life" would not matter to this Veritometre column at all since those were unquantitative claims! But those claims probably would matter to voters! On the other hand, Trump's pledge to spend at least $100 million of his own money on his campaign would have been counted as a lie, albeit one only detectable afterwards.

This criticism might be overcomable by also obtaining fact-checking data from Le Monde's "Les decodeurs." Unfortunately Le Monde only has made Decodeurs archives available online starting soon after the 2012 election ended, so I haven't been able to investigate that. And Desintox did not provide tallies and hence also was not very useful. Fortunately an independent (?) collection of facts were checked one by one by Veritometre using a "Pinocchio scale" over several months and their "average Pinocchio" tallies were computed by Vincent Flores – albeit unfortunately the number of these facts checked for each candidate remains 3-10 times fewer than Politifact's corresponding fact-check-counts for the major USA 2016 candidates.

First try at a Conclusion

If we naively multiplicatively combine the USA 2012 coincidence with the USA year-2016 coincidence (we also optionally could expunge the Santorum and Perry data from the 2016 race to make 2012 and 2016 more-independent; this has little effect on our conclusions), then we deduce combined confidence levels of between 99.94 and 99.996% (depending on how calculated) that the following hypothesis is correct (versus the null hypothesis)

HYPOTHESIS: The highest approvals happen only for candidates with high truthfulness.

If we also throw in Hollande in France 2012 that would bring us a further factor of about 3 nearer to 100% confidence, reaching between 99.98 and 99.999% confidence that highest approval is connected to highest truthfulness.

Incidentally, note that I chose these three elections simply because they were the only major elections in human history for which I could get both approval and fact-checker quantitative truthfulness data for all the major candidates. But for a harder-working scholar than I, it is conceivable that this might also be possible for a couple more elections, such as USA 2008 or Chile 2013 (or in future, France 2017) – or that more truthfulness data from an independent fact-checker could be got for France 2012.

I consider this finding to be a further sign that approval voting is a good idea, because I consider more-truthful presidents to be better. Why are they better? Well, first of all, the whole ideal of "democracy" works better if everybody has more information, and more-honest information. Second, candidates lie either because they are unable to determine, or unwilling to state, the truth. The former is likely to yield poor performance as president. The latter ("strategic lying") might help a president in, e.g. negotiations, but only in circumstances where whoever he is lying to, cannot recognize that it was a lie. If the lie is detected, then it tends to hurt, not help. But now consider that every lie we have tabulated was detected as a lie.

Conclusion revisited in light of "p-hacking" worry

An early reader of this report, with expertise in statistics (who shall remain nameless for the moment, unless he tells me to use his name) claimed "[The] argument seems to be extensively 'p-hacked.' (Unfortunately he didn't say why it 'seemed' that way.) I have not closely investigated this, but I'd guess this is at the level of 'probably but not definitely meaningful' rather than the '99.999% confidence' claimed. For more on p-hacking, google the 'crisis of replication'." He concluded (to slightly paraphrase) this all was "[probably worth publishing] but the exaggerated claims should be toned down."

Ouch. My first reaction to this accusation was to note that I had computed the confidences asociated with the USA 2016 and USA 2012 elections each in two or three different ways. This was done intentionally, not to do "p-hacking," but rather to make it clear the confidences did vary depending on the precise statistical test chosen, but not by enough to hurt us, i.e. we still get a lot of confidence even using the least powerful among the three tests. (And by the way, these were the only tests I tried, all were done manually, not with a computer, and were chosen mainly because it was easy to compute them manually.)

However, I thought about it some more and think the complaint deserves a better response than that. So: what is "p-hacking" and what should one do about it?

Well, "p-hacking" is inherently a somewhat vague concept, but the rough idea is this. Suppose you have some data. You want to deduce some impressive-sounding statistical conclusion from it. So, you create (say) a computer program that tries N different statistical tests on the data (and perhaps N=1000000). You then find the "most amazing" test, from that ensemble, which, viewed in isolation, would produce the greatest confidence, e.g. smallest "p-level." For example, suppose that this most-amazing test produced p≤0.00001, i.e. confidence≥99.999%, in some conclusion. Wow! You publish it and are acclaimed by all as a great scientist who's discovered some Wonderful New Truth.

Unfortunately, if you behave that way, your conclusion is quite likely to be bunk. If the N tests were all independent events, then we'd expect 10 out of the 1000000 tests, and therefore the one you chose, to have p≤0.00001 even for meaningless fake data consisting entirely of random numbers. Hence, your "confidence≥99.999% conclusion" really would mean nothing!

In practice, though, usually such a computer search would have been searching among tests which were not at all "independent," e.g. they probably fell into at most a few families, in which case the situation would not be nearly that bad... but there still would effectively be confidence-exaggeration happening, it just would be highly non-obvious a priori how much. In particular, you could search a 1-parameter continuum family of tests to find the "most amazing" parameter value; this has N=infinity, but certainly in most practical applications that test result would not be an infinite exaggeration! In fact, we later are going to find a maximum-likelihood 1-parameter model, and at the end of that section we shall explain a way to justify it using a finite-N Bonferroni correction.

One way to protect yourself against this is to multiply your p-value by N ("Bonferroni correction") as a "safety factor." Which you trivially can do, if you know N. This is always safe, i.e. always conservative, in the sense that the new p-level is guaranteed to upper bound the truth. However, this often will be overprotection.

Now although my critic was using "p-hacking" in a pejorative way, in fact some of this behavior is what you are supposed to do. I.e. you are supposed to be intelligent in your choice both of statistical tests, and of hypothetical theories of nature that you devise to test versus data. You aren't supposed to be an idiot and choose some irrelevant-sounding test, or devise some obviously dumb theory. (And if you weren't intelligent, then you could use computer searching to effectively become more intelligent.) The problem only arises if you keep N secret and do not employ any Bonferroni (or other appropriate) correction to stay safe.

Now let us return from the abstract to the particular situation here. I devised only N=2 or N=3 tests (all variants of the same test actually, but under some definitional changes of, e.g, who really was a "candidate") for each of the two elections USA 2016 and USA 2012. I chose the "≥51% honesty" and "approval≥disapproval" threshold for 2016 without any "computer search" – those were simply the first values that sprung to my mind as a plausible "theory of nature." E.g, I as a typical(?) voter would disapprove any candidate who couldn't outperform a coin-toss in terms of honesty.

In a simplistic 1-dimensional model where a president, each time he makes a decision based on a false fact, moves his country 1 step "backward" but if based on a true fact moves it "forward"... we see that by approving presidents who outperform coin tosses, we are precisely approving the ones who "tend to move the country forward."

But when I then attempted to redo that same test for USA 2012, it was inapplicable because nobody achieved ≥51% honesty! Therefore I, putting on my "typical(?) voter" hat once again, said to myself, said I, "in that case, let's just lower our standards to: the top two most honest candidates in the race." (I was somewhat biased in my choice of "two" since I knew exactly two candidates tend to get the lion's share of the votes in US presidential elections; plus in USA 2016, exactly two candidates were involved in both "magic pairs.")

Then, after a considerable delay unsuccessfully seeking help from some Frenchmen, I found out how to obtain Veritometre data for the France 2012 election. But then again my initial test was inapplicable because all the 6 candidates I had data on achieved ≥51% honesty! Therefore I, putting on my "typical(?) voter" hat a final time, said to myself, "in that case, let's just raise our approval-threshold to: the top two most honest candidates in the race." Note how I'm trying to keep my "theory of nature" the same for all three elections. In the France 2012 case, however, the theory↔facts match, unfortunately for me, no longer was arguably perfect. But there still was a partial match.

Then I finally was able to obtain a reasonable tranche of data for USA 2008, and again the theory↔facts match was only partial.

The point of all the above was to reconstruct my thought process (which I assure you was not at all deep) in writing. And now that I have done so, we know what "N" is so that we can apply Bonferroni corrections. If for safety we intentionally choose the least-confidence among the two or three listed tests for each election, then no correction is needed at all, and the confidence we get should be a safe lower bound. If we choose the highest, however, then it should again be safe provided we use Bonferroni with N=3 for USA 2012, and N=2 for USA 2016. (For France 2012 I never computed any precise confidence, but merely claimed that overall the French data supported the Hypothesis more than it opposed it.)

Using USA 2016 & 2012 alone, 99.8% and 99% confidence using N=2 and N=3 corrections to each (i.e. 99.5% and 97%) yield ≥99.97% in combination. Here I am not using the naive p-multiplication combining formula "ba", but the more valid and conservative formula (2b-a)a where b is the greater and a the lesser p-level. (Alternatively, 99.5% and 88% confidence combine to yield at least 99.9% confidence, but this is less good so I will not continue with it.) Finally we can get an extra confidence boost in either case by throwing in the French data too, which I believe is enough to reach 99.98%.

I hope this satisfies my critic, and note 99.98% indeed is not as great as 99.999%, i.e. his criticism did indeed accomplish something useful in my [present revised] opinion.

USA 2008 presidential election

The free-access online poll database contains many approval-style (and some score-style) polls. This, together with early fact-checking data, enables me to extend this study to include the USA 2008 election.

 Politifact truthfulness   
CandidateH0 H1 H2 H3 H4 H5FCSummaryApproval/Disapproval [Poll]
Hillary Clinton(D)2 11 15 22 28 22249 50; 1.2246/38 [CBS/NYT 7-14 Jul RV]; 53/43 [Newsweek 21-22 May]
Barack Obama(DN)2 12 12 27 28 2157249; 1.5460/39 [CNN/ORC 10/30-11/1]
John McCain(RN)4 21 17 17 20 2018340; 1.2043/42 [CBS/NYT 10/31-11/2]
Joe Biden(D)5 13 15 28 21 177538; 1.08-1.2313/12 [CBS 15-20 Aug]; 27/22 [CNN/ORC 27-29 July RV]
Mitt Romney(R)9 16 17 28 16 1520631; 1.2841/32 [CNN/ORC 27-29 July]
Rudy Giuliani(R)6 23 19 21 15 154730; 1.1846/39 [CNN/ORC 29-31 Aug RV]
Mike Huckabee(R)10 12 29 22 7 204127; 0.8941/38 [Gallup Feb 8-10]; 24/34 [CBS/NYT 20-24 Feb]

Notes: Warning: for candidates who also ran in other elections, fact-checking data mixed in with facts checked during later years.

Candidates tabulated only if FC≥20 and I have a 2008 approval poll for them. For example Dennis Kucinich is excluded since even though he satisfies the first criterion with FC=25≥20, has no 2008 approval poll for him. Similarly we exclude Ron Paul since there is no 2008 approval poll among those listed by All the other candidates had ≤17 checked facts each.

Ron Paul was approval-polled by Economist/YouGov polls of 22-24 July, 4-6 Aug, and 11-13 Aug 2008 (and possibly more) but unfortunately their polling reports only stated one approval number (which was 302 approvals out of 891 "valid cases" in July, 303 of 978 in early Aug, 284 out of 954 in mid-Aug) despite their question wording "Do you have a favorable or an unfavorable opinion of ..." soliciting two possible answers, plus a probable, albeit unsolicited, "don't know" answer.

Generally we report only the most-recent approval polls we have, prior to the Nov. 2008 election date. However, Biden dropped out of the Presidential race on 3 January 2008, but on 23 August became Obama's (vice presidential) running mate. To avoid Biden's approval being conflated with Obama's, we only use his approval polls before 23 August. "RV" means that poll was among registered voters only. CBS=Columbia Broadcasting System. CNN=Cable News Network. NYT=New York Times. ORC=Opinion Research Center.

"Summary" column of table gives honesty (true + mostly true percentage); and approval/disapproval numerical ratio.

Our Hypothesis would have predicted that H.Clinton and Obama, as the two most-truthful candidates with 50% and 49% honesty (true + mostly true) percentages respectively, ought to be the two most-approved. In fact, Obama was the most-approved, but Romney, not Clinton, was the second-most approved. This could be regarded as a partial success.

A maximum likelihood model, and yet another criticism (by Gary Smith)

Gary Smith (no relation) on p.22 of his enjoyable book Standard deviations (flawed assumptions, tortured data, and other ways to lie with statistics) (Overlook Duckworth 2014) – which is an excellent book to read to get into the right frame of mind to be skeptical about statistics – wrote

It is not sensible to test a theory with the same data that were ransacked to concoct the theory. If a theory was made up to fit the data, of course the data support the theory! Theories should be tested with new data not contaminated by data grubbing.

I'm afraid I must plead guilty to this criticism. Referring to my description of my thought process "concocting the theory" last section, we could argue my original theory was developed without reference to any data, but the fact is, this whole investigation was stimulated by seeing, ten seconds later, that it worked perfectly for the USA 2016 election. If it had not, I probably would have never started this whole project, or perhaps I'd have tried to devise a new theory. Then, when it turned out the original theory was inapplicable to the USA 2012 and France 2012 elections, I modified it a bit, although as it turned out, just one common modification was used for both. Finally, I did indeed test the theory in Gary-Smith-approved style without any modification, on the USA 2008 election.

In some situations, e.g. if I were trying to invent the "ideal gas law" during the era 1600-1850, it would be no problem, after concocting the theory, to test it on new gas samples. However, the present case is quite different because I may have used up 100% of the (truthfulness, approval) election data available to me in the world. It is difficult to get more! (Also, political data is inherently "messier" than gas-measurement data since definitional questions arise.)

Plenty of other statistics books pay no attention to Gary Smith's Dictum. They instead advise something like this:

  1. Concoct a parameterized class of statistical models.
  2. Using all the data you can obtain, construct the "likelihood function" telling you the probability that data would have occurred given any particular model.
  3. Find the "maximum likelihood" model in your class.

And in fact, I agree with those other statistics books to some degree, i.e. do not 100% agree with that Gary Smith quote. Why? Even if your theory is devised entirely by data-grubbing, with no independent ungrubbed data at all available for testing, then a theory that works well enough plainly must be given credence. For example if a coin, tossed 20 times, came up "heads" every time, we suspect even Gary Smith would suspect the coin was biased, even if he devised that theory after seeing that data.

In any case, let's carry out that procedure – which Gary Smith may not like, but at the very least it will give us more understanding and more clarity about exactly what our "hypothesis" is. Here is my 1-parameter class of generative models:

  1. There are C≥2 candidates in some particular election. The input to the model is the honesty percentage for each candidate.
  2. We then sample (without replacement) two candidates from the C. The probability of choosing a candidate is p if that candidate is among the two most-honest, and q otherwise, where 0≤p≤½, 0≤q≤1/C, and 2p+(C-2)q=1. (Our model has exactly one parameter, namely p.)
  3. Those two candidates are predicted to be the two with the greatest approval/disapproval ratios.

If p=q=1/C in each election, then that model is just the uniform sampling model, i.e, the "null hypothesis." We presume/hope, however, that some p>q will deliver greater likelihood, i.e. a candidate by being one of the top two most honest, tends to enjoy more approval.

Now here is a table giving likelihood formulas:

ElectionCLikelihoodNull Hyp.likelihood
USA 2016212p2/(1-p)1/210=2/[(C-1)C]
USA 201292p2/(1-p)1/36=2/[(C-1)C]
France 20126pq/(1-p)+qp/(1-q)1/15=2/[(C-1)C]
USA 20087pq/(1-p)+qp/(1-q)1/21=2/[(C-1)C]
21+9+6+7 5-1 (1-p)-2 p6 (1-2p)2 ×
[1/(1-p)+4/(3+2p)] [1/(1-p)+7/(6+2p)]

The combined likelihood formula (got by multiplying the individual likelihoods under the naive assumption the four elections are independent) is maximized over the domain 0≤p≤½ when p≈0.4003 yielding Likelihood≈0.00667324≈1589.2×NHL where NHL=1/2381400 is the likelihood of the null hypothesis. So according to this calculation, our maximum likelihood model is 1589.2 times more likely than the null hypothesis, corresponding to confidence≈99.937% versus it.

But my first critic presumably would denounce this whole process as "p-hacking" while Gary Smith would call it "data grubbing."

I believe both my critics here do have a point. I believe I had answered the first critic, in the preceding section. However, the question is what to do about Gary Smith's related complaint. If we approximated the situation as "50% of our theory was 'concocted via data grubbing' while the other 50% was tested on new data" then perhaps our p-level should be squarerooted in which case the "99.937% confidence" would shrink to 97.5%. If 2/3 of the theory were concocted, then taking the cuberoot presumably would be wanted (91.4% confidence) or if 3/4 then the 4th root (84.5%).

At this point I'm feeling somewhat statistically-battered, and it evidently is not so easy to do statistics well under these circumstances. But whichever way we do it, the confidences always come out high enough that it still seems probable that there is some validity to all this.

Final Conclusion

So what is a valid conclusion, taking advantage of the information obtainable through likelihood maximization using all 4 elections worth of data – but immune to the criticisms? Let me now try to state one.

The final (?) theory, in a formulation that seems obnoxiously specific considering its crudity, is:

Hypothesis: "If we associate probability 0.4 with each of the two most-honest candidates, with the remaining 0.2 probability equidistributed among the remaining candidates (everything restricted to candidates important enough to have substantial fact-checking done on them and for pollsters to determine their approval) then: the two most-approved candidates are predicted to be got by sampling two candidates from that non-uniform distribution without replacement."

We now deduce a final confidence as follows. This hypothesis arose by finding the max-likelihood model within a 1-parameter class of models. The parameter was called "p." Its likelihoods range everywhere from zero (when p=0 or p=0.5) up to the maximum attained when p≈0.4003. If we had tried every p in the 7-element equispaced finite set {0.04, 0.11, 0.18, 0.25, 0.32, 0.39, 0.46} then the top two most-likely models would have been with p=0.46 and p=0.39 which are respectively 815 and 1571 times more likely than the null hypothesis. It then seems valid to apply a Bonferroni correction with N=7 to deduce a valid confidence>1-7/1571≈99.55% lower bound.

If we had instead employed the 8-element equispaced finite set {0.01, 0.08, 0.15, 0.22, 0.29, 0.36, 0.43, 0.5} then the top two most-likely models would have been with p=0.43 and p=0.36, which are respectively 1406 and 1363 times more likely than the null hypothesis. Even the worse among these two (p=0.36) with a Bonferroni N=8 correction would yield a valid confidence>1-8/1363≈99.41% lower bound.

In view of both the fact that Bonferroni is often overprotection, and the fact the true max-likelihood parameter choice p=0.4003 ought to be better than either 0.36 or 0.39, the true confidence is greater than these lower bounds; and furthermore our model obviously is crude, so presumably there is some better and truer (but unknown) model of the truth↔approval connection which would enjoy greater confidence still.

It also is worth noting that in all four of the elections we've examined, the two top finishers with the official voting system, were never the same pair as either the two most-approved or the two most-honest among the candidates we considered:

Election#canddtsTop 2 officiallyTop two approvalTop two most-honest
USA 201621H.Clinton & TrumpSanders & KasichKasich & Sanders
USA 20129Obama & RomneyRon Paul & ObamaObama & Ron Paul
France 20126Hollande & SarkozyHollande & BayrouMelenchon & Hollande
USA 20087Obama & McCainObama & RomneyH.Clinton & Obama

In other words, approval voting is genuinely different than the official USA & French systems. And it's probably better, since in USA 2016 and USA 2012, Approval chose the top 2 most honest candidates both times, while Official got 0 and 1 respectively; while in France 2012 and USA 2008, Approval and Official each got 1.

Regarding the latter two races: Bayrou overall seems to have been slightly more honest than Sarkozy, indicating Approval was a slightly better voting system, although each beat the other on one of the two Veritometre honesty scores. Which of {McCain, Romney} was more honest depends how one determines that, e.g. McCain had greater H4 and H5 indicating McCain was more honest (and H4+H5 had been my main honesty measure for USA 2016 and USA 2012), but H3+H4+H5=59 was greater for Romney than McCain's 57, indicating Romney was more honest – so in that sense their honesties are not comparable. (Despite this I still would overall regard McCain as the more honest.) In that same sense Sanders & Clinton had noncomparable honesties in USA 2016, but both dominated Trump at every politifact honesty level. In USA 2012, Obama dominated Pawlenty, who in turn dominated Ron Paul, who in turn dominated everybody else except for being noncomparable with Huntsman and Romney (and despite that I still would regard Paul as overall more honest than either). As the bottom line, Approval seems better than the Official systems based on honesty data from all 4 elections, but about the same based on the earliest two alone.

In all of these 4 elections, both the official and approval winner were "effectively most honest" candidates in the sense that the true max-honesty candidate was less than one statistical standard deviation error (assuming checked facts are chosen randomly from a much larger fact-pool for each candidate) above Politfact's estimate of the official- and approval-winners' honesty percentages. We could summarize that as "this is evidence that Democracy works!" But we warn the reader that actually, fact-checkers do not choose which facts to check by random selection from a large pool – they instead concentrate on particularly prominent, interesting and/or unclear ones – and even a single Big Lie can (and probably should) cause many voters not to vote for a candidate.

The plots below present our data tables in graphical form. Blue points are low-data candidates (<10 checked facts) and red ones are "candidates" who either never actually ran, or dropped out well before the first vote was cast. All other candidates represented by green points. Least-squares regression lines also shown (with the red Pawlenty point omitted from the USA 2012 fit), demonstrating that greater honesty and greater approval tend to correlate, for all candidates, not merely the ones obtaining the highest approval. Note that in all 4 cases the line's slope has the "correct" sign (and this remains true whether or not we include red and/or blue points in the linefits), which alone is an event of probability=2-4=1/16=6.25% under the null hypothesis.

Implications for Democracy

The important book Democracy for Realists by C.H.Achen and L.M.Bartels (Princeton Univ. Press 2016) begins

In the conventional view, democracy begins with the voters. Ordinary people have preferences about what their government should do. They choose leaders who will do those things, or they enact their preferences directly in referendums. In either case, what the majority wants becomes government policy – a highly attractive prospect... That way of thinking about democracy... constitutes a kind of "folk theory" of democracy, a set of accessible, appealing ideas assuring people that they live under an ethically defensible form of government that has their interests at heart.
    Unfortunately, while the folk theory of democracy has flourished as an ideal, its credibility has been severely undercut by a growing body of scientific evidence presenting a different and considerably darker view of democratic politics... the great majority of citizens pay little attention to politics. At election time, they are swayed by how they feel about "the nature of the times," especially the current state of the economy, and by political loyalties typically acquired in childhood. Those loyalties, not the facts of political life and government policy, are the primary drivers of political behavior. Election outcomes turn out to be largely random events from the viewpoint of contemporary democratic theory. That is, elections are well determined by powerful forces, but those forces are not the ones that current theories of democracy believe should determine how elections come out. Hence the old frameworks will no longer do.
    We want to persuade the reader to think about democracy in a fundamentally different way... the mental framework they bring to democratic life, while it may once have seemed defensible, can now be maintained only by willful denial of a great deal of credible evidence. However disheartening the task, intellectual honesty requires all of us to grapple with the corrosive implications of that evidence for our understanding of democracy. That is what this book aims to do.

Unfortunately, Achen & Bartels' book tells us very little about what to do to improve this "disheartening" and "unfortunate" state of affairs. Their book does not even mention, much less consider the effect of, improved forms of voting like approval, score, or even Borda, Condorcet systems, instant runoff, or proportional representation. However, the picture suggested by our data here differs from the gloomy one painted by Achen & Bartels. With approval voting, the voters work in a largely fact-based manner to determine and elect the candidates (from a wide democratic marketplace, not just two) who are the most honest – or they somehow accomplish something equivalent to that despite working in some different manner. And if by "random events" Achen & Bartels meant what we have been calling "the null hypothesis" then we have refuted this claim (with whatever confidence the reader chooses to believe we attained – we contend >99.5%), for approval voting based democracy.

This suggests that there is a very simple very easy cure for the ills of democracy pointed out by Achen & Bartels – just switch to approval (or, probably even better, score) voting – and that the underlying cause of the disease was our ultra-poor plurality voting system.

Furthermore, some of the symptoms A&B point out, such as "political [party] loyalties typically acquired in childhood" (also on p.26 they complain about artificial "unidimensionality") are caused by the fact the USA has a 2-party domination, which probably is the result of its foolish use of plurality voting. (E.g. Score and approval voting were used in Ancient Sparta and renaissance Venice both for longer than the entire history of the USA, with no record of political 2-party dominance ever developing.) And the lack of voter knowledge A&B complain about may have something to do with the lack of media coverage, which in turn is caused by the plurality voting system. Specifically, right now (mid-September 2016), few Americans know anything much about Gary Johnson, even though he, if reckoned by approval/disapproval ratio, is the clear frontrunner in the USA presidential race. (38% of Americans do not even recognize Johnson's name, according to a Huffpost/YouGov survey 25-26 August 2016.) Why do few Americans know about Johnson? Because the media has told them almost nothing about him. Why has the media said almost nothing about him? Because the USA is using plurality voting, not approval voting, so that the fact Johnson is the leader with the latter system, implies virtually nothing about his chances of winning the presidency, and indeed Johnson's chances are near zero – therefore media has no motivation to cover him. Why are Johnson's chances near zero? Because with plurality voting any vote for anybody who is not one of the two leaders, is "wasted." Few are willing to waste their vote, causing Johnson to get few votes (tremendously fewer than he "deserves" in view of his approval), causing his chances to be near zero, in a self-reinforcing cycle.

This Johnson discrepancy also makes it completely clear that USA democracy, because of the voting system, does not elect the candidates voters (tell pollsters they) regard as best. The same effect, over historical time, has caused only two parties to dominate US politics totally, even though their current presidential candidates are the two most-disapproved major party nominees in the entire history of approval-style polling (dating back to 1956 using Gallup data, and dating back to 1980 using CBS/New York Times poll data; the closest competitor in terms of low approval was Barry Goldwater in 1964) and even though the US congress as a whole – almost entirely occupied by the Democratic and Republican parties – is currently only 18% approved. This in turn has caused the development of fairly mindless "party loyalties" because under such circumstances it is pointless to be a free-thinker.

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