Not simple enough

 Once again, I have gotten diverted into another topic.  Yesterday morning, Donald Trump tweeted about a paper by John Lott called "A simple test for vote fraud with absentee ballots in the 2020 Presidential election", which according to Trump "estimates 11,350 absentee votes lost to Trump in Georgia. Another 289,000 'excess (fraudulent) votes' across GA, AZ, MI, NV, PA, and WI."*  Lott is an economist best known for work arguing against gun control.  He's been involved in a number of controversies, but has published extensively in scholarly journals, and I happen to have read one of his papers recently and it seemed solid.  So he's not someone like L. Lin Wood or Sidney Powell who can be dismissed out of hand.  How much of a case does he have?

One obvious point is that his statistical evidence isn't very strong.  He has two separate analyses, and the key estimates in both have p-values of about .07.  He doesn't acknowledge that, and disguises it in various ways, including the classic "statistically significant at the .05 level for a one-tailed test."  There's also a big jump from statistical analysis to conclusions--the abstract speaks of "fraudulent votes" rather than something like "excess votes beyond those predicted by the model."  But could you at least say there's suggestive evidence, something that calls for further investigation?  One analysis, the one about "excess" votes in certain counties in swing states, just isn't convincing in principle.  The model was basically about increase in turnout over 2016, and it appears that the increase was greater in certain counties where fraud had been alleged.  Lott interprets that increase as evidence that there was ballot box stuffing in those counties, but another possibility is that the Democrats just made special efforts to get out the vote in them.  Given that those counties were heavily Democratic and located in states that were expected to be close, that seems very plausible.  The other analysis was more complex.  It involved differences between the distribution of absentee votes and in-person votes in Georgia, comparing Fulton county (Atlanta) to surrounding areas.   Lott argued that it was easier to manipulate or fake absentee votes than regular ones, so fraud would show up as a large discrepancy between different kinds of votes.  For example, suppose that outside of Fulton county, 60% of the in-person votes and 50% of absentee votes were for Trump, and in Fulton county, 40% of the in-person votes and 30% of the absentee votes were for Trump.  Then you'd just say that Trump voters preferred to vote in person, as surveys suggested would happen.  But suppose that in Fulton county, 40% of the in-person votes and only 10% of the absentee ballots were for Trump.  Lott says that would be evidence of fraud (votes being switched, Biden votes manufactured, or Trump votes discarded), on the grounds that "while Democrats were pushing their voters to vote by absentee ballot, there is no reason to expect that rate to differ between two precincts that are next to each other and are similar in terms of their in-person voting support and their demographics." And I can't immediately think of a reason why it would, so this approach seems more promising.  

But rather than just computing differences between absentee and in-person votes, he regressed the Trump share of absentee votes on the Trump share of in-person votes plus a dummy variable for Fulton county.  The problem with doing that is that there is no sense in which the in-person share is a cause of, or even prior to, the absentee share (or vice-versa)--they are just two variables that are correlated with each other.  In this situation,  if the Trump share of the vote is lower in Fulton county, then the Fulton county variable is in effect correlated with the error term in the regression, producing spurious results.**  Lott didn't make his data available, but a look at his regression estimates suggests that this misspecification accounts for his "Fulton county effect."

I include a little simulation at the end, in which both in-person and absentee votes are a function of a precinct-level propensity to support Trump plus a random error term, and "Fulton county" precincts have a lower propensity to vote for Trump.  If you do a regression with a Fulton county dummy, it is highly significant, but if you compute the gap between absentee and in-person ballots, it is the same in and out of "Fulton county."

*The phrase "excess (fraudulent) votes" doesn't appear in Lott's paper, although "excess votes" and "fraudulent votes" do.  

**I'm simplifying, because he looks at differences between pairs of precincts, but the basic point remains the same.

> propensity<-c(4000:7000)/100
> min(propensity)
[1] 40
> length(propensity)
[1] 3001
> person<-propensity+3*rnorm(3001)
> absent<-propensity-10+3*rnorm(3001)
> fulton<-propensity<50
> m1<-lm(absent~person)
> summary(m1)

Call:
lm(formula = absent ~ person)

Residuals:
    Min      1Q  Median      3Q     Max
-14.191  -2.785   0.017   2.793  14.553

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -3.736393   0.459804  -8.126 6.41e-16 ***
person       0.886926   0.008248 107.530  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.15 on 2999 degrees of freedom
Multiple R-squared:  0.794,     Adjusted R-squared:  0.794
F-statistic: 1.156e+04 on 1 and 2999 DF,  p-value: < 2.2e-16

> m2<-lm(absent~person+fulton)
> summary(m2)

Call:
lm(formula = absent ~ person + fulton)

Residuals:
     Min       1Q   Median       3Q      Max
-12.6099  -2.6932   0.0678   2.6976  14.7024

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  6.63930    0.74541   8.907   <2e-16 ***
person       0.72331    0.01234  58.621   <2e-16 ***
fultonTRUE  -4.13916    0.24041 -17.217   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.959 on 2998 degrees of freedom
Multiple R-squared:  0.8126,    Adjusted R-squared:  0.8125
F-statistic:  6499 on 2 and 2998 DF,  p-value: < 2.2e-16

> gap<-absent-person
> t.test(gap~fulton)

        Welch Two Sample t-test

data:  gap by fulton
t = -0.078946, df = 1987.1, p-value = 0.9371
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.3386609  0.3124506
sample estimates:
mean in group FALSE  mean in group TRUE
          -9.958066           -9.944960

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