Tuesday, April 7, 2015

Kansas Election Fraud

Statisticians need to be careful in the way we communicate with the public.

Last week, a Wichita State University statistician filed a lawsuit regarding the Kansas 2014 election.  She is trying to get access to vote machine tallies, to rule out the potential of voter fraud.  Here's a link to the article in the Wichita Eagle.

The statistician, who works as a QA engineer, has found some "voting anomalies"... essentially that republicans receive larger than expected vote shares in larger precincts.  Keep in mind that QA (Quality Assurance) engineers are trained to look for anomalies, things you wouldn't expect in data, and make a big deal out of them so that systems don't fail. 

Of note from the article:

“This is not just an anomaly that occurred in one place,” Clarkson said. “It is a pattern that has occurred repeatedly in elections across the United States.”

Read more here: http://www.kansas.com/news/politics-government/article17139890.html#storylink=cpy
AND


The pattern could be voter fraud or a demographic trend that has not been picked up by extensive polling, she said.
On face, as a research statistician, this doesn't seem like that big of deal.  Just a researcher looking into anomalies.  I did think that putting "voter fraud" out there as a possibility seems a little aggressive at this point, but didn't see this as an issue that would get a lot of attention.  But consider the political climate of Kansas:


  1. I've posted on this before, but the political climate in Kansas right now is really tense, largely due to a highly contested election.
  2. Same post from before, but progressives are especially upset because they just lost an election, which their leaders told them would be a fairly easy win.
  3. I've seen the article above posted by many progressive friends as fodder, evidence, and proof that, statistically speaking, Brownback was probably re-elected due to election fraud.
This seems to be a heating-up massive conspiracy theory.  But let's be calm and analyze:

What we know


First, from Clarkson's comments, she has no evidence of voter fraud.  She has found a small statistical anomaly, that exists nationwide and wants to use Kansas to verify that it isn't due to fraud or voting machine issues.

But what is that anomaly? 


The anomaly is that after a certain size threshold (500), there is a positive correlation between precinct size and percent republican votes.  

Why is that an issue?


Clarkson is a QA engineer and in anomaly detection mode.  She's starting from an a priori premise that precinct size should not determine results, and thus, statistically significant correlations should not exist.

Is Clarkson's analysis of the data correct?


Though disagreeing with her conclusions, I tend to think the mechanics of her analysis are correct.  In fact, I was able to replicate, using the 2010 Kansas Gubernatorial Election.  The relationship is weak statistically speaking, but statistically significant which indicates that something non-random is happening in the data. Regression stats and visual plot below.


And here's what a sample of Clarkson's work with an Ohio example looks like:

So the correlations exist, and then people must be acting nefariously in those large districts right?


Here we go.  Absolutely not.  And here's why: covariates.  In the real world, multiple variables often correlate with one another, causing us to find relationships, that are really measuring something else.  Clarkson's comments allude to this when she talks about potentially underlying and undetermined demographic factors.  

There are many what-if's here.  What if other variables also correlate with precinct size?  Age, Race, Wealth, Urbanity, etc, etc, etc.  In these cases we are latently measuring other factors through measuring precinct size.   Another specific issue, what if more conservative populaces somehow push for fewer, larger precincts and less division?  Keep in mind, this is a WEAK relationship we are trying to explain.

The authors of this analysis already drop the smallest precincts as a whole, because they tend to be more rural, and thus more Republican.  In essence, the authors tacitly admit underlying demographic factors can impact the correlation between precinct size and voting behavior.  We haven't gone through the steps to exclude all other demographic factors, so why are we making vague accusations of fraud and making a big deal of this in the press?  

Partially speculation here, but it is much more likely that the correlation found is due to underlying demographics and other covariates, rather than something nefarious going on with voter machines.

Conclusion

This is an interesting area of research, and I will likely post on this again when I get access to more data.  I think it is quite likely, that this is due to underlying demographic drivers.

I absolutely think Clarkson should have access to the vote machine records, as well as the software itself for testing.  But, the way this has been handled by Clarkson and the press at this point is pretty reckless.  In the political environment that is Kansas, where many believe the election was stolen or rigged, this "evidence" is being handled by many people as more evidence of fraud by the administration, while we really have no real evidence for that.

19 comments:

  1. Democratic-leaning precincts in Kansas are anomalous. In 2010, Brownback received 63.28% of the statewide vote. You won't find many precincts of any size where he received 25% more than that. But you can locate precincts, where Brownback received 25% less than that. The median and mode are more Republican than the mean.

    If you replicate Clarkson's effort (but don't discard the precinct identification), you will see the same decline in the cumulative support for Brownback as precincts with more votes cast are added in. But if you look closely, this is not monotonic. You will observe a sudden drop, when the precinct is in Kansas City, or Lawrence, or Topeka, or some parts of Wichita, as a precinct that is 35% Brownback is added in. And then you will see it creep upward as precincts that a more Republican than the mean are added in. But then this pattern stops. In particular, there are no more precincts from Wyandotte (Kansas City) and Shawnee (Topeka). There are some from Douglas (Lawrence) and Sedgwick (Wichita). But the Republican precincts do not stop.

    For a precinct with votes cast = X, calculate the frequency of precincts in the interval (X*80%, X*120%) that were less than 50% D. For very small X, this is quite small, it reaches close to 20%, and then drops off.

    If this is due to counting fraud, it is not vote flipping but precinct flipping. That is instead of changing a few percentage of the vote in most precincts (say 20 votes among 1000 votes); we would have precincts where 20% of the votes were flipped. But that would be quite obvious to any knowledgeable observer.

    The Wichita Eagle article claims that no demographic relationship has been found by "statisticians", but it does not appear that Clarkson even attempted to find whether there was such a relationship. She accepted that it was well known that smaller precincts in rural communities were more Republican. A claim that polls have never found a relationship is not surprising. "How many votes will be cast in your election precinct? Press 1 for more than 1000; Press 2 for between 500 and 1000; Press 3 for between 200 and 500; Press 4 for less than 200; Press 5 if you don't know; Press 6 if you have no earthly idea; Press 7 if you are thinking about hanging up; Press 8 if you want to pass on this question; Press 9 to hear the entire inanity repeated"

    Election precincts don't change that much. If you have a reasonable number of voters, and an established polling place, why confuse voters. If there is some legal limit, say 1000, and a precinct reaches 1200, you split the precinct in half, rather than trying to keep 1000 in a major part.

    In long-developed portions of cities, the precincts may have been established based on being able to walk to the poll. As family sizes have declined, and housing has been converted to other purposes, the population shrinks. That is, in the areas most likely to be Democratic, the precincts are smaller. Merging of precinct will be resisted, just as consolidation of schools or post offices will be resisted.

    The largest precincts are not in the urban center. They are more suburban, and some in small towns. In Sedgwick County the largest precincts are outside Wichita, or on the outskirts of Wichita, particularly west of the Arkansas River or on the east side of town. Butler County has many large precincts. And some large precincts are in large towns. The largest precinct in the state is in McPherson, which only has four precincts. It turns out that the largest precinct that went 72.2% for Brownback, is the least Republican precinct in town.

    As you noted, the number of votes cast is not dependent solely on the number of voters registered in the precinct. If Republican voters are more likely to turn out, then more Republican precincts will have more votes cast. in the more rural areas, it doesn't matter. The voters will vote in every election. There are fewer competing distractions, and people are more connected to their community.

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    1. Thank you for your response, some great ideas on here and I'll certainly look into them when I have some time to grab some additional data. Your comment that "but it does not appear that Clarkson even attempted to find whether there was such a relationship." is especially poignant because it's the reason I felt it was necessary to look deeper into this subject.

      I also appreciate your last paragraph's analysis on turnout, will likely try to spin it in with my future endogeneity exploration.

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    2. In Kansas at least, there appears to be different styles of precinct delineation. It appears that townships and municipalities must have there own precincts. There also may be a contiguity requirement.In 2010, there were over 300 precincts with zero votes cast. Most had names that indicated they were "exclaves", "enclaves", "parts" etc. This is also true among the very smallest precincts.

      Around 7 votes cast or so, the precincts are mostly townships. 7 votes cast in an area that has around 36 square miles is quite rural.

      As the numbers increase, there will be more precincts with the words "precinct" or "ward" in their name. There will also be ever so often be a precinct in a larger city. Perhaps these correspond to isolated neighborhoods, or are split by a legislative district.

      Precincts serve two purposes: (1) convenience for voting; (2) ballot preparation. It is not desirable to have different ballots at the same polling location. A mix-up could permit voters to vote for someone they aren't permitted to vote for. Kansas has a strong policy preference for towns with four wards. So smaller towns will be divided into 4 wards with one precinct per ward. The size of the precincts depends on the population of the town divided by four.

      Townships do not elect a government from districts. So a suburban township might have a single fairly large precinct. If the population is dispersed, it may not be convenient to divide into precincts.

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    3. In larger cities, wards will be divided into multiple precincts. As an example, this is a map of the wards and precincts of Overland Park.

      http://www.opkansas.org/wp-content/uploads/downloads/overland-park-ward-boundaries.pdf

      Overland Park is the second largest city in Kansas. In considering where we might expect larger precincts, 4 of the 9 largest cities are suburbs of Kansas City, Missouri in Johnson County. The other 3 are Olathe (5th), Shawnee (7th), and Lenexas (9th). The larger cities in Johnson County are quite lengthy, as they have annexed territory to prevent be blocked in.

      Overland Park stretches out to the south, roughly 19 miles long and 5-6 miles wide. Ward 1 in the far north is characterized by streets on a rectangular city grid (16 streets x 8 streets per section, with a city block. Moving south, the street grid becomes curvy, suggesting development in the 1960s, but the Public Land Survey System grid of section lines and half section lines is clearly visible. Precincts are typically quarter sections. They evidently have a convenient number of voters, and the half section streets provide clear boundaries.

      South of I-435, the grid disappears other than section lines. Residential area are loops and cul-de-sacs designed to discourage cars from driving through. Precincts are larger and more irregular in shape.

      The precinct with the most votes cast was in Ward 6 in the south. It was 57% larger than the second largest precinct, and was 79.9% for Brownback, The largest precinct in Ward 2 was 17th overall, and the largest precinct in Ward 1 as 26th overall. Both of these larger precincts in the north were 1/2 sections. In essence, they had more votes cast because they were atypically large for their area of Overland Park.

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    4. In more established parts of cities, there will be a regular rectangular street grid, and precincts may be quite regular in size. It might no be feasible to merge precincts. Even if there was a room with enough space for twice the voting machines at one of the polling places, there might not be twice the parking space. So you might keep two precincts with 350 voters rather than merging into one with 700 votes cast (and remember this might close to 1000 in a presidential election). Meanwhile, if a precinct with 700 votes already exists, you might keep it. If a single voting machine can handle 50 voters in a day, the only difference between polling places that can handle 500 voters and one that can handle 700 is space for four voting machines, some more parking, and some more election judges. It should scale quite linearly. Larger precincts might be a bit more efficient because the likelihood of the head election judge being competent is greater.

      We can rank the precincts in Overland Park by votes cast, and divide them into quartiles based on cumulative votes cast. Brownback received 62.26% citywide. In the smallest precincts, he did worse than that in 31 of 42 precincts (74%) (< 455 votes). In precincts with 456-559 votes cast he did worse in 19 of 30.(63%). In precincts with 560-658, he did worse in 15 of 24 precincts (62%). In precincts with 659+. he did worse in 5 of 20 (only 25%).

      Overall, Brownback did worse than his citywide percentage in 70 of 116 precincts (60%). That he could do worse in most precincts just reinforces that larger precincts tend to be more Republican. In Overland Park, traveling south correlates with newer developments, larger precincts, and being more Republican. One would expect newer developments in a suburb to be more Republican.

      Olathe is the 5th largest city. A comparable analysis shows Brownback underperforming in 13/28 of the smallest precincts, 9/17 of the next quartile, 6/14 of the next quartile, and 4/12 of the largest precinct. The variation is less than Overland Park. But Olathe overall was 67.41% for Brownback. Olathe had 4.4% of the total Kansas population, but 7.4% of precincts with more than 500 votes cast.

      Shawnee is the 7th largest city. Low performing precincts by quartile: 11/16; 6/10; 4/8; 2/7. Smaller precincts are particularly concentrated in Ward 2 in the long settled eastern part of the city. The east-west extent of Shawnee is about 2-1/2 times the north-south extent, with the areas to the west having a lot of develop-able and developing land.

      Lenexa is the 9th largest city. Lower performing precincts by quartile 9/12, 4/7, 2/7, 2/5.

      Long skinny cities such as Overland Park, Shawnee, and Lenexa show a strong relationship between precinct size and Republican performance. These cities have a long distance between areas likely developed after WWII and newly developed areas. Olathe which is more square, does not show such a strong relationship, but it overall more Republican.

      There is a cluster of small cities in the northeast corner of the county: Fairview, Merriam, Mission, Mission Hills, Mission Woods, Prairie Village, Roeland Park, Westwood, and Westwood Hills. These cities are locked in and have been longed developed. Democrats are competitive here.

      But only 18/55 (32.7%) of precincts had more than 500 votes. Among Leawood, Lenexa, Olathe, Overland Park, and Shawnee.153/290 (52.7%) had more than 500 votes.

      The fundamental question is not why larger precincts tend to be stronger for Republicans, but why do newer developing areas tend to have larger precincts.

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  2. I'm not sure it was Clarkson who introduced the "large precinct = urban precinct" bias. It may have been the newspaper reporter. It sounds like a mistake a reporter would make, or an editor would introduce into a story, because journalists would: 1) not understand statistics and 2) hear "larger precincts did not perform as expected" and interpret that as "urban precincts did not produce as many Democratic votes as expected" and then create a "nut paragraph" for the story that attempts to explain the researcher's trend but instead just gets it wrong. Clarkson's work is based on this: http://madisonvoices.com/pdffiles/2008_2012_ElectionsResultsAnomaliesAndAnalysis_V1.5.pdf. Because she's following their work, I suspect she caught these researchers' explanation that precinct "size" (or is it population? total vote count? voter turnout?) does not predict urban v. rural location. I see arguments here, however, that offer plausible scenarios for why greater total precinct vote tallies might correlate to greater GOP vote percentages.

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    1. http://bethclarkson.com/?page_id=46&paged=25

      "This is not surprising as it’s well known that rural areas (smaller precincts) lean more republican than the large city precincts."

      Here she was explaining the dropoff in the cumulative percentage as larger precincts are added in.

      "This second and unexpected trend is evidence of a previously unknown relationship between the number of votes cast and the percent of vote in favor of the Republican candidate, one that is in the opposite direction of what was expected due to the known relationship between rural and urban precincts."

      And here she was noting that the opposite effect occurred for the largest precincts.

      Chaquette/Johnson state:

      "Using precinct-level results, especially when thousands of precincts are available per county, you remove virtually all effects of voter demographics which could influence the county’s chart results. There are many
      more precincts in cities, which tend to keep precinct sizes (vote tally) reasonably even. It is very unlikely that demographics vary appreciably as a function of precinct size although there could be exceptions if the
      precinct sizes were kept large and re-districting based on demographics was used. The fraud, if any will come out in the charts."

      There is no reason to suppose that having more precincts in a city would cause the precincts to be more equal in size.

      Chaquette/Johnson's work is particularly problematic with regard to primaries. The number of votes cast in a Republican primary is extremely correlated with the number of Republican voters.


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    2. Thanks for the informed response. Apologies for the speculation.

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    3. I took a look at the 2012 presidential primary in Orange County, California. It is true that there is a relationship between votes cast in the Republican primary and the Romney performance, but this a consequence of other phenomena.

      Chaquette/Johnson seemed to think that election precincts are redistricted to make them equal in population. While precincts are grossly of the same size, there is quite a bit of variation. This may reflect sensibilities about efficiency when the precincts were established. If paper ballots were used, it might be based on the number of ballots than can be efficiently counted by the election officials at a precinct. If the voting machines were mechanical behemoths, there might be some minimum number of voters to make their use effective. If punch cards were used, you can scale the number of punching stations to the number of voters. The same may be true for modern electronic voting machines, where you can deploy 5 machines for one precinct, and 6 for another that is somewhat larger.

      Registration data was included with the election data for the 2012 primary. Combining the Republican and Democratic registration gives a rough estimate of the number of registered voters in a precinct, even though it excludes independent (NPP) voters and minor party registrants. Current registration data shows that NPP voters are between 22% and 25% of registration among the 5 Orange County supervisors districts, even though the number of voters registered in the districts is quite different, as is the ratio of Republicans to Democrats. So Republican + Democratic registration is a reliable measure of (relative) total registration.

      In 2012, the average precinct had 636 R+D registrants, but the standard deviation was 194, which was 31% of the average.

      If we compare the Republican registration share to the R+D registration:
      R/(R+D) :: R+D, we get a correlation of -0.150. That is, precinct size is relatively neutral with respect to political affiliation. If anything, Democratic leaning precincts are slightly larger.

      If we arrange the precincts by R+D registration, the cumulative Romney percentage reaches close to the final value and stays there. The smallest precincts are slightly more favorable to Romney (83.1% for the 1st decile of Republican vote), and the largest slightly less favorable (81.3% for the last decile, versus an 82.4% overall percentage.

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    4. So where did Chaquette/Johnson go wrong? They used the Republican vote total. Consider: Precinct A has 1000 voters, Precinct B has 1000 voters. What can we conclude about the demographics or political orientation of the two precincts? Absolutely nothing.

      Now consider Precinct C has 600 registered Republican voters, Precinct D has 400 registered Republican voters. What can we conclude about the the two precincts? Possibilities:

      1) Precinct C is more Republican, and the precincts are of the same size.
      2) Precinct C has more voters, and the precincts are equally Republican.
      3) Precinct D has more voters, but Precinct C is more Republican.
      4) Precinct C has more voters, but Precinct D is more Republican.
      5) Vote flipping has occurred in Precinct C.

      We can not foreclose any of these possibilities, from the information given. The number of Republican voters is a function of both the size of the precinct and the registration share of the precinct. If we are considering votes cast, we also have to include the turnout percentage.

      If Precinct E is 60% Republican and Precinct F is 40% Republican, what can we conclude? Precinct E is likely to be whiter, older, more married, more home-owning, and more economically stable.

      Since the number of votes in the Republican primary is dependent in part on how Republican the precinct is, it can not be considered to be politically or demographically neutral.

      Correlation between the Republican registration percentage: R/(R+D) and the total registration: R+D is quite weak (-0.150), and is any case quite small: For each 100 additional registrants, the percentage of Republican registrants declines 0.94%. But since overall, the Republican registration percentage is 57.3%, when 100 registrants are added to a precinct the number of Republicans will increase by around 50.

      The correlation between Republican registration percentage: R/(R+D) and Republican registrants is 0.543. This is quite unremarkable, For every 1% increase in Republican strength, we add 5.76 Republicans. Since the average precinct has 643 registrants, this is about what we would expect.

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  3. Correlation between the Republican registration percentage, R/(R+D) and the turnout R-ballots/R-registrants is 0.543. This is of interest. In areas of higher Republican concentration, there are not only more Republicans, but a larger share of them vote. As noted above, areas with a higher Republican percentage are likely to be whiter, older, more married, more home-owning, and economically stable. This makes it more likely that they will vote, particularly in a primary election. They may also be more likely to be permanent vote-by-mail voters, who were sent the primary ballot. And they are more likely to be living at the same place they were in 2008 or 2010. A counter-example is a precinct on the UC-Irvine campus (59019), Of 1277 registered Democrats and Republicans, 28 (2.19%) voted in June. Turnout among Republicans (9 of 343, or 2.69%) was actually higher than for Democrats. Students who had registered for the 2008 presidential election as freshmen may have graduated, or moved off campus. The primary was in June, during the summer.

    Republicans in Republican-leaning areas might have been more motivated in the primary since they were accustomed to the primary being decisive. If you are actually choosing your legislator, rather than just deciding on a token opponent, you would be more likely to vote.

    Since voters in more Democratic areas are more likely to move, the potential number of actual voters is reduced. If a Republican has moved to Arizona, she won't vote in California, even though she is still on the registration rolls. It is a rare individual who will move to another state and inform their previous state to remove them from the voting rolls.

    So it is plausible that Republicans turnout will be greater in more Republican leaning areas. The increase in turnout is 0.47% per 1% increase in Republican share of registration.

    So let's consider two precincts with 643 registered R+D, which is the average for Orange County. Precinct G is 70% Republican, Precinct H is 40% Republican. Precinct G has 450 Republican registrants, Precinct H has 257 Democratic registrants. Turnout is 39.58% in Precinct G resulting in 178 Republican voters. Turnout is 25.43% in Precinct H resulting in 66 Republican voters. Two precincts with the same number of registrants, but Precinct G would have 2.7 times as many Republican voters.

    Chaquette/Johnson used the number of votes cast for President, which is slightly less than the number of Republican voters. About 5% of voters skipped the presidential race. The presidential race had been decided by then, but the votes for other races were meaningful. There is a weak relationship between the presidential undervote and the Republican registration share, but the effect is slight. Correlation is 0.403 and the reduction in the undervote is only 0.11% per 1% increase in the Republican registration share.

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  4. The Romney share is mildly correlated with the Republican registration share, with a correlation of 0.543. The Romney share increases 0.33% for every 1% increase in the Republican registration share. As noted earlier, areas with higher Republican share, are whiter, older, more married, more home-owning, and more economically stable. They may be Republican for social reasons (eg all the neighbors are Republican), and may have been more inclined to vote for the more conventional candidate, or more interested in victory in November, than expressing their disaffection.

    82.4% of all Republican votes were for Romney. A vote for Paul, Santorum, or Gingrich could be considered eccentric. It is unsurprising that there are more eccentric votes in areas where being a Republican in the first place is more uncommon. But even in the more Democratic areas, 75% of votes were for Romney.

    Let's compare correlation between the Romney%, the Republican%, and the Republican votes cast;

    Romney% v Republican% 0.526
    Republican% v Republican votes 0.685
    Romney% v Republican votes 0.475

    The weakest relationship is that observed by Chaquette/Johnson. The strongest relationship is between Republican registration share and Republican votes. But this is totally expected. More Republican registrants causes more Republican votes and a greater Republican registration share.

    The relationship between Republican registration share and the Romney vote share is plausible, and the Republican registration share is definitely linked to demographic and political difference.

    The relationship between Romney share and Republican votes is an effect of the other two stronger phenomena.

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  5. The first two digits of the precinct number indicate the city it is located in. We can therefore calculate the R/(R+D) registration, Republican turnout, and the Romney%. For example, Anaheim was 50.2% Republican registration, 28.36% Republican turnout, and 80.64% Romney.

    We can also project the Romney% based on the Republican registration share. Most cities had an actual Romney share that was within 5% of the projected Romney share. Outliers include:

    Laguna Woods, 51.2% Republican, 88.6% Romney, 78.4% projected Romney. Laguna Woods is largely a retirement community, distinctly older (79.5% over 65, and distinctly female (100 female per 55 males). This female bias likely accounts for the low Republican percentage, but the age bias would account for the high Romney support among Republicans. Turnout was 56%, the highest in the county, while Paul support was distinctly low.

    Laguna Beach 48.9% Republican, 84.4% Romney, 77.7% Romney projected. The majority Democratic registration is likely culturally based rather than economically based, but the Republican registrants are likely to be wealthier than the county as a whole.

    Seal Beach 56.3% Republican, Romney 84.5%, Romney projected. The median age is 57, and there are 100 adult females per 78 adult males. Like in Laguna Woods, the large share of women reduces the Republican share, but the older population increases the Romney share.

    Irvine, 52.5% Republican 83.1% Romney, 78.9% Romney projected. The UC-Irvine campus pushes the Republican share downward, but many of these students have graduated, and students don't vote in primaries held in June. Ron Paul had 44% of the vote in a campus precinct, but that was on a 2% turnout.

    La Habra 50.2% Republican, 82.1% Romney, 78.1% Romney projected. La Habra is majority Hispanic, which is likely reflected in the registratioin share, but may not be reflective of voters in the Republican primary.
    '
    The Republican registration share is an imperfect measure of the underlying social and economic factors that lead to Republican primary voters to support Romney, but it may be the best available based on election statistics alone.

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  6. She has not accused anyone of fraud at this point. She mentioned fraud as ONE possible explanation, not the only one.

    Let's not put words in the mouth of a statistician who is at this point trying to do research to identify the explanation for something she has observed.

    She is not jumping to conclusions; this author took it a step too far and is accusing her of jumping to conclusions.

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  7. I don't believe she is accusing anyone of anything as of yet. She has said she would like to investigate, maybe you could offer your assistance as to oversee fairness? Also, the Gov Race precinct results have not been released still. Certainly the response from the SOS office has not helped curb speculation of rigging.

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  8. To confirm Clarkson’s results, I downloaded 2014 Kansas Senate precinct data for each county. Cumulative vote shares (CVS) were calculated for the five largest: Sedgwick, Johnson, Saline, Shawnee and Wyandotte and the Total for all counties.
    https://richardcharnin.wordpress.com/2015/04/02/12370/

    Note the Republican state total cumulative share margin is in steady decline for the first 500,000 votes, but then becomes flat. Since the largest counties show the GOP cumulative share increasing with precinct size, it confirms that they were the counties where the anomalies occurred. In other words, the Independent Orman may have caught the Republican Roberts if the trend was not halted by election fraud (vote switching, disenfranchisement, etc.) in the larger (presumably more Democratic) precincts.

    https://docs.google.com/spreadsheets/d/1D087y0AlsFiITeypDEk3W_c4P-O2iytQRCp85wFIw-Q/edit#gid=1367668624

    Clarkson’s analysis confirms my previous CVS analysis of the 2014 Wisconsin, Florida, Maryland and South Dakota governor elections, all of which showed the same counter-intuitive, mathematically anomalous trend: cumulative vote shares increased in favor of the Republican candidate in large precincts. One would expect that the cumulative vote shares should move slightly in favor of the Democrats as larger (urban) precinct votes are added to the total.

    https://richardcharnin.wordpress.com/2015/02/27/proving-election-fraud-cumulative-vote-share-analysis/

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