Meta-Meta pollster analysis of the 2016 election
Complaint about confusion
OK, maybe someone can explain this to me because it’s bothering me too much. At this point there are enough polls that most people look at meta-analysis of polls. It used to be that was just the Princeton election consortium.. But now there are several different models: from the Upshot to fivethirtyeight.
Upshot and fivethirtyeight seem like the most elaborate models, but it’s driving me crazy that they are highly-autocorrelated. That is to say: if their predictions of the November outcome for Clinton decline today, it’s likely they’ll decline tomorrow, and vice versa.
I find that idea extremely confusing. Poll results can have momentum because they are supposed to represent a real thing that can change. But if you can predict where a prediction is going, the prediction should already be there. If it’s not, it’s clearly leaving information on the table unused. Put more crassly; if you used the fivethirtyeight or upshot model to make a bet about the election each day, I would be able to bet against you and make money consistently. This suggests that the loess curves fivethirtyeight (and Upshot?) are using are not a good job at being adequately responsive.
Ridiculously, this implies that there is a place in this election for meta-analysis of the meta-analysis; the meta-analyzers are leaving information on the table that could be corrected without even looking at the underlying polls, just like they can meta-analyze the polls without looking at data. Fivethirtyeight gives its predictions to a tenth of a percentage; but my first model suggests that their trend estimates are off by at least a half percent just assuming their model to be self-consistent. (That’s obviously not a lot. If you look at the red and blue lines, they’re still almost the same.)
But maybe I’m wrong here. If this were analysis of stock prices, for instance, I’d assume that I was simply not accounting for some tail probability where massive drops are possible but not present in the test data. Maybe being able to predict least-squares error isn’t the same as being able to make money on a bet. (I tried to simulate what I would bet: but I don’t actually know how betting works. I could beat an over-under probability 2/3 of the time.) I’ve done a little hold-one-out testing, but not enough to be sure. I certainly don’t know the right way to correct the problem, although I can make a stab at using linear regression (below). (This an adjustment but not a solution: linear regression on the predictions against 4 days earlier, also suffers the same flaw; another round of regression produces statistically significant corrections in the same direction.)
On the other hand; there are certain things about this kind of predicting that make me think meta-analysts might get have incentives to get it wrong. So much of the commercial point of these things is to be ‘soothing’; wild, random-walk adjustments would not look right. And there’s a heritage here where the prediction lines are expected to have the same gradual slopes and rises as do polling trend averages.
Anyhow, if anyone wants to create that meta-meta-analysis website, I for one promise to refresh it about 6 times a day for the next 4 months. As a first stab, here are some “adjusted” trendlines for the various models out there using the 2016 upshot and fivethirtyeight data; the 2012 fivethirtyeight data; and, as a control, the 2016 electionbettingodds data (which shows some autocorrelation, but no statistically significant trend; my ‘predicted’ line looks basically the same.) The method here is a linear model on logits trained against the current value and the change from 4 days ago.
Code
First the function that creates the model. It’s pretty straightforward: a linear model trained on the change from four days ago. (Most numbers between 1 and 7 are statistically significant for most time series.)
library(jsonlite)
library(dplyr)
library(ggplot2)
library(lubridate)
replot = function(timeseries,normalization = function(d) {qlogis(d)}) {
# Function to normalize a timeseries by have it incorporate information about its own future direction.
# Input a df 'timeseries', which should have two columns: prob and date.
# Use logits for probabilities so random walk hypothesis is reasonable.
as_frame = timeseries %>% mutate(logit=normalization(prob))
# Add 'tomorrow' value to regress against.
ts = as_frame %>% arrange(date) %>% mutate(tomorrow = lead(logit,1))
# Add change in odds terms for each of the past 8 days.
for (i in 1:8) {
ts[[paste0("diff",i)]]=ts$logit-lag(ts$logit,i)
# If 8 days ago doesn't exist, use 7; if seven doesn't exist, use 6; etc.
if (i==1) {
ts[[paste0("diff",i)]][1:i] = 0
} else {
ts[[paste0("diff",i)]][1:i] = ts[[paste0("diff",i-1)]][1:i]
}
}
# I use just 4 days ago as the sole predictor. That seems to work the best on
# this year's upshot and 2012's fivethirtyeight. More terms leads to greater predictive power,
# but seems better to stick to just a single day for simplicity.
# For the same reason, I fix the intercept at zero and the weight
# of today's score at one rather than letting them vary.
odd_model = (lm(tomorrow~ offset(logit) + diff4 + 0,ts))
# Predict from the model, and undo the logit transform.
ts$improved = predict(odd_model,newdata=ts) %>% plogis
plot = ggplot(ts) + geom_line(aes(x=date,y=prob),color='red') + geom_line(aes(x=date,y=improved),color="blue") +
labs(title="old (red) and new (blue) trendlines")
list(plot=plot,ts=ts,model=odd_model)
}Using this function on the upshot model, Hillary’s chances a couple days ago are about two points higher.
# Upshot: to get data, with d3 loaded on the page:
# JSON.stringify(d3.selectAll(".line--dem").data()[0].map(function(d) {return [d.date,d.dem]}))
upshot_ = fromJSON('[["2016-06-01",0.5835],["2016-06-02",0.5892],["2016-06-03",0.5975],["2016-06-04",0.5983],["2016-06-05",0.6079],["2016-06-06",0.6144],["2016-06-07",0.6198],["2016-06-08",0.6259],["2016-06-09",0.6332],["2016-06-10",0.6423],["2016-06-11",0.6505],["2016-06-12",0.6597],["2016-06-13",0.6714],["2016-06-14",0.6745],["2016-06-15",0.6827],["2016-06-16",0.704],["2016-06-17",0.7105],["2016-06-18",0.7164],["2016-06-19",0.7233],["2016-06-20",0.7363],["2016-06-21",0.7473],["2016-06-22",0.7584],["2016-06-23",0.7627],["2016-06-24",0.7638],["2016-06-25",0.7681],["2016-06-26",0.7782],["2016-06-27",0.782],["2016-06-28",0.7708],["2016-06-29",0.7711],["2016-06-30",0.7718],["2016-07-01",0.7693],["2016-07-02",0.7678],["2016-07-03",0.7724],["2016-07-04",0.7737],["2016-07-05",0.7691],["2016-07-06",0.7569],["2016-07-07",0.7525],["2016-07-08",0.7717],["2016-07-09",0.7706],["2016-07-10",0.7611],["2016-07-11",0.766],["2016-07-12",0.7649],["2016-07-13",0.7601],["2016-07-14",0.7513],["2016-07-15",0.7532],["2016-07-16",0.7537],["2016-07-17",0.754],["2016-07-18",0.7545],["2016-07-19",0.7521],["2016-07-20",0.751],["2016-07-21",0.7434],["2016-07-22",0.7436],["2016-07-23",0.7438],["2016-07-24",0.7417],["2016-07-25",0.6887],["2016-07-26",0.6861],["2016-07-27",0.6896],["2016-07-28",0.6862]]')
# I'm working with logits instead of raw percentages. Not much of a difference, but it lets the random walk assumption stand.
upshot = data.frame(date=date(upshot_[,1]),prob = as.numeric(upshot_[,2]))
upshot_improved = replot(upshot)
print(upshot_improved$plot + labs(title="Raw Upshot percents (red)\nand improved predictions (blue)"))
summary(upshot_improved$model)##
## Call:
## lm(formula = tomorrow ~ offset(logit) + diff4 + 0, data = ts)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.252997 -0.008330 0.006242 0.023501 0.123345
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## diff4 0.15830 0.05139 3.081 0.0032 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.04688 on 56 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.9979, Adjusted R-squared: 0.9978
## F-statistic: 2.602e+04 on 1 and 56 DF, p-value: < 2.2e-16Next I load the fivethirtyeight data from 2012. That’s not shown in a block, because it’s huge. It’s easy enough to find in the old nytimes site.
old_38 = data.frame(date=date(five_thirty_eight_2012[,1]),prob = as.numeric(five_thirty_eight_2012[,4])/100)
better38 = old_38 %>% replot
better38$plot + labs(title="Old (red) and 'corrected' (blue) trendlines for 538's forecast from 2012.")
summary(better38$model)##
## Call:
## lm(formula = tomorrow ~ offset(logit) + diff4 + 0, data = ts)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.31892 -0.03869 0.00480 0.05559 0.45021
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## diff4 0.14129 0.03205 4.409 1.91e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09026 on 158 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.9918, Adjusted R-squared: 0.9918
## F-statistic: 1.92e+04 on 1 and 158 DF, p-value: < 2.2e-16I couldn’t quickly find the fivethirty data in their javascript files, so instead I just pulled the svg locations from their plots and translated those back into percentages using the extreme values.
# This creates the JSON below from fivethirtyeight.com
# JSON.stringify(d3.selectAll("g.candidate.D").selectAll("path.line").attr("d").split(/(?=[LMC])/).slice(2).map(function(string) {return string.replace(/L|M/,"").split(",").map(function(d) {return +d})}))
fivethirty_eight_2016 = fromJSON("[[240.32157070841072,86.65400000000001],[235.29742637304327,85.32200000000002],[230.27328203767578,77.1635],[225.2491377023083,78.329],[220.22499336694082,76.44200000000001],[215.20084903157337,73.6115],[210.17670469620586,70.58675],[205.1525603608384,70.892],[200.12841602547095,66.267],[195.10427169010347,64.75925],[190.08012735473602,63.7695],[185.05598301936854,61.66975000000001],[180.03183868400106,62.52074999999999],[175.00769434863358,53.723999999999975],[169.9835500132661,41.75450000000002],[164.95940567789864,42.20774999999999],[159.93526134253116,40.82025],[154.9111170071637,40.82025],[149.88697267179623,40.76475000000003],[144.86282833642878,41.467749999999995],[139.8386840010613,41.56025],[134.81453966569381,41.56025],[129.79039533032633,41.47700000000001],[124.76625099495887,40.54274999999999],[119.7421066595914,40.76475000000003],[114.71796232422392,40.699999999999996],[109.69381798885647,38.4245],[104.66967365348899,36.46350000000001],[99.64552931812153,38.628],[94.62138498275404,38.184000000000005],[89.59724064738657,40.08025],[84.5730963120191,41.190250000000006],[79.54895197665164,41.71749999999999],[74.52480764128417,41.9025],[69.50066330591669,39.81199999999999],[64.47651897054922,40.85725],[59.45237463518174,43.01250000000001],[54.428230299814274,44.631249999999994],[49.4040859644468,47.508],[44.37994162907933,48.32200000000002],[39.35579729371186,49.93150000000003],[34.33165295834439,51.383750000000006],[29.30750862297692,53.99225000000001],[24.283364287609444,53.52050000000002],[19.259219952241974,52.780499999999996],[14.235075616874504,55.44449999999999],[9.210931281507031,57.68299999999999],[4.186786946139559,61.44774999999999],[-0.8373573892279118,62.20625000000001]]")
plottable = data.frame(x=fivethirty_eight_2016[,1],y=-fivethirty_eight_2016[,2])
maxx = max(plottable$y)
minx = min(plottable$y)
translate = function(x) {
# I'm getting SVG coordinates, but why not translate to probabilies
maxx = max(plottable$y)
minx = min(plottable$y)
slope = (80.3-53.2)/(maxx-minx)
#maxx * slope +80.3 = intercept
intercept = 80.3 - slope*maxx
minx * slope + intercept
x*slope+intercept
}
ts = plottable %>% arrange(x) %>% mutate(
prob=translate(y)/100)
ts$date = (today()-nrow(ts)+1):today()
fivethirtyeight_improved = ts %>% replot
fivethirtyeight_improved$plot + labs(title="Old (red) and improved (blue) trendlines for the fivethirtyeight forecast.")
summary(fivethirtyeight_improved$model)##
## Call:
## lm(formula = tomorrow ~ offset(logit) + diff4 + 0, data = ts)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.33510 -0.03166 0.00724 0.03528 0.09652
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## diff4 0.12887 0.05307 2.428 0.019 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.07449 on 47 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.9949, Adjusted R-squared: 0.9948
## F-statistic: 9232 on 1 and 47 DF, p-value: < 2.2e-16#foo = replot(upshot)
#foo = replot(foo$ts %>% mutate(prob=improved))
#summary(foo$model)
#foo$plotFinally, an online aggregation of election odds. This one I’m not hiding because there’s nothing at the
electionbettingodds.com = fromJSON('
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')
as_frame = data.frame(date=date(electionbettingodds.com[,1]),prob = as.numeric(electionbettingodds.com[,2])/100) %>% distinct(date,.keep_all=TRUE)
internet_odds = replot(as_frame)
internet_odds$plot + labs(title="This model isn't able to do anything statistically significant with election betting odds.\n(If it could, I probably should go make some money.")
summary(internet_odds$model)##
## Call:
## lm(formula = tomorrow ~ offset(logit) + diff4 + 0, data = ts)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.201621 -0.019811 -0.000132 0.022523 0.242148
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## diff4 0.03243 0.02715 1.195 0.233
##
## Residual standard error: 0.04299 on 286 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.9951, Adjusted R-squared: 0.9951
## F-statistic: 5.862e+04 on 1 and 286 DF, p-value: < 2.2e-16