The Data Defect Index for Samples that May not be IID.
d.d.i. (Data Defect Index) for non i.i.d. Samples
A simple set of functions to implement the Data Defect Index (d.d.i.), described in:
Xiao-Li Meng. 2018. “Statistical Paradises and Paradoxes in big data (I): Law of Large Populations, Big Data Paradox, and the 2016 US Presidential Election.” Annals of Applied Statistics 12:2, 685–726. doi:10.1214/18-AOAS1161SF.
Install
# install.packages("devtools")
remotes::install_github("kuriwaki/ddi")
Usage
With a dataframe with columns for a group’s estimates and components of the formula, ddc computes the data defect correlation (ρ).
An example dataset from the 2016 US Presidential Election is included (this also serves as the replication dataset for the AOAS article). The dataset compares official election results with estimates the Cooperative Congressional Election Study (CCES), the largest political survey in the US. The CCES micro-data is fully public and accessible at its website. Here, we produce state-level estimates which are documented with help(g2016).
library(ddi)
library(tidyverse)
data(g2016)
g2016
## # A tibble: 51 x 10
## state st pct_djt_voters cces_pct_djt_vv cces_pct_djtrun… votes_djt
## <chr> <chr> <dbl> <dbl> <dbl> <dbl>
## 1 Alab… AL 0.621 0.408 0.428 1318255
## 2 Alas… AK 0.513 0.306 0.319 163387
## 3 Ariz… AZ 0.487 0.423 0.445 1252401
## 4 Arka… AR 0.606 0.416 0.434 684872
## 5 Cali… CA 0.316 0.285 0.305 4483810
## 6 Colo… CO 0.433 0.350 0.371 1202484
## 7 Conn… CT 0.409 0.294 0.318 673215
## 8 Dela… DE 0.419 0.329 0.349 185127
## 9 Dist… DC 0.0409 0.0575 0.0690 12723
## 10 Flor… FL 0.490 0.403 0.422 4617886
## # … with 41 more rows, and 4 more variables: tot_votes <dbl>, cces_n_vv <dbl>,
## # vap <dbl>, vep <dbl>
We can compute the data defect correlation just by plugging in some numbers. For example
ddc(mu = 62984824/136639786, muhat = 12284/35829, N = 136639786, n = 35829)
## [1] -0.003837163
and the d.d.i. is the square of that, about 0.0000147.
we got these numbers by
select(g2016, cces_pct_djt_vv, cces_n_vv, tot_votes, votes_djt) %>%
summarize_all(sum)
## # A tibble: 1 x 4
## cces_pct_djt_vv cces_n_vv tot_votes votes_djt
## <dbl> <dbl> <dbl> <dbl>
## 1 17.5 35829 136639786 62984824
where
cces_totdjt_vv: The count of Trump voters (among validated voters)cces_n_vv: The count of CCES validated voters (sample size)votes_djt: Total votes for Trumptot_votes: Total turnoutcces_pct_djt_vv: Estimated vote share,cces_totdjt_vv / cces_n_vvpct_djt_voters: Estimated vote share,votes_djt / tot_votes
The function also takes vectors as inputs:
with(g2016, ddc(mu = pct_djt_voters,
muhat = cces_pct_djt_vv,
N = tot_votes,
n = cces_n_vv))
## [1] -0.0059541279 -0.0062341071 -0.0023488019 -0.0061097707 -0.0009864919
## [6] -0.0025746344 -0.0035362241 -0.0033951165 0.0014015382 -0.0029747918
## [11] -0.0038228152 -0.0001757426 -0.0073716139 -0.0036437192 -0.0069956521
## [16] -0.0058255411 -0.0059093759 -0.0057837854 -0.0040533230 -0.0047893714
## [21] -0.0024905368 -0.0028280876 -0.0050296619 -0.0043292576 -0.0056626724
## [26] -0.0069305025 -0.0046563153 -0.0075840944 -0.0047785897 -0.0037497506
## [31] -0.0028289070 -0.0025619899 -0.0031936586 -0.0051968951 -0.0078308914
## [36] -0.0057088185 -0.0065654840 -0.0030642004 -0.0039137353 -0.0039907269
## [41] -0.0040871158 -0.0069019981 -0.0050741833 -0.0044884762 -0.0059634270
## [46] -0.0034491625 -0.0040918085 -0.0024121681 -0.0075404659 -0.0051378753
## [51] -0.0086086072
so can be implemented in a tibble as well:
transmute(g2016, st,
ddc = ddc(mu = pct_djt_voters,
muhat = cces_pct_djt_vv,
N = tot_votes,
n = cces_n_vv))
## # A tibble: 51 x 2
## st ddc
## <chr> <dbl>
## 1 AL -0.00595
## 2 AK -0.00623
## 3 AZ -0.00235
## 4 AR -0.00611
## 5 CA -0.000986
## 6 CO -0.00257
## 7 CT -0.00354
## 8 DE -0.00340
## 9 DC 0.00140
## 10 FL -0.00297
## # … with 41 more rows
A negative ρ means ρ = Cor(Respond, 1(Trump Supporter)) < 0, i.e. Trump supporters were less likely to respond.