Modeling Social Data, Lecture 2: Introduction to Counting

Transcript

1. Introduction to Counting APAM E4990 Modeling Social Data Jake Hofman Columbia University February 1, 2019 Jake Hofman (Columbia University) Intro to Counting February 1, 2019 1 / 30
2. Why counting? Jake Hofman (Columbia University) Intro to Counting February 1, 2019 2 / 30
3. Why counting? http://bit.ly/august2016poll p( y support | x age ) Jake Hofman (Columbia University) Intro to Counting February 1, 2019 3 / 30
4. Why counting? http://bit.ly/ageracepoll2016 p( y support | x1, x2 age, race ) Jake Hofman (Columbia University) Intro to Counting February 1, 2019 3 / 30
5. Why counting? ? p( y support | x1, x2, x3, . . . age, sex, race, party ) Jake Hofman (Columbia University) Intro to Counting February 1, 2019 3 / 30
6. Why counting? How many responses do we need to estimate p(y) with a 5% margin of error? Jake Hofman (Columbia University) Intro to Counting February 1, 2019 4 / 30
7. Why counting? How many responses do we need to estimate p(y) with a 5% margin of error? What if we want to split this up by age, sex, race, and party? Assume ≈ 100 age, 2 sex, 5 race, 3 party Jake Hofman (Columbia University) Intro to Counting February 1, 2019 4 / 30
8. Why counting? Problem: Traditionally difficult to obtain reliable estimates due to small sample sizes or sparsity (e.g., ∼ 100 age × 2 sex × 5 race × 3 party = 3,000 groups, but typical surveys collect ∼ 1,000s of responses) Jake Hofman (Columbia University) Intro to Counting February 1, 2019 5 / 30
9. Why counting? Potential solution: Sacrifice granularity for precision, by binning observations into larger, but fewer, groups (e.g., bin age into a few groups: 18-29, 30-49, 50-64, 65+) Jake Hofman (Columbia University) Intro to Counting February 1, 2019 5 / 30
10. Why counting? Potential solution: Develop more sophisticated methods that generalize well from small samples (e.g., fit a model: support ∼ β0 + β1age + β2age2 + . . .) Jake Hofman (Columbia University) Intro to Counting February 1, 2019 5 / 30
11. Why counting? (Partial) solution: Obtain larger samples through other means, so we can just count and divide to make estimates via relative frequencies (e.g., with ∼ 1M responses, we have 100s per group and can estimate support within a few percentage points) Jake Hofman (Columbia University) Intro to Counting February 1, 2019 6 / 30
12. Why counting? International Journal of Forecasting 31 (2015) 980–991 Contents lists available at ScienceDirect International Journal of Forecasting journal homepage: www.elsevier.com/locate/ijforecast Forecasting elections with non-representative polls Wei Wanga,⇤ , David Rothschildb, Sharad Goelb, Andrew Gelmana,c a Department of Statistics, Columbia University, New York, NY, USA b Microsoft Research, New York, NY, USA c Department of Political Science, Columbia University, New York, NY, USA a r t i c l e i n f o Keywords: Non-representative polling Multilevel regression and poststratification Election forecasting a b s t r a c t Election forecasts have traditionally been based on representative polls, in which randomly sampled individuals are asked who they intend to vote for. While representative polling has historically proven to be quite effective, it comes at considerable costs of time and money. Moreover, as response rates have declined over the past several decades, the statistical benefits of representative sampling have diminished. In this paper, we show that, with proper statistical adjustment, non-representative polls can be used to generate accurate election forecasts, and that this can often be achieved faster and at a lesser expense than traditional survey methods. We demonstrate this approach by creating forecasts from a novel and highly non-representative survey dataset: a series of daily voter intention polls for the 2012 presidential election conducted on the Xbox gaming platform. After adjusting the Xbox responses via multilevel regression and poststratification, we obtain estimates which are in line with the forecasts from leading poll analysts, which were based on aggregating hundreds of traditional polls conducted during the election cycle. We conclude by arguing that non-representative polling shows promise not only for election forecasting, but also for measuring public opinion on a broad range of social, economic and cultural issues. © 2014 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. 1. Introduction At the heart of modern opinion polling is representative sampling, built around the idea that every individual in a The wide-scale adoption of representative polling can be traced largely back to a pivotal polling mishap in the 1936 US presidential election campaign. During that campaign, the popular magazine Literary Digest W. Wang et al. / International Journal of Forecasting 31 (2015) 980–991 981 pollsters, including George Gallup, Archibald Crossley, and Elmo Roper, used considerably smaller but representative samples, and predicted the election outcome with a reasonable level of accuracy (Gosnell, 1937). Accordingly, non-representative or ‘‘convenience sampling’’ rapidly fell out of favor with polling experts. So, why do we revisit this seemingly long-settled case? Two recent trends spur our investigation. First, ran- dom digit dialing (RDD), the standard method in modern representative polling, has suffered increasingly high non-response rates, due both to the general public’s grow- ing reluctance to answer phone surveys, and to expand- ing technical means of screening unsolicited calls (Keeter, Kennedy, Dimock, Best, & Craighill, 2006). By one mea- sure, RDD response rates have decreased from 36% in 1997 to 9% in 2012 (Kohut, Keeter, Doherty, Dimock, & Chris- tian, 2012), and other studies confirm this trend (Holbrook, Krosnick, & Pfent, 2007; Steeh, Kirgis, Cannon, & DeWitt, 2001; Tourangeau & Plewes, 2013). Assuming that the ini- tial pool of targets is representative, such low response rates mean that those who ultimately answer the phone and elect to respond might not be. Even if the selection is- sues are not yet a serious problem for accuracy, as some have argued (Holbrook et al., 2007), the downward trend in response rates suggests an increasing need for post- sampling adjustments; indeed, the adjustment methods we present here should work just as well for surveys ob- tained by probability sampling as for convenience samples. The second trend driving our research is the fact that, with recent technological innovations, it is increasingly conve- nient and cost-effective to collect large numbers of highly non-representative samples via online surveys. The data that took the Literary Digest editors several months to col- lect in 1936 can now take only a few days, and, for some surveys, can cost just pennies per response. However, the challenge is to extract a meaningful signal from these un- conventional samples. In this paper, we show that, with proper statistical ad- justments, non-representative polls are able to yield ac- curate presidential election forecasts, on par with those based on traditional representative polls. We proceed as follows. Section 2 describes the election survey that we conducted on the Xbox gaming platform during the 45 days leading up to the 2012 US presidential race. Our Xbox sample is highly biased in two key demographic dimen- how to transform voter intent into projections of vote share and electoral votes. We conclude in Section 5 by discussing the potential for non-representative polling in other domains. 2. Xbox data Our analysis is based on an opt-in poll which was avail- able continuously on the Xbox gaming platform during the 45 days preceding the 2012 US presidential election. Each day, three to five questions were posted, one of which gauged voter intention via the standard query, ‘‘If the elec- tion were held today, who would you vote for?’’. Full de- tails of the questionnaire are given in the Appendix. The respondents were allowed to answer at most once per day. The first time they participated in an Xbox poll, respon- dents were also asked to provide basic demographic in- formation about themselves, including their sex, race, age, education, state, party ID, political ideology, and who they voted for in the 2008 presidential election. In total, 750,148 interviews were conducted, with 345,858 unique respon- dents – over 30,000 of whom completed five or more polls – making this one of the largest election panel studies ever. Despite the large sample size, the pool of Xbox respon- dents is far from being representative of the voting pop- ulation. Fig. 1 compares the demographic composition of the Xbox participants to that of the general electorate, as estimated via the 2012 national exit poll.1 The most strik- ing differences are for age and sex. As one might expect, young men dominate the Xbox population: 18- to 29-year- olds comprise 65% of the Xbox dataset, compared to 19% in the exit poll; and men make up 93% of the Xbox sam- ple but only 47% of the electorate. Political scientists have long observed that both age and sex are strongly correlated with voting preferences (Kaufmann & Petrocik, 1999), and indeed these discrepancies are apparent in the unadjusted time series of Xbox voter intent shown in Fig. 2. In contrast to estimates based on traditional, representative polls (in- dicated by the dotted blue line in Fig. 2), the uncorrected Xbox sample suggests a landslide victory for Mitt Romney, reminiscent of the infamous Literary Digest error. 3. Estimating voter intent with multilevel regression and poststratification 3.1. Multilevel regression and poststratification http://bit.ly/nonreppoll Jake Hofman (Columbia University) Intro to Counting February 1, 2019 7 / 30
13. Why counting? The good: Shift away from sophisticated statistical methods on small samples to simpler methods on large samples Jake Hofman (Columbia University) Intro to Counting February 1, 2019 8 / 30
14. Why counting? The bad: Even simple methods (e.g., counting) are computationally challenging at large scales (1M is easy, 1B a bit less so, 1T gets interesting) Jake Hofman (Columbia University) Intro to Counting February 1, 2019 8 / 30
15. Why counting? Claim: Solving the counting problem at scale enables you to investigate many interesting questions in the social sciences Jake Hofman (Columbia University) Intro to Counting February 1, 2019 8 / 30
16. Learning to count We’ll focus on counting at small/medium scales on a single machine Jake Hofman (Columbia University) Intro to Counting February 1, 2019 9 / 30
17. Learning to count We’ll focus on counting at small/medium scales on a single machine But the same ideas extend to counting at large scales on many machines (Hadoop, Spark, etc.) Jake Hofman (Columbia University) Intro to Counting February 1, 2019 9 / 30
18. Counting, the easy way Split / Apply / Combine1 • Load dataset into memory • Split: Arrange observations into groups of interest • Apply: Compute distributions and statistics within each group • Combine: Collect results across groups 1http://bit.ly/splitapplycombine Jake Hofman (Columbia University) Intro to Counting February 1, 2019 10 / 30
19. Examples How much time and space do we need to compute per-group averages? Jake Hofman (Columbia University) Intro to Counting February 1, 2019 11 / 30
20. Examples How much time and space do we need to compute per-group averages? What about per-group variances? Jake Hofman (Columbia University) Intro to Counting February 1, 2019 11 / 30
21. The generic group-by operation Split / Apply / Combine for each observation as (group, value): place value in bucket for corresponding group for each group: apply a function over values in bucket output group and result Jake Hofman (Columbia University) Intro to Counting February 1, 2019 12 / 30
22. The generic group-by operation Split / Apply / Combine for each observation as (group, value): place value in bucket for corresponding group for each group: apply a function over values in bucket output group and result Useful for computing arbitrary within-group statistics when we have required memory (e.g., conditional distribution, median, etc.) Jake Hofman (Columbia University) Intro to Counting February 1, 2019 12 / 30
23. Why counting? Jake Hofman (Columbia University) Intro to Counting February 1, 2019 13 / 30
24. Example: Anatomy of the long tail Dataset Users Items Rating levels Observations Movielens 100K 10K 10 10M Netflix 500K 20K 5 100M Jake Hofman (Columbia University) Intro to Counting February 1, 2019 14 / 30
25. Example: Anatomy of the long tail Dataset Users Items Rating levels Observations Movielens 100K 10K 10 10M Netflix 500K 20K 5 100M Jake Hofman (Columbia University) Intro to Counting February 1, 2019 14 / 30
26. Example: Movielens How many ratings are there at each star level? 0 1,000,000 2,000,000 3,000,000 1 2 3 4 5 Rating Number of ratings Jake Hofman (Columbia University) Intro to Counting February 1, 2019 15 / 30
27. Example: Movielens 0 1,000,000 2,000,000 3,000,000 1 2 3 4 5 Rating Number of ratings group by rating value for each group: count # ratings Jake Hofman (Columbia University) Intro to Counting February 1, 2019 16 / 30
28. Example: Movielens What is the distribution of average ratings by movie? 1 2 3 4 5 Mean Rating by Movie Density Jake Hofman (Columbia University) Intro to Counting February 1, 2019 17 / 30
29. Example: Movielens group by movie id for each group: compute average rating 1 2 3 4 5 Mean Rating by Movie Density Jake Hofman (Columbia University) Intro to Counting February 1, 2019 18 / 30
30. Example: Movielens What fraction of ratings are given to the most popular movies? 0% 25% 50% 75% 100% 0 3,000 6,000 9,000 Movie Rank CDF Jake Hofman (Columbia University) Intro to Counting February 1, 2019 19 / 30
31. Example: Movielens 0% 25% 50% 75% 100% 0 3,000 6,000 9,000 Movie Rank CDF group by movie id for each group: count # ratings sort by group size cumulatively sum group sizes Jake Hofman (Columbia University) Intro to Counting February 1, 2019 20 / 30
32. Example: Movielens What is the median rank of each user’s rated movies? 0 2,000 4,000 6,000 8,000 100 10,000 User eccentricity Number of users Jake Hofman (Columbia University) Intro to Counting February 1, 2019 21 / 30
33. Example: Movielens join movie ranks to ratings group by user id for each group: compute median movie rank 0 2,000 4,000 6,000 8,000 100 10,000 User eccentricity Number of users Jake Hofman (Columbia University) Intro to Counting February 1, 2019 22 / 30
34. Example: Anatomy of the long tail Dataset Users Items Rating levels Observations Movielens 100K 10K 10 10M Netflix 500K 20K 5 100M What do we do when the full dataset exceeds available memory? Jake Hofman (Columbia University) Intro to Counting February 1, 2019 23 / 30
35. Example: Anatomy of the long tail Dataset Users Items Rating levels Observations Movielens 100K 10K 10 10M Netflix 500K 20K 5 100M What do we do when the full dataset exceeds available memory? Sampling? Unreliable estimates for rare groups Jake Hofman (Columbia University) Intro to Counting February 1, 2019 23 / 30
36. Example: Anatomy of the long tail Dataset Users Items Rating levels Observations Movielens 100K 10K 10 10M Netflix 500K 20K 5 100M What do we do when the full dataset exceeds available memory? Random access from disk? 1000x more storage, but 1000x slower2 2Numbers every programmer should know Jake Hofman (Columbia University) Intro to Counting February 1, 2019 23 / 30
37. Example: Anatomy of the long tail Dataset Users Items Rating levels Observations Movielens 100K 10K 10 10M Netflix 500K 20K 5 100M What do we do when the full dataset exceeds available memory? Streaming Read data one observation at a time, storing only needed state Jake Hofman (Columbia University) Intro to Counting February 1, 2019 23 / 30
38. The combinable group-by operation Streaming for each observation as (group, value): if new group: initialize result update result for corresponding group as function of existing result and current value for each group: output group and result Jake Hofman (Columbia University) Intro to Counting February 1, 2019 24 / 30
39. The combinable group-by operation Streaming for each observation as (group, value): if new group: initialize result update result for corresponding group as function of existing result and current value for each group: output group and result Useful for computing a subset of within-group statistics with a limited memory footprint (e.g., min, mean, max, variance, etc.) Jake Hofman (Columbia University) Intro to Counting February 1, 2019 24 / 30
40. Example: Movielens 0 1,000,000 2,000,000 3,000,000 1 2 3 4 5 Rating Number of ratings for each rating: counts[movie id]++ Jake Hofman (Columbia University) Intro to Counting February 1, 2019 25 / 30
41. Example: Movielens for each rating: totals[movie id] += rating counts[movie id]++ for each group: totals[movie id] / counts[movie id] 1 2 3 4 5 Mean Rating by Movie Density Jake Hofman (Columbia University) Intro to Counting February 1, 2019 26 / 30
42. Yet another group-by operation Per-group histograms for each observation as (group, value): histogram[group][value]++ for each group: compute result as a function of histogram output group and result Jake Hofman (Columbia University) Intro to Counting February 1, 2019 27 / 30
43. Yet another group-by operation Per-group histograms for each observation as (group, value): histogram[group][value]++ for each group: compute result as a function of histogram output group and result We can recover arbitrary statistics if we can afford to store counts of all distinct values within in each group Jake Hofman (Columbia University) Intro to Counting February 1, 2019 27 / 30
44. The group-by operation For arbitrary input data: Memory Scenario Distributions Statistics N Small dataset Yes General V*G Small distributions Yes General G Small # groups No Combinable V Small # outcomes No No 1 Large # both No No N = total number of observations G = number of distinct groups V = largest number of distinct values within group Jake Hofman (Columbia University) Intro to Counting February 1, 2019 28 / 30
45. Examples (w/ 8GB RAM) Median rating by movie for Netflix N ∼ 100M ratings G ∼ 20K movies V ∼ 10 half-star values V *G ∼ 200K, store per-group histograms for arbitrary statistics (scales to arbitrary N, if you’re patient) Jake Hofman (Columbia University) Intro to Counting February 1, 2019 29 / 30
46. Examples (w/ 8GB RAM) Median rating by video for YouTube N ∼ 10B ratings G ∼ 1B videos V ∼ 10 half-star values V *G ∼ 10B, fails because per-group histograms are too large to store in memory G ∼ 1B, but no (exact) calculation for streaming median Jake Hofman (Columbia University) Intro to Counting February 1, 2019 29 / 30
47. Examples (w/ 8GB RAM) Mean rating by video for YouTube N ∼ 10B ratings G ∼ 1B videos V ∼ 10 half-star values G ∼ 1B, use streaming to compute combinable statistics Jake Hofman (Columbia University) Intro to Counting February 1, 2019 29 / 30
48. The group-by operation For pre-grouped input data: Memory Scenario Distributions Statistics N Small dataset Yes General V*G Small distributions Yes General G Small # groups No Combinable V Small # outcomes Yes General 1 Large # both No Combinable N = total number of observations G = number of distinct groups V = largest number of distinct values within group Jake Hofman (Columbia University) Intro to Counting February 1, 2019 30 / 30