What my Undergraduate Research experience was like in Statistics

I am entering my third and final year of my undergraduate degree.  I have been doing research since almost day 1, and I wanted to share what my experience was like. As a statistician, I feel like I have to mention this is from a sample size of 1 and may not reflect all undergraduate research experiences.

First, I want to give a little background.  The summer before my senior year of high school, I was chosen to participate in an NSF (National Science Foundation) funded REU (Research Experience for Undergraduates)  at Texas Tech.  There I was exposed to what research was like.  We had a series of workshops each led by different researchers over a two week period. I loved the Texas Tech math department and decided to attend Texas Tech for my undergraduate degree. I meet my current research advisor Dr. Ellingson at the REU.

Right after classes started during my freshman year, I decided to email Dr. Ellingson and see if could do research with him.  I started work on image analysis (Dr. Ellingson’s specialty).  I was also following the GOP nomination because it was interesting to me.  I had an idea to predict the nomination using Bayesian statistics, similar to how Five Thirty Eight predicts elections.  I had talked with Dr. Ellingson about political science statistics before and how there was a need for a statistically sound open source academic model.  He agreed to help guide me through the process of building a model to predict the GOP nomination process.

At the time of the GOP nomination my math background was pretty limited, so I decided to just use Baye’s theorem and used the normal distribution to estimate likelihood.  I did all the calculations in excel and I downloaded csv files from Huffington Post Pollster with the poll data.  I used previous voting results from similar states as the prior in my model.  More info about my model can be found here. What I found the most challenging was making a lot decisions about how I was going to predict the election.  I also struggled with making the decisions about the delegate assignments which often involved breaking the results down by congressional districts, even when the poll data was state wide.  After the first Super Tuesday (March 1st) I began to realize that how difficult it is to find a good prior state and reassign support of candidates who dropped out of the race.  The nomination process taught me that failure is inevitable in research, especially in statistics, where everything is at least slightly uncertain.

In the summer of 2016, I started gearing up for the general election. I decided to use Scipy (a python package for science and stats) to make my predictions.  Making the programs was incredibly difficult.  I had over a dozen variations to match different combinations of poll data.  I had the programs up and running by early October, but I discovered a couple of bugs that invalidated my early test predictions.  The original plan was to run the model on the swing states two or three times before the real election. In the middle of October I discovered a bug in one of my programs.  I had to then fix the bug in every program.  I then finally did some manual calculations to confirm the programs worked.  It was difficult to have to admit that my early predictions were totally off, but I am glad I found it before the election.  Research isn’t like a homework assignment with answers in a solution manual.  You don’t know what is exactly going to happen and it is easy to make mistakes.

I ended up writing a paper on my 2016 general election model.  Writing an paper on your research is very different than writing a paper on other peoples research.  My paper was 14 pages (and over 6500 words) long, and only about one or two pages were about what other people’s research on the topic.  It took a very long time to write, and I had 17 drafts.  I hated writing the paper at first, but when I finished it felt amazing. It was definitely worth the effort.

Undergraduate research is difficult, but I loved the entire process.  I got to work with real data to solve a real problem.  I learned how to read a research paper, and eventually I got to write my own.  I got to give presentations to both general audiences and mathematicians and statisticians.  I got to use my research to  inform others about statistics. If you are thinking about doing undergraduate research, you definitely should.

 

Data Sharing

Last semester I took a research ethics class.  I wrote a paper on preregistration and data sharing in academic research. I decided to modify the paper into two blog posts. Here is the first part on data sharing.

Statistics is the study of uncertainty.   Any research study not involving the entire population of group will not be able to provide a definite conclusion with 100% certainty.   Conclusions can be made with a high degree of certainty (95-99%) but false positives and false negatives are inevitable in any large statistical analysis.  This means that studies can fail to make the right call, and after multiple replications the original conclusion may be overturned.

One way to improve the statistical integrity of research is to have a database of the data from non-published studies.  Ideally, this database would be accessible to all academic researchers.   A research would then be able to see the data from other similar studies.   The research would then be able to compare his data with the data from the other studies.  At a significance level of .05,  approximately 1 in 20 studies that were statistically significant were a false positive.    This number applies to theoretically perfect studies that meet all the statistically assumptions used.   Any modelling error increases that rate.  With each external replication of a study the probability of a false positive or a false negative greatly decreases.   Grants from the National Science Foundation1, and the National Institute of Health2 currently require that data from the funded studies be made available to the public after the study was completed.  But not all grants and funding sources require this disclosure.    Without an universal requirement for data disclosure, it can be difficult to confirm that the study and the results are legitimate.

Advocates of open data say that data sharing saves time and reduces false positives and false negatives.  A research can look at previously conducted studies and try to replicate the results.   The results of the data can then be recalculated by another research to confirm accuracy.   In a large study with lots of data it is very easy to make a few mistakes.  These mistakes could cause the results to be misinterpreted.   Open data can even help discover fraudulent studies.  There are methods to estimate the probability the data is fraudulent by looking at the relative frequency of the digits.   The distributions of the digits should be pretty uniform and in one case the data didn’t look quite right.  In 2009, Strategic Vision (a polling company) came under fire from potentially falsifying polls, after a Five Thirty Eight analysis3  discovered that something didn’t look quite right.  This isn’t an academic example, but open access data could prevent fraudulent studies from being accepted as fact as in the infamous vaccines cause autism study.  The statistical analysis of the randomness isn’t definite, but they can raise questions that prompt further investigations of the data.   Open data makes replication easier. False positives and false negatives can cause harm in some cases.  Easier replication can help confirm findings quicker.

 

Works Cited

[1] Public Access To the Results of NSF-Funded Research. (n.d.). Retrieved April 28, 2017, from https://www.nsf.gov/news/special_reports/public_access/

[2] NIH’s Commitment to Public Accountability. (n.d.). Retrieved April 28, 2017, from https://grants.nih.gov/grants/public_accountability/

 

[3] Silver, N. (2014, May 07). Strategic Vision Polls Exhibit Unusual Patterns, Possibly Indicating Fraud. Retrieved April 28, 2017, from https://fivethirtyeight.com/features/strategic-vision-polls-exhibit-unusual/

My New Project

Update 09/23/17:  I am switching to two proportion Z tests.  I am setting the population proportion to .5 to prevent an underestimation of variance.

 

This is a bit of a technical post,  I will have a better explanation later.

Post election, I have been working on a paper and thinking about what to do next.  I am really interested in breaking down voter behavior in the swing states. I have collected exit poll data from the 11 swing states.  I want to test if voter behavior across the swing states was consistent with the national vote or the swing state average.

For phase 1 of this experiment, I will run Chi-Square Test of Homogeneity between a swing state compared to the average of the other swing state and the national vote.  I will look at each category four different ways: Trump vs. not Trump,  Clinton vs. not Clinton, Other vs Clinton and Turmp, and overall.  This will probably be around 1500 tests.   I will have an initial alpha level of 0.05.  I will then run a two proportion z-tests on the tests were the p value was less than 0.05.  I will do the z-tests on the direction that matches the data.

For phase 2, I will collect data from 2008 and 2012 in states that have a statistically significant portion of significant tests.  Then I will compare voting behavior with Chi-Square Test of Homogeneity on: 2008 vs 2012,  2008 vs 2016, and 2012 vs 2016.  Then significant results will be tested using a two proportion z-test.

I am going with the Chi-Square test first for two reasons.  The Chi-Square test is not subject to errors in the direction of an effect, and the Chi-Square test is less sensitive than a two proportion z-test.  I have to be very careful in my interpretation of the results since an analysis this large means that there is a big  potential for false positives and false negatives. This analysis will probably take me most of next year. I’ll give an update on my progress in December.

My Comments on the Special Elections in 2017

I thought I would provide my perspective on the special election that occurred last week in Kansas, and the upcoming special house election in Georgia.

For full disclosure, I am a republican who is against some of the President’s policies on immigration, and health care.

I do not think Trump’s performance will have a major affect the voting behavior of people with strong party ties.  Republicans vote Republican most of the time, and Democrats vote Democrat most of the time.  Independents and moderates are more of a wild card.  Independents may not vote the same as they did in 2016.

The districts in question are in no way representative of the whole country. They  Any result from these elections cannot be applied to the whole country or  “predict” the entire midterm election outcome.  You could maybe use the results to for certain districts, but certainly not the entire country.  For statistical analysis to work properly, the samples need to be reasonably representative.

Special elections are all about who turns out.  In the Kansas election, Democrats spent a lot of money and attention on the race since there are only a few races this year.   The money and a lack of an incumbent is probably why the race was closer than the 2016 race.  The 2017 Kansas race had about half the votes compared to the 2016 race,  this big of a change can affect the outcome.  In Georgia, I expect a race  that is closer than usual for that district, but still with a Republican win. I doubt that a Democrat will win a majority of the votes in the primary.

These special elections need to be interpreted in context.  They are two races in House districts that haven’t been competitive in years.  We should not even try to extrapolate to the entire country from these races.  Favorability polls are a much better indicator of political sentiment  However, I think that the favorability polls like the general election polls could be underestimating Trump’s support.  It has been difficult to get Republicans to respond to the polls, and this may affect the accuracy of polls.  After the midterms in 2018,  there will be a clearer picture of support for the Republican party.  Until then we can only guess.

 

 

We Don’t Live in Statsland

Statsland is a magical world that exists only in (certain) Statistics textbooks. In Statsland,  statistics is easy.  We can invoke Central Limit theorem and use the normal distribution when n is larger than 30.   In Statsland we either know or can easily determine the correct distribution.  In Statsland 95% confidence intervals have a 95% chance of containing the real value.  But we don’t live in Statsland.

The point of doing statistics is that it would be too difficult (or impossible) to find the true value of a population.  You aren’t likely to find  the exact value, but you can be pretty close.   In a statistics textbook problem, you probably have enough information to do a good job of estimating the desired value. But in applied statistics you may not have as much information.  If you know the mean and standard deviation of a population you do not need to do much (if any) statistics.  Any time you have to estimate or substitute information, your model will not perform as well as a theoretically perfect model.

Statistics never was and never will be an exact science.   In most cases, your model will be wrong.  There are no perfect answers.  Your confidence intervals will rarely perform as they theoretically should.  The requisite sample size to invoke Central Limit Theorem is not clear cut.  Your approach should vary on the individual problem.   There is no universal formula to examine data.   Applied Statistics should be flexible and instead of rigid.   The world is not a statistics textbook problem, and should never be treated as such.

 

A Non-Technical Overview of My Research

Recently I have been writing up a draft of a research article on my general election model to submit for academic publication.  But that paper is technical and requires you to have some exposure to statistical research to understand.  I wanted to explain my research without going into all the technical details.

Introduction

The President of the United States is elected every four years.  The Electoral College decides the winner,  by the votes of electors chosen by their home state.  Usually the electors are chosen based on the winner of that state and they vote for the winner of that state. Nate Silver correctly predicted the winner of the 2008 election with Bayesian statistics.  Silver got 49 out of 50 states correct.   Silver certainly wasn’t the first person to predict the election, but he received a lot of attention for his model.   Silver’s runs Five Thirty Eight  which talks about statistics and current events.  Bayesian statistics is a branch of statistics that uses information you already know (called a prior) and adjusts the model as more information comes in.  My model like Nate Silver’s used  Bayesian statistics. We do not know the details of the Silver model, besides that it used Bayesian statistics.  To the best of my knowledge, my method is the first publicly available model that used poll data from other states as the prior.  A prediction was made for 2016, where I correctly predicted 6 states.  Then the model was applied to 2008 and 2012, where my prediction of state winners matched the prediction of Five Thirty Eight.

Methodology

I took poll data from Pollster, which provided me csv files for the 2016 and 2012 election.  For 2008 I had to create the csvs by hand.  I had a series of computer programs in Python (a common programming language) to analyze.  My model, used the normal distribution.  My approach divided the 50 states into 5 regional categories: swing states,  southern red states,  midwestern red states, northern blue states,  and western blue states.  The poll data source used as the prior were National,  Texas,  Nebraska,  New York, and California respectively.  This approach is currently believed to be unique, but since multiple models are proprietary it is unknown if this has been used before.  I only used polls if they were added to pollster before the Saturday before election date.   For the 2016 election analysis this meant November 5th.  I posted my predictions on November 5th.

I outline more of my method here.

Results and Discussion

My model worked pretty well compared to other models.  Below is a table of other models and their success rate at predicting the winning candidate in all 50 states plus (and Washington D.C.).

Race Real Clear Politics Princeton Election Consortium Five Thirty Eight (Polls Plus) PredictWise (Fundamental) Sabato’s Crystal Ball My Model
2008 Winner Accuracy 0.96078 0.98039 0.98039  N/A 1 0.98039
2012 Winner Accuracy 0.98039 0.98039 1 0.98039 0.96078 1
2016 Winner Accuracy 0.92157 0.90196 0.90196 0.90196 0.90196 0.88235
Average Accuracy 0.95425 0.95425 0.96078 0.94118 0.95425 0.95425

As you can see all the models do a similar job at picking the winner in each state, which predicts the electoral college.  There are other ways to compare accuracy, but I don’t want to discuss this here since it gets a little technical.   No one was right for every state in every election.  It would probably be impossible to create a model that would consistently predict the winner in all states, because of the variability of political opinions.   Election prediction is not an exact science.  But there is the potential to apply polling analysis to estimate public opinion on certain issues and politicians.  Right now the errors in polls are too large determine public opinion on close issues.   But further research could determine ways to reduce error in polling analysis.

Only You can Prevent Bad Political Polls

My research relies heavily on polls.  So I understand why it is important to do polls.   If I see a poll and determine it’s well written, I do it.  But I think this position is rare because people don’t know the importance of polls. I want to explain why I think polls are important.   Pre-election polls are commonly used to predict elections, and favorability polls are often used to judge a politician’s  popularity. Polls are an important part of American politics.

I get that polls are annoying.  I know it takes time and you are probably busy (like me).  But doing 1 political poll a year can greatly help improve the accuracy of polls.   You don’t have to always answer a poll, but increased participation in polls improves accuracy.   Now there are a lot of bad polls, and it’s difficult to tell if a phone poll is good based of the phone number.  Some people have “polls” that really are marketing calls.  I understand if you are hesitant to do phone polls.  But internet polling provides a good alternative.  I think the future of polling is quality internet polls.  When you do an good internet poll you know more about the quality of the poll then a poll phone call. But Internet polls from scientific polling agencies require a large base of people to create accurate samples.  You can randomly call 1000 phones, but you really can’t send 1000 random internet users a poll. To combat this problem polling agencies have databases of users to send polls. Polling agencies send surveys to certain users to create a good sample. Joining a survey panel with political polls is a way to get your voice heard.

My view on participating in political polls is you can’t complain if you don’t participate.  Polls need a diverse sample to be accurate.  If you feel your political stance is not heard in the polls, then you should do more polls instead of less.  We need all kinds of people to do good polls.  Not everyone may have internet access, but enough voters do to create a good sample.  What you can do is join a poll panel.  My two recommendations are https://today.yougov.com/ or https://www.i-say.com/.  They also do non-political polls and market research which are also important (I might do a post later on this). I recommend them because they are user friendly and statistically sound.  I am not receiving anything for recommending these agencies, I just think they are good.

If you want polls to be more accurate, the best (and easiest) thing to do is participate in polls.  As a statistician, I value good data.  But for data to be good it needs a representative sample.  Regardless of your politics, you should participate in political polls.

 

A look at Alternatives to the Current Electoral College Process

First, I want to be clear, that there is no universally fair way to elect a president. All methods have pros and cons, and you can have your own opinion about which way is the best.

Current System

Right now with the exception of Nebraska and Maine, the electoral college is decided by whoever has the most support in a state.  The winner usually has a majority of votes, but sometimes no single candidate was a majority. This method also helps smaller states as they have a lower ratio of voters to electors than larger states.

Pros

This method makes it easy to determine the winner on election night.  You don’t necessarily need all the votes to come in if you have enough information to predict the winner.

Cons

Most states have a clear winner party.  So most of the attention goes to swing states who do not have a regular winner.

Popular Vote

The popular vote method is based on the winner of the popular vote.  Whoever gets the most votes wins.  This method can be implemented if enough states change their laws to award their electors to the popular vote winner.

Pros

Every vote counts the same.  Larger states would have more power than the current system.

Cons

Smaller states lose some electoral power compared to the current system.

Congressional District System

This system awards 2 electors to the state winner and 1 elector to the winner of every congressional district.  This is the method Maine and Nebraska use.

Disclaimer:  This is my personally prefered system.

Pros

It’s a compromise between the current system and the popular vote system.  The electoral college would probably mimic the congressional makeup.

Cons

Like the current system,  could elect a president that didn’t win the popular vote.

 

All of these systems have pros and cons.  There isn’t necessarily a “best” way to pick the president.

Here is a Five Thirty Eight article about different methods of deciding the electoral college.

 

If I were a Senator

Donald Trump is officially the 45th president of the United States. Next the cabinet nominees will be voted on by the senate for confirmation. Republicans have a majority, but it would take only three Republican senators to prevent the appointment of a nominee. Technically a Democrat might vote for a nominee, but considering how many Democrats aren’t participating in the inauguration no Democrats will probably vote yes on the more controversial nominees. The question is if you were a senator that doesn’t like Trump or a certain nominee should you vote for them anyway to protect your position in the senate?

This is a complicated decision given what we know about Trump’s low favorability rating. A YouGov/Economist poll asks questions about how voters view Trump and his cabinet picks. Some picks aren’t as controversial like mates who has the highest favorability among non-Trump voters in the poll. But the Secretary of State nominee Rex Tillerson and Attorney General nominee Jeff Session have the lowest favorability among non-Trump voters. You have to consider that the majority of voters didn’t vote for Trump in the election, and the majority of voters have neutral or negative opinions in most polls. This decision is difficult if you don’t like the nominees, but as a Republican senator feel obligated to support your party.

Here is what I would do if I were a Republican Senator. I don’t think most of the cabinet picks are qualified or good candidates for their positions. I know that independent and democratic and a portion of Republican voters don’t like some of the cabinet picks. Not voting for a nominee would hurt me, it would probably anger my colleagues and lower my favorability with my constituents. Not voting for my party’s nominee would probably make national news and may not be beneficial for me. However, if I vote against a nominee and it turns out they don’t get confirmed, it probably wouldn’t hurt me that much. If I run for reelection and Trump and his cabinet are unpopular my dissent could help. If I don’t vote for a cabinet candidate, but they still get confirmed I would have risked my position for nothing. This scenario is complicated and an example of a prisoner’s dilemma game (more info here). The idea in this case is voting against a candidate is only worth it if it blocks the confirmation of that candidate and it turns out that Trump is not favorable at the time of my reelection. But the payoff is higher if I vote for the nominee regardless of the actions of the 51 other Republican senators. I also know that the rejection of this nominee doesn’t mean that the next nominee going to be a better nominee. Knowing these things the only nominees that I would consider voting against being Sessions and Tillerson because those are high power positions and are unpopular enough to increase the chance that my vote would prevent their confirmation. So it wouldn’t surprise me if almost all the senate nominees get confirmed.

Coincidences: A Lesson in Expected Value

As I followed the election I noticed the frequent mentions counties (or cities) that have been known “predict” the presidential election winner. The idea is that a the winner of a certain county has matched the winner of the election for multiple elections. Let’s look at county A for an example. To simplify things lets assume the odds of predicting a winner in a presidential election are 50-50. This would mean that the probability of getting 8 elections right would be 1 in 256. This means that it is unlikely that county A would predict the election by chance. But what about the rest of the counties in America? There are over 3,000 counties in America (according to an economist article found here: http://www.economist.com/blogs/economist-explains/2016/11/economist-explains), so we can expect about 12 of these counties would have “predicted” the winner of the presidential election.

Rare events happen all the time. Rare are rare, but rare is not impossible. Let’s say that there is a (hypothetical) free sweepstakes with a 1 in 100 chance of winning $100. You may not think that you wouldn’t know anyone who won, but if a sweepstakes like this exists you might be surprised about the likely outcome. It may not be likely that you specifically win, but if all your Facebook friends enter the contest someone you know is probably going to win. If you have at least 99 Facebook friends it is likely that you or someone you know will win the sweepstakes. You may think its a coincidence or luck, but it is really math. Expected value can’t tell you who is going to win, but it can tell you someone you know is likely to win. Now expected value is not a magic bullet. You may have 0 friends win or 2 friends win, but the most likely event is that someone will win. Unfortunately (legit) sweepstakes like this don’t exist, but it is a good example of how your perception of probability may not match reality. Another example is it probably going to rain 1 in 10 days where the probability of rain is 10%, but it is easy to pretend like it never rains when the probability of rain is 10%.

You may wonder why expected value matters. But it’s actually quite important when looking at everyday events. Sometimes it is easy to underestimate the chance that something odd or rare would happen. You may think it’s odd that runs when the meteorologist says the chance of that happening is 10%. Or that it only takes 23 people to have a 50% chance of there being, two people with the same birthday (details here). It is easy to forget that once in a lifetime event do happen once in a lifetime. How you think about probability is important. So before you yell at the TV meteorologist that said there was a 10% chance of rain but it rained, try to remember that unlikely does not equal impossible.