Forecasting large collections of time series

With the recent launch of Amazon Forecast, I can no longer procrastinate writing about forecasting "at scale"!

Quantitative forecasting of time series has been used (and taught) for decades, with applications in many areas of business such as demand forecasting, sales forecasting, and financial forecasting. The types of methods taught in forecasting courses tends to be discipline-specific:

• Statisticians love ARIMA (auto regressive integrated moving average) models, with multivariate versions such as Vector ARIMA, as well as state space models and non-parametric methods such as STL decompositions.
• Econometricians and finance academics go one step further into ARIMA variations such as ARFIMA (f=fractional), ARCH (autoregressive conditional heteroskedasticity), GARCH (g=general), NAGARCH (n=nonlinear, a=asymmetric), and plenty more
• Electrical engineers use spectral analysis (the equivalent of ARIMA in the frequency domain)
• Machine learning researchers use neural nets and other algorithms

In practice, it is common to see 3 types of methods being used by companies for forecasting future values of a time series : exponential smoothing, linear regression, and sometimes ARIMA. 

Image from https://itnext.io

Why the difference? Because the goal is different! Statistical models such as ARIMA and all its econ flavors are often used for parameter estimation or statistical inference. Those are descriptive goals  (e.g., "is this series a random walk?", "what is the volatility of the errors?"). The spectral approach by electrical engineers is often used for the descriptive goals of characterizing a series' frequencies (signal processing), or for anomaly detection. In contrast, the business applications are strictly predictive: they want forecasts of future values. The simplest methods in terms of ease-of-use, computation, software availability, and understanding, are linear regression models and exponential smoothing. And those methods provide sufficiently accurate forecasts in many applications - hence their popularity! 

ML algorithms are in line with a predictive goal, aimed solely at forecasting. ARIMA and state space models can also be used for forecasting (albeit using a different modeling process than for a descriptive goal). The reason ARIMA is commonly used in practice, in my opinion, is due to the availability of automated functions.

For cases with a small number of time series to forecast (a typical case in many businesses), it is usually worthwhile investing time in properly modeling and evaluating each series individually in order to arrive at the simplest solution that provides the required level of accuracy. Data scientists are sometimes over-eager to improve accuracy beyond what is practically needed, optimizing measures such as RMSE, while the actual impact is measured in a completely different way that depends on how those forecasts are used for decision making. For example, forecasting demand has completely different implications for over- vs. under-forecasting; Users might be more averse to certain directions or magnitudes of error.

But what to do when you must forecast a large collection of time series? Perhaps on a frequent basis? This is "big data" in the world of time series. Amazon predict shipping time for each shipment using different shipping methods to determine the best shipping method (optimized with other shipments taking place at the same/nearby time); Uber forecasts ETA for each trip; Google Trends generates forecasts for any keyword a user types in near-realtime. And... IoT applications call for forecasts for time series from each of their huge number of devices. These applications obviously cannot invest time and effort into building handmade solutions. In such cases, automated forecasting is a practical solution. A good "big data" forecasting solution should

• be flexible to capture a wide range of time series patterns 
• be computationally efficient and scalable
• be adaptable to changes in patterns that occur over time
• provide sufficient forecasting accuracy

In my course "Business Anlaytics Using Forecasting" at NTHU this year, teams have experienced trying to forecast hundreds of series from a company we're collaborating with. They used various approaches and tools. The excellent forecast package in R by Rob Hyndman's team includes automated functions for ARIMA (auto.arima), exponential smoothing (ets), and a single-layer neural net (nnetar). Facebook's prophet algorithm (and R package) runs a linear regression. Some of these methods are computationally heavier (e.g. ARIMA) so implementation matters. 

While everyone gets excited about complex methods, in time series so far evidence is that "simple is king": naive forecasts are often hard to beat! In the recent M4 forecasting contest (with 100,000 series), what seemed to work well were combinations (ensembles) of standard forecasting methods such as exponential smoothing and ARIMA combined using a machine learning method for the ensemble weights. Machine learning algorithms were far inferior. The secret sauce is ensembles.

Because simple methods often work well, it is well worth identifying which series really do require more than a naive forecast. How about segmenting the time series into groups? Methods that first fit models to each series and then cluster the estimates are one way to go (although can be too time consuming for some applications). The ABC-XYZ approach takes a different approach: it divides a large set of time series into 4 types, based on the difficulty of forecasting (easy/hard) and magnitude of values (high/low) that can be indicative of their importance.

Forecasting is experiencing a new "split personality" phase, of small-scale tailored forecasting applications that integrate domain knowledge vs. large-scale applications that rely on automated "mass-production" forecasting. My prediction is that these two types of problems will continue to survive and thrive, requiring different types of modeling and different skills by the modelers.

For more on forecasting methods, the process of forecasting, and evaluating forecasting solutions see Practical Time Series Forecasting: A Hands-On Guide and the accompanying YouTube videos. 

Election polls: description vs. prediction

My papers To Explain or To Predict and Predictive Analytics in Information Systems Research contrast the process and uses of predictive modeling and causal-explanatory modeling. I briefly mentioned there a third type of modeling: descriptive. However, I haven't expanded on how descriptive modeling differs from the other two types (causal explanation and prediction). While descriptive and predictive modeling both share the reliance on correlations, whereas explanatory modeling relies on causality, the former two are in fact different. Descriptive modeling aims to give a parsimonious statistical representation of a distribution or relationship, whereas predictive modeling aims at generating values for new/future observations.

The recent paper Election Polls—A Survey, A Critique, and Proposals by Kenett, Pfeffermann & Steinberg gives a fantastic illustration of the difference between description and prediction: they explain the different goals of election surveys (such as those conducted by Gallup) as compared to survey-based predictive models such as those by Nate Silver's FiveThirtyEight:

"There is a subtle, but important, difference between reflecting current public sentiment and predicting the results of an election. Surveys [election polls] have focused largely on the former—in other words, on providing a current snapshot of voting preferences, even when asking about voting preference as if elections were carried out on the day of the survey. In that regard, high information quality (InfoQ) surveys are accurately describing current opinions of the electorate. However, the public perception is often focused on projecting the survey results forward in time to election day, which is eventually used to evaluate the performance of election surveys. Moreover, the public often focuses solely on whether the polls got the winner right and not on whether the predicted vote shares were close to the true results."

In other words, whereas the goal of election surveys is to capture the public perception at different points in time prior to the election, they are often judged by the public as failed because of low predictive power on elections day. The authors continue to say:

"Providing an accurate current picture and predicting the ultimate winner are not contradictory goals. As the election approaches, survey results are expected to increasingly point toward the eventual election outcome, and it is natural that the success or failure of the survey methodology and execution is judged by comparing the final polls and trends with the actual election results."

Descriptive models differ from predictive models in another sense that can lead to vastly different results: in a descriptive model for an event of interest we can use the past and the future relative to that event time. For example, to describe spikes in pre-Xmas shopping volume we can use data on pre- and post-Xmas days. In contrast, to predict pre-Xmas shopping volume we can only use information available prior to the pre-Xmas shopping period of interest.

As Kenett et al. (2017) write, description and prediction are not contradictory. They are different, yet the results of descriptive models can provide leads for strong predictors, and potentially for explanatory variables (which require further investigation using explanatory modeling).

The new interesting book Everybody Lies by Stephens-Davidowitz is a great example of descriptive modeling that uncovers correlations that might be used for prediction (or even for future explanatory work). The author uncovers behavioral search patterns on Google by examining keywords search volumes using Google Trends and AdWords. For the recent US elections, the author identifies a specific keyword search term that separates between areas of high performance for Clinton vs. Trump:

"Silver noticed that the areas where Trump performed best made for an odd map. Trump performed well in parts of the Northeast and industrial Midwest, as well as the South. He performed notably worse out West. Silver looked for variables to try to explain this map... Silver found that the single factor that best correlated with Donald Trump's support in the Republican primaries was... [areas] that made the most Google searches for ______."

[I am intentionally leaving the actual keyword blank because it is offensive.]
While finding correlations is a dangerous game that can lead to many false discoveries (two measurements can be correlated because they are both affected by something else, such as weather), careful descriptive modeling tightly coupled with domain expertise can be useful for exploratory research, which later should be tested using explanatory modeling.

While I love Stephens-Davidowitz' idea of using Google Trends to uncover behaviors/thoughts/feelings that are hidden otherwise (because what people say in surveys often diverges from what they really do/think/feel), a main question is who do these search results represent? (sampling bias). But that's a different topic altogether.

Key challenges in online experiments: where are the statisticians?

Randomized experiments (or randomized controlled trials, RCT) are a powerful tool for testing causal relationships. Their main principle is random assignment, where subjects or items are assigned randomly to one of the experimental conditions. A classic example is a clinical trial with one or more treatment groups and a no-treatment (control) group, where individuals are assigned at random to one of these groups.

Story 1: (Internet) experiments in industry 

Internet experiments have now become a major activity in giant companies such as Amazon, Google, and Microsoft, in smaller web-based companies, and among academic researchers in management and the social sciences. The buzzword "A/B Testing" refers to the most common and simplest design which includes two groups (A and B), where subjects -- typically users -- are assigned at random to group A or B, and an effect of interest is measured. A/B tests are used for testing anything from the effect of a new website feature on engagement to the effect of a new language translation algorithm on user satisfaction. Companies run lots of experiments all the time. With a large and active user-base, you can run an internet experiment very quickly and quite cheaply. Academic researchers are now also beginning to use large scale randomized experiments to test scientific hypotheses about social and human behavior (as we did in One-Way Mirrors in Online Dating: A Randomized Field Experiment).

Based on our experience in this domain and on what I learned from colleagues and past-students working in such environments, there are multiple critical issues challenging the ability to draw valid conclusions from internet experiments. Here are three:

1. Contaminated data: Companies constantly conduct online experiments introducing interventions of different types (such as running various promotions, changing website features, and switching underlying technologies). The result is that we never have "clean data" to run an experiment, and we don't know how they are dirty. The data are always somewhat contaminated by other experiments that are taking place in parallel, and in many cases we do not even know which or when such experiments have taken place.
2. Spill-over effects: in a randomized experiment we assume that each observation/user experiences only one treatment (or control). However, in experiments that involve an intervention such as knowledge sharing (eg, the treatment group receives information about a new service while the control group does not), the treatment might "spill over" to control group members through social networks, online forums, and other information-sharing platforms that are now common. For example, many researchers use Amazon Mechanical Turk to conduct experiments, where, as DynamoWiki describes, "workers" (the experiment subjects) share information, establish norms, and build community through platforms like CloudMeBaby, MTurk Crowd, mTurk Forum, mTurk Grind, Reddit's /r/mturk and /r/HITsWorthTurkingFor, Turker Nation, and Turkopticon. This means that the control group can be "contaminated" by the treatment effect.
3. Gift effect: Treatments that benefit the treated subjects in some way (such as a special promotion or advanced feature) can confuse the effect of the treatment with the effect of receiving a special treatment. In other words, the difference between the outcome for the treatment and control groups might be not due to the treatment per-se but rather due to the "special attention" the treatment group received by the company or researcher.

Story 2: Statistical discipline of Experimental Design 

Design of Experiments (DOE or DOX) is a subfield of statistics that is focused on creating the most efficient designs for an experiment, and the most appropriate analysis. Efficient here refers to a context where each run is very expensive or resource consuming in some way. Hence, the goal in DOE methodology is to answer the causal questions of interest with the smallest number of runs (observations). The statistical methodological development of DOE was motivated by agricultural applications in the early 20th century, led by the famous Ronald Fisher. DOE methodology gained further momentum in the context of industrial experiments (today it is typically considered part of "industrial statistics").  Currently, the most active research area within DOE is "computer experiments" that is focused on constructing simulations to emulate a physical system for cases where experimentation is impossible, impractical, or terribly expensive (e.g. experimenting on climate).

Do the two stories converge? 

With the current heavy use of online experiments by companies, one would have thought the DOE discipline would flourish: new research problems, plenty of demand from industry for collaboration, troves of new students. Yet, I hear that the number of DOE researchers in US universities is shrinking. Most "business analytics" or "data science" programs do not have a dedicated course on experimental design (with focus on internet experiments). Recent DOE papers in top industrial statistics journals (eg Technometrics) and DOE conferences indicate the burning topics from Story 1 are missing. Academic DOE research by statisticians seems to continue focusing on the scarce data context and on experiments on "things" rather than human subjects. The Wikipedia page on DOE also tells a similar story. I tried to make these points and others in my recent paper Analyzing Behavioral Big Data: Methodological, practical, ethical, and moral issues. Hopefully the paper and this post will encourage DOE researchers to address such burning issues and take the driver's seat in creating designs and analyses for researchers and companies conducting  "big experiments".

Experimenting with quantified self: two months hooked up to a fitness band

It's one thing to collect and analyze behavioral big data (BBD) and another to understand what it means to be the subject of that data. To really understand. Yes, we're all aware that our social network accounts and IoT devices share our private information with large and small companies and other organizations. And although we complain about our privacy, we are forgiving about sharing it, most likely because we really appreciate the benefits.

So, I decided to check out my data sharing in a way that I cannot ignore: I started wearing a fitness band. I bought one of the simplest bands available - the Mi Band Pulse from Xiaomi - this is not a smart watch but "simply" a fitness band that counts steps, measures heart rate and sleep. It's cheap price means that this is likely to spread to many users. Hence, BBD at speed on a diverse population.

I had two questions in mind:

1. Can I limit the usage so that I only get the benefits I really need/want while avoiding generating and/or sharing data I want to keep private?
2. What do I not know about the data generated and shared?

I tried to be as "private" as possible, never turning on location, turning on bluetooth for synching my data with my phone only once a day for a few minutes, turning on notifications only for incoming phone calls, not turning on notifications for anything else (no SMS, no third party apps notifications), only using the Mi Fit app (not linking to other apps like Google Fit), and only "befriending" one family member who bought the same product.

While my experiment is not as daring as a physician testing his/her developed drugs on themselves, I did learn a few important lessons. Here is what I discovered:

• Despite intentionally turning off my phone bluetooth right after synching the band's data every day (which means the device should not be able to connect to my phone, or the cloud), the device and my phone did continue having secret "conversations". I discovered this when my band vibrated on incoming phone calls when bluetooth was turned off. This happened on multiple occasions. I am not sure what part of the technology chain to blame - my Samsung phone? the Android operating system? the Mi Band? It doesn't really matter. The point is: I thought the band was disconnected from my phone, but it was not.
• Every time I opened the Mi Fit app on my phone, it requested access to location, which I had to decline every time. Quite irritating, and I am guessing many users just turn it on to avoid the irritation.
• My family member who was my "friend" through the Mi Fit app could see all my data: number of steps, heart rate measurements (whenever I asked the band to measure heart rate), weight (which I entered manually), and even... my sleep hours. Wait: my family member now knew exactly what time I went to sleep and what time I got up every morning. That's something I didn't consider when we decided to become "friends". Plus, Xiaomi has the information about our "relationship" including "pokes" that we could send each other (causing a vibration).
• The sleep counter claims to measure "deep sleep". It was showing that even if I slept 8 hours and felt wonderfully refreshed, my deep sleep was only 1 hour. At first I was alarmed. With a bit of search I learned that it doesn't mean much. But the alarming effect made me realize that even this simple device is not for everyone. Definitely not for over-worried people.

After 2 months of wearing of the band 24-by-7, what have I learned about myself that I didn't already know? That my "deep sleep" is only one hour per night (whatever that means). And that the feeling of the vibrating band when conquering the 8000 steps/day limit feels good. Addictively good, despite being meaningless. All the other data was really no news (I walk a lot on days when I teach, my weight is quite stable, my heart rate is reasonable, and I go to sleep too late).

To answer my two questions:

1. Can I limit the usage to only gain my required benefits? Not really. Some is beyond my control, and in some cases I am not aware of what I am sharing.
2. What do I not know? Quite a bit. I don't know what the device is really measuring (eg "deep sleep"), how it is sharing it (when my bluethooth is off), and what the company does with it.

Is my "behavioral" data useful? To me probably not. To Xiaomi probably yes. One small grain of millet (小米 = XiaoMi) is not much, but a whole sack is valuable.

So what now? I move to Mi Band 2. Its main new feature is a nudge every hour when you're sitting for too long.

A non-traditional definition of Big Data: Big is Relative

I've noticed that in almost every talk or discussion that involves the term Big Data, one of the first slides by the presenter or the first questions to be asked by the audience is "what is Big Data?" The typical answer has to do with some digits, many V's, terms that end with "bytes", or statements about software or hardware capacity.

I beg to differ.

"Big" is relative. It is relative to a certain field, and specifically to the practices in the field. We therefore must consider the benchmark of a specific field to determine if today's data are "Big". My definition of Big Data is therefore data that require a field to change its practices of data processing and analysis.

On the one extreme, consider weather forecasting, where data collection, huge computing power, and algorithms for analyzing huge amounts of data have been around for a long time. So is today's climatology data "Big" for the field of weather forecasting? Probably not, unless you start considering new types of data that the "old" methods cannot process or analyze.

Another example is the field of genetics, where researchers have been working with an analyzing large-scale datasets (notably from the Human Genome Project) for some time. The "Big Data" in this field is about linking different databases and integrating domain knowledge with the patterns found in the data ("As big-data researchers churn through large tumour databases looking for patterns of mutations, they are adding new categories of breast cancer.")

On the other extreme, consider studies in the social sciences, in fields such as political science or psychology that have traditionally relied on 3-digit sample sizes (if you were lucky). In these fields, a sample of 100,000 people is Big Data, because it challenges the methodologies used by researchers in the field.  Here are some of the challenges that arise:

• Old methods break down: the common method of statistical significance tests for testing theory no longer works, as p-values will tend to be tiny irrespective of practical significance (one more reason to carefully consider the recent statement by the American Statistical Association about the danger of using the "p-value < 0.5" rule.
• Technology challenge: the statistical software and hardware used by many social science researchers might not be able to handle these new data sizes. Simple operations such as visualizing 100,000 observations in a scatter plot require new practices and software (such as state-of-the-art interactive software packages). 
• Social science researchers need to learn how to ask more nuanced questions, now that richer data are available to them. 
• Social scientists are not trained in data mining, yet the new sizes of datasets can allow them to discover patterns that are not hypothesized by theory

In terms of "variety" of data types, Big Data is again area-dependent. While text and network data might be new to engineering fields, social scientists have long had experience with text data (qualitative researchers have analyzed interviews, videos, etc. for long) and with social network data  (the origins of many of the metrics used today are in sociology).

In short, what is Big for one field can be considered small for another field. Big Data is field-dependent, and should be based on the "delta" (the difference) between previous data analysis practices and ones called for by today's data.

What's in a name? "Data" in Mandarin Chinese

The term "data", now popularly used in many languages, is not as innocent as it seems. The biggest controversy that I've been aware of is whether the English term "data" is singular or plural. The tone of an entire article would be different based on the author's decision.

In Hebrew, the word is in plural (Netunim, with the final "im" signifying plural), so no question arises.

Today I discovered another "data" duality, this time in Mandarin Chinese. In Taiwan, the term used is 資料 (Zīliào), while in Mainland China it is 數據 (Shùjù). Which one to use? What is the difference? I did a little research and tried a few popularity tests:

1. Google Translate from Chinese to English translates both terms to "data". But Chinese-to-English translates data to 數據 (Shùjù) with the other term appearing as secondary. Here we also learn that 資料 (Zīliào) means "material".
   
2. Chinese Wikipedia's main data article (embarrassingly poor) is for 數據 (Shùjù) and the article for 資料 (Zīliào) redirects you to the main article.
3. A Google search of each term leads to surprising results on number of hits:

Search results for "data" term Zīliào

Search results for "data" term Shùjù

I asked a few colleagues from different Chinese-speaking countries and learned further that 資料 (Zīliào) translates to information. A Google Images search brings images of "Information". This might also explain the double hit rate. A duality between data and information is especially interesting given the relationship between the two (and my related work on Information Quality with Ron Kenett).

So what about Big Data?  Here too there appear to be different possible terms, yet the most popular seems to be 大数据 (Dà shùjù), which also has a reasonably respectable Wikipedia article.

Thanks to my learned colleagues Chun-houh Chen (Academia Sinica), Khim Yong Goh (National University of Singapore), and Mingfeng Lin (University of Arizona) for their inputs.

What does "business analytics" mean in academia?

But what exactly does this mean?

In the recent ISIS conference, I organized and moderated a panel called "Business Analytics and Big Data: How it affects Business School Research and Teaching". The goal was to tackle the ambiguity in the terms "Business Analytics" and "Big Data" in the context of business school research and teaching. I opened with a few points:

1. Some research b-schools are posting job ads for tenure-track faculty in "Business Analytics" (e.g., University of Maryland; Google "professor business analytics position" for plenty more). What does this mean? what is supposed to be the background of these candidates and where are they supposed to publish to get promoted? ("The Journal of Business Analytics"?)
2. A recent special issue of the top scholarly journal Management Science was devoted to "Business Analytics". What types of submissions fall under this label? what types do not?
3. Many new "Business Analytics" programs have been springing up in business schools worldwide. What is new about their offerings? 

Panelists Anitesh, Ram and David - photo courtesy of Ravi Bapna

The panelist were a mix of academics (Prof Anitesh Barua from UT Austin and Prof Ram Chellapah from Emory University) and industry (Dr. David Hardoon, SAS Singapore). The audience was also a mixed crowd of academics mostly from MIS departments (in business schools) and industry experts from companies such as IBM and Deloitte.

The discussion took various twists and turns with heavy audience discussion. Here are several issues that emerged from the discussion:

• Is there any meaning to BA in academia or is it just the use of analytics (=data tools) within a business context? Some industry folks said that BA is only meaningful within a business context, not research wise.
• Is BA just a fancier name for statisticians in a business school or does it convey a different type of statistician? (similar to the adoption of "operations management" (OM) by many operation research (OR) academics)
• The academics on the panel made the point that BA has been changing the flavor of research in terms of adding a discovery/exploratory dimension that does not typically exist in social science and IS research. Rather than only theorize-then-test-with-data, data are now explored in further detail using tools such as visualization and micro-level models. The main concern, however, was that it is still very difficult to publish such research in top journals.
• With respect to "what constitutes a BA research article", Prof. Ravi Bapna said "it's difficult to specify what papers are BA, but it is easy to spot what is not BA".
• While machine learning and data mining have been around for some good time, and the methods have not really changed, the application of both within a business context has become more popular due to friendlier software and stronger computing power. These new practices are therefore now an important core in MBA and other business programs. 
• One type of b-school program that seems to lag behind on the BA front is the PhD program. Are we equipping our PhD students with abilities to deal with and take advantage of large datasets for developing theory? Are PhD programs revising their curriculum to include big data technologies and machine learning capabilities as required core courses?

Some participants claimed that BA is just another buzzword that will go away after some time. So we need not worry about defining it or demystifying it. After all, the software vendors coin such terms, create a buzz, and finally the buzz moves on. Whether this is the case with BA or with Big Data is yet to be seen. In the meanwhile, we should ponder whether we are really doing something new in our research, and if so, pinpoint to what exactly it is and how to formulate it as requirements for a new era of researchers.

Linear regression for a binary outcome: is it Kosher?

Regression models are the most popular tool for modeling the relationship between an outcome and a set of inputs. Models can be used for descriptive, causal-explanatory, and predictive goals (but in very different ways! see Shmueli 2010 for more).

The family of regression models includes two especially popular members: linear regression and logistic regression (with probit regression more popular than logistic in some research areas). Common knowledge, as taught in statistics courses, is: use linear regression for a continuous outcome and logistic regression for a binary or categorical outcome. But why not use linear regression for a binary outcome? the two common answers are: (1) the linear regression can produce predictions that are not binary, and hence "nonsense" and (2) inference based on the linear regression coefficients will be incorrect.

I admit that I bought into these "truths" for a long time, until I learned never to take any "statistical truth" at face value. First, let us realize that problem #1 relates to prediction and #2 to description and causal explanation. In other words, if issue #1 can be "fixed" somehow, then I might consider linear regression for prediction even if the inference is wrong (who cares about inference if I am only interested in predicting individual observations?). Similarly, if there is a fix for issue #2, then I might consider linear regression as a kosher inference mechanism even if it produces "nonsense" predictions.

The 2009 paper Linear versus logistic regression when the dependent variable is a dichotomy by Prof. Ottar Hellevik from Oslo University de-mystifies some of these issues. First, he gives some tricks that help avoid predictions outside the [0,1] range. The author identifies a few factors that contribute to "nonsense predictions" by linear regression:

• interactions that are not accounted for in the regression
• non-linear relationships between a predictor and the outcome

The suggested remedy for these issues is including interaction terms for categorical variables, and if numerical predictors are involved, then bucket them into bins and include those as dummies + interactions. So, if the goal is predicting a binary outcome, linear regression can be modified and used.

Now to the inference issue. "The problem with a binary dependent variable is that the homoscedasticity assumption (similar variation on the dependent variable for units with different values on the independent variable) is not satisfied... This seems to be the main basis for the widely held opinion that linear regression is inappropriate with a binary dependent variable". Statistical theory tells us that violating the homoscedasticity assumption results in biased standard errors for the coefficients, and that the coefficients might not be the most precise in terms of variance. Yet, the coefficients themselves remain unbiased (meaning that with a sufficiently large sample they are "on target"). Hence, with a sufficiently large sample we need not worry! Precision is not an issue in very large samples, and hence the on-target coefficients are just what we need.

I will add that another concern is that the normality assumption is violated: the residuals from a regression model on a binary outcome will not look very bell-shaped... Again, with a sufficiently large sample, the distribution does not make much difference, since the standard errors are so small anyway.

Chart from Hellevik (2009)

Hellevik's paper pushes the envelope further in an attempt to explore "how small can you go" with your sample before getting into trouble. He uses simulated data and compares the results from logistic and linear regression for fairly small samples. He finds that the differences are minuscule.

The bottom line: linear regression is kosher for prediction if you take a few steps to accommodate non-linear relationships (but of course it is not guaranteed to produce better predictions than logistic regression!). For inference, for a sufficiently large sample where standard errors are tiny anyway, it is fine to trust the coefficients, which are in any case unbiased.

Policy-changing results or artifacts of big data?

The New York Times article Big Study Links Good Teachers to Lasting Gain covers a research study coming out of Harvard and Columbia on "The Long-Term Impacts of Teachers: Teacher Value-Added and Student Outcomes in Adulthood". The authors used sophisticated econometric models applied to data from a million students to conclude:

"We find that students assigned to higher VA [Value-Added] teachers are more successful in many dimensions. They are more likely to attend college, earn higher salaries, live in better neighborhoods, and save more for retirement. They are also less likely to have children as teenagers."

When I see social scientists using statistical methods in the Big Data realm I tend to get a little suspicious, since classic statistical inference behaves differently with large samples than with small samples (which are more typical in the social sciences). Let's take a careful look at some of the charts from this paper to figure out the leap from the data to the conclusions.

How much does a "value added" teacher contribute to a person's salary at age 28?

Figure 1: dramatic slope? largest difference is less than $1,000

The slope in the chart (Figure 1) might look quite dramatic. And I can tell you that, statistically speaking, the slope is not zero (it is a "statistically significant" effect). Now look closely at the y-axis amounts. Note that the data fluctuate only by a very small annual amount! (less than $1,000 per year). The authors get around this embarrassing magnitude by looking at the "lifetime value" of a student ("On average, having such a [high value-added] teacher for one year raises a child's cumulative lifetime income by $50,000 (equivalent to $9,000 in present value at age 12 with a 5% interest rate)."


Here's another dramatic looking chart:

What happens to the average student test score as a "high value-added teacher enters the school"?

The improvement appears to be huge! But wait, what are those digits on the y-axis? the test score goes up by 0.03 points!

Reading through the slides or paper, you'll find various mentions of small p-values, which indicate statistical significance ("p<0.001" and similar notations). This by no means says anything about the practical significance or the magnitude of the effects.

If this were a minor study published in a remote journal, I would say "hey, there are lots of those now." But when a paper covered by the New York Times and is published as in the serious National Bureau of Economic Research Working Paper series (admittedly, not a peer-reviewed journal), then I am worried. I am very worried.

Unless I am missing something critical, I would only agree with one line in the executive summary: "We find that when a high VA teacher joins a school, test scores rise immediately in the grade taught by that teacher; when a high VA teacher leaves, test scores fall." But with one million records, that's not a very interesting question. The interesting question which should drive policy is by how much?

Big Data is also becoming the realm in social sciences research. It is critical that researchers are aware of the dangers of applying small-sample statistical models and inference in this new era. Here is one place to start.

Big Data: The Big Bad Wolf?

"Big Data" is a big buzzword. I bet that sentiment analysis of news coverage, blog posts and other social media sources would show a strong positive sentiment associated with Big Data. What exactly is big data depends on who you ask. Some people talk about lots of measurements (what I call "fat data"), others of huge numbers of records ("long data"), and some talk of both. How much is big? Again, depends who you ask.

As a statistician who's (luckily) strayed into data mining, I initially had the traditional knee-jerk reaction of "just get a good sample and get it over with", and later recognizing that "fitting the data to the toolkit" (or, "to a hammer everything looks like a nail") is straight-jacketing some great opportunities.

The LinkedIn group Advanced Business Analytics, Data Mining and Predictive Modeling reacted passionately to a the question "What is the value of Big Data research vs. good samples?" posted by a statistician and analytics veteran Michael Mout. Respondents have been mainly from industry - statisticians and data miners. I'd say that the sentiment analysis would come out mixed, but slightly negative at first ("at some level, big data is not necessarily a good thing"; "as statisticians, we need to point out the disadvantages of Big Data"). Over time, sentiment appears to be more positive, but not reaching anywhere close to the huge Big Data excitement in the media.

I created a Wordle of the text in the discussion until today (size represents frequency). It highlights the main advantages and concerns of Big Data. Let me elaborate:

• Big data permit the detection of complex patterns (small effects, high order interactions, polynomials, inclusion of many features) that are invisible with small data sets
• Big data allow studying rare phenomena, where a small percentage of records contain an event of interest (fraud, security)
• Sampling is still highly useful with big data (see also blog post by Meta Brown); with the ability to take lots of smaller samples, we can evaluate model stability, validity and predictive performance
• Statistical significance and p-values become meaningless when statistical models are fitted to very large samples. It is then practical significance that plays the key role.
• Big data support the use of algorithmic data mining methods that are good at feature selection. Of course, it is still necessary to use domain knowledge to avoid "garbage-in-garbage-out"
• Such algorithms might be black-boxes that do not help understand the underlying relationship, but are useful in practice for predicting new records accurately
• Big data allow the use of many non-parametric methods (statistical and data mining algorithms) that make much less assumptions about data (such as independent observations)

Thanks to social media, we're able to tap on many brains that have experience, expertise and... some preconceptions. The data collected from such forums can help us researchers to focus our efforts on the needed theoretical investigation of Big Data, to help move from sentiments to theoretically-backed-and-practically-useful knowledge.