Reporting Regression Results Responsibly

We’re in a Golden Age for access to data, which unfortunately also means we’re in a Golden Age for the potential to misinterpret data. Though the absurdity of gated academic journals persists, academic research is more accessible now than ever before. We’ve also seen a rapid growth in the use of arguments based on statistics in the popular media in the last several years. This is potentially a real boon to our ability to understand the world around us, but it carries with it all of the potential for misleading statistical arguments.

My request is pretty simple. All statistical techniques, particularly the basic parametric statistical techniques that are most likely to show up in data journalism, require the satisfaction of assumptions and checking of diagnostic measures to ensure that hidden bias isn’t misleading us. Many of these assumptions and diagnostics are ultimately judgment calls, relying on practitioners to make informed decisions about what degree of wiggle room is appropriate given the research scenario. There are, however, conventions and implied standards that people can use to guide their decisions. The most important and useful kind of check, though, is the  eyes of other researchers. Given that the ability to host graphs, tables, and similar kinds of data online is simple and nearly free, I think that researchers and data journalists alike should provide links to their data and to the graphs and tables they use to check assumptions and diagnostic measures. In the digital era, it’s crazy this is still a rare practice. I don’t expect to find these graphs and tables sitting square in the center of a blog post, and I expect that 90% of readers wouldn’t bother to look. But there’s nothing to risk in having them available, and transparency, accountability, and collaboration to gain.

That’s the simple part, and you can feel free to close tab. For a little more:

What kind of assumptions and diagnostics am I talking about? Let’s consider the case of one of the most common types of parametric methods, linear regression. Whether we have a single predictor for simple linear regression or multiple predictors for multilinear regression, fundamentally regression is a matter of assessing the relationship between quantitative (continuous) predictor variables and a quantitative (continuous) outcome variable. For example, we might ask how well SAT scores predict college GPA; we might ask how well age, weight, and height predict blood pressure. When someone talks about how one number predicts another, the strength of their relationship, and how we might attempt to change one by changing the other, they’re probably making an appeal to regression.

The types of regression analysis, and the issues therein, are vast, and there are many technical issues at play that I’ll never understand. But I think it’s worthwhile to talk about some of the assumptions we need to check and some problems we have to look out for. Regression has come in for a fair amount of abuse lately from sticklers and skeptics, and not for no reason; it’s easy to use the techniques irresponsibly. But we’re inevitably going to ask basic questions of how X and Y predict Z, so I think we should expand public literacy about these things. I want to talk a little bit about these issues not because I think I’m qualified to teach statistics to others, or because regression is the only statistical process that we need to see assumptions and diagnostics for. Rather, I think regression is an illustrative example through which to explore why we need to check this stuff, to talk about both the power and pitfalls of public engagement with data.

There are four assumptions that need to be true to run a linear (least squares) regression: independence of observations, linearity, constancy of variance, and normality. (Some purists add a fifth, existence, which, whatever.)

Independence of Observations

This is the biggie, and it’s why doing good research can be so hard and expensive. It’s the necessary assumption that one observation does not affect another. This is the assumption that requires randomness. Remember that in statistics error, or necessary and expected variation, is inevitable, but bias, or the systematic influence on observations, is lethal.

Suppose you want to see how eating ice cream affects blood sugar level. You gather 100 students into the gym and have them all eat ice cream. You then go one by one through the students and give them a blood test. You dutifully record everyone’s values. When you get back to the lab, you find that your data does not match that of much of the established research literature. Confused, you check your data again. You use your spreadsheet software to arrange the cells by blood sugar. You find a remarkably steady progression of results running higher to lower. Then it hits you: it took you several hours to test the 100 students. The highest readings are all from the students who were first to be tested, the lowest from those who were tested last. Your data was corrupted by an uncontrolled variable, time-after-eating-to-test. Your observations were not truly independent of each other – one observation influenced another because taking one delayed taking the other. This is an example that you’d hope most people would avoid, but the history of research is the history of people making oversights that were, in hindsight, quite obvious.

Independence is scary because threats to it so often lurk out of sight. And the presumption of independence often prohibits certain kind of analysis that we might find natural. For example, think of assigning control and test conditions to classes rather than individual students in educational research. This is often the only practical way to do it; you can’t fairly ask teachers to only teach half their students one technique and half another. You give one set of randomly-assigned classes a new pedagogical technique, while using the old standard with your control classes. You give a pre- and post-test to both and pop both sets of results in an ANOVA. You’ve just violated the assumption of independence. We know that there are clustering effects of children within classrooms; that is, their results are not entirely independent of each other. We can correct for this sort of thing using techniques like hierarchical modeling, but first we have to recognize that those dangers exist!

Independence is the assumption that is least subject to statistical correction. It’s also the assumption that is the hardest to check just by looking at graphs. Confidence in independence stems mostly from rigorous and careful experimental design. You can check a graph of your observations (your actual data points) against your residuals (the distance between your observed values and the linear progression from your model), which can sometimes provide clues. But ultimately, you’ve just got to know your data was collected appropriately. On this one, we’re largely on our own. However, I think it’s a good idea for academic researchers to provide online access to a Residuals vs. Observations graph when they run a regression. This is very rare, currently.

Here’s a Residuals vs. Observations graph I pulled off of Google Images. This is what we want to see: snow. Clear nonrandom patterns in this plot are bad.


The name of the technique is linear regression, which means that observed relationships should be roughly linear to be valid. In other words, you want your relationship to fall along a more or less linear path as you move across the x axis; the relationship can be weaker or it can be stronger, but you want it to be more or less as strong as you move across the line. This is particularly the case because curvilinear relationships can appear to regression analysis to be no relationship. Regression is all about interpolation: if I check  my data and find a strong linear relationship, and my data has a range from A to B, I should be able to check any x value within A and B and have a pretty good prediction for y. (What “pretty good” means in practice is a matter of residuals and r-squared, or the portion of the variance in y that’s explained by my xs.) If my relationship isn’t linear, my confidence in that prediction is unfounded.

Take a look at these scatter plots. Both show close to zero linear relationship according to Pearson’s product-moment coefficient:

And yet clearly, there’s something very different going on from one plot to the next. The first is true random variance; there is no consistent relationship between our x and y variables. The second is a very clear association; it’s just not a linear relationship. The degree and direction of y varying along x changes over different values for x. Failure to recognize that non-linear relationship could compel us to think that there is no relationship at all. If the violation of linearity is as clear and consistent as in this scatter plot, it can be cleaned up fairly easily by transforming the data.

Regression is fairly robust to violations of linearity, and it’s worth noting that any relationship that is sufficiently lower than 1 will be non-linear in the strict sense. But clear, consistent curves in data can invalidate our regression analyses.

Readers could check data for linearity if scatter plots are posted for simple linear regression. For multilinear regression, it’s a bit messier; you could plot every individual predictor, but I would be satisfied if you just mention that you checked linearity.

Constancy of variance

Also known by one of my very favorite ten-cent words, homoscedasticity. Constancy of variance means that, along your range of x predictors, your y varies about as much; it has as much spread, as much error. Remember, when I’m doing inferential statistics, I’m sampling, and sampling means sampling error – even if I’m getting quality results, I’m inevitably going to get differences in my data from one collection of samples to the next. But if our assumptions are true, we can trust that those samples will vary in predictable intervals relative to the true mean. That is, if an SAT score predicts freshman year GPA with a certain degree of consistency for students scoring 400, it should be about as consistent for students scoring 800, 1200, and 1600, even though we know that from one data set to the next, we’re not going to get the exact same values even if we assume that all of the variables of interest are the same. We just need to know that the degree to which they vary for a given is constant over our range.

Why is this important? Think again about interpolation. I run a regression because I want to understand a relationship between various quantitative variables, and often because I want to use my predictor variables to… predict. Regression is useful insofar as I can move along the axes of my x values and produce a meaningful, subject-to-error-but-still-useful value for y. Violating the assumption of constant variance means that you can’t predict y with equal confidence as you move around x(s); the relationship is stronger at some points than others, making you vulnerable to inaccurate predictions.

Here’s a residuals plot showing the dreaded megaphone effect: the error (size of residuals, difference between observations and results expected from the regression equation) increases as we move from low to high values of x. The relationship is strong at low values of x and much weaker at high values.

We could check homoscedasticity by having access to residual plots. Violations of constant variance can often be fixed via transformation, although it may often be easier to use techniques that are more inherently robust to this violation, such as quantile regression.


The concept of the normal distribution is at once simple and counterintuitive, and I’ve spent a lot of my walks home trying to think of the best way to explain it. The “parametric” in parametric statistics refers to the assumption that there is a given underlying distribution for most observable data, and frequently this distribution is the normal distribution or bell curve. Think of yourself walking down the street and noticing that someone is unusually tall or unusually short. The fact that you notice is in and of itself a consequence of the normal distribution. When we think of someone that is unusually tall or short, we are implicitly assuming that we will find fewer and fewer people as we move further along the extremes of the height distribution. If you see a man in North American who is 5’10, he is above average height, but you wouldn’t bat an eye; if you see a man who is 6’3, you might think yourself, that’s a tall guy; when you see someone who is 6’9, you say, wow, he is tall!, and when you see a 7 footer, you take out your cell phone. This is the central meaning of the normal distribution: that the average is more likely to occur than extremes, and that the relationship between position on the distribution and probability of occurrence is predictable.

Not everything in life is normally distributed. Poll 1,000 people and ask how much money they received in car insurance payments last year and it won’t look normal. But a remarkable amount of naturally occurring phenomena are normally distributed, simply thanks to the reality of numbers and extremes, and the central limit theorem teaches us that essentially all averages are normally distributed. (That is, if I take a 100 person sample of a population for a given quantitative trait, I will get a mean; if I take another 100 person sample, I will get a similar but not exact mean, and so on. If I plot those means, they will be normal even if the overall distribution is not.)

The assumption of normality in regression requires our data to be roughly normally distributed; in order to assess the relationship of y as it moves across x, we need to know the relative frequency of extreme observations to observations close to the mean. It’s a fairly robust assumption, and you’re never going to have perfectly normal data, but too strong of a violation will invalidate your analysis. We check normality with what’s called a qq plot. Here’s an almost-perfect one, again scraped from Google Images:

That strongly linear, nearly 45 degree angle is just what we want to see. Here’s a bad one, demonstrating the “fat tails” phenomenon – that is, too many observations clustered at the extremes relative to the mean:

Usually the rule is that unless you’ve got a really clear break from a straightish 45 degree angle, you’re probably alright. When the going gets tough, seek help from a statistician.


OK, so 2000 words into this thing, we’ve checked out four assumptions. Are we good? Well, not so fast. We need to check a few diagnostic measures, or what my stats instructor  used to call “the laundry list.” This is a matter of investigating influence. When we run an analysis like regression, we’re banking on the aggregate power of all of our observations to help us make responsible observations and inferences. We never want to rely too heavily on individual or small numbers of observations because that increases the influence of error in our analysis. Diagnostic measures in regression typically involve using statistical procedures to look for influential observations that have too much sway over our analysis.

The first thing to say about outliers is that you want a systematic reason for eliminating them. There are entire books about the identification and elimination of outliers, and I’m not qualified to say what the best method is in any given situation. But you never want to toss an observation simply because it would help your analysis. When you’ve got that one data point that’s dragging your line out of significance, it’s tempting to get rid of it, but you want to analyze that observation for a methodology-internal justification for eliminating it. On the other hand, sometimes you have the opposite situation: your purported effect is really the product of a single or small number of influential outliers that have dragged the line in your favor (that is, to a p-value you like). Then, of course, the temptation is simply to not mention the outlier and publish anyway. Especially if a tenure review is in your future…

Some examples of influential observation diagnostics in regression include examining leverage, or outliers in your predictors that have a great deal of influence on your overall model; Cook’s Distance, which tells you how different your model will be if you delete a given observation; DFBetas, which tells you how a given predictor observation influences on a particular parameter estimate; and more. Most modern statistical packages like SAS or R have commands for checking diagnostic measures like these. While offering numbers would be nice, I would mostly like it if researchers reassured readers that they had run diagnostic measures for regression and found acceptable results. Just let me know: I looked for outliers and influential observations and things came back fairly clean.


Regression is just one part of a large number of techniques and applications that are happening in data journalism right now. But essentially any statistical techniques are going to involve checking assumptions and diagnostic measures. A typical ANOVA, for example, the categorical equivalent of regression, will involve checking some of the same assumptions. In the era of the internet, there is no reason not to provide a link to a brief, simple rundown of what quality controls were pursued in  your analysis.

None of these things are foolproof. Sums of squares are spooky things; we get weird results as we add and remove predictors from our models. Individual predictors are strongly significant by themselves but not when added together; models are significant with no individual predictors significant; individual predictors are highly significant without model significance; the order you put your predictors in changes everything; and so on. It’s fascinating and complicated. We’re always at the mercy of how responsible and careful researchers are. But by sharing information, we raise the odds that what we’re looking at is a real effect.

This might all sound like an impossibly high bar to clear. There are so many ways things can go wrong. And it’s true that, in general, I worry that people today are too credulous towards statistical arguments, which are often advanced without sufficient qualifications. There are some questions where statistics more often mislead than illuminate. But there is a lot we can and do know. We know that age is highly predictive of height in children but not in adults; we know that there is a relationship between SAT scores and freshman year GPA; we know point differential is a better predictor of future win-loss record than past win-loss record. We can learn lots of things, but we always do it better together. So I think that academic researchers and data journalists should share their work to a greater degree than they do now. That requires a certain compromise. After all, it’s scary to have tons of strangers looking over your shoulder. So I propose that we get more skeptical and critical on our statistical arguments as a media and readership, but more forgiving of individual researchers who are, after all, only human. That strikes me as a good bargain.

And one I’m willing to make myself, so please email me to point out the mistakes I’ve inevitably made in this post.