Regression Statistics

Regression Is NOT Easy

Regression is NOT easy – to do competently. Regression is arguably the workhorse of statistics and widely-used in many disciplines but, unfortunately, often poorly understood. Many misconceptions surround regression analysis which can have important implications for businesses as well as public welfare.

At the end of the day, when using regression, it’s the thought that counts most. Regression is easy to use with today’s software…and easy to misuse with today’s software. To demonstrate that regression analysis is NOT easy to do competently, and to provide a few tips and guidelines for those who want to use it more effectively, below are some brief excerpts from the popular textbook Data Analysis Using Regression and Multilevel/Hierarchical Models by Andrew Gelman and Jennifer Hill.

The book is more than 600 pages in total and covers a wide range of topics that include ANOVA, the Generalized Linear Model, multilevel models, missing data imputation, sample size and power calculations, Bayesian methods and causal analysis. A revised and expanded edition of the book is under preparation but, as of this writing, a publication date has not been announced.

All copy/paste and editing errors are mine.

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Assumptions of the regression model

Some of the most important assumptions rely on the researcher’s knowledge of the subject area and may not be directly testable from the available data alone.

We list the assumptions of the regression model in decreasing order of importance.

1. Validity. Most importantly, the data you are analyzing should map to the research question you are trying to answer. This sounds obvious but is often overlooked or ignored because it can be inconvenient. Optimally, this means that the outcome measure should accurately reflect the phenomenon of interest, the model should include all relevant predictors, and the model should generalize to the cases to which it will be applied.

For example, with regard to the outcome variable, a model of earnings will not necessarily tell you about patterns of total assets. A model of test scores will not necessarily tell you about child intelligence or cognitive development. Choosing inputs to a regression is often the most challenging step in the analysis. We are generally encouraged to include all “relevant” predictors, but in practice it can be difficult to determine which are necessary and how to interpret coefficients with large standard errors.

A sample that is representative of all mothers and children may not be the most appropriate for making inferences about mothers and children who participate in the Temporary Assistance for Needy Families program. However, a carefully selected subsample may reflect the distribution of this population well. Similarly, results regarding diet and exercise obtained from a study performed on patients at risk for heart disease may not be generally applicable to generally healthy individuals. In this case assumptions would have to be made about how results for the at-risk population might relate to those for the healthy population.

Data used in empirical research rarely meet all (if any) of these criteria precisely. However, keeping these goals in mind can help you be precise about the types of questions you can and cannot answer reliably.

2. Additivity and linearity. The most important mathematical assumption of the regression model is that its deterministic component is a linear function of the separate predictors: y = β1×1 + β2×2 + · · · .

If additivity is violated, it might make sense to transform the data (for example, if y = abc, then log y = log a + log b + log c) or to add interactions. If linearity is violated, perhaps a predictor should be put in as 1/x or log(x) instead of simply linearly. Or a more complicated relationship could be expressed by including both x and x^2 as predictors.

For example, it is common to include both age and age^2 as regression predictors. In medical and public health examples, this allows a health measure to decline with higher ages, with the rate of decline becoming steeper as age increases. In political examples, including both age and age^2 allows the possibility of increasing slopes with age and also U-shaped patterns if, for example, the young and old favor taxes more than the middle-aged.

In such analyses we usually prefer to include age as a categorical predictor. Another option is to use a nonlinear function such as a spline or other generalized additive model. In any case, the goal is to add predictors so that the linear and additive model is a reasonable approximation.

3. Independence of errors. The simple regression model assumes that the errors from the prediction line are independent. We will return to this issue in detail when discussing multilevel models. [NOT SHOWN]

4. Equal variance of errors. If the variance of the regression errors are unequal, estimation is more efficiently performed using weighted least squares, where each point is weighted inversely proportional to its variance. In most cases, however, this issue is minor. Unequal variance does not affect the most important aspect of a regression model, which is the form of the predictor Xβ.

5. Normality of errors. The regression assumption that is generally least important is that the errors are normally distributed. In fact, for the purpose of estimating the regression line (as compared to predicting individual data points), the assumption of normality is barely important at all. Thus, in contrast to many regression textbooks, we do not recommend diagnostics of the normality of regression residuals.

If the distribution of residuals is of interest, perhaps because of predictive goals, this should be distinguished from the distribution of the data, y. For example, consider a regression on a single discrete predictor, x, which takes on the values 0, 1, and 2, with one-third of the population in each category. Suppose the true regression line is y = 0.2 + 0.5x with normally distributed errors with standard deviation 0.1. Then a graph of the data y will show three fairly sharp modes centered at 0.2, 0.7, and 1.2. Other examples of such mixture distributions arise in economics, when including both employed and unemployed people, or the study of elections, when comparing districts with incumbent legislators of different parties.

Further assumptions are necessary if a regression coefficient is to be given a causal interpretation as we discuss in Chapters 9 and 10. [NOT SHOWN]

General principles for model building

Our general principles for building regression models for prediction are as follows:

Include all input variables that, for substantive reasons, might be expected to be important in predicting the outcome.

It is not always necessary to include these inputs as separate predictors—for example, sometimes several inputs can be averaged or summed to create a “total score” that can be used as a single predictor in the model. For inputs that have large effects, consider including their interactions as well.

We suggest the following strategy for decisions regarding whether to exclude a variable from a prediction model based on expected sign and statistical significance (typically measured at the 5% level; that is, a coefficient is “statistically significant” if its estimate is more than 2 standard errors from zero):

(a) If a predictor is not statistically significant and has the expected sign, it is generally fine to keep it in. It may not help predictions dramatically but is also probably not hurting them.

(b) If a predictor is not statistically significant and does not have the expected sign (for example, incumbency having a negative effect on vote share), consider removing it from the model (that is, setting its coefficient to zero).

(c) If a predictor is statistically significant and does not have the expected sign, then think hard if it makes sense. Try to gather data on potential lurking variables and include them in the analysis.

(d) If a predictor is statistically significant and has the expected sign, then by all means keep it in the model.

D

These strategies do not completely solve our problems but they help keep us from making mistakes such as discarding important information. They are predicated on having thought hard about these relationships before fitting the model. It’s always easier to justify a coefficient’s sign after the fact than to think hard ahead of time about what we expect. On the other hand, an explanation that is determined after running the model can still be valid. We should be able to adjust our theories in light of new information.

It is common to fit a regression model repeatedly, either for different datasets or to subsets of an existing dataset. For example, one could estimate the relation between height and earnings using surveys from several years, or from several countries, or within different regions or states within the United States. As discussed in Part 2 of this book [NOT SHOWN], multilevel modeling is a way to estimate a regression repeatedly, partially pooling information from the different fits.

Quick tips to improve your regression modeling

Fit many models

Think of a series of models, starting with the too-simple and continuing through to the hopelessly messy. Generally it’s a good idea to start simple. Or start complex if you’d like, but prepare to quickly drop things out and move to the simpler model to help understand what’s going on. Working with simple models is not a research goal—in the problems we work on, we usually find complicated models more believable—but rather a technique to help understand the fitting process.

Do a little work to make your computations faster and more reliable

This sounds like computational advice but is really about statistics: if you can fit models faster, you can fit more models and better understand both data and model. But getting the model to run faster often has some startup cost, either in data preparation or in model complexity.

Data subsetting

Related to the “multiple model” approach are simple approximations that speed the computations. Computers are getting faster and faster—but models are getting more and more complicated! And so these general tricks might remain important.

A simple and general trick is to break the data into subsets and analyze each subset separately. The advantage of working with data subsets is that computation is faster on data subsets, for two reasons: first, the total data size n is smaller, so each regression computation is faster; and, second, the number of groups J is smaller, so there are fewer parameters, and the Gibbs sampling [used in Bayesian analysis] requires fewer updates per iteration.

The two disadvantages of working with data subsets are: first, the simple inconvenience of subsetting and performing separate analyses; and, second, the separate analyses are not as accurate as would be obtained by putting all the data together in a single analysis. If computation were not an issue, we would like to include all the data, not just a subset, in our fitting.

In practice, when the number of groups is large, it can be reasonable to perform an analysis on just one random subset, for example one-tenth of the data, and inferences about the quantities of interest might be precise enough for practical purposes.

Graphing the relevant and not the irrelevant

Graphing the fitted model

Graphing the data is fine but it is also useful to graph the estimated model itself. A table of regression coefficients does not give you the same sense as graphs of the model. This point should seem obvious but can be obscured in statistical textbooks that focus so strongly on plots for raw data and for regression diagnostics, forgetting the simple plots that help us understand a model.

Don’t graph the irrelevant

Are you sure you really want to make those quantile-quantile plots, influence diagrams, and all the other things that spew out of a statistical regression package? What are you going to do with all that? Just forget about it and focus on something more important. A quick rule: any graph you show, be prepared to explain.

Transformations

Consider transforming every variable in sight:

• Logarithms of all-positive variables (primarily because this leads to multiplicative models on the original scale, which often makes sense)

• Standardizing based on the scale or potential range of the data (so that coefficients can be more directly interpreted and scaled); an alternative is to present coefficients in scaled and unscaled forms

• Transforming before multilevel modeling (thus attempting to make coefficients more comparable, thus allowing more effective second-level regressions, which in turn improve partial pooling)

Plots of raw data and residuals can also be informative when considering transformations.

In addition to univariate transformations, consider interactions and predictors created by combining inputs (for example, adding several related survey responses to create a “total score”). The goal is to create models that could make sense (and can then be fit and compared to data) and that include all relevant information.

Consider all coefficients as potentially varying

Don’t get hung up on whether a coefficient “should” vary by group. Just allow it to vary in the model, and then, if the estimated scale of variation is small, maybe you can ignore it if that would be more convenient.

Practical concerns sometimes limit the feasible complexity of a model—for example, we might fit a varying-intercept model first, then allow slopes to vary, then add group-level predictors, and so forth. Generally, however, it is only the difficulties of fitting and, especially, understanding the models that keeps us from adding even more complexity, more varying coefficients, and more interactions.

Estimate causal inferences in a targeted way, not as a byproduct of a large regression

Don’t assume that a regression coefficient can be interpreted causally. If you are interested in causal inference, consider your treatment variable carefully and use the tools of Chapters 9, 10, and 23 [NOT SHOWN] to address the difficulties of comparing comparable units to estimate a treatment effect and its variation across the population. It can be tempting to set up a single large regression to answer several causal questions at once; however, in observational settings (including experiments in which certain conditions of interest are observational), this is not appropriate.

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