Why I’m Not a Fan of R-Squared

The Big Message

People sometimes use \(R^2\) as their preferred measure of model fit. Unlike quantities such as MSE or MAD, \(R^2\) is not a function only of model’s errors, its definition contains an implicit model comparison between the model being analyzed and the constant model that uses only the observed mean to make predictions. As such, \(R^2\) answers the question: “does my model perform better than a constant model?” But we often would like to answer a very different question: “does my model perform worse than the true model?”

In theoretical examples, it is easy to see that the answers to these two questions are not interchangeable. We can construct examples in which our model performs no better than a constant model, even though it also performs no worse than the true model. But we can also construct examples in which our model performs much better than a constant model, even when it also performs much worse than the true model.

As with all model comparisons, \(R^2\) is a function not only of the models being compared, but also a function of the data set being used to perform the comparison. For almost all models, there exist data sets that are wholly incapable of distinguishing between the constant model and the true model. In particular, when using a dataset with insufficient model-discriminating power, \(R^2\) can be pushed arbitrarily close to zero — even when we are measuring \(R^2\) for the true model. As such, we must always keep in mind that \(R^2\) does not tell us whether our model is a good approximation to the true model: \(R^2\) only tells us whether our model performs noticeably better than the constant model on our dataset.

A Theoretical Example

To see how comparing a proposed model with a constant model could lead to opposite conclusions than comparing the same proposed model with the true model, let’s consider a simple example: we want to model the function \(f(x)\), which is observed noisily over an equally spaced grid of \(n\) points between \(x_{min}\) and \(x_{max}\).

To start, let’s assume that:

  • \(f(x) = \log(x)\).
  • \(x_{min} = 0.99\).
  • \(x_{max} = 1.01\).
  • At 1,000 evenly spaced values of \(x\) between \(x_{min}\) and \(x_{max}\), we observe \(y_i = f(x_i) + \epsilon_i\), where \(\epsilon_i \sim \mathbf{N}(0, \sigma^2)\).

Based on this data, we’ll attempt to learn a model of \(f(x)\) using univariate OLS regression. We’ll fit both a linear model and a quadratic model. An example realization of this modeling process looks like the following:

In this graph, we can see that \(f(x)\) is well approximated by a line and so both our linear and quadratic regression models come close to recovering the true model. This is because \(x_{min}\) and \(x_{max}\) are very close and our target function is well approximated by a line in this region, especially relative to the amount of noise in our observations.

We can see the quality of these simple regression models if we look at two binary comparisons: our model versus the constant model and our model versus the true logarithmic model. To simplify these calculations, we’ll work with an alternative \(R^2\) calculation that ignores corrections for the number of regressors in a model. Thus, we’ll compute \(R^2\) for a model \(m\) (versus the constant model \(c\)) as follows:

$$
R^2 = \frac{\text{MSE}_c – \text{MSE}_m}{\text{MSE}_c} = 1 – \frac{\text{MSE}_m}{\text{MSE}_c}
$$

As with the official definition \(R^2\), this quantity tells us how much of the residual errors left over by the constant model are accounted for by our model. To see how this comparison leads to different conclusions than a comparison against the true model, we’ll also consider a variant of \(R^2\) that we’ll call \(E^2\) to emphasize that it measures how much worse the errors are than we’d see using the true model \(t\):

$$
E^2 = \frac{\text{MSE}_m – \text{MSE}_t}{\text{MSE}_t} = \frac{\text{MSE}_m}{\text{MSE}_t} – 1
$$

Note that \(E^2\) has the opposite sense as \(R^2\): better fitting models have lower values of \(E^2\).

Computing these numbers on our example, we find that \(R^2 = 0.007\) for the linear model and \(R^2 = 0.006\) for the true model. In contrast, \(E^2 = -0.0008\) for the linear model, indicating that it is essentially indistinguishable from the true model on this data set. Even though \(R^2\) suggests our model is not very good, \(E^2\) tells us that our model is close to perfect over the range of \(x\).

Now what would happen if the gap between \(x_{max}\) and \(x_{min}\) grew larger? For a monotonic function like \(f(x) = \log(x)\), we’ll see that \(E^2\) will constantly increase, but that \(R^2\) will do something very strange: it will increase for a while and then start to decrease.

Before considering this idea in general, let’s look at one more specific example in which \(x_{min} = 1\) and \(x_{max} = 1000\):

In this case, visual inspection makes it clear that the linear model and quadratic models are both systematically inaccurate, but their values of \(R^2\) have gone up substantially: \(R^2 = 0.760\) for the linear model and \(R^2 = 0.997\) for the true model. In contrast, \(E^2 = 85.582\) for the linear model, indicating that this data set provides substantial evidence that the linear model is worse than the true model.

These examples show how the linear model’s \(R^2\) can increase substantially, even though most people would agree that the linear model is becoming an increasingly unacceptably bad approximation of the true model. Indeed, although \(R^2\) seems to improve when transitioning from one example to the other, \(E^2\) gets strictly worse as the gap between \(x_{min}\) and \(x_{max}\) increases. This suggests that \(R^2\) might be misleading if we interpret it as a proxy for the generally unmeasurable \(E^2\). But, in truth, our two extreme examples don’t tell the full story: \(R^2\) actually changes non-monotonically as we transition between the two cases we’ve examined.

Consider performing a similar analysis to the ones we’ve done above, but over many possible grids of points starting with \((x_{min}, x_{max}) = (1, 1.1)\) and ending with \((x_{min}, x_{max}) = (1, 1000)\). If we calculate \(R^2\) and \(E^2\) along the way, we get graphs like the following (after scaling the x-axis logarithmically to make the non-monotonicity more visually apparent).

First, \(R^2\):

Second, \(E^2\):

Notice how strange the graph for \(R^2\) looks compared with the graph for \(E^2\). \(E^2\) always grows: as we consider more discriminative data sets, we find increasingly strong evidence that our linear approximation is not the true model. In contrast, \(R^2\) starts off very low (precisely when \(E^2\) is very low because our linear model is little worse than the true model) and then changes non-monotonically: it peaks when there is enough variation in the data to rule out the constant model, but not yet enough variation in the data to rule out the linear model. After that point it decreases. A similar non-monotonicity is seen in the \(R^2\) value for the quadratic model. Only the true model shows a monotonic increase in \(R^2\).

Conclusion

I wrote down all of this not to discourage people from ever using \(R^2\). I use it sometimes and will continue to do so. But I think it’s important to understand both that (a) the value of \(R^2\) is heavily determined by the data set being used and that (b) the value of \(R^2\) can decrease even when your model is becoming an increasingly good approximation to the true model. When deciding whether a model is useful, a high \(R^2\) can be undesirable and a low \(R^2\) can be desirable.

This is an inescapable problem: whether a wrong model is useful always depends upon the domain in which the model will be applied and the way in which we evaluate all possible errors over that domain. Because \(R^2\) contains an implicit model comparison, it suffers from this general dependence on the data set. \(E^2\) also has such a dependence, but it at least seems not to exhibit a non-monotonic relationship with the amount of variation in the values of the regressor variable, \(x\).

Code

The code for this post is on GitHub.

Retrospective Edits

In retrospect, I would like to have made more clear that measures of fit like MSE and MAD also depend upon the domain of application. My preference for their use is largely that the lack of an implicit normalization means that they are prima facie arbitrary numbers, which I hope makes it more obvious that they are highly sensitive to the domain of application — whereas the normalization in \(R^2\) makes the number seem less arbitrary and might allow one to forget how data-dependent they are.

In addition, I really ought to have defined \(E^2\) using a different normalization. For a model \(m\) compared with both the constant model \(c\) and the true model \(t\), it would be easier to work with the bounded quantity:

$$
E^2 = \frac{\text{MSE}_m – \text{MSE}_t}{\text{MSE}_c – \text{MSE}_t}
$$

This quantity asks where on a spectrum from the worse defensible model’s performance (which is \(\text{MSE}_c – \text{MSE}_t\)) to the best possible model’s performance (which is \(\text{MSE}_t – \text{MSE}_t\) = 0), one lies. Note that in the homoscedastic regression setup I’ve considered, \(\text{MSE}_t = \sigma^2\), which can be estimated from empirical data using the identifying assumption of homoscedasticity and repeated observations at any fixed value of the regressor, \(x\). (I’m fairly certain that this normalized quantity has a real name in the literature, but I don’t know it off the top of my head.)

I should also have noted that some additional hacks are required to guarantee this number lies in \([0, 1]\) because the lack of model nesting means that violations are possible. (Similar violations are not possible in the linear regression setup because the constant model is almost always nested in the more elaborate models being tested.)

Retrospective Edits Round 2

After waking up to see this on Hacker News, I should probably have included a graphical representation of the core idea to help readers who don’t have any background in theoretical statistics. Here’s that graphical representation, which shows that, given three models to be compared, we can always place our model on a spectrum from the performance of the worst model (i.e. the constant model) to the performance of the best model (i.e. the true model).

r2_e2

When measuring a model’s position on this spectrum, \(R^2\) and \(E^2\) are complementary quantities.

Retrospective Edits Round 3

An alternative presentation of these ideas would have focused on a bias/variance decompositions of the quantities being considered. From that perspective, we’d see that:

$$
R^2 = 1 – \frac{\text{MSE}_m}{\text{MSE}_c} = 1 – \frac{\text{bias}_m^2 + \sigma^2}{\text{bias}_c^2 + \sigma^2}
$$

Using the redefinition of \(E^2\) I mentioned in the first round of “Retrospective Edits”, the alternative to \(R^2\) would be:

$$
E^2 = \frac{\text{MSE}_m – \text{MSE}_t}{\text{MSE}_c – \text{MSE}_t} = \frac{(\text{bias}_m^2 + \sigma^2) – (\text{bias}_t^2 + \sigma^2)} {(\text{bias}_c^2 + \sigma^2) – (\text{bias}_t^2 + \sigma^2)} = \frac{\text{bias}_m^2 – \text{bias}_t^2}{\text{bias}_c^2 – \text{bias}_t^2} = \frac{\text{bias}_m^2 – 0}{\text{bias}_c^2 – 0} = \frac{\text{bias}_m^2}{\text{bias}_c^2}
$$

In large part, I find \(R^2\) hard to reason about because of the presence of the \(\sigma^2\) term in the ratios shown above. \(E^2\) essentially removes that term, which I find useful for reasoning about the quality of a model’s fit to data.

A Variant on “Statistically Controlling for Confounding Constructs is Harder than you Think”

Yesterday, a coworker pointed me to a new paper by Jacob Westfall and Tal Yarkoni called “Statistically controlling for confounding constructs is harder than you think”. I quite like the paper, which describes some problems that arise when drawing conclusions about the relationships between theoretical constructs using only measurements of observables that are, at best, approximations to those theoretical constructs.

Given their intended audience, I think Westfall and Yarkoni’s approach is close to optimal: they motivate their concerns using examples that depend only upon the reader having a basic understanding of linear regression and measurement error. That said, the paper made me wish that more psychologists were familiar with Judea Pearl’s work on graphical techniques for describing causal relationships.

To see how a graphical modeling approach could be used to articulate Westfall and Yarkoni’s point, let’s assume that the true causal structure of the world looks like the graph shown in Figure 1 below:

Figure 1

For those who’ve never seen such a graph before, the edges are directed in a way that indicates causal influence. This graph says that the objective temperature on any given day is the sole cause of (a) the subjective temperature experienced by any person, (b) the amount of ice cream consumed on that day, and (c) the number of deaths in swimming pools on the day.

If this graph reflects the true causal structure of the world, then the only safe way to analyze data about the relationship between ice cream consumption and swimming pool deaths is to condition on objective temperature. If highlighting represents conditioning on a variable, the conclusion is that a correct analysis must look like the following colored-in graph:

Figure 2

Westfall and Yarkoni’s point is that this correct analysis is frequently not performed because the objective temperature variable is not observed and therefore cannot be conditioned on. As such, one conditions on subjective temperature instead of objective temperature, leading to this colored-in graph:

Figure 3

The problem with conditioning on subjective temperature instead of objective temperature is that you have failed to block the relevant path in the graph, which is the one that links ice cream consumption and swimming pool deaths by passing through objective temperature. Because the relevant path of dependence is not properly blocked, an erroneous relationship between ice cream consumption and swimming pool deaths leaks through. It is this leaking through that causes the high false positive rates that are the focus of Westfall and Yarkoni’s paper.

One reason I like this graphical approach is that it makes it clear that the opposite problem can occur as well: the world might have also have a causal structure such that one must condition on subjective temperature, but conditioning on objective temperature is always insufficient.

To see how that could occur, consider this alternative hypothetical causal structure for the world:

Figure 4

If this alternative structure were correct, then a correct analysis would need to color in subjective temperature:

Figure 5

An incorrect analysis would be to color in objective temperature rather than subjective temperature:

Figure 6

But, somewhat surprising, coloring in both would be fine:

Figure 7

It’s taken me some time to master this formalism, but I now find it quite easy to reason about these kinds of issues thanks to the brevity of graphical models as a notational technique. I’d love to see this approach become more popular in psychology, given that it has already become quite widespread in other fields. Of course, Westfall and Yarkoni are already advocating for something very similar by advocating for the use of SEM’s, but the graphical approach is strictly more general than SEM’s and, in my personal opinion, strictly simpler to reason about.

Understanding the Pseudo-Truth as an Optimal Approximation

Introduction

One of the things that set statistics apart from the rest of applied mathematics is an interest in the problems introduced by sampling: how can we learn about a model if we’re given only a finite and potentially noisy sample of data?

Although frequently important, the issues introduced by sampling can be a distraction when the core difficulties you face would persist even with access to an infinite supply of noiseless data. For example, if you’re fitting a misspecified model \(m_1\) to data generated by a model \(m_2\), this misspecification will persist even as the supply of data becomes infinite. In this setting, the issues introduced by sampling can be irrelevant: it’s often more important to know whether or not the misspecified model, \(m_1\), could ever act as an acceptable approximation to the true model, \(m_2\).

Until recently, I knew very little about these sorts of issues. While reading Cosma Shalizi’s draft book, “Advanced Data Analysis from an Elementary Point of View”, I learned about the concept of pseudo-truth, which refers to the version of \(m_1\) that’s closest to \(m_2\). Under model misspecification, estimation procedures often converge to the pseudo-truth as \(n \to \infty\).

Thinking about the issues raised by using a pseudo-true model rather than the true model has gotten me interested in learning more about quantifying approximation error. Although I haven’t yet learned much about the classical theory of approximations, it’s clear that a framework based on approximation errors allows one to formalize issues that would otherwise require some hand-waving.

A General Framework for Quantifying Approximation Error

Suppose that I want to approximate a function \(f(x)\) with another function \(g(x)\) that comes from a restricted set of functions, \(G\). What does the optimal approximation look like when \(g(x)\) comes from a parametric class of functions \(G = \{g(x, \theta) | \theta \in \mathbb{R}^{p}\}\)?

To answer this question with a single number, I think that one natural approach looks like:

  1. Pick a point-wise loss function, \(L(f(x), g(x))\), that evaluates the gap between \(f(x)\) and \(g(x)\) at any point \(x\).
  2. Pick a set, \(S\), of values of \(x\) over which you want to evaluate this point-wise loss function.
  3. Pick an aggregation function, \(A\), that summarizes all of the point-wise losses incurred by any particular approximation \(g(x, \theta)\) over the set \(S\).

Given \(L\), \(S\) and \(A\), you can define an optimal approximation from \(G\) to be a function, \(g \in G\), that minimizes the aggregated point-wise loss function over all values in \(S\).

An Analytically Tractable Special Case

This framework is so general that I find it hard to reason about. Let’s make some simplifying assumptions so that we can work through an analytically tractable special case:

  1. Let’s assume that \(f(x) = x^2\). Now we’re trying to find the optimal approximation of a quadratic function.
  2. Let’s assume that \(g(x, \theta) = g(x, a, b) = ax + b\), where \(\theta = \langle a, b \rangle \in \mathbb{R}^{2}\). Now we’re trying to find the optimal linear approximation to the quadratic function, \(f(x) = x^2\).
  3. Let’s assume that the point-wise loss function is \(L(f(x), g(x, \theta)) = [f(x) – g(x, \theta)]^2\). Now we’re quantifying the point-wise error using the squared error loss that’s commonly used in linear regression problems.
  4. Let’s assume that \(S\) is a closed interval on the real line, that is \(S = [l, u]\).
  5. Let’s assume that the aggregation function, \(A\), is the integral of the loss function, \(L\), over the interval, \([l, u]\).

Under these assumptions, the optimal approximation to \(f(x)\) from the parametric family \(g(x, \theta)\) is defined as a solution, \(\theta^{*}\), to the following minimization problem:

$$
\theta^{*} = \arg \min_{\theta} \int_{l}^{u} L(f(x), g(x, \theta)) dx = \arg \min_{\theta} \int_{l}^{u} [f(x) – g(x, \theta)]^2 dx = \arg \min_{\theta} \int_{l}^{u} [x^2 – (ax + b)]^2 dx.
$$

I believe that this optimization problem describes the pseudo-truth towards which OLS univariate regression converges when the values of the covariate, \(x\), are drawn from a uniform distribution over the interval \([l, u]\). Simulations agree with this belief, although I may be missing some important caveats.

Solving the Problem Analytically

To get a feeling for how the pseudo-truth behaves in this example problem, I found it useful to solve for the optimal linear approximation analytically by computing the gradient of the cost function with respect to \(\theta\) and then solving for the roots of the resulting equations:

$$
\frac{\partial}{\partial a} \int_{l}^{u} [x^2 – (ax + b)]^2 dx = 0 \\
\frac{\partial}{\partial b} \int_{l}^{u} [x^2 – (ax + b)]^2 dx = 0 \\
$$

After some simplification, this reduces to a matrix equation:

$$
\begin{bmatrix}
\frac{2}{3} (u^3 – l^3) & u^2 – l^2 \\
u^2 – l^2 & 2 (u – l) \\
\end{bmatrix} \begin{bmatrix}
a \\
b \\
\end{bmatrix} = \begin{bmatrix}
\frac{1}{2} (u^4 – l^4) \\
\frac{2}{3} (u^3 – l^3) \\
\end{bmatrix}.
$$

In practice, I suspect it’s much easier to solve for the optimal approximation on a computer by using quadrature to approximate the aggregated point-wise loss function and then minimizing that aggregation function using a quasi-Newton method. I experimented a bit with this computational approach and found that it reproduces the analytic solution for this problem. The computational approach generalizes to other problems readily and requires much less work to get something running.

Some Optimal Linear Approximations to a Quadratic Function over Different Intervals

You can see examples of the pseudo-truth versus the truth below for a few intervals \([l, u]\):

approximations

I don’t think there’s anything very surprising in these plots, but I nevertheless find the plots useful for reminding myself how sensitive the optimal approximation is to the set \(S\) over which it will be applied.

Formally Defining Asymptotic Bias

One reason that I find it useful to think formally about the quality of optimal approximations is that it makes it easier to rigorously define some of the problems that arise in machine learning.

Consider, for example, the issues raised in this blog post by Paul Mineiro:

In statistics the bias-variance tradeoff is a core concept. Roughly speaking, bias is how well the best hypothesis in your hypothesis class would perform in reality, whereas variance is how much performance degradation is introduced from having finite training data.

I would say that this definition of bias is quite far removed from the definition used in statistics and econometrics, although it can be formalized in terms of the expected point-wise loss.

The core issue, as I see it, is that, for a statistician or econometrician, bias is not an asymptotic property of a model class, but rather a finite-sample property of an estimator. Variance is also a finite-sample property of an estimator. Some estimators, such as the ridge estimator for a linear regression model, decrease variance by increasing bias, which induces a trade-off between bias and variance in finite samples.

But Paul Mineiro is not, I think, interested in these finite-sample properties of estimators. I believe he’s concerned about the intrinsic error introduced by approximating one function with another. And that’s a very important topic that I haven’t seen discussed as often as I’d like.

Put another way, I think Mineiro’s concern is much more closely linked to formalizing an analogue for regression problems of the concepts of VC dimension and Rademacher complexity that come up in classification problems. Because I wasn’t familiar with any such tools, I found it helpful to work through some problems on optimal approximations. Given this framework, I think it’s easy to replace Mineiro’s concept of bias with an approximation error formalism that might be called asymptotic bias or definitional bias, as Efron does in this paper. These tools let one quantify the costs incurred by the use of a pseudo-true model rather than the true model.

I’d love to learn more about this issue. As it stands, I know too much about sampling and too little about quantifying the quality of approximations.

Some Observations on Winsorization and Trimming

Over the last few months, I’ve had a lot of conversations with people about the use of winsorization to deal with heavy-tailed data that is positively skewed because of large outliers. After a conversation with my friend Chris Said this past week, it became clear to me that I needed to do some simulation studies to understand the design space of techniques for dealing with outliers.

In this post, I’m going to write up my current understanding of the topic. To motivate my perspective, I’m going to describe a specific inferential problem. Trying to solve that problem will require me to explore a small portion of the design space of algorithms for dealing with outliers. It is not at all clear to me how well the conclusions I reach will generalize to other settings, but my hope is that a thorough examination of one specific problem will touch on some of the core issues that would arise in other settings. I’ve also put the code for this simulation study on GitHub so that others can easily build variants of it.

Defining Outliers

To simplify things, I’m going to define an outlier as any value greater than a certain percentile of all of the observed data points. For example, I might say that all observations above the 90th percentile are outliers. In this case, I will say that the “outlier window” is 10%, because I will consider all observations above the (100 – 10)th percentile to be outliers. This outlier window parameter is the first dimension of the design space that I want to understand. Most applications employ a very gentle outlier window, which labels no more than 5% of observations as outliers. But sometimes one sees more radical windows used. In this post, I will consider windows ranging from 1% up to 40%.

Outlier Removal

Given that we’ve flagged some observations as outliers, what do we do with them? I’ll consider three options:

  • Do nothing and leave the data unadjusted.
  • Discard all of the outliers. The removal of extreme values is usually called trimming or truncation.
  • Replace all of the outliers with the largest value that is not considered an outlier. The replacement of extreme values is usually called winsorization.

The Data Generation Process and Inferential Problem

As a further simplification, I’m only going to consider a single inferential problem: estimating the difference in means between two groups of observations. In this problem, I will generate data as follows:

  • Draw \(n\) IID observations from a distribution \(D\). Label this set of observations as Group A.
  • Draw \(n\) additional IID observations from the same distribution \(D\) and add a constant, \(\delta\), to every one of them. Label this set of observations as Group B.

In this post, the distribution, \(D\), that I will use is a log-normal distribution with mean 1.649 and standard deviation 2.161. These strange values correspond to a mean of 0 and a standard deviation of 1 on the pre-exponentiation scale used by my RNG.

Given that distribution, \(D\), I’ll consider three values of \(\delta\): 0.00, 0.05 and 0.10. Each value of \(\delta\) defines a new distribution, \(D^{\prime}\).

To contrast these distributions, I want to consider the quantity: \(\delta = \mathbb{E}[D^{\prime}] – \mathbb{E}[D]\). The outlier approaches I’ll consider will lead to several different estimators. I’ll refer to each of these estimators as \(\hat{\delta}\). From each estimator, I’ll construct point estimates and interval estimates; I’ll also conduct hypothesis tests based on the estimators using t-tests.

Selection of Outlier Windows

Given that we have two data sets, \(A\) and \(B\), applying outlier detection requires us to make another decision about our position in design space: which data sets do we use to compute the percentile that defines the outlier threshold?

I’ll consider three options:

  • Compute a single percentile from a new data set formed by combining A and B. In the results, I’ll call this the “shared threshold” rule.
  • Compute a single percentile using data from A only. In the results, I’ll call this the “threshold from A” rule.
  • Compute two percentiles: one using data from A only and another using data from B only. We will then apply the threshold from A to A’s data and apply the threshold from B to B’s data. In the results, I’ll call this the “separate thresholds” rule.

Combining this new design decision with the others already mentioned, the design space I’m interested in has three dimensions:

  • The outlier window.
  • The method for addressing outliers, which is either (a) no adjustment, (b) trimming, or (c) winsorization.
  • The data set used to compute the outlier threshold, which is a shared threshold, a separate threshold or a threshold derived from A’s observations only.

Results

I consider the quality of inferences in terms of several metrics:

  • Point Estimation:
    • The bias of \(\hat{\delta}\) as an estimator of \(\delta\).
    • The RMSE of \(\hat{\delta}\) as an estimator of \(\delta\).
  • Interval Estimation:
    • The coverage probability of nominal 95% CI’s for \(\delta\) built around \(\hat{\delta}\).
  • Hypothesis Testing:
    • The false positive rate of a t-test when \(\delta = 0.00\).
    • The false negative rates of t-tests when \(\delta = 0.05\) or \(\delta = 0.10\).

Point Estimation

The results for the point estimation problem are shown graphically below. I begin by plotting KDE’s of the sampling distributions of the seven estimators I’m considering, which give an intuition for all of the core results shown later. The columns in these plots reflect the different values of \(\delta\), which are either 0.00, 0.05 or 0.10.

delta_kde

Next I plot the bias of these estimators as a function of the outlier window. I split these plots according to the true value of \(\delta\):

delta_bias

Finally, I plot the RMSE of these estimators as a function of the outlier window, again splitting the plots according to the true value of \(\delta\):

delta_rmse

Interval Estimation

To assess the quality of interval estimates derived from these estimators of \(\hat{\delta}\), I plot the empirical coverage probability of nominal 95% CI’s:

coverage

Hypothesis Testing

In this last set of analyses, I plot the false positive and false negative rates as a function of the outlier window:

fp_rates

fn_rates

And, for one final analysis, I include a plot of false positive rate when \(\delta = 0.00\) vs false negative rate when \(\delta = 0.05\). This makes it easier to see which methods are strictly dominated by others:

fp_fn

Take-Aways

  • Both winsorization and trimming can improve the quality of point estimates of \(\delta\). But they can make inferences worse because they can introduce bias into these point estimates. The only way to prevent bias is to compute outlier thresholds in Groups A and B separately.
  • There is sometimes an optimal outlier window that minimizes the RMSE of point estimates of \(\delta\). It is not clear how to calculate this without knowing the ground truth parameter that you are trying to infer. Perhaps cross-validation would work. Importantly, choosing a bad value for the outlier window makes both winsorization and trimming inferior to doing no adjustment.
  • Both winsorization and trimming can improve the FP/FN tradeoffs made in a system, although one must use outlier thresholds that are shared across the A and B groups to reap these benefits
  • CI’s only provide the desired coverage probabilities when the outlier window is very small.

Putting all this together, my understanding is that there is no universally best method without a more complex approach to outlier control. Whether winsorization or trimming improves the quality of inference depends on the structure of the data being analyzed and the inferential approach being used. Some methods work well for point estimation and interval estimation, others work well for hypothesis testing. No method universally beats out the option of doing no adjustment for outliers.

A Note about Likely Failures of Generalization

If you consider a different kind of difference between the A and B sets of observations than the constant additive difference rule that I’ve considered, I am fairly sure that the conclusions I’ve reached will not apply.

Why Julia’s DataFrames are Still Slow

Introduction

Although I’ve recently decided to take a break from working on OSS for a little while, I’m still as excited as ever about Julia as a language.

That said, I’m still unhappy with the performance of Julia’s core data analysis infrastructure. The performance of code that deals with missing values has been substantially improved thanks to the beta release of the NullableArrays package, which David Gold developed during this past Julia Summer of Code. But the DataFrames package is still a source of performance problems.

The goal of this post is to explain why Julia’s DataFrames are still unacceptably slow in many important use cases — and will remain slow even after the current dependency on the DataArrays package is replaced with a dependency on NullableArrays.

Problematic Interactions with Julia’s Compiler

The core problem with the DataFrames library is that a DataFrame is, at its core, a black-box container that could, in theory, contain objects of arbitrary types. In practice, a DataFrame contains highly constrained objects, but those constraints are (a) hard to express to the compiler and (b) still too weak to allow the compiler to produce the most efficient machine code.

The use of any black-box container creates the potential for performance problems in Julia because of the way that Julia’s compiler works. In particular, Julia’s compiler is able to execute code quickly because it can generate custom machine code for every function call — and this custom machine code is specialized for the specific run-time types of the function’s arguments.

This run-time generation of custom machine code is called specialization. When working with black-box containers, Julia’s approach to specialization is not used to full effect because machine code specialization based on run-time types only occurs at function call sites. If you access objects from a black-box container and then perform extended computations on the results, those computations will not be fully specialized because there is no function call between (a) the moment at which type uncertainty about the contents of the black-box container is removed and (b) the moment at which code that could benefit from type information is executed.

A Minimal Example

To see this concern in practice, consider the following minimal example of a hot loop being executed on values that are extracted from a black-box container:

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function g1(black_box_container)
    x, y = black_box_container[1], black_box_container[2]
    n = length(x)
    s = 0.0
    for i in 1:n
        s += x[i] * y[i]
    end
    s
end
 
function hot_loop(x, y)
    n = length(x)
    s = 0.0
    for i in 1:n
        s += x[i] * y[i]
    end
    s
end
 
function g2(black_box_container)
    x, y = black_box_container[1], black_box_container[2]
    hot_loop(x, y)
end
 
container = Any[randn(10_000_000), randn(10_000_000)];
 
@time g1(container)
# 2.258571 seconds (70.00 M allocations: 1.192 GB, 5.03% gc time)
 
@time g2(container)
# 0.015286 seconds (5 allocations: 176 bytes)

g1 is approximately 150x slower than g2 on my machine. But g2 is, at a certain level of abstraction, exactly equivalent to g1 — the only difference is that the hot loop in g1 has been put inside of a function call. To convince yourself that the function call boundary is the only important difference between these two functions, consider the following variation of g2 and hot_loop:

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@inline function hot_loop_alternative(x, y)
    n = length(x)
    s = 0.0
    for i in 1:n
        s += x[i] * y[i]
    end
    s
end
 
function g3(black_box_container)
    x, y = black_box_container[1], black_box_container[2]
    hot_loop_alternative(x, y)
end
 
@time g1(container)
# 2.290116 seconds (70.00 M allocations: 1.192 GB, 4.90% gc time)
 
@time g2(container)
# 0.017835 seconds (5 allocations: 176 bytes)
 
@time g3(container)
# 2.250301 seconds (70.00 M allocations: 1.192 GB, 5.08% gc time)

On my system, forcing the hot loop code to be inlined removes all of the performance difference between g1 and g2. Somewhat ironically, by inlining the hot loop, we’ve prevented the compiler from generating machine code that’s specialized on the types of the x and y values we pull out of our black_box_container. Inlining removes a function call site — and function call sites are the only times when machine code can be fully specialized based on run-time type information.

This problem is the core issue that needs to be resolved to make Julia’s DataFrames as efficient as they should be. Below I outline three potential solutions to this problem. I do not claim that these three are the only solutions; I offer them only to illustrate important issues that need to be addressed.

Potential Solutions to the Under-Specialization Problem

One possible solution to the problem of under-specialization is to change Julia’s compiler. I think that work on that front could be very effective, but the introduction of specialization strategies beyond Julia’s current "specialize at function call sites" would make Julia’s compiler much more complex — and could, in theory, make some code slower if the compiler were to spend more time performing compilation and less time performing the actual computations that a user wants to perform.

A second possible solution is to generate custom DataFrame types for every distinct DataFrame object. This could convert DataFrames from black-box containers that contain objects of arbitrary type into fully typed containers that can only contain objects of types that are fully known to the compiler.

The danger with this strategy is that you could generate an excessively large number of different specializations — which would again run the risk of spending more time inside the compiler than inside of the code you actually want to execute. It could also create excessive memory pressure as an increasing number of specialized code paths are stored in memory. Despite these concerns, a more aggressively typed DataFrame might be a powerful tool for doing data analysis.

The last possible solution I know of is the introduction of a high-level API that ensures that operations on DataFrames always reduce down to operations on objects whose types are known when hot loops execute. This is essentially the computational model used in traditional databases: take in a SQL specification of a computation, make use of knowledge about the data actually stored in existing tables to formulate an optimized plan for performing that computation, and then perform that optimized computation.

I think this third option is the best because it will also solve another problem Julia’s data infrastructure will hit eventually: the creation of code that is insufficiently generic and not portable to other backends. If people learn to write code that only works efficiently for a specific implementation of DataFrames, then their code will likely not work when they try to apply it to data stored in alternative backends (e.g. traditional databases). This would trap users into data structures that may not suit their needs. The introduction of a layer of appropriate abstractions (as in dplyr and Ibis) would resolve both issues at once.

Take-Aways

  • Making Julia’s DataFrames better is still a work-in-progress.
  • The core issue is still the usage of data structures that are not amenable to Julia’s type inference machinery. One of the two main issues is now resolved; another must be addressed before things function smoothly.
  • Several solutions to this remaining are possible; we will probably see one or more of these solutions gain traction in the near-term future.

Rereading Meehl

Lately, I’ve been rereading a lot of Meehl’s papers on the epistemological problems with research in psychology. This passage from “The Problem Is Epistemology, Not Statistics: Replace Significance Tests by Confidence Intervals and Quantify Accuracy of Risky Numerical Predictions” strikes me as an almost perfect summary of his concerns, although it’s quite abstract and assumes familiarity with his “crud factor principle”, which asserts that all variables in psychology are correlated with all other variables.

I said that the crud factor principle is the concrete empirical form, realized in the sciences, of the logician’s formal point that the third figure of the implicative (mixed hypothetical) syllogism is invalid, the error in purported deductive reasoning termed affirming the consequent. Speaking methodologically, in the language of working scientists, what it comes to is that there are quite a few alternative theories \(T’\), \(T”\), \(T”’\), \(\ldots\) (in addition to the theory of interest \(T\)) that are each capable of deriving as a consequence the statistical counternull hypothesis \(H^{*}: \delta = (\mu_1 – \mu_2) > 0\), or, if we are correlating quantitative variables, that \(\rho > 0\). We might imagine (Meehl, 1990e) a big pot of variables and another (not so big but still sizable) pot of substantive causal theories in a specified research domain (e.g., schizophrenia, social perception, maze learning in the rat). We fantasize an experimenter choosing elements from these two pots randomly in picking something to study to get a publication. (We might impose a restriction that the variables have some conceivable relation to the domain being investigated, but such a constraint should be interpreted very broadly. We cannot, e.g., take it for granted that eye color will be unrelated to liking introspective psychological novels, because there is evidence that Swedes tend to be more introverted than Irish or Italians.) Our experimenter picks a pair of variables randomly out of the first pot, and a substantive causal theory randomly out of the second pot, and then randomly assigns an algebraic sign to the variables’ relation, saying, “\(H^{*}: \rho > 0\), if theory \(T\) is true.” In this crazy example there is no semantic-logical-mathematical relation deriving \(H^{*}\) from \(T\), but we pretend there is. Because \(H_0\) is quasi-always false, the counternull hypothesis \(~H_0\) is quasi-always true. Assume perfect statistical power, so that when \(H_0\) is false we shall be sure of refuting it. Given the arbitrary assignment of direction, the directional counternull \(H^{*}\) will be proved half the time; that is, our experiment will “come out right” (i.e., as pseudo-predicted from theory \(T\)) half the time. This means we will be getting what purports to be a “confirmation” of \(T\) 10 times as often as the significance level \(\alpha = .05\) would suggest. This does not mean there is anything wrong with the significance test mathematics; it merely means that the odds of getting a confirmatory result (absent our theory) cannot be equated with the odds given by the \(t\) table, because those odds are based on the assumption of a true zero difference. There is nothing mathematically complicated about this, and it is a mistake to focus one’s attention on the mathematics of \(t\), \(F\), chi-square, or whatever statistic is being employed. The population from which we are drawing is specified by variables chosen from the first pot, and one can think of that population as an element of a superpopulation of variable pairs that is gigantic in size but finite, just as the population, however large, of theories defined as those that human beings will be able to construct before the sun burns out is finite. The methodological point is that \(T\) has not passed a severe test (speaking Popperian), the “successful” experimental outcome does not constitute what philosopher Wesley Salmon called a “strange coincidence” (Meehl, 1990a, 1990b; Nye, 1972; Salmon, 1984), because with high power \(T\) has almost an even chance of doing that, absent any logical connection whatever between the variables and the theory.

Put another way: what’s wrong with psychology is that the theories being debated make such vague predictions as to be quasi-unfalsifiable. NHST is bad for the field because it encourages researchers to make purely directional predictions — that is, to make predictions that are so vague that their success is, to use an old-fashioned word, vain-glorious.

What’s Wrong with Statistics in Julia?

Introduction

Several months ago, I promised to write an updated version of my old post, “The State of Statistics in Julia”, that would describe how Julia’s support for statistical computing has evolved since December 2012.

I’ve kept putting off writing that post for several reasons, but the most important reason is that all of my attention for the last few months has been focused on what’s wrong with how Julia handles statistical computing. As such, the post I’ve decided to write isn’t a review of what’s already been done in Julia, but a summary of what’s being done right now to improve Julia’s support for statistical computing.

In particular, this post focuses on several big changes to the core data structures that are used in Julia to represent statistical data. These changes should all ship when Julia 0.4 is released.

What’s Wrong with Statistics in Julia Today?

The primary problem with statistical computing in Julia is that the current tools were all designed to emulate R. Unfortunately, R’s approach to statistical computing isn’t amenable to the kinds of static analysis techniques that Julia uses to produce efficient machine code.

In particular, the following differences between R and Julia have repeatedly created problems for developers:

  • In Julia, computations involving scalars are at least as important as computations involving vectors. In particular, iterative computations are first-class citizens in Julia. This implies that statistical libraries must allow developers to write efficient code that iterates over the elements of a vector in pure Julia. Because Julia’s compiler can only produce efficient machine code for computations that are type-stable, the representations of missing values, categorical values and ordinal values in Julia programs must all be type-stable. Whether a value is missing or not, its type must remain the same.
  • In Julia, almost all end-users will end up creating their own types. As such, any tools for statistical computing must be generic enough that they can be extended to arbitrary types with little to no effort. In contrast to R, which can heavily optimize its algorithms for a very small number of primitive types, Julia developers must ensure that their libraries are both highly performant and highly abstract.
  • Julia, like most mainstream languages, eagerly evaluates the arguments passed to functions. This implies that idioms from R which depend upon non-standard evaluation are not appropriate for Julia, although it is possible to emulate some forms of non-standard evaluation using macros. In addition, Julia doesn’t allow programmers to reify scope. This implies that idioms from R that require access to the caller’s scope are not appropriate for Julia.

The most important way in which these issues came up in the first generation of statistical libraries was in the representation of a single scalar missing value. In Julia 0.3, this concept is represented by the value NA, but that representation will be replaced when 0.4 is released. Most of this post will focus on the problems created by NA.

In addition to problems involving NA, there were also problems with how expressions were being passed to some functions. These problems have been resolved by removing the function signatures for statistical functions that involved passing expressions as arguments to those functions. A prototype package called DataFramesMeta, which uses macros to emulate some kinds of non-standard evaluation, is being developed by Tom Short.

Representing Missing Values

In Julia 0.3, missing values are represented by a singleton object, NA, of type NAtype. Thus, a variable x, which might be either a Float64 value or a missing value encoded as NA, will end up with type Union(Float64, NAtype). This Union type is a source of performance problems because it defeats Julia’s compiler’s attempts to assign a unique concrete type to every variable.

We could remove this type-instability by ensuring that every type has a specific value, such as NaN, that signals missingness. This is the approach that both R and pandas take. It offers acceptable performance, but does so at the expense of generic handling of non-primitive types. Given Julia’s rampant usage of custom types, the sentinel values approach is not viable.

As such, we’re going to represent missing values in Julia 0.4 by borrowing some ideas from functional languages. In particular, we’ll be replacing the singleton object NA with a new parametric type Nullable{T}. Unlike NA, a Nullable object isn’t a direct scalar value. Rather, a Nullable object is a specialized container type that either contains one value or zero values. An empty Nullable container is taken to represent a missing value.

The Nullable approach to representing a missing scalar value offers two distinct improvements:

  • Nullable{T} provides radically better performance than Union(T, NA). In some benchmarks, I find that iterative constructs can be as much as 100x faster when using Nullable{Float64} instead of Union(Float64, NA). Alternatively, I’ve found that Nullable{Float64} is about 60% slower than using NaN to represent missing values, but involves a generic approach that trivially extends to arbitrary new types, including integers, dates, complex numbers, quaternions, etc…
  • Nullable{T} provides more type safety by requiring that all attempts to interact with potentially missing values explicitly indicate how missing values should be treated.

In a future blog post, I’ll describe how Nullable works in greater detail.

Categorical Values

In addition to revising the representation of missing values, I’ve also been working on revising our representation of categorical values. Working with categorical data in Julia has always been a little strange, because the main tool for representing categorical data, the PooledDataArray, has always occupied an awkward intermediate position between two incompatible objectives:

  • A container that keeps track of the unique values present in the container and uses this information to efficiently represent values as pointers to a pool of unique values.
  • A container that contains values of a categorical variable drawn from a well-defined universe of possible values. The universe can include values that are not currently present in the container.

These two goals come into severe tension when considering subsets of a PooledDataArray. The uniqueness constraint suggests that the pool should shrink, whereas the categorical variable definition suggests that the pool should be maintained without change. In Julia 0.4, we’re going to commit completely to the latter behavior and leave the problem of efficiently representing highly compressible data for another data structure.

We’ll also begin representing scalar values of categorical variables using custom types. The new CategoricalVariable and OrdinalVariable types that will ship with Julia 0.4 will further the efforts to put scalar computations on an equal footing with vector computations. This will be particularly notable for dealing with ordinal variables, which are not supported at all in Julia 0.3.

Metaprogramming

Many R functions employ non-standard evaluation as a mechanism for augmenting the current scope with the column names of a data.frame. In Julia, it’s often possible to emulate this behavior using macros. The in-progress DataFramesMeta package explores this alternative to non-standard evaluation. We will also be exploring other alternatives to non-standard evaluation in the future.

What’s Next

In the long-term future, I’m hoping to improve several other parts of Julia’s core statistical infrastructure. In particular, I’d like to replace DataFrames with a new type that no longer occupies a strange intermediate position between matrices and relational tables. I’ll write another post about those issues later.

The Lesser Known Normal Forms of Database Design

-1st Normal Form: The database contains at least one table that is an exact copy of another table, except with additional columns.

-2nd Normal Form: The database contains at least one table that is a corrupt, out-of-date copy of another table, except with additional columns. It is impossible to determine if these additional columns can be trusted.

-3rd Normal Form: The database contains at least one table whose name contains the string _v1 and another table whose name contains the string _v2. At least one column in the _v1 table must be an undeclared foreign key that refers to rows that no longer exist in any other table.

-4th Normal Form: The database contains at least one table that has two or more columns whose contents are exactly the same, except that one of the columns has a name that is a misspelling of the other column’s name.

-5th Normal Form: The database contains (A) at least one table whose name contains the string do_not_use and (B) at least one other table whose name contains the string do_not_ever_ever_use. In each of these tables separately, at least two columns must only contain NULL’s repeated for every single row. In addition, every string in these tables must have random amounts of whitespace padded on the left- and right-hand sides. Finally, at least one row must contain the text, “lasciate ogni speranza, voi ch’entrate.”

Values vs. Bindings: The Map is Not the Territory

Many newcomers to Julia are confused by the seemingly dissimilar behaviors of the following two functions:

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julia> a = [1, 2, 3]
3-element Array{Int64,1}:
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 3
 
julia> function foo!(a)
           a[1] = 10
           return
       end
foo! (generic function with 1 method)
 
julia> foo!(a)
 
julia> a
3-element Array{Int64,1}:
 10
  2
  3
 
julia> function bar!(a)
           a = [1, 2]
           return
       end
bar! (generic function with 1 method)
 
julia> bar!(a)
 
julia> a
3-element Array{Int64,1}:
 10
  2
  3

Why does the first function successfuly alter the global variable a, but the second function does not?

To answer that question, we need to explain the distinction between values and bindings. We’ll start with a particularly simple example of a value and a binding.

In Julia, the number 1 is a value:

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julia> 1
1

In contrast to operating on a value, the Julia assignment operation shown below creates a binding:

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julia> a = 1
1

This newly created binding is an association between the symbolic name a and the value 1. In general, a binding operation always associates a specific value with a specific name. In Julia, the valid names that can be used to create bindings are symbols, because it is important that the names be parseable without ambiguity. For example, the string "a = 1" is not an acceptable name for a binding, because it would be ambiguous with the code that binds the value 1 to the name a.

This first example of values vs. bindings might lead one to believe that values and bindings are very easy to both recognize and distinguish. Unfortunately, the values of many common objects are not obvious to many newcomers.

What, for example, is the value of the following array?

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julia> [1, 2, 3]
3-element Array{Int64,1}:
 1
 2
 3

To answer this question, note that the value of this array is not defined by the contents of the array. You can confirm this by checking whether Julia considers two objects to be exactly identical using the === operator:

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julia> 1 === 1
true
 
julia> [1, 2, 3] === [1, 2, 3]
false

The general rule is simple, but potentially non-intuitive: two arrays with identical contents are not the same array. To motivate this, think of arrays as if they were cardboard boxes. If I have two cardboard boxes, each of which contains a single ream of paper, I would not claim that the two boxes are the exact same box just because they have the same contents. Our intuitive notion of object identity is rich enough to distinguish between two containers with the same contents, but it takes some time for newcomers to programming languages to extend this notion to their understanding of arrays.

Because every container is distinct regardless of what it contains, every array is distinct because every array is its own independent container. An array’s identity is not defined by what it contains. As such, its value is not equivalent to its contents. Instead, an array’s value is a unique identifier that allows one to reliably distinguish each array from every other array. Think of arrays like numbered cardboard boxes. The value of an array is its identifier: thus the value of [1, 2, 3] is something like the identifier “Box 1”. Right now, “Box 1” happens to contain the values 1, 2 and 3, but it will continue to be “Box 1” even after its contents have changed.

Hopefully that clarifies what the value of an array is. Starting from that understanding, we need to re-examine bindings because bindings themselves behave like containers.

A binding can be thought of as a named box that can contain either 0 or 1 values. Thus, when a new Julia session is launched, the name a has no value associated with it: it is an empty container. But after executing the line, a = 1, the name has a value: the container now has one element in it. Being a container, the name is distinct from its contents. As such, the name can be rebound by a later operation: the line a = 2 will change the contents of the box called a to refer to the value 2.

The fact that bindings behave like containers becomes a source of confusion when the value of a binding is itself a container:

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a = [1, 2, 3]

In this case, the value associated with the name a is the identifier of an array that happens to have the values 1, 2, and 3 in it. But if the contents of that array are changed, the name a will still refer to the same array — because the value associated with a is not the contents of the array, but the identifier of the array.

As such, there is a very large difference between the following two operations:

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a[1] = 10
a = [1, 2]
  • In the first case, we are changing the contents of the array that a refers to.
  • In the second case, we are changing which array a refers to.

In this second case, we are actually creating a brand new container as an intermediate step to changing the binding of a. This new container has, as its initial contents, the values 1 and 2. After creating this new container, the name a is changed to refer to the value that is the identifier of this new container.

This is why the two functions at the start of this post behave so differently: one mutates the contents of an array, while the other mutates which array a name refers to. Because variable names in functions are local, changing bindings inside of a function does not change the bindings outside of that function. Thus, the function bar! does not behave as some would hope. To change the contents of an array wholesale, you must not change bindings: you must change the contents of the array. To do that, bar! should be written as:

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function bar!(a)
    a[:] = [1, 2]
    return
end

The notation a[:] allows one to talk about the contents of an array, rather than its identifier. In general, you should not expect that you can change the contents of any container without employing some indexing syntax that allows you to talk about the contents of the container, rather than the container itself.

That Way Madness Lies: Arithmetic on data.frames

tl;dr

Please do not use arithmetic on data.frame objects when programming in R. It’s a hack that only works if you know everything about your datasets. If anything happens to change the order of the rows in your data set, previously safe data.frame arithmetic operations will produce incorrect answers. If you learn to always explicitly merge two tables together before performing arithmetic on their shared columns, you’ll produce code that is both more reliable and more powerful.

Arithmetic between tables: getting wrong answers quickly

You may not be aware of it, but R allows you to do arithmetic on data.frame objects. For example, the following code works in R as of version 3.0.2:

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> df1 <- data.frame(ID = c(1, 2), Obs = c(1.0, 2.0))
> df2 <- data.frame(ID = c(1, 2), Obs = c(2.0, 3.0))
> df3 <- (df1 + df2) / 2
> df3
  ID Obs
1  1 1.5
2  2 2.5

If you discover that you can do this, you might think that it’s a really cool trick. You might even start using data.frame arithmetic without realizing that your specific example had a bunch of special structure that was directly responsible for you getting the right answer.

Unfortunately, other examples that you didn’t see would have produced rather less pleasant outputs and led you to realize that arithmetic operations on data.frame objects don’t really make sense:

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> df1 <- data.frame(ID = c(1, 2), Obs = c(1.0, 2.0))
> df2 <- data.frame(ID = c(2, 1), Obs = c(3.0, 2.0))
> df3 <- (df1 + df2) / 2
> df3
   ID Obs
1 1.5   2
2 1.5   2

What happened here is obvious in retrospect: R added all of the columns together and then divided the result by two. The problem is that you didn’t actually want to add all of the columns together and then divide the result by two, because you had forgotten that the matching rows in df1 and df2 were not in the same index positions in the two tables.

Getting right answers with just a little more typing

Thankfully, it turns out that doing the right thing just requires a few more characters. What you should have done was to call merge before doing any arithmetic:

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> df1 <- data.frame(ID = c(1, 2), Obs = c(1.0, 2.0))
> df2 <- data.frame(ID = c(2, 1), Obs = c(3.0, 2.0))
> df3 <- merge(df1, df2, by = "ID")
> df3 <- transform(df3, AvgObs = (Obs.x + Obs.y) / 2)
> df3
  ID Obs.x Obs.y AvgObs
1  1     1     2    1.5
2  2     2     3    2.5

What makes merge so unequivocally superior to data.frame arithmetic is that it still works when the two inputs have different numbers of rows:

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> df1 <- data.frame(ID = c(1, 2), Obs = c(1.0, 2.0))
> df2 <- data.frame(ID = c(1, 2, 3), Obs = c(5.0, 6.0, 7.0))
> df3 <- merge(df1, df2, by = "ID")
> df3 <- transform(df3, AvgObs = (Obs.x + Obs.y) / 2)
> df3
  ID Obs.x Obs.y AvgObs
1  1     1     5      3
2  2     2     6      4

Knowledge is half the battle

Now that you know why performing arithmetic operations on data.frame objects is generally unsafe, I implore you to stop doing it. Learn to love merge.