Evaluating discrete choice prediction models when the evaluation data is corrupted: analytic results and bias corrections for the area under the ROC

Abstract

There has been a growing recognition that issues of data quality, which are routine in practice, can materially affect the assessment of learned model performance. In this paper, we develop some analytic results that are useful in sizing the biases associated with tests of discriminatory model power when these are performed using corrupt (“noisy”) data. As it is sometimes unavoidable to test models with data that are known to be corrupt, we also provide some guidance on interpreting results of such tests. In some cases, with appropriate knowledge of the corruption mechanism, the true values of the performance statistics such as the area under the ROC curve may be recovered (in expectation), even when the underlying data have been corrupted. We also provide estimators of the standard errors of such recovered performance statistics. An analysis of the estimators reveals interesting behavior including the observation that “noisy” data does not “cancel out” across models even when the same corrupt data set is used to test multiple candidate models. Because our results are analytic, they may be applied in a broad range of settings and this can be done without the need for simulation.

Appendices

Appendix A: Some comments on data corruption in the development sample

Although beyond the scope of this paper, we note that the case of data corruption of the development sample represents a more involved problem as it can potentially affect the model parameters themselves. Fortunately, a number of aspects of this problem have been studied extensively in the statistics and econometrics literature, so in many cases, the effects of such mislabeling on model estimation are well understood and fix-ups have been developed for many common problems.

For example, it is well known that in general, random noise to the independent variables in regression problems will serve to bias downward the estimates of the model coefficients since the sum of squares of the independent variables ($\mathbf {X'X}$) will increase faster than the sum of squares of the dependent and independent variables ($\mathbf {X'y}$) implying that the ratio of the two will decline. This type of bias is often termed regression dilution.

For standard regression, corruption of the dependent variable will not bias the estimates of model coefficients, however it will reduce the efficiency of the estimation by adding additional error to the right-hand side. Unbiasedness does not hold though when the dependent variable is limited in certain ways, as it is in the case of a logit or probit specification where ${y}_{i}{\in }\{0,1\}$. In such cases, the coefficients estimated via maximum likelihood will be biased.

As discussed in Sect. 4.1, there are known methods for correcting for this bias by simultaneously estimating both the model coefficients and the rate of mislabeling in the sample (cf., Hausman et al. 1998).

Finally, if there are “bad” records missing from the development sample, the situation is that of a sample base rate that differs from the true base rate (and is lower). In this case, estimation is still feasible, but the predicted probabilities will be biased downward due to the “lower” prevalence in the estimation data. Elkan (2001) provides a proof that when the base rates of the development sample and the evaluation sample differ, the observed probability ${p}_{i}^{*}$, for the $i\mathrm{th}$ observation in the evaluation sample is calculated as:

$$\begin{aligned} {p}_{i}^{*}={\pi }_{T}\frac{{p}_{i}-{p}_{i}{\pi }_{S}}{{\pi }_{S}-{p}_{i}{\pi }_{S}+{{p}_{i}\pi }_{T}-{\pi }_{S}{\pi }_{T}}, \end{aligned}$$

where ${p}_{i}$ is the raw probability produced by the model, ${\pi }_{S}$ is the baseline prevalence rate in the development sample, and ${\pi }_{T}$ is the baseline prevalence in the evaluation sample (Note that we adopt the notation used in Bohn and Stein (2009) in preference to that of Elkan (2001)).

Bohn and Stein (2009) provides a discussion of this approach as well as examples.

Note that the observations in most of this Appendix is premised on the data corruption being CAR or MCAR. If it is not the case that one of these assumption holds, alternative approaches are available in some cases, but the estimators typically become much more involved. This is also a problem area that has been studied extensively (see, e.g., (York 1969) for an early reference).

Appendix B: Recovering an approximate ROC curve (under parametric assumptions)

Some readers may find it useful to construct ROC curves in addition to calculating AUC statistics. It is feasible to generate representative ROC curves that are, under certain assumptions, consistent with the recovered values of the AUC, $\widehat{A_r}$. Because, in general, we do not know which records are corrupted (if we did, we would correct them), we cannot easily determine the recovered values of $(\widehat{FN}_r, \widehat{TP}_r)$, at individual points on the ROC. However, subject to assumptions about the distribution of the model scores, we can generate a parametric ROC.

If we make the assumption that G and B are distributed normally with means $\mu _G, \mu _B$ and variances $\sigma ^2_G, \sigma ^2_B$, respectively, it can then be shown that

$$\begin{aligned} ROC(u)={\varPhi }\left( a+b{\varPhi }^{-1}(u)\right) ,~~0\le u \le 1, \end{aligned}$$

(15)

where $a=(\mu _B- \mu _G)/\sigma ^2_B, b=\sigma ^2_G/\sigma ^2_B$ and ${\varPhi }(\cdot )$ and ${\varPhi }^{-1}(\cdot )$ are the cumulative and inverse cumulative normal distribution functions, respectively. It then follows that

$$\begin{aligned} A={\varPhi }\left( \frac{a}{\sqrt{1+b^2}}\right) . \end{aligned}$$

(16)

Finally, we can solve for each of either a or b in terms of the other (and A):

$$\begin{aligned} a= & {} {\varPhi }^{-1}(A) \times \sqrt{1+b^2} \end{aligned}$$

(17)

and

$$\begin{aligned} b=\sqrt{\left( \frac{a}{{\varPhi }^{-1}(A)}\right) ^2-1}. \end{aligned}$$

(18)

We do not know which records are corrupted so there is little guidance on how to adjust the estimates of the $\mu$ and $\sigma ^2$ parameters. Though there is no theoretical motivation for them, we present two heuristic approaches for plotting the recovered ROC curve. Both focus on estimating a based on assumptions about b.

The first is the simpler of the two and assumes that the variances of G and B do not change as a result of the corruption. Thus, we would plug in the estimates of $\sigma ^2_G$ and $\sigma ^2_B$, from the corrupted data.

The second, and more restrictive, assumes that the variances are equal, and thus that $b=\sigma ^2_G/\sigma ^2_B=1$. Then

$$\begin{aligned} a={\varPhi }^{-1}(A) \times \sqrt{2}. \end{aligned}$$

(19)

In either case, the ROC curve may be generated by assuming a value for b, solving for a, and then calculating Eq. 15 for values of u in [0, 1].

Example 10

(Recovering an estimated ROC from corupted data using an estimate of $\hat{A}_r$) Consider again the analyst from Example 4 who is evaluating a bankruptcy model using a corrupted dataset of anonymized financial statement data and corresponding default flags. In Example 4 the analyst calculated that $\widehat{A}_r= 0.814$ (based on the original estimate of $\hat{A}_c=0.73$).

If the analyst wished to produce an estimated ROC curve, and were comfortable making the assumption that the variances of the model scores for the defaulting and non-defaulting firms were equal, he could use Eq. 19 to estimate a, and then use Eq. 15 with $b=1$ to generate the ROC curve.

$$\begin{aligned} a= & {} {\varPhi }^{-1}(A) \times \sqrt{2}.\\= & {} {\varPhi }^{-1}(0.814) \times \sqrt{2}.\\\approx & {} 1.26\\ ROC(u)= & {} {\varPhi }\left( 1.26+{\varPhi }^{-1}(u)\right) ,~~0\le u \le 1. \end{aligned}$$

The resulting curve is shown in Fig. 6. For comparison, the same curve, plotted for the original estimate of $\widehat{A_c}=0.73$ is shown as a dashed line.

Fig. 6

Recovered estimate of ROC curve based on analysis shown in Example 10. Solid line shows ROC based on recovered value of A ($\widehat{A}_r$), dashed line shows ROC based on original estimate ($\widehat{A}_c$). Assumes constant variance ($\sigma ^2_G=\sigma ^2_B$)

We note that because of the various assumptions, ROC curves generated in this fashion will likely differ from those that would have been recovered, e.g., nonparametrically, were the corruption mechanism known. However, for some applications, and with caveats about the strong assumptions above, this approach may still provide helpful visual guidance.