Are Logistic Models Really Interpretable?

1 Introduction

In high-stakes domains such as finance and healthcare, there is renewed interest in inherently interpretable models [Lipton, 2018; Molnar, 2022; Gunning, 2019], where the model form is such that it admits useful explanations of its output without any post-processing. Calls for transparency of algorithms being used on the public are widespread [Veale et al., 2018]. Within finance there is already increased regulatory scrutiny [OCC, 2021; Commission, 2021; Register, 2021] being introduced with regards to the usage of AI, which could extend in the future - within certain contexts - to require full algorithmic transparency, i.e. sharing model coefficients.

One of the prototypical inherently interpretable classification models often used as a baseline is Logistic Regression (LR) [Molnar, 2022]. Other additive models such as Generalised Additive Models (GAMs) and variants thereof [Lou et al., 2013; Gkolemis et al., 2023] generalise LR to have more flexibility [Hastie et al., 2009] and are also considered to be interpretable. As with LR, such models entail evaluating a real-valued function of the input data in logit, or log-odds space, that is subsequently transformed into probability space via a non-linear logistic link function. We call this class of models logistic models - a rigorous definition is given in Definition 2.1. In some sense, logistic models can be thought of as reasoning in logit space, in that the model weights naturally find their interpretation in terms of log-odds rather than probabilities, the units of the eventual model output.

Logistic models are now ubiquitous within Explainable AI (XAI) [Lou et al., 2013; Sudjianto and Zhang, 2021; Vaughan et al., 2018] but the literature is scant on evaluating the interpretability of these models. Indeed, there is a small amount of evidence to the contrary [Harris, 2017; von Hippel, 2015], with no more thorough study to the authors' knowledge. To what extent is this family of models truly interpretable? The present work aims to (at least partially) answer this question. We show via a User Study that for at least one definition of interpretability, based on Human-Grounded Evaluation [Doshi-Velez and Kim, 2017], such models provide limited and misleading explanations. For contexts where a certain level of interpretability is required we propose a remedy, Linearised Additive Models (LAM), that largely keeps the properties of any base logistic model the same, while dispensing with the non-linearities that cause confusion when using model weights as explanations.

Contributions. We outline the primary contributions below.

1. Identification of interpretability limitations for LR. A concrete motivating example demonstrating that model explanations provided in log-odds can be difficult for humans to interpret.
2. Linearised Additive Models (LAM). An efficient procedure to convert any trained logistic additive model that reasons in log odds to one that reasons directly about probabilities, without any retraining. For the special case of LR, LAM is rigorously proved to be the optimal approximation out of a large class of possible models.
3. Empirical evaluation of performance preservation. On a collection of public datasets from credit modelling, we establish that there is only a very small penalty in classification performance and a somewhat larger - but still small - penalty in calibration incurred for using

LAMs versus logistic models.

1. User evaluation. We conduct a user study with N = 36 participants, concluding via Human-Grounded Evaluation that LAMs are more interpretable than logistic models, as suggested by the motivating example. The measured outcomes of the user study are statistically significant.

Related Work. There is a vast literature on defining and evaluating performance of inherently interpretable models in general, a non-exhaustive group of which is [Dash et al., 2018; Lou et al., 2013; Yang et al., 2021; Vaughan et al., 2018; Kraus and Feuerriegel, 2019; De Bock and De Caigny, 2021; Vidal and Schiffer, 2020; Alaa and van der Schaar, 2019]. The evaluation of interpretability itself is still an open question, with detailed discussion of this issue in the references [Halliwell et al., 2022; Chen et al., 2022b; Narayanan et al., 2018; Lipton, 2018]. There are several different approaches, with the current preferred (and most expensive) approach being User Studies, as this allows one to directly measure resulting outcomes from explanations [Doshi-Velez and Kim, 2017]. Indeed, measuring explanation quality via measured outcomes when humans are asked to simulate the action of an algorithm predates most of the AI interpretability literature, for instance works such as [Kulesza et al., 2013]. [Rong et al., 2022] provide a recent survey paper on User Studies for evaluating explanations and interpretability of AI models.

The closest works in the literature to this paper are [Abdul et al., 2020; Poursabzi-Sangdeh et al., 2021]. In [Abdul et al., 2020], the authors measure interpretability of sparse linear models and GAMs via user studies that measure cognitive load on participants carrying out tasks with these models and their associated explanations. In the work by [Poursabzi-Sangdeh et al., 2021], users are presented with the coefficients of 2 and 8-variable linear models. The quality of the explanations from each model is measured by how adept users are at simulating the action of the model. Crucially, both these works evaluate regression models and are not concerned with issues arising due to the non-linearity of the sigmoid transformation required for logistic classification models.

The present work highlights that experts are not immune from misinterpreting certain model explanations, as also observed in the works: [Kaur et al., 2020; Bhatt et al., 2020].

Low-degree polynomial approximations to the sigmoid are employed in Private Machine Learning, e.g. [Kim et al., 2018a; Chen et al., 2018; Kim et al., 2018b], but to our knowledge the piecewise linear approximation in this work is unique.

Structure. In Section 2 we provide the motivating example and present the LAM definition and optimality results. Section 3 contains the performance evaluation of LAMs against their logistic counterparts and Section 4 details the User Study. We include most detail in the main text for the User Study, providing derivations, proofs and experimental details in the Supplementary Material (SM). We conclude with limitations and future work in Section 5.

Notation. This work considers binary classification, and we use {(x(j) , y(j) )}Mj=1 to denote the data, where x ∈ Rd is a vector of numerical features in a given dataset. The binary labels are indicators of some (usually bad) event such as a loan default: yi ∈ {0, 1}. We assume data points are drawn i.i.d. For a point x ∈ Rd , we denote the ith element of x by xi. The ith Euclidean basis vector is denoted by ei. The sigmoid function is defined as σ(z) := (1 + e-z )-1 for z ∈ R. We follow credit modelling terminology, where risk ˆ y(x(j) ) is a model's subjective probability in [0, 1] for a data point x(j) to be of positive class, that is y(j) = 1. The set [d] := {1, . . . , d} for d ∈ N.

2 Logistic and Linear ProbabilityModelling

GAMs [Hastie et al., 2009; Molnar, 2022] are a widely-known and long standing class of models considered to be inherently interpretable. In this work we restrict attention to GAMs without feature interactions. We formalise the notion of additive models as understood in this paper in Definition 2.1.

Definition 2.1 (Logistic Additive Model). Let x ∈ Rd . We call ˆ y : Rd → [0, 1] a logistic additive model if takes the form ˆ y(x) = σ(f(x)) := σ(β0 + ∑di=1 βifi(xi)), where the bias β0 ∈ R and for all i ∈ [d], βi ∈ R and the fi : R → R are univariate shape functions.

We refer to logistic additive models as defined in Definition 2.1 simply as logistic models or additive models when it is clear from context. The simplest and most common additive model in wide usage is LR, where fi(xi) = xi for all i ∈ [d] [Hastie et al., 2009]. Another example would be an Explainable Boosting Machine of [Lou et al., 2013] for classification, where the fi are piecewise constant functions (when there are no feature interactions).

2.1 Logistic Modelling and Interpretability

Historically, linear probability modelling, i.e. linear regression on dichotomous variables, was used prior to the advent of efficient methods for fitting LR models; see [Aldrich and Nelson, 1984; Hastie et al., 2009]. It is generally accepted that the application of linear regression to binary classification problems is unwise due to the propensity of the model returning probability estimates outside the [0, 1] interval and sensitivity to outliers [Ng, 2011; Hastie et al., 2009; Molnar, 2022]. In certain circumstances, these issues are not observed, with linear regression obtaining similar classification performance to LR [Hellevik, 2009; von Hippel, 2015]. LR models have superceded linear probability models.

The LR model coefficients are typically interpreted as follows [Molnar, 2022; Hastie et al., 2009]: a unit change in variable xi leads to a multiplicative increase in odds for the positive class of exp(βi). However, as observed by [Harris, 2017; von Hippel, 2015] this interpretation can be unwieldy for experts and nigh-on impossible for non-experts to reason with when we are concerned with probabilities, which is how the model outputs are typically presented and thought about.

Motivating Example. Suppose there is a LRmodel ˆ y used to predict some negative outcome and coefficients are shared with downstream users of the model. Inputs with risk ≥ 0.5 are considered "high-risk" and "low-risk" otherwise. Referring to Figure 1, Alice has a predicted risk of ˆ y(x(A) ) = 0.1 and Bob has a predicted risk of ˆ y(x(B) ) = 0.25. Both Alice and Bob are interested in what happens to their risk if they increase the value of feature xi by one unit, all else equal. They are told the model coefficient βi = 1.61 ≈ ln 5, a common form of transparent model explanation. First, both exponentiate βi which gives 5. They now know that increasing feature xi by increases their odds by a factor 5. The model outputs a risk score in units of probability, so they now have to compute what increasing their odds corresponds to in probabilities.

In this instance for Alice, odds(A) = ˆ y(x(A) )/(1 - ˆ y(x(A) )) = 0.1/(1 - 0.1) ≈ 0.111. Alice then multiplies her odds by 5, yielding 0.556. Converting back to probabilities we have ˆ y(x(A) + ei) ≈ 0.556/(1 + 0.556) ≈ 0.357. Subtracting her original risk score, we have that increasing xi by one unit increases the risk by a probability of 0.257 and Alice remains low-risk. A similarly laborious computation gives an increase in risk for Bob to approximately 0.625. Bob would be considered high-risk under a unit increase in xi. This was not obvious on first inspection before carrying out the computation explicitly.

The nonlinearity of odds as a function of probabilities (and vice versa) means that users with different risk scores cannot attribute logistic regression model outputs to the model coefficients in the same way. Moreover, the necessary computations are such that one cannot easily reason about the model's inputoutput relationship without a significant amount of practice. On the contrary, the coefficients βi of a linear probability model admit the more direct interpretation of the increase in output model probability arising from a unit increase in xi, regardless of the risk value of the user in question.

2.2 Linearised Additive Models (LAMs)

Given the preceding example, we wish to keep the interpretability characteristics of linear probability modelling while simultaneously sidestepping the issues arising from using a non-linear link function as in LR. To this end, we present the Linearised Additive Model (LAM), which is defined with respect to an already trained logistic model, σ ◦ f. Informally, a LAM replaces the sigmoid link function with a clipping function and scales f by an affine transformation. Denote by Π[0,1] the projector from R onto the unit interval, that is Π[0,1](z) = max(0, min(1, z)).

Definition 2.2 (Linearised Additive Models (LAM)). Let x ∈ Rd and let ˆ y(x) = σ(f(x)) be an additive model per Definition 2.1. Moreover, set α⋆ := 80000 30773 ≈ 2.5996 as a universal constant. Then, the Linearised Additive Model, ˆ yLAM, relative to f is given by

For brevity we refer to ˆ yLAM as a linearised model and say that ˆ yLAM is the LAM induced by ˆ y.

The LAM as defined in Definition 2.2 is derived by considering the optimal 3-piece piecewise linear approximation (see Figure 2) to the sigmoid function in terms of squared error, using this function as a link function for f(x) and invoking linearity. We choose this family of approximating functions as 3 pieces gives the simplest non-trivial approximation to the logistic sigmoid, while simulataneously allowing for a similar interpretation to a linear probability model. Squared error is chosen as it is a common metric for function approximation [Hastie et al., 2009] for which we are able to derive a universal tractable approximator.

As an example, an LR model is written as ˆ y(x) = σ(∑di=0 βixi), where we fix x0 = 1 according to convention. Then, the linearised version will be ˆ yLAM(x) = Π[0,1]( 12 + ∑di=0 βi 2α⋆ xi). In the case of linearised LR we can interpret the coefficients βi/2α⋆ as the contribution to model output in probability space for a unit increase in xi, as with a linear probability model.

Remark 2.3. To train a LAM, all that is required is to train the underlying logistic additive model, then apply Definition 2.2 using the trained coefficients {βi}di=0 .

Motivating Example Revisited. Under an LR model, Alice and Bob had to interpret a unit increase in feature xi as having different and unintuitive effects on their respective risk scores. Using the induced LAM, a unit increase in xi gives rise to the same change in model output, regardless of the input, i.e. βi 2α⋆ = 1.61 2×2.5996 ≈ 0.310, modulo outputs greater than unity which will be clipped. Alice and Bob's respective inputs of 0.1 and 0.25 can be readily seen to change to 0.41 and 0.56. Note this gives the same qualitative result as the non-linearised model, namely Alice stays low-risk and Bob becomes highrisk. We surmise that this direct interpretation of the LAM coefficients incurs smaller cognitive overhead in answering questions of this type, in constrast to LR.

The following optimality result for the LAM approximation to LR lends theoretical support to our definition of LAMs.

Theorem 2.4 (LAMOptimality). Let PL3 be the space of 3piece piecewise linear functions of one variable and X = Rd . For any LR model ˆ y(x) = σ(f(x)) = σ(β0 + ∑di=1 βixi) on X, an approximator ˜ σ(f(x)) is defined for all ˜ σ ∈ PL3. Then, ˆ yLAM is the squared-error optimal approximator for arbitrary f, that is,

Proof (Sketch). One can show via symmetry arguments that the minimising approximator comes from a one-parameter function family, when d = 1 with f(x) = x. The error is a convex function of this parameter α, which is minimised at α⋆ . The argument can then be extended to d-dimensional, affine f(x) by considering the error integral explicitly, fromwhich the result follows.

3 Performance Comparison

In this section we detail our experiments comparing the model performance of logistic additive models against their linearised (LAM) counterparts.

3.1 Experimental Setup

Models. We compare all models to XGBoost [Chen et al., 2022a], with shorthand XGB, and XGB with monotone constraints imposed (MonoXGB). XGB classification performance serves as an effective upper-bound on the competing models. As state-of-the-art baseline logistic additive models we consider the Additive Risk Models (ARMs) of [Chen et al., 2022a], since they are GAMs that explicitly incorporate monotone constraints that are often required in sensitive domains such as finance [Reserve, 2011]. These models come in 1-layer (ARM1) and 2-layer (ARM2) variants. Further baselines include NNLR (Non-negative LR [Chen et al., 2022a]) with raw feature inputs as an effective lower bound on model performance. For an additive model with shorthand M, we denote its linearised version (in the sense of Definition 2.2) by LAM-M. We include the linearised models LAM-NNLR, LAM-ARM1 and LAM-ARM2 in our experiments. LAMARM2 has both the individual subscale models and global NNLRmodel linearised.

Datasets. In this work we are principally interested in the consumer credit domain. Bankruptcy prediction datasets are also included due to their similar problem structure and origin. We consider publically available datasets from the UCI repository [Kelly et al.], namely, the German Credit dataset [Hofmann, 1994], Australia credit approvals [Quinlan, 1987], Taiwanese bankruptcy [Liang et al., 2020] prediction, Japanese credit screening [Sano, 1992] and the Polish companies bankruptcy [Tomczak, 2016] dataset. We consider also the FICO Home Equity Line of Credit dataset (HELOC) [FICO, 2018], Give Me Some Credit (GMSC) and Lending Club (LC) [Kaggle, 2019] datasets.

3.2 Performance Metrics

We are chiefly interested to what extent linearising relative to logistic models introduces degredation (if any) of both classification performance and calibration - the latter being of interest as we are modifying the probability estimates of a trained logistic model. For each metric the 10-fold stratified cross-validation score is computed for every (classifier, dataset) combination.

Classification Performance. To measure of classification performance, we use the area under the curve of the receiver operating characteristic [Bradley, 1997; Hanley and McNeil, 1983], denoted as AUC.

Calibration. We consider two widely-used numerical summary statistics for the calibration, Expected Calibration Error (ECE) and Maximum Calibration Error, (MCE). Lower values of ECE and MCE correspond to better calibration of a particular model, with the idealised model having a value of zero for both.

Statistical Methodology. For the purposes of discussion here, consider a graph where for each classification algorithm A we draw a node. We draw an edge between any nodes corresponding to algorithm pairs (A, A′ ) such that the performance of A cannot be distinguished from the performance A′ with significance α = 0.05 according to a a Wilcoxon signed-rank test [Wilcoxon, 1945] conducted over the considered datasets. We display this graph, where the nodes A are arranged according to their average rank RA in Figure 3. These are the Critical Difference (CD) diagrams of [Demˇsar, 2006] for AUC, ECE and MCE . Not only are we interested in whether an observed difference in cross validated score between two algorithms is statistically significant, but also the size of this difference. In Table 1 for the AUC metric. The quantity ˆ θHL is a robust point estimate 1 computed across the datasets.

Results for Classification Performance. We highlight several observations from the CD diagram for AUC (Figure 3). Unconstrained XGB is the strongest model in terms of AUC, as we may expect. The weakest performing model families are {NNLR, LAM-NNLR}, presumably due to their simplicity as compared with the other models. LAM-NNLR models are indistinguishable in AUC performance from their logistic counterparts, NNLR. Interestingly, LAM-ARM1 is connected to MonoXGB in the CD graph. MonoXGB models are considered state of the art for monotone constrained models on tabular data and LAM-ARM1 is more interpretable according to a number of criteria, yet here they display statistically indistinguishable classification performance. However, LAMARM1 can be distiguished with statistical significance from its logistic counterpart, ARM1. Nonetheless they are in the same connected component of the CD graph indicating a very similar level of proficiency. The AUC performance of LAMARM2 can be separated from the performance of ARM2. In this instance we hypothesise that this is happening due to linearisation being applied in two layers as opposed to just one, allowing errors to accumulate. In terms of absolute numerical difference in AUC performance, linearisation incurs a very small penalty, as shown in Table 1. The AUC penalties (point estimate) for linearising NNLR, ARM1 and ARM2 are 0.000, 0.003 and 0.003 respectively across the datasets.

1 The Hodges-Lehmann estimator associated to Wilcoxon's signed rank test [Wilcox, 2022].

Results for Calibration. Inspecting the CDdiagrams for the ECE and MCE calibration metrics in Figure 3, we see there is a penalty incurred on model calibration for linearising. Indeed for both metrics, only the linearisation of NNLR models gives a penalty in calibration that cannot be distinguished from the logistic model with statistical significance. However, we note the numerical difference across the datasets is small,being of order ∼0.005 for ECE and ∼0.03 for MCE. We believe this discrepancy in calibration is likely due to "model certainty" evinced by the linearised models, namely output values lying in {0, 1}. Notably, the LC datasets get a very large fraction of predictions with certainty (∼80%) using LAM-ARM1 and LAM-ARM2 models. The MCE ranks are all fairly close to one another, meaning all of the classifiers are more closely matched on this metric as compared with AUC and ECE.

4 User Survey

The motivating example in Section 2.1 suggests that linearised models will be easier to interpret and reason about by users as opposed to logistic models. To substantiate this, we conduct a user study where the aim is to ascertain which class of models is more interpretable: logistic models or LAMs. Our proxy for interpretability is how capable users are at carrying out basic reasoning tasks about the models' outputs, given the model coefficients. This falls within the paradigm of Human-Grounded Evaluation [Doshi-Velez and Kim, 2017], where empirically measured human performance on a simplified task serves as a proxy for explanation quality. Using human simulation of model outputs as a proxy for model interpretability is a similar strategy to that used in the work by [Poursabzi-Sangdeh et al., 2021].

4.1 Setup

The study is structured as a questionnaire, wherein participants are shown a small LR model alongside its linearised counterpart and are asked to predict how the output of each model will change in both direction and magnitude from some initial (input, output) pairs. Participants are shown several instantiations, which we call scenarios. More formally, a scenario consists of the following elements:

• A tuple of model coefficients (A0,A1,A2).
• A tuple of model coefficients (B0,B1,B2).
• An input to the models x := (x1, x2)
• The result of applying models A and B to x, ˆ yA(x) ∈ [0, 1] and ˆ yB(x) ∈ [0, 1].
• A modified input to the models x′ = x + δem, where m ∈ {1, 2}. Both x′ and δ ∈ R are shown to participants. The feature m being modified is also highlighted.

Crucially, participants are given no indication as to what kind of model the coefficients represent. This choice was made so as to not prime participants with any expectation of the models' behaviour [Natesan et al., 2016]. Nor are participants led to believe the models are related to one another. In fact, for all scenarios Model A was the linearisation of Model B. Model B is the LR model ˆ yB(x) = σ(B0 + B1x1 + B2x2), with the Model A coefficients computed using Definition 2.2. Upon being shown a particular scenario a user is directed to carry out the following tasks:

Direction Task. For both models A and B give the change of direction in model output, that is, compute sign(ˆ yA(x′ ) - ˆ yA(x)) and sign(ˆ yB(x′ ) - ˆ yB(x)). The options given to participants are {"OUTPUT INCREASES", "OUTPUT DECREASES"}.

Magnitude Task. For both models A and B give the magnitude of the change in model output, that is, compute |ˆ yA(x′ ) - ˆ yA(x)| and |ˆ yB(x′ ) - ˆ yB(x)|. The options given to participants are {≈ 0.05,≈ 0.1,≈ 0.2,≈ 0.3, "DON'T KNOW"}.

Respondents are shown six scenarios, not including three training scenarios shown at the beginning of the survey. The training scenarios show the same input x with progressively increasing (and decreasing) inputs x + δuem such that |δ1| < |δ2| < · · · along with the corresponding outputs, so that the user can learn the behaviour of each model class. The six test scenarios were designed to be balanced, in the sense that positive and negative changes were included as well as positive and negative values for the xm, (B0,B1,B2), so as to not skew the results. The scenarios were randomly shuffled three times and each permutation assigned to a different subgroup of participants at random. As a final task after all scenarios are presented to study participants, we query the following.

Preference Task. This is a question asked upon completion of the survey about which class of model was easiest to use. The choices are: {"MODEL A", "MODEL B", "NEITHER"}.

The survey was sent to 101 possible respondents via the SurveyMonkey platform [Inc., 2022], who are AI researchers and practitioners in the authors' firm. Participation was anonymous and on a voluntary basis. There were 46 respondents in total, from which 36 successfully completed the training scenarios and gave answers to more than 35% of scenarios. This was the data that was analysed. Despite the small number of participants, the design of the experiment, providing multiple scenarios per-respondent, meant that we can still report results with statistical significance.

4.2 Results and Analysis

We summarise the findings of the User Survey in Figure 4. In the Direction Task we see that a slightly greater proportion of correct answers are given for the LAM as opposed to the logistic model. In the Magnitude Task we see that the logistic models received a response of "DON'T KNOW" most often, followed by an incorrect response and a small number of correct responses. The linearised counterparts of these models yielded mostly correct answers, followed by "DON'T KNOW", then incorrect answers. In the Preference Task, the majority of users declared neither model easiest to use, followed closely by the LAM. Only one respondent declared logistic models easiest to use.

Statistical Analysis. The Direction and Magnitude Tasks are instances of a clustered matched-pair binary data experiment, with the matched pairs being the logistic model and corresponding LAM within each scenario, the binary data comprising a correct vs not correct response to an individual task on each scenario and each cluster corresponding to an individual respondent's answers to multiple scenarios. We use the statistical test for non-inferiority in clustered matchedpair binary data of [Yang et al., 2012], which accounts for within cluster correlated responses. In our setting this corresponds to an individual's responses possibly being correlated with one another, but independent from the responses of other individuals. Suppose that plog is the success probability of a respondent for a given task on a logistic model and pLAM the corresponding success probability for the linearised version. The true difference between the two classes of model is δ = pLAM - plog. Choose a small non-inferiority margin 2 δ0 > 0. Then the hypothesis test we are conducting is

H0 : pLAM -plog ≤ δ0; vs H1 : pLAM -plog > δ0. Yang et al.'s ZMO test statistic asymptotically follows a normal distribution assuming the null hypothesis H0. For the Direction Task we observe ZMO = 2.822 corresponding to a p-value of 0.0024, which is significant at the α = 0.05 level. Moreover, the 95% confidence interval for the difference in success rates between LAMs and logistic models was δ ∈ (0.03, 0.16). This small value of the performance difference in the Direction Task is expected, as for both logistic models and LAMs we can easily inspect the signs of the model coeficients to get the directions, a strategy which many respondents guessed from the training scenarios. For the Magnitude Task we observe ZMO = 4.55 corresponding to a p-value of 2.72×10-6 , which is significant at the α = 0.05 level. The 95% confidence interval for the difference in success rates between LAMs and logistic models was δ ∈ (0.23, 0.50), a significant gap. We attribute this gap to inherent non-interpretability of reasoning in log-odds space vs reasoning directly in probabilities.

2 We choose δ0 = 0.001

For the Preference Task, the data correspond to matched pairs, each pair belonging to one participant and corresponding to a preference of logistic models vs LAMmodels, that is, possible responses are {A ≺ B, A ≻ B, A ∼ B}, with Model A corresponding to LAM and Model B to logistic. As there is no numerical or ranked comparison, the usual appropriate statistical test is the Sign Test [Dixon and Mood, 1946]. Given there are many ties A ∼ B, we use the Trinomial Test of [Bian et al., 2011], specially developed for this regime. Let pA denote the probability a randomly chosen participant prefers LAMs to logistic models and let pB denote the converse. Moreover, p0 is the probability neither is preferred. Then our null hypothesis is H0 : pA = pB and alternative is H1 : pA > pB. Let NA, NB and N0 be the random variables denoting the observed counts corresponding to Model A preferred, Model B preferred and neither respectively, with N := NA + NB + N0. Assuming H0, the test statistic Nd = NA - NB has critical value at significance α = 0.05 of C0.05 = 6, which is exceeded in our data with nd = 12, corresponding to a p-value of 0.0009, which is statistically significant. This supports there being a user preference for LAMs over logistic models, although the impact of this is somewhat dampened by the comparatively large number of responses preferring neither.

Findings. We interpret these findings as follows: as measured by performance in basic reasoning tasks about model behaviour, logistic models are less interpretable than their linearised counterparts. This difference is more pronounced when it comes to reasoning about actual numerical model outputs in response to varying input variables as opposed to reasoning just about the general direction of change. Interestingly, in spite of the gulf in respondents' performance between the LAMs and logistic models this difference is not necessarily felt strongly by the respondents themselves, a large number of whom stated that neither class of models was easier to reason about. In this study, consumers of LAM explanations were correctly interpreting them without even knowing what the underlying model was, but were not generally confident in their interpretations. For logistic models, participants did not correctly interpret the explanations and did not declare them easy to use, in spite of their preexisting modelling expertise.

5 Conclusion

This work introduces techniques for improving the interpretability characteristics of existing models while incurring only very small penalties in classification performance. For an additive logistic model such as ARM, one can use our linearisation scheme (LAM) with the model and incur only a very small reduction in ROC-AUC and a small increase in calibration error. Via the User Study, we showed that when participants are required to simulate the output of a model, that is, predict its behaviour, their performance was far better with linearised models than their logistic counterparts.

Lessons Learned. In this work we closely examined a common tacit assumption within the XAI literature, that there is no reduction in interpretability of GAMs when moving from regression to classification via a non-linear link function (here, the logistic function). We showed that this assumption in fact does not hold in general, with the commonly used explanation method of sharing model weights or shape functions proving to be misleading to human respondents. Our LAM construction is shown to mostly overcome this issue while largely preserving an underlying model's behaviour. The implication here is that when providing model coefficients as explanations, it may be worth paying a small price in performance by linearising a trained logistic model to ensure explanations are correctly understood.

Limitations and Future Work. With this work we must take the following into consideration.

• Model Certainty. For a LAM ˆ yLAM induced by a logistic model ˆ y, inputs x such that ˆ y(x) ̸∈ [0.07, 0.93] correspond to LAM outputs ˆ yLAM(x) ∈ {0, 1}. Risk scores of ≤ 7% and ≥ 93% correspond to confident predictions. LAMs effectively round these risk scores to certainty. If a difference in risk score between, say, 97% and 99.99% is important, then using a LAMmay not be appropriate. Possible mitigations are: i. clip to the interval [ϵ, 1 - ϵ] for some small ϵ > 0; or ii. increase α in the approximation to the sigmoid, thereby growing the set of model inputs with output in (0, 1), while incurring a penalty in approximation.
• Alternate Linearisation Schemes. One could consider alternate methods of linearising as opposed to LAMs. As an example, Average Marginal Effects (AME) [Scholbeck et al., 2024; Bartus, 2005] are local model explanations based on evaluating model prediction function derivatives. The AME values could potentially be interpreted as coefficients of a clipped linear model, although they would lack the theoretical guarantees afforded by LAM.
• User Study Scope. One could increase the scope of the User Study in terms of tasks and information provided to users, considering interpretation of the model coefficients integrated as part of a downstream task, as opposed to predicting model outputs being the tasks' focus. A potentially fruitful investigation is measuring the effect of linearisation on interpretability of general shape functions in GAMs, since LR models have linear shape functions.

6 Disclaimer

This paper was prepared for informational purposes by the Artificial Intelligence Research group of JPMorgan Chase & Co. and its affiliates ("JP Morgan"), and is not a product of the Research Department of JP Morgan. JP Morgan makes no representation and warranty whatsoever and disclaims all liability, for the completeness, accuracy or reliability of the information contained herein. This document is not intended as investment research or investment advice, or a recommendation, offer or solicitation for the purchase or sale of any security, financial instrument, financial product or service, or to be used in any way for evaluating the merits of participating in any transaction, and shall not constitute a solicitation under any jurisdiction or to any person, if such solicitation under such jurisdiction or to such person would be unlawful.