How do you calculate SHAP values for a logistic regression PD model?

For a logistic regression the SHAP value has a closed form, adds exactly to the logit, and gives reason codes a decline letter can defend.

Julian R. Sterling · from Machine Learning for Finance

For a logistic regression the SHAP value of each feature has a closed form in log-odds: coefficient × (applicant's value − portfolio mean), assuming the features are independent. On an illustrative four-variable PD model, an applicant with a PD of 29.4 per cent against a base of 3.97 per cent owes 0.990 of the 2.310 log-odds gap to revolving utilisation and 0.625 to recent enquiries, so those are the two reasons to give for the decline, not the debt-to-income ratio that the raw coefficient-times-value reading would put second.

Worked in full in Machine Learning for Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →

Lenders need per-applicant explanations for adverse action notices, model validation and challenge from the business. For a tree ensemble SHAP needs an algorithm. For the logistic regression that still runs most credit decisions it needs one line of arithmetic, and getting that line wrong produces the wrong reason codes.

The assumptions

Illustrative PD model, logit(PD) = intercept + Σ coefficient × value. Intercept −5.2.
Feature	Coefficient	Portfolio mean	Applicant
Debt-to-income, %	0.035	32	45
Revolving utilisation, %	0.022	40	85
Years at address	−0.06	6	2
Credit enquiries, last 6 months	0.25	1.5	4

How the PD itself comes out of the logistic function is worked in calculating PD from a logistic regression. Here the question is how to split that PD between the four inputs.

Step 1: the base value

SHAP explains the difference between the applicant's output and the expected output. For a linear model in log-odds the expected output is the model evaluated at the portfolio means.

Base = −5.2 + 0.035 × 32 + 0.022 × 40 − 0.06 × 6 + 0.25 × 1.5

= −5.2 + 1.120 + 0.880 − 0.360 + 0.375 = −3.185 log-odds, a PD of 3.97 per cent.

Step 2: one SHAP value per feature

SHAP_i = β_i × (x_i − mean_i)

Debt-to-income: 0.035 × (45 − 32) = 0.035 × 13.0 = 0.455

Utilisation: 0.022 × (85 − 40) = 0.022 × 45.0 = 0.990

Years at address: −0.06 × (2 − 6) = 0.240

Enquiries: 0.25 × (4 − 1.5) = 0.25 × 2.5 = 0.625

Excel, coefficients in B2:B5, means in C2:C5, applicant in D2:D5: =B2*(D2-C2), filled down.

Step 3: check the additivity

Sum of SHAP values = 0.455 + 0.990 + 0.240 + 0.625 = 2.310

Base + sum = −3.185 + 2.310 = −0.875 = the applicant's logit. PD = 1 ÷ (1 + e^0.875) = 29.4 per cent.

The four values add exactly to the gap between the applicant and the average, which is the property that makes SHAP usable for reason codes. In shares of the gap, utilisation explains 43 per cent, enquiries 27, debt-to-income 20 and years at address 10.

Applicant A: SHAP in log-odds and in scorecard points (PDO 20, factor 28.85).
Feature	SHAP, log-odds	Points lost	Naive β × x
Revolving utilisation	0.990	28.6	1.870
Credit enquiries	0.625	18.0	1.000
Debt-to-income	0.455	13.1	1.575
Years at address	0.240	6.9	−0.120
Total	2.310	66.7

The points column multiplies each SHAP value by PDO ÷ ln 2 = 20 ÷ 0.693 = 28.85, the same factor used to turn a model into scorecard points. Credit teams usually find “28.6 points lost on utilisation” easier to act on than a log-odds figure.

What if: three applicants

SHAP values in log-odds against the same base of −3.185 (PD 3.97%).
Applicant	DTI	Utilisation	Address	Enquiries	PD	Main reason
A: 45, 85, 2, 4	+0.455	+0.990	+0.240	+0.625	29.4%	Utilisation
B: 24, 15, 11, 0	−0.280	−0.550	−0.300	−0.375	0.9%	None needed
C: 50, 30, 8, 1	+0.630	−0.220	−0.120	−0.125	4.7%	Debt-to-income

Applicant C is the useful case. Three of four features pull the PD below average, and one, a 50 per cent debt-to-income ratio, pushes it above. If C is declined at a cut-off below 4.7 per cent, debt-to-income is the only honest reason, and a model monitor can see that from the signs alone.

The common mistakes

Using β × x instead of β × (x − mean). Without centring, the ranking for applicant A becomes utilisation, debt-to-income, enquiries, and years at address shows as reducing risk (−0.120) when two years is well below the six-year average and adds 0.240 to the log-odds. The decline letter would cite the wrong second reason.
Attributing in PD instead of log-odds. The logistic link is not linear, so a feature's PD effect depends on what has been added before it. Utilisation added first moves the PD from 3.97 to 10.0 per cent, 6.0 points; added last it moves it from 13.4 to 29.4, 16.0 points. Rank and report in log-odds or points, where the split is exact.
Reading the base value as the portfolio default rate. 3.97 per cent is the PD of an applicant at the mean of every feature. Because the logistic curve is convex at low PDs, the average PD across the book is usually higher.

The independence caveat. The closed form is the interventional SHAP value, which treats the features as independent. Utilisation and enquiries are often correlated, and a conditional SHAP implementation will share credit between them differently. State which one the reason codes use, and keep it fixed between validations.

Takeaway

For a logistic PD model, SHAP needs no library: coefficient times distance from the mean, in log-odds, adding exactly to the applicant's logit. Applicant A's 29.4 per cent PD is 43 per cent utilisation and 27 per cent enquiries, and those are the reason codes. The book builds its scorecard from counts and carries every coefficient into points in the free workbook for this case.

Questions readers ask

Are SHAP values for logistic regression the same as the coefficients?

No. A coefficient is the effect of one unit; the SHAP value is the coefficient times the applicant's distance from the portfolio mean. A utilisation coefficient of 0.022 gives a SHAP value of 0.990 for an applicant at 85 per cent against a mean of 40, and a negative value for one below the mean.

Should SHAP values be in probability or log-odds for a credit model?

Log-odds, or scorecard points derived from them. In log-odds the values add exactly to the applicant's logit. In probability the split depends on order: in the illustrative case utilisation is worth 6.0 PD points if added first and 16.0 if added last, so the ranking of reasons can change.

How do you turn SHAP values into scorecard points?

Multiply each log-odds SHAP value by the scorecard factor, PDO divided by ln 2. With a PDO of 20 the factor is 28.85, so a SHAP value of 0.990 is 28.6 points lost and the applicant's four features cost 66.7 points in total against the average applicant.

Read the whole case

This article is one calculation from Machine Learning for Finance. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.

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Also on this site

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