Key driver analysis
Shapley value regression vs Johnson's relative weights
Shapley value regression and Johnson's relative weights both answer the key driver question “how much of the outcome does each driver explain?” by splitting the model's R² among correlated drivers. Shapley averages each driver's gain in R² over every order in which drivers could enter the model; Johnson's method gets nearly the same split in one step from an orthogonal version of the drivers. On survey data the two usually agree to within a percentage point, so the choice matters less than not ranking drivers by raw regression coefficients.
How results are checked
Named statistical review pending. Selected automated reference cases are published on the validation page.Why key driver analysis needs relative importance
Satisfaction ratings move together. A respondent who rates quality highly tends to rate service and price highly too, so the drivers in a key driver model are correlated, sometimes strongly. Ordinary regression still fits well, but its coefficients split the shared variance arbitrarily: a driver that matters can get a tiny or negative beta, and the ranking can flip when one driver is added or removed.
Relative importance methods solve this by asking a different question. Instead of “what is the effect of this driver holding the others fixed?”, they ask “what share of the explained variance belongs to this driver, including its fair part of what it shares with the others?”. The answer is a set of non-negative shares that sum to R², which is what a stakeholder chart needs.
How the two methods differ
Shapley value regression (also called LMG after Lindeman, Merenda and Gold, or dominance analysis's general dominance weights) fits the regression for every subset of drivers. A driver's importance is its increase in R² when it joins a subset, averaged over all orders of entry. With p drivers that is 2p regressions: 64 for six drivers, 32,768 for fifteen. The result is exact and has a clean game-theory justification.
Johnson's relative weights (Johnson, 2000) replace the drivers with their closest set of uncorrelated variables, regress the outcome on those, and map the squared coefficients back to the original drivers through the squared loadings. It is a single eigen-decomposition, fast for any number of drivers, and its shares track Shapley's closely in published comparisons and in the example below.
The method
Formula
Shapleyj = ΣS ⊆ D∖{j} [|S|! (p − |S| − 1)! / p!] × [R²(S ∪ {j}) − R²(S)]
Johnson: Rxx = VΛV′, Λxz = VΛ½V′, β* = Λxz−1 rxy, RWj = Σk λjk² β*k²
where
- D, p
- the set of drivers and how many there are
- R²(S)
- R² of the regression of the outcome on the drivers in subset S (R²(∅) = 0)
- R_xx, r_xy
- driver correlation matrix and driver–outcome correlations
- λ_jk
- loading of driver j on orthogonal variable k; both methods' weights sum to the full-model R²
- Share
- 100 × weight / R², the percentage shown on a driver chart
Worked example
Sample data: the synthetic snack-brand tracker that the relative importance tool loads with “Use sample data” (600 respondents, generated, not collected). The outcome is overall satisfaction (0–10) and the drivers are six satisfaction ratings. Together they explain R² = 0.705 of the variance in overall satisfaction.
| Driver | r | Beta | Shapley share | Johnson share |
|---|---|---|---|---|
| Quality | 0.697 | 0.393 | 30.6% | 30.7% |
| Service | 0.667 | 0.320 | 24.9% | 24.9% |
| Price | 0.611 | 0.230 | 18.4% | 18.6% |
| Speed | 0.523 | 0.043 | 9.6% | 9.3% |
| Range | 0.487 | 0.115 | 9.2% | 9.6% |
| Website | 0.478 | −0.004 | 7.3% | 7.0% |
The two methods agree closely: the largest difference between a driver's Shapley share and its Johnson share is 0.36 percentage points. The top three drivers (Quality, Service, Price) take the same places under both methods. Further down the list the ranking is not identical: Speed and Range swap places: their Shapley shares (9.6% and 9.2%) are 0.36 points apart. Read them as tied.
The betas show why neither method ranks drivers by coefficient: Website correlates r = 0.48 with overall satisfaction, yet its beta is −0.004 because the other ratings already carry what it shares with them. Shapley and Johnson still give it 7.3% and 7.0% of the explained variance.
Using it well
When to use it
Use it when
- Use: You want a defensible ranking of correlated ratings (attributes, touchpoints, image statements) against an overall measure.
- Use: Stakeholders need shares that add up to 100% of what the model explains.
- Use: Use Shapley for up to about 15 drivers; relative weights when you have more, or both as a cross-check.
Not the right tool when
- Avoid: You need the effect of changing one driver while holding the others fixed: report regression coefficients with their intervals instead.
- Avoid: The outcome is yes/no (bought, churned): the tool's linear R² decomposition is not a logistic model; use logistic regression for effects.
- Avoid: Drivers are near-duplicates (|r| above about 0.8): merge them or report them as one theme, because any split between them is fragile.
- Avoid: The sample is small relative to the number of drivers; check the bootstrap intervals before ranking.
How to interpret it
Rule of thumb
Read the shares as “how much of the explained variation travels with this driver”, not as effects. Treat drivers whose shares are within a point or two as tied, and pair importance with performance (the driver's mean rating) to decide where improvement pays off.
Next step: run it on your own drivers
The free relative importance analysis tool computes correlations, betas, exact Shapley shares and Johnson's relative weights side by side from a CSV, Excel or SPSS file, with an optional weight column, bootstrap intervals for the Shapley shares, collinearity warnings and an importance–performance quadrant map. It runs in your browser: no account, and the file is not uploaded.
Questions
Frequently asked questions
Do Shapley value regression and Johnson's relative weights give the same answer?
Usually almost. Both split the model R² among the drivers and both shares sum to 100%. Shapley averages each driver's R² gain over every possible order of entry; Johnson's method approximates the same idea from an orthogonal transformation of the drivers. On typical survey data the shares differ by well under a percentage point, but close drivers can swap rank.
Why not just rank drivers by their regression coefficients?
Because satisfaction ratings are correlated. When drivers overlap, the standardized betas share the overlap arbitrarily, can be close to zero or even negative for a driver that clearly matters on its own, and change a lot when one driver is added or removed. Shapley and relative weights give every driver its fair share of the overlap, so the ranking is stable.
Which one should I report?
Either, with the method named. Shapley is exact and easy to explain (the average gain in R² across all orders of entry) but needs 2^p regressions, so it slows down beyond about 15 drivers. Relative weights are a one-step calculation that scales to many drivers. When the two disagree on the order of two drivers, report them as tied.
Does a high importance share mean the driver causes the outcome?
No. Both methods describe how the outcome's variance is shared among correlated ratings in this sample. A driver can rank high because it is a proxy for something else, or because respondents rate everything through the same halo. Treat the result as a priority list for further work, not proof of cause.
How many respondents do I need?
Enough for a stable correlation matrix: the free tool warns when there are fewer than 10 complete rows per driver and reports bootstrap intervals for the Shapley shares, so you can see whether neighbouring drivers are distinguishable.
Sources and next steps
References & further reading
- [1]Johnson, J. W. (2000). A heuristic method for estimating the relative weight of predictor variables in multiple regression. Multivariate Behavioral Research, 35(1), 1–19.
- [2]Lipovetsky, S., & Conklin, M. (2001). Analysis of regression in game theory approach. Applied Stochastic Models in Business and Industry, 17(4), 319–330.
- [3]Grömping, U. (2007). Estimators of relative importance in linear regression based on variance decomposition. The American Statistician, 61(2), 139–147.
- [4]Tonidandel, S., & LeBreton, J. M. (2011). Relative importance analysis: A useful supplement to regression analysis. Journal of Business and Psychology, 26(1), 1–9.
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