How Many Independent Signals? A Principal Component Analysis Of An 8-Factor Relative Stock Scoring Model

The question

DUEL scores every duel on eight fundamental metrics, each with a fixed weight: Revenue Growth (16%), Operating Cash Margin (15%), ROIC (15%), Sloan Ratio (12%, inverted), Asset Turnover (10%), Receivables Turnover (8%), Operating Margin (12%), and FCF Margin (12%). The winner-take-all design implicitly treats each metric as an independent piece of evidence.

Financial ratios rarely cooperate with that assumption. Pinches, Mingo, and Caruthers (1973) were the first to show, using factor analysis, that a large set of financial ratios collapses onto a handful of underlying dimensions — not the number of ratios you started with. Chen and Shimerda (1981) pushed the point further with a clean illustration: net worth/debt, debt/net worth, net worth/assets, and debt/assets look like four ratios, but carry exactly one piece of information between them. If two of DUEL's eight metrics move together across companies, a nominal 24–27% combined weight might really be voting once, not twice.

Research question: how many statistically independent signals are actually encoded in DUEL's eight metrics, and how do they group?

Data and method

Panel: 30 fresh duels run this cycle, none repeated from earlier coverage — 60 companies across technology, healthcare, industrials, consumer staples, retail, energy, telecom, semiconductors, materials, and travel & leisure. Sector diversity here is a feature, not noise: it tests whether the correlation structure holds across very different business models, not just within one industry.

Variables: the eight raw metric values per company, taken directly from each Battle Report's calculation detail (not the win/loss outcome).

Missing data: ROIC and Operating Margin were unavailable for OXY and COP (sector-specific line-item gaps), and Receivables Turnover was unavailable for NFLX and PEP. Pearson and Spearman correlations use pairwise-complete observations (standard practice); PCA uses the 56 companies with complete records across all eight metrics.

Correlation: both Pearson (linear) and Spearman (rank-based) were computed. Spearman matters here because several metrics — Sloan Ratio, FCF Margin, ROIC in a couple of cases — carry outliers and skew (Ford's (F) -34% ROIC year, MO's 82x Receivables Turnover), and Pearson alone can be distorted by a handful of extreme values.

PCA: standardized (z-scored) inputs, correlation-based principal component analysis, Kaiser criterion (eigenvalue > 1) as a first cut, cumulative variance explained as a second, and a varimax rotation (Kaiser, 1958) on the components needed to reach 80% variance, for a more interpretable loading structure.

Sample size, honestly: 56–60 observations is still short of textbook comfort, though it has moved meaningfully closer since the first pass at this analysis (46–50). Comrey and Lee (1992) rate N=50 "very poor" and N=100 "poor" for factor-analytic work; Nunnally's (1978) 10-observations-per-variable rule would call for 80. What we do have going for us: Guadagnoli and Velicer (1988) found that required sample size drops sharply when loadings are strong and clearly separated — and, as the results below show, ours mostly are. This is an exploratory pilot, in the same spirit as our earlier Bradley-Terry sector study — a first pass meant to be extended, not a confirmatory result. (It's worth noting upfront that adding these 10 companies changed the structure of the answer — see below. That's not a flaw in the method; it's exactly the kind of instability a small-N pilot is supposed to surface.)

Results

Correlation structure

The strongest pairwise relationships in the 60-company panel (Pearson / Spearman):

Three of DUEL's margin-based metrics — Op. Cash Margin, Operating Margin, and FCF Margin — correlate with each other at r ≈ 0.66–0.76. That is not a coincidence of naming: all three are, mechanically, different numerators (CFO, operating income, CFO minus CAPEX) divided by the same denominator (revenue). A company that runs a wide operating margin tends to convert that into cash at a similar rate. Separately, Asset Turnover and Receivables Turnover — both revenue-over-a-balance-sheet-item ratios — move together at r ≈ 0.56, exactly the kind of "same-denominator" clustering Chen and Shimerda (1981) documented forty years ago in a different dataset. ROIC now shows a clearer relationship to both the margin cluster and the turnover cluster than it did in the smaller sample — a point we come back to below.

The one metric that stands apart in both matrices, at both sample sizes: Sloan Ratio. Its strongest correlation with anything else in the panel is -0.30 (Spearman, vs. Receivables Turnover) — weaker than nearly every pair in the table above. Sloan Ratio measures something the other seven metrics don't: whether accounting earnings are being generated with unusually large accruals rather than cash. That is a genuinely different question from "how profitable" or "how efficient," and the correlation matrix reflects it at both 50 and 60 companies.

How many components? (and why the answer moved)

By the Kaiser criterion (eigenvalue > 1), two components still matter — consistent with the first pass. But the 80%-cumulative-variance cutoff now lands at four components (76.6% at three, 85.5% at four), not three. Adding 10 companies didn't just add precision — it changed which cutoff the third component clears.

This is worth being transparent about rather than quietly smoothing over. It's a textbook illustration of exactly the instability Guadagnoli and Velicer (1988) and Comrey and Lee (1992) warn about at this sample size: small-N component solutions can shift as data accumulates, and a component sitting right at a threshold (PC3 was at 80.0% with 46 companies, 76.6% with 56) is the first thing to move. We're reporting the 4-component solution below because it's the one that actually satisfies the stated 80% criterion on the larger, more current sample — not because it tells a tidier story.

What the components actually measure

Varimax rotation of the four components needed for 80%+ variance, on the 56 complete-record companies:

The bigger sample separated Revenue Growth from the margin cluster instead of merging with it — with 50 companies, Revenue Growth's dominant loading was 0.42 on the margin factor; with 60, it's 0.94 on its own factor and essentially zero everywhere else. In hindsight, that makes economic sense: a company can be growing fast with thin margins (many of the newer software and travel names in this update — CCL, RCL, XYZ — are exactly that mix), so growth and margin quality don't have to move together, and with more companies in the panel, the data made that distinction clear instead of blurring it into one factor.

ROIC's picture also sharpened: it now cross-loads on both the Cash Conversion factor (0.41) and Capital Efficiency (0.58, still dominant) — a reminder that ROIC is, by construction, a blend of a margin-like element (return) and a turnover-like element (capital intensity), so some cross-loading is expected rather than a data artifact.

Mapped onto DUEL's actual scoring weights, the eight metrics now resolve into four effective blocks instead of three:

39/33/16/12 — noticeably more balanced than the 55/33/12 three-factor split from the smaller sample, and it got there on its own, from more data, not from any adjustment to the model.

Discussion

The finding cuts in a direction that's easy to miss if you only think of "redundancy" as a flaw. Two things are true at once:

First, roughly 72% of the model's nominal weight (the 39% Cash Conversion block plus the 33% Capital Efficiency block) is riding on two empirical signals, not six. When Operating Margin, Op. Cash Margin, and FCF Margin all favor the same company — which the correlation structure says will happen more often than chance — that company isn't collecting three independent votes. It's collecting one strong vote, counted three times. This doesn't necessarily produce a "wrong" winner; margin quality genuinely is more informative than any single ratio measuring it. But it does mean the practical weight distribution behaves like a 4-factor model (39/33/16/12) rather than the 8-factor model the scorecard displays.

Second, and this is the part worth sitting with: the metric a casual reader might assume is the most "redundant add-on" — Sloan Ratio, the one most people don't recognize by name — is statistically the least redundant metric in the entire model, at both sample sizes we've now tested. It shares almost no variance with anything else DUEL measures. In our earlier duel coverage, Sloan Ratio showed up as the single metric responsible for flagging earnings-quality risk in an otherwise strong-looking company. This analysis says that's not a coincidence of one case: structurally, Sloan Ratio is doing a job none of the other seven metrics are equipped to do, which is exactly why it's uncorrelated with them.

A practical note, for readers who see "39/33/16/12" and think "just fix the weights"

Two intuitive reactions to a finding like this are worth addressing directly, because they're both wrong in the same subtle way.

Reaction one: "If the real structure is four factors, just cut the model down to four metrics." This confuses statistical overlap with waste. When three margin metrics agree on a company, that's not three metrics saying nothing new — it's three independently-calculated numbers (from CFO, from operating income, from CFO-minus-CAPEX) converging on the same conclusion. That convergence is closer to a second and third opinion from different tests than to a single test asked three times. A doctor who orders blood pressure, resting heart rate, and an ECG isn't wasting two tests just because all three usually move together in a healthy patient — the agreement itself is informative, and on the rare company where they disagree, that disagreement is exactly the kind of signal a single-metric model would miss entirely. Dropping to four metrics would remove that cross-check, not just remove redundancy.

Reaction two: "If Sloan Ratio is the least redundant, give it ~30% weight so the factors land at roughly 33/33/33." This is a more sophisticated-sounding version of the same mistake. PCA tells you how much a metric overlaps with the others — it says nothing about how reliable or predictive that metric is on its own. Sloan Ratio is calculated from just one accounting relationship (net income vs. cash flow vs. total assets), it's more sensitive to the quarter/year period-mismatch issue flagged in every Battle Report's own disclaimer, and — as this same panel shows (SNDK's Sloan Ratio of +27.6%, driven by an unusual asset-sale year, is the most extreme value in the entire dataset) — it can swing hard on one-off accounting events. Tripling its weight because it happens to be statistically uncorrelated with everything else would mean betting a third of the final score on the single noisiest, least-diversified metric in the model. Being "independent" and being "trustworthy at high weight" are not the same property, and conflating them is a bigger error than the redundancy this article started with.

What this actually argues for, concretely:

• No change to the eight published metrics or their weights on duelstocks.com. Every metric is still a real, distinct, SEC-derived calculation, visible in full in every report's calculation detail — nothing here suggests any number is fabricated, duplicated, or wrong. This is a finding about statistical behavior across many companies, not a defect in any individual duel.

• The honest reading of "39/33/16/12" is a map, not a verdict. It tells you that if you want to meaningfully change a duel's outcome, moving one of the three Cash Conversion metrics or one of the three Capital Efficiency metrics will do less than you'd expect (their sibling metrics pull toward the same conclusion), while Sloan Ratio and Revenue Growth each carry their full, undiluted nominal weight.

• The real next test is a robustness check, not a reweighting. The open, testable question this raises for future coverage: does actually halving the combined weight of the three Cash Conversion metrics change any past duel's winner? If the answer is "rarely," that's reassuring — it means the overlap is corroboration, not manipulation of the score through triple-counting. If it changes several outcomes, that's worth knowing too. Either way, that's a question for a follow-up piece with backtesting infrastructure this project doesn't have yet — not a reason to hand-adjust a formula based on one exploratory PCA pass.

Limitations

This is a pilot, and it should be read as one:

• Sample size. 56–60 companies is closer to comfortable than our first pass (46–50) but still below Comrey and Lee's (1992) "poor" threshold of 100. The component structure moved once already between 50 and 60 companies (three factors → four) — direct evidence that it could still move again with the next batch. Treat the 39/33/16/12 split as the current best estimate, not a settled number.

• Cross-sectional, not longitudinal. This is one snapshot. Whether the same four-factor structure holds a year from now, across a full market cycle, is an open question — and probably the natural next step for this line of work.

• Mixed reporting periods. Several companies in this panel mix quarterly and annual periods for individual metrics (as flagged in each Battle Report). We did not apply a TTM-harmonization correction here; if that correction shifts individual company values meaningfully, it could also shift the correlation structure at the margin. That's a separate methodological question, one we may take up directly in a future piece.

• Four dropped-metric companies. OXY and COP were excluded from the PCA (missing ROIC/Operating Margin data), and NFLX/PEP were included in correlations but not PCA (missing Receivables Turnover). None were dropped for their results — only for data gaps.

None of this changes DUEL's actual scoring in the product. It's a methodology finding, not a product decision — same distinction we've tried to hold to in every piece in this series.

Bottom line

Eight metrics, one fixed weighting scheme, and — on this now-60-company pilot — four empirically distinguishable signals underneath it: cash conversion, capital efficiency, accrual quality, and growth, each behaving separately from the others. The nominal 16/15/15/12/10/8/12/12% split is closer, in practice, to a 39/33/16/12% split once you account for what the metrics are actually measuring versus what they're named — and that number moved toward more balance, not less, as the sample grew, without anyone touching the formula. That's a reason to keep collecting data, not a reason to start hand-tuning weights based on one pass.

As always: run the numbers yourself at duelstocks.com — every duel report shows its full calculation detail, which is exactly what made this analysis possible in the first place.

References

• Chen, K. H., & Shimerda, T. A. (1981). An empirical analysis of useful financial ratios. Financial Management, 10(1), 51–60.

• Comrey, A. L., & Lee, H. B. (1992). A First Course in Factor Analysis (2nd ed.). Erlbaum.

• Gombola, M. J., & Ketz, J. E. (1983). A note on cash flow and classification patterns of financial ratios. The Accounting Review, 58(1), 105–114.

• Guadagnoli, E., & Velicer, W. F. (1988). Relation of sample size to the stability of component patterns. Psychological Bulletin, 103(2), 265–275.

• Jolliffe, I. T. (2002). Principal Component Analysis (2nd ed.). Springer.

• Kaiser, H. F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23(3), 187–200.

• Nunnally, J. C. (1978). Psychometric Theory (2nd ed.). McGraw-Hill.

• Pinches, G. E., Mingo, K. A., & Caruthers, J. K. (1973). The stability of financial patterns in industrial organizations. Journal of Finance, 28(2), 389–396.

• Sloan, R. G. (1996). Do stock prices fully reflect information in accruals and cash flows about future earnings? The Accounting Review, 71(3), 289–315.

Not investment advice. All data sourced from public SEC EDGAR filings.