Opinion for the Court filed by Circuit Judge WILLIAMS.
STEPHEN F. WILLIAMS, Circuit Judge:In 1993 Williams Natural Gas Company,1 a natural gas pipeline company within the jurisdiction of the Federal Energy Regulatory Commission, filed for a general rate increase under § 4 of the Natural Gas Act, 15 U.S.C. § 717c. The proceeding closed in 1999 with the Commission’s third rehearing order. Williams Natural Gas Co., 86 FERC ¶ 61,323 (1999) (“Third Rehearing”). That and the underlying orders are attacked from two sides. A host of Kansas cities, the Missouri Public Service Commission and others, which we will collectively call the “Public Service Commission,” attack the allowed rate of return. They argue that the Commission wrongly refused (a) to impute to Williams the capital structure of its corporate parent, or alternatively, (b) to adjust Williams’s return on equity downward to reflect its subsidiary status and the “thickness” of its equity ratio in comparison to that of firms in the proxy group used by the Commission to calculate the return on equity. The pipeline itself attacks on an unrelated issue, objecting to the Commission’s method of projecting the costs for cleaning up PCB (polychlorinated biphenyl).
We cannot say that the Commission’s use of Williams’s capital structure and the median return on equity for the proxy group was arbitrary and capricious. As to clean-up costs, the Commission no longer defends the $1.4 million annual cost recovery as a figure representative of actual cost, and its decision does not purport to rely on any procedural default by Williams; we therefore grant Williams’s petition and remand for further proceedings.
Capital structure and rate of return on equity
The Public Service Commission’s brief offers a non-exclusive, but here uncontested, explanation of the role of capital structure and equity rate of return. It points out that a firm’s return on equity must be higher than the return on debt because (1) any dividends are paid out of after-tax earnings, whereas the firm can deduct interest on debt, and (2) equity is riskier. Because the overall cost of equity is the product of the equity share of capital and the equity rate of return, these factors imply that an increase in the equity-debt ratio tends to increase a firm’s allowable overall rate of return. But there is an offset: Because debt service has priority, the higher the proportion of equity capital, the lower the financial risk for the firm’s stock, and thus, in this respect, the lower the necessary rate of return. See also Richard J. Pierce, Jr. & Ernest Gellhorn, Regulated Industries 136-37 (3d ed.1994).
Williams is a wholly owned subsidiary of The Williams Companies (“TWC”). Williams’s own capital structure is 35.71% debt and 64.29% equity, while TWC’s is 50% debt, 3% preferred equity, and 47% *3common equity. Assuming use of the same equity rate of return, FERC’s use of TWC’s ratio would be an advantage for Williams’s customers.
In calculating the equity rate of return of a wholly owned subsidiary, the Commission has a special problem. Since its shares are not traded in the market, they have no market price from which to infer their rate of return. So the Commission looks instead to a proxy group of supposedly similar firms whose stock is traded, calculates their return on equity with the “DCF” or “discounted cash flow” method, and then tacks the resulting number onto the equity of the subsidiary. See generally Williston Basin Interstate Pipeline Co. v. FERC, 165 F.3d 54, 56-57 (D.C.Cir.1999); North Carolina Utilities Comm’n v. FERC, 42 F.3d 659, 661 (D.C.Cir.1994).
Here the Commission used Williams's capital structure. It found the company's business risk average, and, though not explicitly so labelling its financial risk, held that its overall risk (the amalgam of the two) was not outside the "broad middle range of average risk." Third Rehearing, 86 FERC ¶ 61,323, at 61,860-61. It thus allowed Williams the median rate of return of the proxy group. In doing so, it made no adjustment to reflect the fact that Williams's equity ratio was a good deal thicker than the average of the proxy group (and therefore presumably less risky). Indeed, Williams's ratio was higher than the highest equity ratio of the proxy group-64%, compared with 42% and 62% for the average and highest ratio of the proxy group, respectively.2
We review the challenge under the arbitrary and capricious standard of the Administrative Procedure Act. 5 U.S.C. § 706(2)(A). The Commission must consider the relevant factors and draw “a rational connection between the facts found and the choice made.” Williston Basin, 165 F.3d at 60 (citation and quotation marks omitted). On the technical aspects of ratemaking FERC’s decisions necessarily enjoy considerable deference. Public Service Comm’n v. FERC, 813 F.2d 448, 451 (D.C.Cir.1987).
The attack on the Commission’s refusal to use TWC’s capital structure opens with the “double leveraging” theory. The theory’s basic concept is that the true cost of a subsidiary’s equity capital is the overall cost of the parent’s capital. Accordingly, the cost of the subsidiary’s equity should be computed as the weighted average of the parent’s debt and equity costs. Otherwise, says the theory, shareholders of the parent receive not only the higher equity returns associated with the parent’s equity, but an artificial (doubly leveraged) return on the subsidiary’s equity.
Although the Commission in the first rehearing order opted in favor of using TWC’s capital structure, Williams Natural Gas Co., 80 FERC ¶ 61,158 (1997) (“First Rehearing”), even then it rejected double leveraging as a rationale: “The rate of return to a pipeline should not depend on who owns the pipeline, nor on how that owner, whether a holding company or individual stockholders, financed its investment.” Id. at 61,682; see also Third Rehearing, 86 FERC ¶ 61,323, at 61,858-59. The double leveraging theory would in principle be applicable to a pipeline owned by a single individual, or by a group of investors, requiring the Commission to pursue its inquiry into these owners’ finances. Further, an expert quoted by the Commission makes the point that the pipeline investment’s true opportunity cost does not depend on the capital structure of the investor, but rather on the foregone risk-adjusted returns of alternative investments. See James E. Brown, “Double Leverage: Indisputable Fact or Precarious Theory,” Public Utilities Fortnightly *426, 29 (May 9,1974), cited at First Rehearing, 80 FERC ¶ 61,158, at 61,682 n. 21.
It is not for us to say whether these arguments have put the kibosh on the double leverage theory. We can, however, say that the Public Service Commission's quick response — individual investors would never directly own a FERC-regulated pipeline, and if they did, they would not stand for such high equity ratios — is not a serious intellectual answer to them. On this record we have no basis to disturb FERC’s refusal to apply the double leveraging theory.
The Commission nevertheless briefly flirted with the idea of using TWC’s capital structure. First Rehearing, 80 FERC ¶ 61,158, at 61,683. But on the next lap it dropped that approach, with the reasoning stated in a chronologically connected case:
Traditionally, the Commission has preferred to utilize the applicant’s own capital structure and will continue to do so if the applicant issues its own non-guaranteed debt and has its own bond rating. But the Commission will utilize an imputed capital structure (most often that of the corporate parent) if the record in a particular case reveals that the pipeline’s own equity ratio is so far outside the range of other equity ratios approved by the Commission and the range of proxy company equity ratios that it is unreasonable.
Transcontinental Gas Pipe Line Corp., 84 FERC ¶ 61,084, at 61,413 (1998) (“Order 414-A”), affirmed North Carolina Utilities Comm’n v. FERC, No. 99-1037, 2000 WL 158898 (D.C.Cir. Feb.7, 2000) (unpublished opinion). The Commission applied this policy to Williams on the second rehearing. Williams Natural Gas Company, 84 FERC ¶ 61,080, at 61,356 (1998) (“Second Rehearing”). As Williams issued its own non-guaranteed debt and had its own bond rating, the normal pre-conditions for using Williams’s own capital structure were satisfied.
We now turn to the Public Service Commission’s argument that Williams’s equity ratio is so out of line that the Commission should either have applied the caveat in the excerpt quoted above (calling for use of an imputed capital structure in cases of anomalous equity ratios), or should have adjusted Williams’s equity rate of return down from that of the proxy group. The common sense of this attack is clear. Given that a high equity ratio reduces financial risk (everything else being equal), it would make no sense for. the Commission to use a rate of return inferred from the market experience of a proxy group with much thinner equity ratios.
But how thick is “too thick,” and how much difference in thickness is too much? Here the issue is whether 64% equity is “anomalous,” bearing in mind that it is 22% above the proxy average but only 2% above the highest in the proxy group. See supra note 2. Judges are hardly in a position to play this numbers game. Such numerical limits cannot readily be derived by judicial reasoning, Hoctor v. USDA, 82 F.3d 165, 170 (7th Cir.1996), though to be sure courts are driven to it occasionally, as in enforcement of the Administrative Procedure Act’s mandate to ensure that agency action is not “unreasonably delayed.” 5 U.S.C. § 706(1). The ultimate choice may partake of arbitrariness — not in the sense of being “arbitrary and capricious,” but in the sense that, while numerical lines sometimes must be drawn, it is impossible to give a reasoned distinction between numbers just a hair on the OK side of the line and ones just a hair on the not-OK side.
Here, it seems clear at the outset that the Commission was on firm ground in rejecting the idea that an equity ratio outside the bounds of the proxy group must automatically require an adjustment. See Second Rehearing, 84 FERC ¶ 61,080, at 61,355. Assume a proxy group with ratios varying from 40% to 44%. Insisting on an adjustment for a firm with one of 45% would surely impute an improbable refinement to the rough inferences derived from *5capital markets, as well as raising the question just how great the adjustment should be.
The Public Service Commission suggests in its brief that Commission precedent can provide some guidance. In Transcontinental Gas Pipeline Corp., 60 FERC ¶ 61,-246 (1992), reh’g denied 64 FERC ¶ 61,039 (1998), FERC found that Transco Energy-Corporation’s equity ratio (the pipeline proposed using its parent’s equity ratio) was 22% below the proxy average and required a different imputed capital structure to boost pipeline returns. See North Carolina Utilities Comm’n, 42 F.3d at 661, 663. FERC does not really respond to this argument, although it did observe in the Second Rehearing that the proxy group average here is brought down by the 16% equity ratio for one of the proxy firms (Transco, interestingly). 84 FERC ¶ 61,080, at 61,358 n. 31. Indeed, Transco’s presence lowered the proxy group average over 5% (47.4% to 42.2%). But as petitioners point out, if one outlier is to be removed, why not another (Sonat, at 62%)? And the double removal would put the average at 43.8%, which would leave Williams still well above the average and even more above the new top (48%). Further, the Commission gives no explanation as to why any outlier should be removed, see United States Telephone Assn. v. FCC, 188 F.3d 521, 525 (D.C.Cir.1999) (agency eliminating outlying data points must explain “why the outliers were unreliable or their use inappropriate”), much less why a low outlier should be removed and a high one retained. Had the Transco precedent been properly raised, FERC’s failure to offer a distinction might well have required a remand. See Greater Boston Television Corp. v. FCC, 444 F.2d 841, 852 (D.C.Cir.1970) (“[I]f an agency glosses over or swerves from prior precedents without discussion it may cross the line from the tolerably terse to the intolerably mute.”). But petitioners’ exceptions before the Commission did not cite the North Carolina Utilities Comm’n case or make such precedent-based arguments.3
Nor is there much force to petitioners’ argument that creeping stare decisis will inch equity ratios ever-higher, as each new peak in equity ratio will justify another, still higher peak. The Commission swears off any such progression, see, e.g., Second Rehearing, 86 FERC ¶ 61,232, at 61,858, and petitioners can identify nothing in the record to undercut its commitment. A slippery slope argument is almost always available. “Judges and lawyers live on the slippery slope of analogies; they are not supposed to ski it to the bottom.” Robert H. Bork, The Tempting of America: The Political Seduction of the Law 169 (1990). Especially with the Commission’s explicit pledge, the slope risk provides no basis for us to upset the Commission’s judgment.
Petitioners also claim that in looking in part to pipeline companies outside the proxy group in determining the reasonableness of Williams’s equity ratio, the Commission failed to provide adequate notice and thus failed to allow them an opportunity to offer evidence distinguishing the companies outside the proxy group. But after the Commission considered pipelines outside the proxy group in the Second Rehearing, petitioners made no request to supplement the record.
All that said, this case is somewhat puzzling. No one contests the Public Service Commission’s point that a thick equity ratio implies less risk and thus a lower rate of return, everything else being equal. Yet the Commission selected a proxy group with widely dispersed equity ratios, from 16% to 62%, as opposed to a proxy group nearer to Williams’s capital structure. Further, it is unclear why the Commission has taken such an interest, both in its orders and in its brief here, in explaining that Williams’s 64% ratio is in the *6mainstream of ratios in the pipeline industry generally. The rate of return is inferred from the rate of return of the proxy group, so the non-anomalous character of Williams’s equity ratio by the standards of the industry generally is not self-evidently pertinent.
But there are also gaps in the petitioners’ attacks, which undermine any inference that FERC’s looking to the industry generally had any material effect. Given the supposed relation between equity ratio, risk, and rate of return, we should expect to see some effort to show it at work within the proxy group, or broadly among publicly traded companies generally. Yet petitioners offer no such analysis. We know the direction of the effect of equity thickness on equity rate of return (as no one contests it), but we have nothing on the degree. Accordingly, we have no basis for thinking that relationship to be so strong as to make a material difference, business risk being held constant. On this record, then, we cannot find anything arbitrary and capricious in the Commission’s use of Williams’s capital structure and the proxy group’s median return on equity.
PCB removal cost estimates
Before the administrative law judge Williams presented evidence of $4.2 million in past unamortized costs for cleaning up PCB (polychlorinated biphenyl), plus projections of future costs. The ALJ allowed the company to amortize the $4.2 million over three years, with a procedure for refunding any amounts Williams recovered from third parties responsible for the PCB. ALJ Opinion, 73 FERC ¶ 63,015, at 65,074-75. Because Williams made a new § 4 rate filing in 1995, the issue of PCB cost recovery in the present case became “locked in” for the period of November 1, 1993 through July 31,1995.
For this locked-in period, the Commission rejected amortization in favor of the “test period” method. Williams Natural Gas Co., 77 FERC ¶ 61,277, at 62,182 (1996) (“First Order”); see generally Southwestern Public Service Co. v. FERC, 952 F.2d 555 (D.C.Cir.1992). This method takes actual costs of the most recent 12-month period (the “base period”), subject under some circumstances to adjustment on the basis of data from a nine-month period following the base period (the “adjustment period”), and absent some anomaly projects them into the period covered by the rate filing. See 18 CFR § 154.303.
Without looking to whether Williams had offered such test period figures, the Commission declared that “the $1.4 million annual amount the participants and the ALJ arrived at using an amortization method is a reasonable equivalent of WNG’s actual PCB-related test period costs.” First Order, 77 FERC ¶ 61,277, at 62,182. It also asserted that Williams had projected 10-year costs of $20 million; “this averages to $2 million a year which is reasonably close to the $1.4 million annual amount the ALJ permitted [Williams] to recover.” Id. at 62,183.
Williams did not object to use of the test period method. But on rehearing it did strenuously argue that the Commission was wrong to convert the amortization figures into test period figures. Williams pointed to Exhibit 216 in the record, which stated “Test Period Actuals” and a total of“$3,990,768.” The Commission rejected the $3.9 million figure, however, claiming it inappropriately covered a 22-month period; but in so doing the Commission cited Exhibit 24 not Exhibit 216. First Rehearing, 80 FERC ¶ 61,158, at 61,680 & n. 11. As for Exhibit 216, the Commission stated it provided “no distinct record evidence” of the level of PCB costs “incurred over any annual period during the test period.” Id. at 61,680 & n. 13. The Commission offered no explanation as to the precise flaw in Exhibit 216’s statement of “Test Period Actuals.”
Sticking by $1.4 million as “representative of annual PCB cost figures,” the Commission chastised Williams for disputing the figure, saying that this was the figure *7initially proposed by Williams (albeit as a figure for amortization). Id. at 61,679-80.
At oral argument the Commission abandoned any claim of equivalence between the $1.4 million accepted by Williams under the amortization theory and imposed by the Commission as a “representative” test period amount. Clearly a number that emerges from talcing past aggregate costs and amortizing them over an arbitrarily chosen future period is not necessarily useful for applying past experience to project future expenses — the basic principle of the test period method. It could hardly satisfy the Natural Gas Act’s requirement of substantial evidence for facts found by the Commission, 15 U.S.C. § 717r(b). See also Public Service Comm’n, 813 F.2d at 451.
Instead, the Commission at oral argument seemed to defend the use of $1.4 million as a response to what it claimed was Williams’s failure to place correct test period figures into the record. The Commission’s brief points out that $3.2 million of the total test period $3.9 million were incurred in two months, proving (in its current view) that Williams’s test period figures were “unrepresentative.” And for the very first time in this seven-year saga, the Commission at oral argument claimed that Williams’s data failed to satisfy a regulatory requirement of monthly cost figures during the test period.
Williams maintains that Exhibit 216’s “Test Period Actuals” was sufficient evidence. Nothing said by the Commission up until oral argument has supplied a reason to believe that that was inadequate. That $3.2 million in costs were incurred during two months of the test period may show that PCB removal costs come in lumps. But it hardly shows that the $3.9 million annual aggregate figure was unrepresentative, a theory in any event never invoked in the Commission’s orders.
When the Commission at oral argument asserted a requirement of monthly data, Williams questioned the existence of any such requirement and said that in fact it had supplied such data. The new Commission theory is in any event the purest form of “appellate counsel’s post hoc rationalization,” which in the usual case we do not accept. North Carolina Utilities Comm’n, 42 F.3d at 663. Since the Commission no longer defends the $1.4 million figure as representative, and in its orders never sought to justify it as a solution to some procedural default by Williams, we grant the petition and remand the case for the Commission to address this issue.
* * *
The petitions of Public Service Commission are denied; the petition of Williams is granted, that part of the order is vacated, and the case is remanded to the Commission.
So ordered.
. In the course of the proceedings Williams Natural Gas Company became Williams Gas Pipelines Central, Inc. We use "Williams” as shorthand.
. The proxy companies and their equity ratios were: Sonat, Inc. (62%), TWC (47%), Enron Corporation (43%), Panhandle Eastern Corporation (45%), Coastal Corporation (39%), and Transcontinental Energy Corporation (16%).
. Petitioners have not been punctilious in their use of precedent, mistaking in their brief the facts of North Carolina Utilities Comm’n, 42 F.3d at 663, for those of Public Service Comm’n v. FERC, 642 F.2d 1335 (D.C.Cir.1980). See Petitioners’ Opening Br. at 32.