Unified equations for the slope, intercept, and standard errors of the best straight line

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1 Unfed equatons for the slope, ntercept, and standard errors of the best straght lne Derek York a) and Norman M. Evensen Department of Physcs, Unversty of Toronto, 60 Sant George St., Toronto, Ontaro M5S A7, Canada Margarta López Martínez and Jonás De Basabe Delgado b) Departamento de Geología, CICESE, km 07 Carr. Tjuana-Ensenada Ensenada, Baja Calforna, Méxco 860 Receved January 003; accepted 7 October 003 It has long been recognzed that the least-squares estmaton method of fttng the best straght lne to data ponts havng normally dstrbuted errors yelds dentcal results for the slope and ntercept of the lne as does the method of maxmum lkelhood estmaton. We show that, contrary to prevous understandng, these two methods also gve dentcal results for the standard errors n slope and ntercept, provded that the least-squares estmaton expressons are evaluated at the least-squares-adjusted ponts rather than at the observed ponts as has been done tradtonally. Ths unfcaton of standard errors holds when both x and y observatons are subject to correlated errors that vary from pont to pont. All known correct regresson solutons n the lterature, ncludng varous specal cases, can be derved from the orgnal York equatons. We present a compact set of equatons for the slope, ntercept, and newly unfed standard errors. 004 Amercan Assocaton of Physcs Teachers. DOI: 0.9/ I. INTRODUCTION Despte ts seemng smplcty, the problem of fndng the best straght lne through an expermentally determned set of ponts on the x y plane has trggered the publcaton of hundreds of papers n a host of felds snce Gauss orgnal development of the method of least-squares estmaton LSE. Ths stuaton led Press et al. to comment that Be aware that the lterature on the seemngly straghtforward subject of ths secton Straght-Lne Data wth Errors n Both Coordnatess generally confusng and sometmes plan wrong... York and Reed 3 usefully dscuss the smple case of a straght lne as treated here... p Although the tradtonal regresson of y on x s wdely used and ncluded n most hand calculators and spreadsheets, only rarely are the x values actually error-free. In realty, both x and y errors may be sgnfcant, and may vary from pont to pont. In addton, the errors n the two coordnates may be hghly correlated. For nstance, n radometrc age determnaton, use s unversally made of sotope rato plots n whch a common normalzng sotope forms the denomnator of both the x and y coordnates. Ths practce mposes an often very sgnfcant correlaton between the errors n x and y. York gave the general solutons for the LSE best straght lne n terms of ts slope and y ntercept and errors n these two parameters, when both observables (X,Y ) are subject to errors whch vary from pont to pont. In Ref. 4, a more general soluton also allowed for the possble correlaton between the x and y errors at each pont. In both papers, t was ponted out that many prevously publshed solutons for example, those of Refs. 5 7 corresponded to dfferng ways of assgnng weghts (X ), (Y )] or error correlatons r to the observed data ponts (X,Y ). Rarely were the publshed specal solutons the correct ones to use n typcal expermental stuatons! The wdespread adopton of York s LSE algorthm led to a great mprovement n ths stuaton n some felds. However the more general treatment n York, 4 publshed n an earth scence journal, has escaped the attenton of many statstcans and others wrtng on least-squares fttng of straght lnes, ncludng Ref. and Reed. 3,8 Although the method of LSE has remaned n wdespread use among expermental scentsts for solvng statstcal problems, members of the statstcal communty have shfted to the use of the maxmum lkelhood estmaton MLE approach. Ttterngton and Hallday 9 recommended ts use for straght lne fttng. They emphaszed that, f all the data errors are normally dstrbuted and all the (X,Y ) pars are ndependent, maxmzng the lkelhood functon was exactly the same as mnmzng the weghted sum of squares n LSE. Therefore, MLE and LSE agree n ther estmates of the best slope and ntercept n ths case. However, they ponted out that MLE s... a general theory that allows us to obtan approxmate varances and covarances for the parameter estmates... It s not possble to say n general whch s best, but the MLE method... seems to be smpler for ths problem, and wth small samples there s probably not much to choose between the methods numercally. The prevous methods such as those of York, 969 were based on Taylor expansons Demng, 966 and t s notceable that MLE theory has tended to supersede ths approach n the statstcal lterature on ths problem. Ttterngton and Hallday 9 carred out numercal comparsons of the MLE and the LSE results 4 for the standard errors of the slopes and ntercepts of a number of data sets. They concluded that In most cases the standard errors from the two methods used here are very smlar... p. 89. Our purpose s to show the exact relatonshp between these two alternatve methods of error calculaton n straght-lne regresson when normal errors are assumed, and when t s presumed that, were t not for random expermental errors, the observed ponts would have been perfectly collnear. We show that despte the apparently very dfferent underlyng analytcal approaches MLE based on secondorder dervatves of the lkelhood functon, LSE based on the frst-order dervatves of the slope and ntercept, the 367 Am. J. Phys. 7 3, March Amercan Assocaton of Physcs Teachers 367 Ths artcle s copyrghted as ndcated n the artcle. Reuse of AAPT content s subject to the terms at: Downloaded to IP: On: Wed, 6 Nov 04 :06:7

2 MLE expressons obtaned n Ref. 9 for the standard errors of the slope and ntercept are algebracally dentcal wth the York 4 LSE standard error estmates when the latter are evaluated, not at the observed ponts (X,Y ) as s usually done n least squares, but at the least-squares-adjusted ponts (x,y ). The substtuton of the observed ponts n the LSE error formulas probably came about because of the addtonal computatonal work nvolved n calculatng the adjusted ponts n the pre-computer era. As early as 943, Demng 0 recognzed that the errors should deally be calculated at the adjusted ponts. The fttng process assumes that the true data ponts would le on a sngle straght lne, so that consstency demands the use of the adjusted ponts, whch satsfy ths crteron, rather than the observed ponts, whch are scattered about the ftted lne. It s now no problem to evaluate the errors at the adjusted ponts as they should be. However, even those authors who have occasonally evaluated ther LSE solutons usng the adjusted ponts 8 were apparently unaware that usng the adjusted ponts made the least-squares solutons for the errors n the ntercept and slope numercally dentcal to the standard MLE solutons. Conversely, the practtoners of MLE dd not realze that they were dentcally reproducng not merely the LSE values of the slope and the ntercept, but also the adjusted-pont LSE standard errors of these parameters. We can thus say that the least-squares and maxmum-lkelhood methods for the fttng of a straght lne have fnally been unfed for the calculaton of all four parameters a, b, a, and b. II. THE UNIFICATION The notaton used n ths paper s summarzed n Table I. In partcular, we adopt the notaton of for the standard error as calculated usng the MLE method and as calculated usng the LSE method of York. 4 Ttterngton and Hallday 9 derved the followng expressons for a and b n terms of the expectaton values x, of the observables X : W a x x x, a W b x x. b The expectaton values (x and y ) are dentcal to the LSE-adjusted values of X and Y, and ndeed Ttterngton and Hallday noted that ther formulas for the expectaton values x and y were dentcal to York s formulas for the adjusted values. The unfcaton proceeds by smplfyng Eq. for a and b. We then show that the more complex LSE expressons for these varances gven n York 4 when evaluated at the least-squares-adjusted ponts reduce exactly to these smplfed versons of the varances n Ref. 9. We smplfy the MLE varances by ntroducng the quantty u, defned n Table I. By transformng to the u from the x see Appendx A, Eqs. a and b become a x b, b u. a b Table I. Summary of our notaton. Note that although the expresson for S, the weghted sum of squared resduals, appears dentcal to that for a standard weghted regresson of y on x, the weght W actually nvolves the weghts n both x and y, as well as the correlatons between the x and y errors. Symbol Turnng now to the LSE errors of York, 4 b was expressed as b X,Y X X r X Y Y b Y, 3 where s the left-hand sde of the least-squares cubc, quadratc, or lnear equatons see Appendx B. From the expressons n York 4 for /X, /Y, and /b t can be shown that X b V, Y, Meanng a, b y ntercept and slope of best lne, yabx a, b Standard errors of a and b X, Y Observed data ponts x, y Least-squares-adjusted ponts, expectaton values of X, Y a (X,Y ), b (X,Y ) LSE standard errors evaluated at the observed ponts (X,Y ) a (x, y ), b (x,y ) LSE standard errors evaluated at the adjusted ponts (x,y ) a, b MLE standard errors (X ), (Y ) Weghts of X, Y (X )(Y ) r Correlaton coeffcent between errors n X and Y W X Y X b Y br X Ȳ V x ȳ u v X Y X X Y Ȳ x y x x y ȳ W Y bv X bv r S (Y bx a) 4a 4b 368 Am. J. Phys., Vol. 7, No. 3, March 004 York et al. 368 Ths artcle s copyrghted as ndcated n the artcle. Reuse of AAPT content s subject to the terms at: Downloaded to IP: On: Wed, 6 Nov 04 :06:7

3 b b W V 4 W b W r b V. 4c The substtuton of these expressons for the three partal dervatves nto Eq. 3 yelds where V Y X r b X,Y D, 5 V D b W V 4 W b W r b V. The expresson for b n Eq. 5 was gven n Ref., except for the omsson of the thrd term on the rght-hand sde of D, whch we presume to be a typographcal error. Tradtonally, n least-squares fttng, Eq. 5 for b would be evaluated by nsertng the correspondng values of the observables (X, Y ) nto t. Instead, let us now evaluate t at the least-squares-adjusted ponts x and y. In Appendx C we show that when we substtute x for X and y for Y, and therefore also substtute u x x for X X and v y ȳ for V Y Ȳ, the numerator of b n Eq. 5 becomes u, and the denomnator becomes ( u ),so that the LSE expresson for b evaluated at the LSE adjusted ponts s b x,y u u u b. York s 4 LSE expresson for a s a X,Y a X where and r a X X a Y Y a Y, a X bw X X b, a Y W X Y b If we substtute the expressons for /X, /Y, and /b n Eq. 4nto the expressons for a/x and a/y, Eqs. 9 and 0, we can then substtute the resultng a/x and a/y nto Eq. 8 to obtan the followng result for a : a X,Y X b X,Y X, D n agreement wth Ref. f ther D s corrected for the mssng term. To evaluate a at the adjusted ponts, we note that n ths case 0 see Appendx C, but D0, X obvously transforms to x and b (x,y ) becomes b. Thus we see mmedately from Eq. that a x,y x b a. Thus Eqs. 7 and yeld the new unfcaton theorem: If the least-squares estmates of York (969) of the errors n slope and ntercept of the best straght lne are evaluated at the least-squares-adjusted ponts nstead of at the observed ponts, the least-squares errors become dentcal to the maxmum-lkelhood errors. It also follows smply that the covarance of the slope and ntercept, cov(a,b), calculated by tradtonal LSE, becomes dentcal wth the MLE estmate of ths covarance, when evaluated at the adjusted ponts. Thus n both cases, cov(a,b)x b, and the correlaton coeffcent of a wth b s r ab x b / ax / x. Ttterngton and Hallday 9 found slght numercal dfferences between the York 4 solutons for a (X,Y ) and b (X,Y ) and ther own detaled MLE results. We now see that ths dfference s due entrely to the York LSE algorthm followng the tradtonal route of evaluatng the expressons for these parameters at the observed ponts rather than the adjusted ponts. These slght dfferences are smply reflectons of the slght dfferences between the observed ponts (X,Y ) and the adjusted ponts (x,y ). Clearly such mnor dfferences between the LSE and MLE values of a and b would be expected to ncrease somewhat as data ponts whch are more scattered about a straght lne are ftted, because greater dfferences would then exst between the observed ponts (X,Y ) and the adjusted ponts (x,y ). Although the LSE evaluated at the observed ponts and MLE error estmates wll generally dffer slghtly, there are two very sgnfcant exceptons: the cases of the regresson weghted f desred of y on x and the regresson weghted f desred of x on y. In each of these regressons, we fnd that the LSE evaluated at the observed ponts and MLE methods automatcally yeld dentcal solutons for a and b, regardless of the scatter of the observed ponts about the best lne. Ths apparent paradox s beautfully resolved when we note that n, say, the case of the regresson of y on x, the LSE expressons for a (X,Y ) and b (X,Y ) reduce exactly to functons only of the X whch are perfectly accurate observables n ths partcular regresson and therefore equal to the x by defnton, so that the a and b are smultaneously evaluated at the observable and the adjusted abscssae, and ther dentcal LSE and MLE values are thus found n one calculaton see Appendx D. By symmetry, the equvalent explanaton apples to optonally weghted regresson of x on y. 369 Am. J. Phys., Vol. 7, No. 3, March 004 York et al. 369 Ths artcle s copyrghted as ndcated n the artcle. Reuse of AAPT content s subject to the terms at: Downloaded to IP: On: Wed, 6 Nov 04 :06:7

4 Whether the above unfcaton of the LSE errors and MLE errors n Eqs. 7 and apples to more general cases than straght-lne fttng n two dmensons, we can only conjecture. But n any case, ths unfcaton of the LSE and MLE errors now obvates any necessty for choosng between these two estmates of error. Henceforth we shall smply use a and b to denote these unfed error estmates, where a a (x,y ) a and b b (x,y ) b. III. CONCISE EQUATIONS FOR THE BEST-FIT LINE We have shown that the equatons of York 4 contan all least-squares and maxmum-lkelhood solutons to the problem of fttng a straght lne to data wth possbly correlated normally dstrbuted errors n x and y. All correct solutons that we are aware of n the lterature can be derved often as specal cases from those equatons. If the newly unfed standard errors of slope and ntercept are used, then the error expressons reduce to partcularly smple forms, yeldng the followng extremely compact set of four equatons: aȳ bx, b V, a x b, 3a 3b 3c b. 3d u Equaton 3b for b must, n the general case, be solved teratvely. A typcal sequence of operatons s Choose an approxmate ntal value of b for nstance, by smple regresson of y on x). Determne the weghts (X ), (Y ) for each pont. If the errors n x and y are known, then normally (X ) / (X ) and (Y )/ (Y ), where (X ) and (Y ) are the errors n the x and y coordnates of the th pont. 3 Use these weghts, wth the value of b and the correlatons r f any between the x and y errors of the th pont, to evaluate W for each pont. 4 Use the observed ponts (X,Y ) and W to calculate X and Ȳ, from whch and V, and hence can be evaluated for each pont. 5 Use W,, V, and n the expresson for b n Eq. 3b to calculate an mproved estmate of b. 6 Use the new b and repeat steps 3, 4, and 5 untl successve estmates of b agree wthn some desred tolerance for example, one part n 0 5 ). 7 From ths fnal value of b, together wth the fnal X and Ȳ, calculate a from Eq. 3a. 8 For each pont (X,Y ), calculate the adjusted values x, where x X. Smlarly, y Ȳ b, although these values are not needed n ths calculaton. 9 Use the adjusted x, together wth W, to calculate x, and thence u. 0 From W, x, and u, calculate b, and then a. Although t s mpossble to guarantee convergence for any arbtrary data set, years of experence have shown that the teraton procedure converges remarkably rapdly, wth about ten teratons for most data sets, and fewer than 50 for pathologcal data sets such as Reed s data set II. 3 The above algorthm s straghtforward to program, and students would fnd t llumnatng to compare the parameters resultng from the above algorthm possbly usng data wth both x and y errors whch they have acqured n a laboratory experment wth the results of the smple regressons of y on x and x on y bult nto most hand calculators and spreadsheet programs. Note that Eq. 3s symmetrcal n x and y ther superfcal appearance to the contrary notwthstandng. They wll therefore produce the dentcal straght lne and correspondng errors f x and y are nterchanged. In our work wth 40 Ar 39 Ar geochronology, where the x ntercept s sgnfcant, we normally nterchange x and y data to obtan the orgnal x ntercept and ts standard error. Of course the slope obtaned after the nterchange s the recprocal of the orgnal slope. If we use Eq. 3 and the defntons of W and n Table I, t s easy to derve smplfed solutons for specal cases, many of whch have been dealt wth n the lterature, sometmes wth closed-form nonteratve solutons. Most of these specal cases use uncorrelated errors (r 0). For example the so-called major-axs soluton 6 s gven smply by settng r 0 and W. Ths soluton corresponds to mnmzng the sum of the squares of the perpendcular dstances of the observed ponts from the ftted lne. Although wdely used, ths soluton s not nvarant under a change of scale. To correct ths defcency, Kermack and Haldane 5 suggested the reduced major-axs soluton whch s nvarant under a change of scale. Ths soluton corresponds to settng r 0 and W /( Y b X ), where X (X X ) /(n) and Y (Y Ȳ ) /(n), that s, X s the varance of the X taken as a group, and smlarly for Y. The ubqutous regresson of y on x s gven smply by settng r 0 and W (Y ), where (Y ) f the regresson s unweghted. An example of regresson wth nonzero error correlatons s gven by Brooks et al. 7 As York ponted out, 4 ther soluton mplctly assumes perfect nverse correlaton of x and y errors, and can be obtaned from Eq. 3 by settng r. IV. MONTE CARLO TESTS OF ACCURACY Now that we have derved unfed LSE MLE estmates of the standard errors of the slope and ntercept, t s reasonable to ask how accurate these unfed error estmates are. To fnd an absolute standard aganst whch to test the above analytcal approxmatons, we must examne the probablstc model of lnear fttng that forms the bass of the above mathematcal analyss. In ths model, we have assumed that there exsts a set of true ponts that le exactly along a straght lne, tself havng a partcular true ntercept and slope. However, we can only measure the postons of these ponts mperfectly. Each pont has assocated measurement errors, expressed as a bnormal dstrbuton parametrzed by x and y errors, and a correlaton between those errors. The measurement process has randomly selected an observed pont from the approprate bnor- 370 Am. J. Phys., Vol. 7, No. 3, March 004 York et al. 370 Ths artcle s copyrghted as ndcated n the artcle. Reuse of AAPT content s subject to the terms at: Downloaded to IP: On: Wed, 6 Nov 04 :06:7

5 Table II. Results of Monte Carlo modelng. Each data set was run for 0 7 Monte Carlo trals. The quanttes ˆ a, ˆ b are the standard devatons of the Monte Carlo dstrbutons of the parameters â and bˆ (y ntercept and slope; â and bˆ are defned n Appendx E. The quantty a s defned as a 00( a ˆ a)/ˆ a ; a smlar defnton holds for b. a, b are the analytcal errors calculated from Eq. 3. Data set 3 s from Ref. 6 wth weghts of York Ref. zero error correlatons. Data set 4 s from Ref. 4. Data set Number of ponts S/(n) y ntercept Slope â ˆ a a % bˆ ˆ b b % mal dstrbuton centered on each true pont. Such measurements have then generated, from the set of true ponts, a set of observed ponts whch wll not, n general, be collnear. Our task n fttng these observed ponts s to reverse ths process; to try to undo the effects of the measurement errors, and thus to recover our best estmate of the slope and ntercept of the orgnal straght lne together wth an estmate of the uncertanty n recoverng those parameters. If t were practcal, the best way to estmate the uncertanty of our estmates would be to repeat the above measurement process a large number of tmes, each tme generatng a new set of observed ponts from the true ponts. Each set of observed ponts would then be used to derve a new best-ft lne. By comparng the resultng large set of ftted slopes and ntercepts obtaned under dentcal expermental condtons, wth the orgnal true values of these parameters, we could then determne the average uncertanty of estmatng the slope or ntercept. In realty, we do not have access to the true parameters of the lne or to the true postons of the data ponts we are attemptng to measure. The whole object of the fttng process s to estmate these quanttes. Furthermore, practcal consderatons lmt the number of possble repettons of the experment. So our best practcal estmate of the true uncertantes n evaluatng the slope and ntercept from a gven data set comes from repeated numercal experments, that s, from a Monte Carlo model of repeated measurement. For a gven observed data set, such a Monte Carlo model begns by fttng the data set wth the frst two members of Eq. 3, and usng the parameters of ths best-ft lne as the true parameters (â,bˆ ), and the least-squares adjusted postons of the n data ponts as the true ponts (xˆ, ŷ ). Each true pont s then assgned the actual errors of measurement ((X ),(Y ),,,...,n) and correlaton coeffcent r assocated wth the correspondng observed pont. As mentoned, both LSE and MLE methods agree on the slope and ntercept of the best-ft lne, and also agree on ther estmates of the true postons of the ftted ponts. In LSE these are termed the adjusted ponts, 0 and n MLE they are the expectaton values of the ponts. 9 Durng ths ntal fttng of the observed data set, we also calculate the uncertantes n the ntercept and slope ( a, b ) from Eq. 3. These are the two uncertantes whose accuracy we wsh to assess. We then conduct a smulated measurement process on ths set of postulated true ponts, by generatng, from the bnormal dstrbutons assocated wth each of the n true ponts, a set of n random observed ponts. These observed ponts can then be used to obtan a best-ft lne, characterzed by a slope and ntercept that wll be dfferent from â and bˆ, the parameters of the orgnal true lne. If we repeat ths smulated measurement process N tmes on our set of true ponts, we obtan N estmated slopes and ntercepts. The dstrbutons of the N slopes and ntercepts about the known true values, as N becomes very large (0 7 n our Monte Carlo models, are measures of the expected errors of estmatng the slope and ntercept n a sngle measurement, such as the actual physcal measurement that we orgnally performed. If the observed dstrbuton of the slopes or ntercepts s Gaussan, then the standard devaton of the dstrbuton s the standard error of the parameter slope or ntercept beng estmated. In fact, for all nne data sets n Table II, the 0 7 pars of a and b values yelded hstograms almost perfectly matchng Gaussan dstrbutons. 3 Note that the standard devatons should be calculated wth respect to the true y-ntercept â and slope bˆ, rather than the means ā and b of the N ntercepts and slopes; that s, ˆ a(a â) /N rather than (a ā) /(N) and smlarly for ˆ b. These quanttes, ˆ a and ˆ b, act as the true values aganst whch we test the estmates ( a, b ) calculated from Eq. 3. We have used ths approach to test the calculated unfed LSE MLE errors n the slope and ntercept aganst the results of Monte Carlo modelng, usng a varety of real, expermentally derved data sets. These nclude data sets havng 7 34 data ponts, showng a range of 0 5 n â and bˆ, a range from to n the correlaton coeffcent r ab, and wth a range of more than a hundred n the goodness-of-ft parameter S/(n) dscussed below. The detals of the Monte Carlo calculatons are gven n Appendx E. In Table II we summarze the results of nne Monte Carlo models. We compare these true errors ˆ a and ˆ b for N 0 7 wth the errors a and b calculated from the orgnal expermental data set usng Eq. 3. The a and b values are the percent dfferences between these calculated and true errors. The maxmum value of observed n the nne data sets was less than.4%, and two-thrds of the values were well under 0.%. In other words, the error estmates calcu- 37 Am. J. Phys., Vol. 7, No. 3, March 004 York et al. 37 Ths artcle s copyrghted as ndcated n the artcle. Reuse of AAPT content s subject to the terms at: Downloaded to IP: On: Wed, 6 Nov 04 :06:7

6 lated usng Eq. 3 wth these data sets are themselves typcally n error by less than a percent. Clearly the approxmatons made n dervng Eq. 3 are exceptonally good n practce. We also wsh to emphasze that our numercal results renforce the conclusons of Ref. 9 that the MLE our adjustedpont LSE and tradtonal LSE our observed-pont LSE errors are very smlar. Although we plan a more elaborate exploraton of ths and other aspects of the Monte Carlo modelng n a later paper, we note here that the devatons between the LSE observed-pont errors and the adjustedpont errors for the nne data sets presented are all less than 0%, and two-thrds are well under %. Ths agreement s surprsngly good for error estmates derved from a relatvely small number of expermental ponts. We conclude that publshed results based on least-squares treatments such as those of York,4 whch use observed-pont errors wll, n general, reman vald for all practcal purposes. V. GOODNESS OF FIT In general, the devatons of the observed ponts from the ftted ponts should be on the order of the assgned errors of the observed ponts. Ths concept can be quantfed by consderng the weghted sum of devatons from the best-ft lne wth error correlatons taken nto account. Ths quantty, S (Y bx a), s the same one mnmzed n the least-squares formulaton of the fttng problem. 4 If n ponts are beng ftted, the expected value of S has a dstrbuton for n degrees of freedom, so that the expected value of S/(n) s unty. Wthout dscussng n detal what to do f S/(n) s apprecably dfferent from unty, we smply note that t can be nterpreted ether as a statstcal fluke wth a probablty obtanable from a table of ), or as a falure of the assumptons for example, the presumed lnear relaton s ncorrect, the errors of the observed ponts are wrongly assgned, or an unaccounted for factor, such as systematc error, has affected the measurements. One technque that s sometmes appled f S/(n) s sgnfcantly larger than unty s to multply the calculated a and b values by S/(n), whch s equvalent to multplyng all the x and y errors, (X ) and (Y ), by the same factor. Ths makes Sn, wthout affectng the computed slope and ntercept. Ths procedure, of course, should not be appled mechancally, wthout gvng some thought to ts approprateness. One can easly magne stuatons where alternatve actons would be more reasonable. ACKNOWLEDGMENTS D.Y. and N.M.E. were supported n ths research by a grant from the Natural Scences and Engneerng Research Councl of Canada. J.D.D. was supported by a M.Sc. grant from the Consejo Naconal de Cenca y Tecnología Méxco. Useful dscussons were enjoyed wth Patrck E. Smth and Enrque Gómez Trevño. Two anonymous referees made helpful comments that have sgnfcantly mproved the presentaton. APPENDIX A: MAXIMUM LIKELIHOOD ESTIMATES OF a AND b The expresson of Ref. 9 for b our Eq. b can obvously be wrtten b x x x, x x and therefore Aa b x x x. Ab Recall that x u x. If we substtute ths expresson nto Eq. Ab for b, we fnd b u x u u. x u But W u W x x Therefore W x x W x W W x x 0. x A A3 b, A4 u as n Eq. b. If we dvde our Eq. a by Eq. b, we fnd for the MLE calculatons of Ref. 9: a W x b. A5 But u x x and u 0, so W x W u x W u x u x W u x W u x W b x W. So x x b and from Eq. A5, a b x b b x b x b, as n Eq. a. A6 A7 A8 37 Am. J. Phys., Vol. 7, No. 3, March 004 York et al. 37 Ths artcle s copyrghted as ndcated n the artcle. Reuse of AAPT content s subject to the terms at: Downloaded to IP: On: Wed, 6 Nov 04 :06:7

7 APPENDIX B: LEAST-SQUARES CUBIC, QUADRATIC, AND LINEAR EQUATIONS WHEN ERRORS IN x AND y ARE CORRELATED In York 4 the followng cubc B, quadratc B, and lnear B3 equatons for the best slope b were gven for the case when X and Y are correlated: b 3 W b X W V X b W W r V W V W r V b W W r W V X 0, B V X r U b W Y V X V Y r V 0, B W b W Y bv b X br bv X r V 0, that s, W V Y bv X r V W Y bv X br W V Y B3. B4 All three of these equatons lnear, quadratc, and cubc yeld dentcal values for best b and the errors a and b. Equaton B4, York s lnear algorthm, was the frst such pseudo-lnear soluton of the general least-squares problem wth correlated errors. It may also be wrtten as b V ba A, B5 where AW V (r / ). By cross multplcaton and collecton of the explct terms n b, we have b V, B6 a form gven n Ref., whch confrmed the result of Ref. 4. APPENDIX C: EVALUATION OF a AND b AT LSE-ADJUSTED POINTS By defnton, all of the LSE-adjusted ponts (x,y ) fall on the LSE best straght lne. 4 Thus, y abx, Ca W y a W b W x, ȳabx, y ȳbx x. Cb Cc Cd That s, v bu for all, C where u x x, C3a v y ȳ. C3b Thus we evaluate b at the adjusted ponts by substtutng u for, v for V and bu v n Eq. 5. The numerator of Eq. 5 becomes numerator W W W W u u u Y Y V X r V v X r u v Y b u X r bu Y b X br W u W W u, when evaluated at the LSE-adjusted ponts. The denomnator n Eq. 5s D, where D b W V 4 W C4 W r b bu V. C5 When we evaluate Eq. C5 at the adjusted ponts, we mmedately see that the thrd term vanshes, because b V transforms to bu v 0 for all. In the second term we have and to transform. Now, Y bv X bv r, C6 by the defnton of. If we evaluate Eq. C6 at the adjusted values (x,y ), we have adjusted u u u u Y bv X bu r Y b X br u u u W u 0, C7 for all. Thus the second term n D also vanshes. Then D, evaluated at the adjusted values (x,y ), becomes adjusted D b W u v b W u bu W u. C8 Then the value of the b, evaluated at the adjusted (x,y ), becomes 373 Am. J. Phys., Vol. 7, No. 3, March 004 York et al. 373 Ths artcle s copyrghted as ndcated n the artcle. Reuse of AAPT content s subject to the terms at: Downloaded to IP: On: Wed, 6 Nov 04 :06:7

8 b x,y W u u b, u by Eq. A4, thus provng Eq. 7. For the case of a, from Eq., C9 a X W X b. C0 D To evaluate ths expresson at the adjusted ponts, we have to evaluate there. We wll show that n ths case 0 by provng that n general x X. We have from York, 4 x X b V b X r. But t s easy to see from the defnton of that W b V Then b X r. x X X X X X. C C C3 Thus, x X, or x X. That s, when evaluated at the adjusted ponts (x,y ), x x 0. If we substtute 0 and X x n Eq. C0 for a, we obtan a x,y x b x,y x b whch proves Eq.. by Eq. C9 a, by Eq. A8, C4 APPENDIX D: PROOF THAT a X,Y Ä a AND b X,Y Ä b IN REGRESSION OF y ON x AND x ON y In the classcal regresson of y on x weghted f desred t s assumed that the X are free of error and all the scatter s attrbuted to errors n Y. Thus r 0 automatcally, and W collapses to (Y ). Furthermore, n ths case, Y bv X Y Y, D because (X )(Y ). Hence, b, evaluated at the observed ponts (X,Y ) as tradtonally done, becomes, from Eq. 5, b X,Y Y Y b Y V Now Eq. B4 becomes Y b Y V. D Y V U Y b Y V, D3 U Y Y Y so that (Y ) V b(y ). We substtute ths value for (Y ) V n Eq. D, b X,Y. D4 Y But for ths regresson the X have no errors, that s, X x and X x, so that u. Then b X,Y Y u b D5 by Eq. b. Smlarly, because X x, 0 for ths regresson. But D0, hence from Eqs. and a a X,Y Y x b X,Y Y x b a. D6 By symmetry, when x s regressed on y, a (X,Y ) and b (X,Y ) are despte both beng evaluated at ther observed values also automatcally dentcal wth a and b, respectvely. APPENDIX E: MONTE CARLO MODELING OF ERROR ESTIMATES The modelng proceeds as follows. Take an expermental data set, consstng of observed ponts (X,Y ), whch are scattered about a lne and have Gaussan errors wth standard devatons (X ) and (Y ), where the errors have a correlaton r,r. Use Eq. 3 and ft ths data set to a lne to obtan the y-ntercept â, the slope bˆ, and the standard errors a and b. Use the adjusted ponts (x,y ), (,...,n), as a set of true ntatng ponts dstrbuted along the ftted straght lne (â,bˆ ) whch s now taken to be the underlyng true straght lne for ths data set. For each collnear pont (x,y ) generate a new (X,Y ) at random from the bnormal dstrbuton functon N (X ),(Y ),r centered on (x,y ). The new set of (X,Y ) wll of course not be collnear, and represents an observed data set n the Monte Carlo model. 3 Use the (X,Y ) data set to calculate wth Eq. 3 a new best-ft lne wth parameters (a j,b j ). 4 Wth the orgnal collnear (x,y ) of step, repeat steps and 3 for j,,...,n, where N s some large number say, 0 7 ) to generate a sequence of ntercepts, a (a,a,...,a N ), and of slopes, b(b,b,...,b N ). 5 Calculate the standard devaton ˆ a of the sequence a from ˆ a (a â) /N, and smlarly for the standard devaton ˆ b of the sequence b. 374 Am. J. Phys., Vol. 7, No. 3, March 004 York et al. 374 Ths artcle s copyrghted as ndcated n the artcle. Reuse of AAPT content s subject to the terms at: Downloaded to IP: On: Wed, 6 Nov 04 :06:7

9 a Electronc mal: b Currently at Facultad de Ingenería, Unversdad Autónoma de Baja Calforna, Km. 07 Carr. Tjuana-Ensenada, Ensenada, Baja Calforna, Méxco 860. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterlng, Numercal Recpes n FORTRAN: The Art of Scentfc Computng Cambrdge U.P., New York, 99, nd ed. Derek York, Least-squares fttng of a straght lne, Can. J. Phys. 44, B. Cameron Reed, Lnear least-squares fts wth errors n both coordnates, Am. J. Phys. 57, D. York, Least squares fttng of a straght lne wth correlated errors, Earth Planet. Sc. Lett. 5, K. A. Kermack and J. B. S. Haldane, Organc correlaton and allometry, Bometrka 37 /, Karl Pearson, On lnes and planes of closest ft to systems of ponts n space, Phlos. Mag. 6, C. Brooks, I. Wendt, and W. Harre, A two-error regresson treatment and ts applcaton to Rb-Sr and ntal 87 Sr/ 86 Sr ratos of younger Varscan grantc rocks from the Schwarzwald Massf, Southwest Germany, J. Geophys. Res. 73, B. Cameron Reed, Lnear least-squares fts wth errors n both coordnates. II. Comments on parameter varances, Am. J. Phys. 60, D. Mchael Ttterngton and Alex N. Hallday, On the fttng of parallel sochrons and the method of maxmum lkelhood, Chem. Geol. 6, W. Edwards Demng, Statstcal Adjustment of Data Wley, New York, 943. J.-F. Mnster, L.-P. Rcard, and C. J. Allègre, 87 Rb- 87 Sr chronology of enstatte meteortes, Earth Planet. Sc. Lett. 44, Grenvlle Turner, 40 Ar- 39 Ar ages from lunar mara, Earth Planet. Sc. Lett., J. De Basabe Delgado, Regresón lneal con ncertdumbres en todas las varables: Aplcacones en geocronología alcálculo de socronas, M.Sc. thess, CICESE, F. Albarède, Introducton to Geochemcal Modelng Cambrdge U.P., Cambrdge, Am. J. Phys., Vol. 7, No. 3, March 004 York et al. 375 Ths artcle s copyrghted as ndcated n the artcle. Reuse of AAPT content s subject to the terms at: Downloaded to IP: On: Wed, 6 Nov 04 :06:7