The Use of the Durbin-Watson d Statistic in Rietveld Analysis

356 J. Appl. Cryst. (1987). 20, 356-361 The Use of the Durbin-Watson Statistic in Rietvel Analysis BY R. J. HILL Division of Mineral Chemistry, CSIRO, PO Box 124, Port Melbourne, Victoria 3207, Australia AND H. D. FLACK Laboratoire e Cristallographie aux Rayons X, Universitk e GenOve, 24 quai Ernest-Ansermet, CH- 1211 GenOve 4, Switzerlan (Receive 31 January 1987; accepte 27 April 1987) Abstract A weighte form of the Durbin-Watson statistic [Durbin & Watson (1971). Biometrika, 58, 1-19] has been use to quantify the serial correlation between ajacent least-squares resiuals in Rietvel refinements of step-scan power iffraction ata. Analyses of X-ray an neutron ata from a range of materials have shown that the statistic: provies a sensitive measure of the progress of a refinement an remains iscriminating when other agreement inices fail, for example, when comparing results at ifferent step withs; provies quantitative information about the significance of serial correlation present in the resiuals; provies a convenient means of assessing the reliability of the estimates of the parameter variances; an provies a basis for the selection of values of step with an intensity corresponing to optimum an/or minimum use of experimental beam time. Introuction The Durbin-Watson statistic (Durbin & Watson, 1950, 1951, 1971) was introuce as a probe of serial correlation in the stuy of the least-squares regression of time series. The value provies a means of quantifying the serial correlation between ajacent least-squares resiuals an of estimating whether this correlation is well behave. Although it can be shown for a moel of autocorrelate errors that parameter estimates are unbiase, the estimates of variances may be consierably either over- or unerestimate. For our purposes, the statistic will be efine in weighte form as _ = (zli/ffi Ai- 1/cri - 1) (Ai/ffi) z, i=2 i where N is the number of observations (steps) an Ai/ai is the usual weighte resiual of the least squares corresponing to the ith element of ata. It is a useful approximation to test against the 0.1% significance point by use of the parameter Q (Theil & Nagar, 1961), Q = 2[(N- 1)/(N - P)- 3.0902/(N + 2)1/2], where P is the number of least-squares parameters estimate. If consecutive resiuals are insignificantly correlate, has a value nearer to 2 than Q. If < Q < 2, then successive values of the resiuals ten to have the same sign an have positive serial correlation, whereas for > 4 - Q > 2 successive resiuals ten to have opposite sign an isplay negative serial correlation. Flack, Vincent & Vincent (1980) have use to test single-crystal intensity ata for time-serial correlation, whilst Bernarinelli & Flack (1985) have mae use of a moifie in the stuy of systematic effects in the etermination of absolute structure from singlecrystal measurements. However, an increasing amount of crystal-structure information is now being erive by Rietvel analysis (Rietvel, 1969) of power iffraction ata. In this metho, the step-scan iffraction pattern is left intact an is irectly fitte, point by point, to a profile erive from the atomic positional an isplacement parameters. It is in just such step-scan ata that serial correlation of the leastsquares resiuals is very likely to be present. Inee, Sakata & Cooper (1979) were perhaps the first to raw attention to the fact that the usual metho of etermining the variance of a parameter in Rietvel analysis is formally invali since it ignores the presence of serial correlation. Following a suggestion by Flack (1985), Hill & Masen (1986) have applie the Durbin-Watson statistic to least-squares resiuals from Rietvel refinements of X-ray power iffraction ata. This preliminary analysis was performe as part of a systematic stuy of the effect of step counting time an step with on the accuracy an precision of structural parameters erive by Rietvel analysis (Hill & Masen, 1984, 1986). It is the purpose of the present stuy to provie a more thorough investigation of the characteristics of the statistic in Rietvel analysis, 0021-8898/87/050356-06501.50 1987 International Union of Crystallography

R. J. HILL AND H. D. FLACK 357 Table 1. Sample an ata collection etails Lea Parameter Fluorite antimonate Corunum Anglesite Formula CaFz PbSb206 ~-A1203 PbSO4 Space group Fm3m P31 m R3c Pbnm Raiation Neutron Neutron Cu Kc~ X-ray Neutron Wavelength(s) (A) 1"500 1"500 1.54056 1"893 1.54439 20 scan range ( ) 20-160 15-160 24-129 27-158 Step interval ( 20) 0"08 0-05 0-01 0"05 Maximum step intensity (counts) 1050" 1130" 56400, 560-t 1270 Number of unique reflections 22 150 50 210 Number of structural parameters + 3 7 6 19 Number of profile parameters 8 9 8 9 *For this neutron iffraction pattern, the step intensities correspon to the average value collecte on each of eight etectors. tcorresponing to step counting times of 5 an 0"05s, respectively. :]:Inclues atomic positional an isplacement parameters an unit-cell imensions. an to expose its power in the interpretation of the estimate stanar eviations (e.s..'s) obtaine by this technique. In using the statistic an testing it against the value of Q, we are using a null hypothesis of no significant serial correlation in the least-squares resiuals. As Prince (1985) points out, least-squares resiuals are always correlate since the calculate quantities obtaine from the process epen on the observe values. The significance points of the statistic (in the approximation use here, the value of Q) allow, of course, for this epenence. The serial correlation teste is aitional to that inherent in resiuals from least squares. The origin of the aitional correlation is not perceive per se from the test an must be euce from other consierations. Thus, for the econometric time series autocorrelation of errors is universally accepte as the source of the serial correlation. With gas electron iffraction it is the intensity observations themselves that are correlate (Morino, Kuchitsu & Murata, 1965; Murata & Morino, 1966). With Rietvel analysis it is generally accepte (Sakata & Cooper, 1979; Hill & Masen, 1984, 1986) that systematic effects are at the origin of the serial correlation an the incorrect estimation of variances. In particular, since the non-linear moel is the prouct of a profile function an an integrate intensity, systematic errors in either of these 'source' functions result in serial correlation of resiuals. Experimental The four samples chosen for analysis are liste in Table 1, together with a summary of the ata collection an refinement proceures use. Further experimental etails, an iffraction profile plots for the lea antimonate an corunum ata sets are provie by Hill (1987) an Hill & Masen (1984, 1986) respectively; the step-scan iffraction ata for fluorite an anglesite have not been publishe.* All least-squares refinements were performe with the Rietvel analysis program LHPM 1 escribe in Hill & Howar (1986). Results an iscussion Fig. l(a) shows the variation in the value of the Durbin-Watson statistic an several other conventional agreement inices (Young, Prince & Sparks, 1982), prouce by the presence of small errors in the scale factor uring Rietvel refinement of the 022 peak in the neutron power iffraction pattern of fluorite, CaFz (Table 1). The peak fits obtaine for the two extreme values of the scale factor are shown in Figs. l(b) an (c), respectively. The conventional Rietvel agreement inices, namely the weighte profile Rwp, the gooness-of-fit parameter GofF an the Bragg RB, all show minima at a scale-factor value of 0.4465. The value, on the other han, isplays a well efine maximum at this best-fit scale value, eclines sharply for an uner- or overestimate scale factor, consistent with the presence of severe positive serial correlation in the least-squares resiuals for ajacent steps in the peak profile. Fig. 2(a) shows that exhibits a similar behaviour when the pseuo-voigt peak shape fitte to the same fluorite 022 peak profile is eliberately altere from its best-fit value of 57.6% Lorentzian character. The fits obtaine for a pure Gaussian an pure Lorentzian peak shape are shown in Figs. 2(b) an (c). Once again, the value eclines sharply on either sie of the least-squares minimum, inicating the presence of positive serial correlation ue to peak shape misfit. *Listings of the step-scan iffraction ata for fluorite an corunum have been eposite with the British Library Document Supply Centre as Supplementary Publication No. SUP 43943 (10 pp.). Copies may be obtaine through The Executive Secretary, International Union of Crystallography, 5 Abbey Square, Chester CH 1 2HU, Englan.

358 THE DURBIN-WATSON STATISTIC The behaviour of for the Rietvel refinement of a full power iffraction profile is shown in Fig. 3 using neutron power iffraction ata collecte on lea antimonate (Table 1; Hill, 1987). In this refinement, the oxygen atom was eliberately isplace a small istance from its true position an the progress of the refinement was monitore, cycle by cycle, as the atom shifte back to its best-fit location. With the oxygen atom locate well away from its true position, as in cycles 0 to 3, the calculate peak intensities are incorrect an the resiuals contain serious serial correlation. As convergence is approache, the value levels off in an analogous way to the other agreement inices. Thus, it is clear that is as sensitive an inicator of the progress of a refinement as are the conventional agreement inices. Furthermore, since Q has a value of 1.90 for this refinement, the value of clearly inicates a significant moel eficiency through the relatively large amount of resiual serial correlation at convergence. Fig. 4 presents results from a systematic stuy of X- ray power ata collecte from corunum, ~-AlzO 3 2-0,,, I'0 0.40 0.44 0.48 I sco,e(::c,o \, 8 RB (%) R=p(%) 4GofF u) E 2"0 I I I I i Rwp (%) RB (%) G of F I-0 4 0 25 50 75 I00 1 Peak Shape (% ~ ( L rentzi n) a ) 1.o 0.5 0-5 E, x.. t.o 6 o.o[ o.o [,. o.o o.o -.... i 20 (~) 20 (~) 40 50 40 50 (b) (c) Fig. 2. Variation in the value of the Durbin-Watson statistic an other conventional agreement inices, prouce by the presence of small errors in the pseuo-voigt peak shape fitte uring Rietvei refinement of the 022 peak in the neutron power iffraction pattern of fluorite, CaF2. (b) an (c) show the peak fits obtaine for a pure Gaussian an a pure Lorentzian shape, respectively; the least-squares minimum correspons to 57-6% Lorentzian character. Rw 1.01 1.ol 1"5 0 40 ( 0.5 u~ 0"0 o.51 E~ x,, 0.01,, _-J '~..._., 1.0 es (xlo 4) 3o 20 R,,p (%) R R (%) o.o 20 (:) (b) ' ' 5'0 0,01... 40 20 (:') 50 Fig. 1. Variation in the value of the Durbin-Watson statistic an the conventional agreement inices Rwp, RB an GofF, prouce by the presence of small errors in the scale factor uring Rietvel refinement of the 022 peak in the neutron power iffraction pattern of fluorite, CaF2. The peak fits obtaine for the two extreme values of the scale factor are shown in (b) an (c), respectively; in these plots, the observe ata are represente as plus signs, the calculate profile as the continuous line overlying them, an the ifference profile as the curve below. (c) o.5 es I0 R,,,p 0 i t = i I, i RB i 0 0 2 4 6 Refinement Cycle Number Fig. 3. Variation in the value of the Durbin-Watson statistic an other conventional agreement inices, as a function of the leastsquares cycle number uring convergence of a Rietvel refinement of neutron power iffraction ata collecte from PbSb206. The example illustrate correspons to the progressive movement of a isplace oxygen atom back to its best-fit location.

R. J. HILL AND H. D. FLACK 359 (Table 1) over a wie range of step withs an two quite ifferent step counting times (Hill & Masen, 1986). The value an the e.s., of the oxygen-atom coorinate are plotte as a function of step with in the range 0-01 to 0-32 for ata collecte at 0.05 an 5 s per step. The shae area correspons to the region Degrees of freeom (N-P) 5224 104642604 1294 639 508 404 512 I I I I I I /~ /....--...--...-.-...-.. ~ :...-...:..--...:~:~:~:~i~,,,,~-...~..~- ~ ~ :: ::: :: :: ::: : ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::: ::~. ::: : ::: z o,,~i::ii::!::!i~iii~::i~t~i~i::::i!::::i::ii::~iiiiiii::!::::!i!!::::i 2o ====================================================================================================....:.:.:.::::::::: ~ 1~"~"~"~':'":'~'~:~:"':~;:~i~!i: 5 -... - :':" es x 103 I.O /T... IO,//=/ _-..." "~ " es J--,~,/' &--""---i ~ O'05s o A ''- Min. FWHM 5s 0 I I I I I I 0 0 0-I0 0"20 0"30 Step with ( ) Fig. 4. Variation in the Durbin-Watson statistic (circles) an the e.s., of the oxygen-atom coorinate (triangles) in corunum, etermine by Rietvel analysis of X-ray ata collecte with counting times of 0"05 (open symbols) an 5 s (fille) per step, as a function of step with. The shae area correspons to the region into which the value must fall if there is no positive or negative serial correlation at the 0"1% significance level, that is, Q < < 4 - Q. The arrow represents the value of the smallest peak FWHM. into which must fall if there is no positive or negative serial correlation at the 0.1% significance level. The refinements at both counting times contain significant levels of positive serial correlation at step withs that are narrower than the smallest peak full with at half maximum (hereafter, FWHM) of about 0.17. As expecte, the serial correlation of the resiuals is more serious for the longer counting-time ata since sample or moel errors, rather than counting statistics, ominate the peak profile in this case. As the step interval wiens, the step intensity resiuals are graually renere inepenent by the fact that only a small number of points lie on each Bragg peak. Thus, for refinements with the 0-05 s ata, serial correlation becomes negligible at a step interval of about 0"16 ~, whereas for the 5 s ata the correlation remains up to a step with of about 0-24. In Fig. 4, the e.s., of the oxygen-atom coorinate [an all other parameters (Hill & Masen, 1984, 1986)] increases with ecreasing step counting time an increasing step with. For a given counting time, the e.s., increases in proportion to the reciprocal square root of the number of steps in the pattern, i.e. a ecrease in the number of steps across each Bragg peak. Similarly, the e.s..'s obtaine at a given step with from the 5 s ata are smaller than those obtaine from the 0.05 s ata, since the integrate peak intensity is etermine with greater precision at the longer step-counting time. We note, in passing, that the accuracy (as oppose to precision) of the crystal structure parameters is not significantly affecte by the changes in step with or counting time 1.0 PB$Oq cn b- c) u 0.5 : :... 0.0, i i., i i i i i i ii,rll I I [ Ii ii Illl iii ii iii i i I0111 JllL. llll I1,111 Ill IIIIIIHIIILI IIIIIIq IIIIIlll el IIIllll Illlll IIIO IIII rlr] IiI11 O llll II 11111 rl ill liar 30 q0 50 60 70 80 90 100 ]1~ 120 I30 1~0 ]50 THO THETR (DEC} Fig. 5. Observe, calculate an ifference neutron power iffraction profiles for anglesite, PbSO4. The observe ata, collecte at a step interval of 0.05, are inicate by plus signs, an the calculate profile is the continuous fine overlying them. The short vertical fines below the pattern represent the positions of all possible Bragg reflections, an the lower curve is the ifference between the observe an calculate intensity at each step.

360 THE DURBIN-WATSON STATISTIC up to, an incluing, a step with of 0.25 (Hill & Masen, 1984, 1986). If it is suppose that the e.s..'s only have relevance when the resiuals from the least-squares refinements show no significant serial correlation, then it is only for those refinements with values in the shae region of Fig. 4 that the e.s..'s can be regare as trustworthy. Thus, the smallest meaningful e.s., obtainable from the 0.05 s ata correspons to that calculate from the refinement with a step interval of 0-15, an the best e.s., for the 5 s ata correspons to the refinement of the 0.25 ata. A similar analysis has been unertaken on neutron power iffraction ata collecte from a sample of anglesite, PbSO4 (Table 1; Fig. 5). This material provies a relatively complex pattern with a minimum peak FWHM of about 0-30. The refinement involves 210 reflections an has a total of 28 structure an profile variables. Fig. 6 shows that the conventional agreement inices o not iscriminate between refinements at ifferent step intervals, even though the e.s..'s necessarily change by a factor of 2.6 over the step interval range 0"05 to 0.35. Furthermore, the crystal structure parameters obtaine from these refinements are inistinguishable from each other, an from the singlecrystal values of Miyake, Minato, Morikawa & Iwai (1978). On the other han, the value oes change with step with. It crosses into the no-serialcorrelation region of Fig. 6 at a step interval of about 0"24, inicating that the calculate e.s..'s are only vali for refinements with step intervals at least as large as this value. This cut-off value for step with is about 80% of the minimum FWHM, close to the corresponing fraction obtaine for the 0-05 s corunum ata refinements, for which the step intensities are of the same orer of magnitue. It is worthy of note that this pattern woul usually be collecte at a step interval of 0.05 or, at most, 0' 10, corresponing to a total experiment time at least three times longer than necessary. Concluing remarks It may be seen that for Rietvel analysis of power iffraction ata, the Durbin-Watson statistic: (i) provies a sensitive measure of the progress of the refinement an remains iscriminating when other agreement inices fail, for example, when comparing results at ifferent step withs; (ii) provies quantitative information about the significance of serial correlation present in the resiuals from the least-squares refinement; (iii) provies a convenient means of assessing the reliability of the erive values of the parameter e.s..'s; an (iv) provies a basis for the choice of step with an intensity corresponing to the optimum an/or minimum use of experimental beam time for a particular application. Its routine implementation in Rietvel analysis is highly recommene. Degrees of freeom (N-P) 2582 1273 840 623 493 406 344 3"0,, ~,,,, 2-0....-...-...-...--..~...-'.. -.-.--.-.-:.-.~.'.::.T. : ~ ~ ================================================================================ ==================================================================== G of g Rwp 8 es x 104 I-O R=p(%) ~//'-"~ ~ ~ GolF %(%) 1 Min.FWHM 0 n n n n n n n 0 0 0.I0 0 20 0.30 0.40 Step with (o) Fig. 6. Variation in the Durbin-Watson statistic, the conventional agreement inices Rwr Re an GolF, an the e.s., of one of the oxygen-atom coorinates in anglesite, etermine by Rietvel analysis of the neutron ata presente in Fig. 5, as a function of step with. The shae area correspons to the region into which the value must fall if there is no positive or negative serial correlation at the 0.1% significance level, that is, Q < < 4 - Q. The arrow represents the value of the smallest peak FWHM. References BERNARDINELLI, G. FLACK, H. D. (1985). Acta Cryst. A41, 500-501. DURBIN, J. & WATSON, G. S. ( 950). Biometrika, 37, 409-428. DURB[N, J. & WATSON, G. S. (1951). Biometrika, 38, 159-178. DURBIN, J. & WATSON, G. S. (1971). Biometrika, 58, 1-19. FLACK, H. D. (1985). Crystallographic Computing 3: Data Collection, Structure Determination, Proteins, an Databases, eite by G. M. SHELDRICK, C. KRUGER & R. GODDARD, pp. 18-27. Oxfor: Clarenon Press. FLACK, n. D., VINCENT, M. G. & VINCENT, J. A. (1980). Acta Cryst. A36, 495-496. HILL, R. J. (1987). J. Soli State Chem. In the press. HILL, R. J. & HOWARD, C. J. (1986). Report No. Ml12. Australian Atomic Energy Commission, Lucas Heights Research Laboratories, PMB, Sutherlan, NSW, Australia. HILL, R. J. & MADSEN, I. C. (1984). J. Appl. Cryst. 17, 297-306. HILL, R. J. & MADSEN, I. C. (1986). J. Appl. Cryst. 19, 10-18. MIYAKE, M., MINATO, ]., MORIKAWA, H. & IWAI, S. (1978). Am. Mineral. 63, 506-510. MOR1NO, Y., KUCHITSU, K. & MURATA, Y. (1965). Acta Cryst. 18, 549-557.

R. J. HILL AND H. D. FLACK 361 MURATA, Y.,~ MORINO, Y. (1966). Acta Cryst. 20, 605-609. PRINCE, E. (1985). Structure an Statistics in Crystallography, eite by A. J. C. WILSON, pp. 95-103. Guilerlan: Aenine Press. RIETVELD, H. M. (1969). J. Appl. Cryst. 2, 65-71. SAKATA, M. & COOPER, M. J. (1979). J. Appl. Cryst. 12, 554 563. THEIL, H. & NAGAR, A. L. (1961). J. Am. Stat. Assoc. 56, 793-806. YOUNG, R. A., PRINCE, E. & SPARKS, R. A. (1982). J. Appl. Cryst. 15, 357-359.