ANALYTICAL CHEMISTRY
1470 Company. I n addition, they are thankful to the U. S. Electric lfanufacturing Corporation for permitting the use of its facilities in certain phases of the work. They are also indebted to D. Killiams, R. C. Hawes, and G. S.Haines for their friendly criticisms while the manuscript was in preparation. LITERATURE CITED (1)
Bastian, R., ANAL.CHEM.,21, 972 (1949).
(2) Bastian, R., Weberling, R., and Palilla, F., Ibid., 22, 160 (1950). (3) Brice, B. A., Reu. Sci. Instruments, 8 , 279 (1937).
(4) Hiskey, C. F., ANAL.CHEM.,21, 1440 (1949). ( 5 ) Hiskey,C. F., Trans. N.1’. Acad. Sci., 11, 223 (1949). RECEIVED May 4 , 1950. The nomenclature and method of approach used in this article are in conformity with those given in a previoua paper ( 4 ) . Reading of the above article will facilitate the understanding of many points which are only briefly treated in this paper. Some of this material was presented before the Fifth Annual Symposium of .halytical Chemistry, Pitts. burgh, February 1950. Other parts mere presented a t the 118th Meeting of the AMERICAX CHEMICAL SOCIETY.Chicago, 111.. September 1950. The erperimental material contained herein has been taken from theses to be s,tbmitted by Jacob Rabinowitz and I. G. Young to the Polytechnic Institute of for the master’s degree in Brooklyn in partial fulfillrnent of the rerj~~irelnent chemistry.
Evaluation of Three Iron Methods Using a Factorial Experiment I,. K. REITZ, A . S. O’BRIEN, AND T. L. D.IVIS Paper Service Department, Eastnian Kodak Company, Rochester, V. Y . The classical method of designing experinients for testing the precision of an analytical nicthod is contrasted with Fisher’s factorial design technique. The advantages of the latter procedure from the point of view of econorn? of experiments and the detection of interactions are presented. The application of an anal!sis of variance to data obtained by a factorial experiment is discussed. These sta-
T
HE value of .staZtistivitl terhiiiqueb to the analytiral vhemist has become increasingly apparent. .I Symposium on Stat istical Methods in Experimental and Industrial Chemistry was SociEw held a t the 113th Meeting of the . ~ M E R I C A N CHEMICAL (8). The progress made in statistics as applied to analytical chemistry has been reviewed by Wernimont ( 9 ) ,who lists the statistieal tools that should be exploited by the analyst. These include control charts, teats of significance, analysis of variance, correlation, and design of experiments. Factorial design and the analysis of variance are described in this article. These techniques when applied to an analytical method yield information on precision but not necessarily on the absolute arcuracy of the p ~ o cedure. If proof of the accuracy of a method is needed, the extrapolation technique of Youden (11)ran sometimes be applied. When a n analytical chemist investigates the precision of a test method, he almost always finds that several factors influence reproducibility. The classical approach to the problem of locating these sources of variability aould bc to alter one factor at a time while holding all others constant. Thus the significance of each factor is determined separately. However, if the factorial design technique originated by Fisher ( 2 )is used, a more reliablemeasure of the significance of factors can hc obtained from a relatively small number of experiments. The technique permits all factors to vary systematically according to a designed pattern. When the classical technique is used, only one estimate of variability is obtained from two cxperinients in which all factors but one are held constant. A repetition of these two experiments would give one more estimate of variability. However, with a factorial design, repeated estimates of the variability attributable to a certain factor are obtained because of the systematic manner in which the factors are varied. The explanation of the illcreased efficiency of a factorial design is given in various discussioiis of the subject (1,6). The classical technique and the factorial design may be contrasted a t another point-that of the detection of “interactions.”
tistical techniques are illustrated by a study of three colorimetric methods for iron-phenanthroline, thiocy anate, and sulfide. The variables included in the study were dictated bj the type of information desired. The o-phenan throline (1,lO-phenanthroline) tnethod proved superior to the thiocyanate method according to the criteria set up. The thiocyanate method was much better than the sulfide procedure.
-4simple interaction is pwseiit ~vhenthe rffect of one variable is not consistent at different levels of a second variable-for esample, the result of a test method may not he influenced by “time of standing” a t one “pH,” but may be strongly influenced at it second “pH.” This would I)e called a p H times time of standing interaction. When data are rollected from a factorially designed experiment, interactions can be detected and evaluated. Using a classical design the interaction is often missed and the variability improperly assigned unless certain experiments happen to he chosen. When an interaction is detected by classical design it is often impossible to measure the magnitude of the effect. Factorial design also systematically detects and evaluates “higher order” internction involving three or more t e r m . These a-ould be missed by rlassical technique or a t most would he detected only hy the most :igilc mind. VARIABLES OF THE-EXPERLMENTS
In this study a factorial experiment and an analysis of variance of the resulting data are illustrated by an evaluation of three colorinietric iron methods. The object was the selection of the most precise of three rapid methods for the testing of iron in raiv 1113 teri als . In the testing of raiv materials for iron it is necessary to measure iron a t several Ion- l c ~ c l s . Some products have very low iron specifications a.nd othrrs may contain considerably more. One variable included in thcsr (valuation experiments was therefore “level of iron.” The three levels scxlected were 0.05, 0.15, and 0.25 mg. of iron, which represent the amounts presrnt in a sample of reasonable size for matrrial ranging from low to high i n iron. Raw materials often contain ot,her heavy metals such its lead, which could possibly intrrfere in an iron method. .1 small quant,ity of lead (0.1 mg.) typical of the amount often cncount,ered was introduced as another factor in the study. The raw material itself, although essentially inert, could influence the results. Sodium sulfittr was selected as a test material and introduced as :mothrr factor in the designed experiments. Five grams of C . P . sotliuni sulf:ite were added to certain experimrnts and t,his is rcft~rrcdto iri this study as inert material.
V O L U M E 22, NO. 12, D E C E M B E R 1 9 5 0 The last factor in the designed experiments would obviously be the three methods being evaluated. The actual design of experiments used to evaluate these factors is best seen by reference to Table I. Three levels of iron were measured by the three methods in both the presence and absence of a trace of lead, which in turn were studied in both the presence and absence of 5 grams of sodium sulfate. DESCRIPTION O F METHODS USED
1471
Table I. Fe Present, Mg. 0.05 0.05 0.05 0.05 0.15 0.15 0.15 0.15 0.25 0.25 0.25 0.25
Comparison of 4Iethods for Iron 1,lO-
pb, Mg. 0
0 0.1 0.1
0
o
0.1 0.1
Potassiiim Thiocyanate. Mg. , Sulfide,JLg.-iir. Duplicates Av. Duplicate3 Av. 0 . 0 5 0 3 0 031 0 , 0 5 0 0.0.505 0 . 0 4 9 0 . 0 4 8 0 . 0 4 8 5 0.0600 0 . 0 4 0 0 . 0 3 7 0 . 0 3 8 5 0.080 0 . 0 8 2 0 . 0 8 1 0 0.0503 0 . 0 3 0 0 . 0 4 9 0 . 0 4 9 5 0 . 0 4 7 0 . 0 4 5 0.0460 0 . 0 3 0 0 0 . 0 3 5 0 . 0 3 6 0 . 0 3 5 5 0.075 0 . 0 7 9 0 . 0 7 7 0 0.1500 0 , 1 5 0 0 . 1 5 1 0 . 1 5 0 5 0 . 1 4 8 0 . 1 4 8 0 . 1 4 8 0 0.1510 n 112 o 113 0.1125 0 , 1 9 1 0 . 1 9 4 0 . 1 0 2 5 0 . 1 3 0 5 0 , 1 4 9 0 . 1 6 0 0 . 1 4 9 3 0 . 0 4 1 0 . 0 3 8 0.0393 0.1500 0 . 1 1 4 0 . 1 1 8 0,1160 0.140 0.107 0.1235 0 . 2 5 0 3 0 . 2 3 2 0 . 2 6 0 0.2510 0 . ~ 5 0 . 2 6 1 0 . 2 3 0 3 0 . 2 3 0 0 0.187 0 . 1 8 7 0 . 1 8 7 0 0 . 2 9 2 0 . 2 8 3 0 . 2 8 7 3 0 . 2 3 1 0 0 . 2 5 0 0 . 2 5 1 0 . 2 5 0 5 0 106 0 102 0 1040 0 . 2 8 0 3 0 . 1 9 0 0 . 1 8 8 0 . 1 8 9 0 0 196 0 . 1 7 7 0 1865
N ~ ~ s o , Phenanthroline, , g& Grams 0 5.0 0 3.0 0 5.0 0 5.0
Duplicates 0 . 0 5 1 0.050 0.050 0 . 0 5 0 0.051 0.050 0.050 0.050 0.150 0.151 0,151 0.150
0.150
0.151 0.150 0,150 0.251 0.251 0.261 0.280
The phenanthroline method has been applied o o 0.250 t o the analysis of a variety of materials (6). A 0 5 0 0.249 0 . 1 0 0.251 systematic study of diverse ions was made by 0.1 5 . 0 0.251 Fortune and Mellon (a), who found relatively -~-___ few ions which seriously interfered and concluded Table 11. Two-way Tables for Phenanthmline Method that, in general, the phenanthroline method is Lead superior to the thiocyanate procedure. hrnoiint of F P Prwent, A I R . present, 11g. 0.03 0.15 0.25 .LY. The thiocyanate method was studied by Woods and JIellon .iverape-: of 4 analyses \\.it11 Sone 0 0303 0 , 1505 0 . 2 5 0 3 0 , 1.503 (IO),who singled out the variables which must, be kept reasonably and without snlfatc 0.1 0.0.50:3 0 . 1 5 0 3 0.2508 0 . 1 6 0 4 constant to ensure reliable results. They also investigated the .i~, n o m 0 liO4 0 2,503 0 . 1304 interference due to diverse ions and found that a large number of Sa2SOa the 57 ions tested introduced some error. Snell and Snell ( ? ) rePresent, G. viewed much of the literature covering the thiocyanate method. Arerages of 4 analyses with 0 0 . 0 5 0 5 0 ,1503 0 2308 0 1,;03 and without lead 5,O 0.0500 0 . 1 5 0 5 0 2,503 0 1.702 The sulfide method is used in some of the test methods of Rosin Ar. 0 , 0 3 0 3 0.1304 0.23io.j 0 . 1 3 0 4 ( 4 ) for the rapid nieasurenient of iron in raw materials. In surh proredures heavy metals if present must be corrected for by the addition of lead to iron standards in an amount equal to that 0 1507 0.1505 .iverages of 6 analyses, 0 0.1603 found by previous test to lie present in the raw material under 0.1502 5 0 0.1303 0 1502 three levels of iron coinbined .4v. 0 , 1 3 0 3 0.1.504 0.1304 test. I t is conceded that the method would give hetter results if a more time-consuming sulfide procedure such as that described 1iy Snell and Snell(7) were used.
Phenanthroline Method. The phenanthroline procedure employed was one recommended by Fortune and Mrllon ( 3 ) . Thc iron in the sample was completely rtduced to the ferrous state with hydroxylamine hydrochloride, after n.hich an excess of 0.1 yo l,l0-phrnanthroline was added. The volunie was brought to 100 nil. in a volumetric flask. Transmit’tariccwas measured at 510 nip. Thiocyanate Method. A standard thiocyanate procedure m-m uscd. The sample was diluted t o approsimately 50 nil. and after the addition of 10 nil. of 6 S hydrochloric acid t,he iron was oxidized completely to the ferric state by the dropwise addition of 0 . 0 4 5 potawium pcrnianganate. -After complete osidation, 10 nil. of lOC% potmsium thiocyanate nerc added and t h o sample ryas diluted to exactly 100 ml. in a volumetric flask. T h r transmittance was measured immediately, using a General 1:lectrir aut,omatic recording spectrophotometer a t a wave length of 465 mp. Sulfide Method. ..in iron sample u-as treat,ed n i t h 10 nil. of saturated hydrogen sulfide water and 10 ml. of 5 K ammoniuni hydroxide. After being diluted to the mark in a volumetric flask, the samples were a,ged 5 to 10 minutes. T h r density wa.s measured at 390 mp. I t as observed that the trxnsmittance of thc samples n-:is niatrrially incrcased after a n aging period of approximately 40 n i i n u t c ~ ~ FACTORIAL DESlGN
The design of the experiment is the moat critical fact,or in assuring the success of any study. In the present case it is desired to find whether the presenc’e of either lead or inert matter n-ill interfere wit,h the determination of iron in a raw mnteria.1. I t is conceivable that the degree of interference might change as the concentration of iron goes from x low level to a higher one; that the interfereric~encountered whrii both lead and inert material are present might not be n.hat n-ould be predicted from results ~ where they were introduced siiigly : : > l i d , finallj., that t h amount of interferenc,e would not he the same ivith each of the three test niethods. In statistical terms, then, n-e wish to evaluate the maiii effects (the average effect produced by changing the condition of a variable) and the interactions (the more specific effect or behavior of one variable as one or more others is being changed). In addition, we wish to evaluate the reproducibility of each method in making replicate analyses on the same material. As was pointed out in the earlier discussion, the most cficicnt type of design for such a study is the factorial experiment. There
were, of course, four factors: test methods, levels of iron coI1ccntration, introduction of inert, and introduction of lead. To run a determination of each conibination of these variables required 36 analyses (3 test methods X 3 levels of iron X 2 conditions of inert X 2 conditions of lead). In order to obtain the desired estimate of reproducibility, duplicate analyses a e r e run a t each condition, thus raising the number of analyses to 72. Duplicate analyses were not run off in pairs. The entire set of 7 2 determinations was made in random order. The results of the experimental work are presented in Tahle I. The tabular form not only illustrates the basic experimental design but is also a convenient arrangement for carrying out the calculations required for a variance analysis of the data. The details of the procedure for m a h n g an analysis of variance are given by Brownlee ( 1 ) and Gnedecor (6) as well as in ot’her statistical texts. The first step is the development of grouped arrangements of the data, which are referred to as two-way tables. They are formed by summing over the variables one a t a timethat, is, the results for all conditions of one variable are summed a t each combination of the remaining two. There will) therefore, be three such tables for each of the three analytical methods (Tables 11, 111, and IT). For purposes of discussion in this report, these tables have bren made up with average values rather than summations--for example, in Table I1 for the phenanthroline method, the result for 0.05 mg. of iron present, no lead present, is 0.0503. It is the average of the four results run under that combination-txo condit,ions of inert, each in duplicate (0.051,0.050 without sodium sulfate and 0,050, 0.050 with sodium sulfate). The results in colunins and rows labeled “av.” are the averages of all the results for a specific condition of onr varia\ilr. IMSCCSSIOS OF RESULTS
Aicareful study of this series of tables will begin to make the advantages of a balanced factorial esperiment readily apparent. Without any further statistical analysis or calculations, t h e following preliminary coiic,lusions can be reached from the results:
ANALYTICAL CHEMISTRY
1472
Table 111.
Two-Wa? Tables for Potassium I’hioc: anate \Iethod Lead preienr 31g
Ax-eranea of 4 analyses with a n d without sulfate
Sone 0 1 Av. SagSOd Pre>ent, G , 0 5 0 .Iv.
Ilnollnt oi r e Pre-ent, 3 I p 0 05 0 1; 0 25 .I\ 0.0443
0.0425 0.0433
0.1315 0 2190 0 1317 0 1328 0 . 2 1 9 8 0.1317 0 I 3 2 1 0 2194 0 1317
0 . 0 D J 0 1300 0 2508 0.0370 0 1143 0 1880 0 0135 0 1321 0 2104 h-a2s~i d m o n n t - o i P b Present. Present, lt~s. Thiocyanate Method. The average of the determinations made in the presence of lead was the same as those made when no lead was present (0.1317 and 0.1317). However, in the first t,wo-u-ay table (first group of three values, Table 111) there are slight inconristencies a t the different iron levels. At the 0.05-mg. level the introduction of lead lowered the average result from 0.0445 to 0.0425, while a t the other two levels the presence of lead raised the average result8 very slightly. However, these differences are sufficiently small that they might well he within experimental error. Thus, the existence of the so-called lead X iron interaction is questionable. It is shown belox that the analysis of variance offers a sound means for rcaching a decision on t,his point. In the second two-way table it can bc seen that the average of t’lie dctrrminations made in the presence of sodium sulfate was niaterislly lower than those n u d e n.hen no inert was present. 111 addition, the effect of thc introduction of inert is not the same a t difl’erent iron levels. At the 0.05-nig. level the average result was lonered 0.0130 unit (0.0500 to 0.0370), a t thr 0.15-mg. level 0.0357 unit, and a t the 0.25-mg. level 0.0628 unit. Tt must be concluded, then, that the interference offcred by the presence of sodium sulfate varies with the concentration of iron and there i q . theirfore, an inert X iron interaction. Sulfide Method. The average of determinations made in the prescnce of lead is lower than those made without lead. From the t two-way table it can be seen that thew is also an interaction w e n ]rad and iron. The interfercnw due to lead becomes much greater as the iron concentration increases. The introduction of inert material increases the iron results as s1lon.n in the second tITo-\vay table. There is also an interaction I)rtn-een inert and iron, brcause the interference is less a t the 1on.er iron concentration than at, the 0.15-mg. and 0.25-mg. levels. The third two-way table indicates that there is also an interaction iietn-een lead and inert. When IrJad is not present t,he introduction of sodium sulfate raiscd the average result from 0.1490 to 0.1570 (0.0380 mg.). The interference due to inert is even greater (0.0658 mg.) when lead is present. Thrse preliminary conclusions from the two-way tables do not, howeverj completely characterize the rc’sults from the sulfide mcthod. It would be incorrect to considcr only the average effect, of a single variable when it behaves differently with changes in a second variable. It is also incorrect to consider only the average effect of a first-order interaction when diflerent behavior is encountered w-ith changes in a third variable. For example, the third tx-o-way table seemed to indicate that the introduction of inert caused a greater effect when lead was also present in the test sample. Reference to the original data, lion-ever, shows clearly that this effect is true only at the higher iron concentrations. The existence of this second-order interaction involving all three variables places a serious limitation on the use of all of the two-way tables from the sulfide procedure. It is shown later that a variance :tnalysis provides a positive means of detecting such a condition. One other ver>-important piece of information can be gathered from the data by reference to the duplicate analyses in the original t:ilile of requltu. Isy a c-nmixii’iwiiot the averagz i’nnge l)et\veen
In many ex-erimental studies the results do riot offer such obvious conclusions. A variance analysis may then become a verjhelpful tool to aid the experimenter in interpreting the data. In order to illustrate the application of this statistical method, varianre analyses have been run on the data from this experiment. The summaries are presented in Table V, VI, and VII. Iii order to simplify the arithmetic involved in the analysis, the original datn were multiplied by a factor of lo3. Thus, the s i ~ moi squares and mean square terms as they appear in the summarie.* are larger 11y :I factor of 106 than if the original data litid hreii used. Table IT.
Tw-o-Way Tables for Sulfide Method Lead present, 31g.
Amrages oi 1 analyses Ivith a n d lvithorit snlfate
Sone 0.1
.4r. NazSOa
Anioiint of Fe Present. 1\10. 0.05 0.15 023 .Iv,
O.OG48 0 . 1 7 0 3 0 2690 0.1680 0.0815 0,1453 0.09Gi
0.061.5 0 0631
0.1259
0.2071
Present, C;. 0 0 0473 0.0938 O.lii3 0 0790 0.1580 0.2370 5.0 Ar. 0.0631 0 12.59 0 . 2 0 7 1 xazso4 Amount of P b Present, Present, G. 0 0.10 Averages of 6 analyse;, 0 0.1490 0.W32 5 0 0.1870 0.1290 three lei-el, uf iron coiiiAv. 0.1680 0.0961 hined hrerageq o f 4 anal;\.-es Tvith and i ~ - i t h o u tlead
0.132i
0.1061 0.1380
0.1321
Ilg. hv. 0 IOGl 0,1580 0.1321
T h e variance ratio column in each table is the yardstick wliich is used t o measure the relative importance of each main effect and
interaction. Each ratio is compared against a value of F (6) whic.11 is derived from the 2 tables compiled b3- Fisher ( 2 ) . Snedecor has tabulated, for certain probability levels and for variousdegrees of freedom, the values of F (variance ratio) which indicate that one variance is significantlj- greater than another. The three probability levels in conimon use are the 5, 1, and 0.1%. -in F result corresponding t o the 5% level indicates that the probahility of differences of this magnitude appearing when the 1 yoit becomes 1 i n no real difference exists is only 1 in 20. 100 times, and a t O.lyoonly 1 in 1000 times. I t is generally accepted that any ratio which does not at least reach the 570 level cannot be considered significant. In this experiment the mean square of each term has been conipared agzrinst the mean square of the “residual.” This residual is a mrxsure of the ability of the analyst to reproduce results n-ith the p:xrticular method. Thus, n e have determined the probabilit,. that variance charged to a given term might have occurred simply because of experimental error, Jf this probabilit3- does not exceed the 5% level, Ti-e cannot say definitely that the effect is real. The following system has been used to designate each ratio: Significant. 1-ariance ratio exceeds 5% level but lower than 1 %.
V O L U M E 22, NO. 12, D E C E M B E R 1 9 5 0
1473
Talrle Y. Yariance . i n a l p i s SLininiary Table, Phen an t h rolin e 3Iethod Srim oi Sqriares
oi \-:il.iani,r
+:IXC
I Ton I.md inert 1roil X lend I r o n X inert I.ead X inert lroi! X lead X i n e r t
lGOl00.2.7 0.01 0 37
0.38 0.76
I?c,:idiial ?'~(I(>.?igned experiment, it is still all too common for them to omit malyses by leaving out certain combinations of test vtiriiiih. This produces serious holes in an>- tabular arrangement of teht results and virtually eliminate. the use of variance :tnnly CIS ' to evaluate the data. Complete and carefully designed esperiments will iila!;e 1ios.iI)le the application of m t n y statistical tools to the rc.sults 01)tained. Objective judgments ('ail then be made on thi, piwi>ioii of a method and on the significant variables and coniliin;itioi~( ~ f vnriahles involved in thP procedure. (x-
Tahlc \ - I .
\-ariarice .Anal>-sis Sumniarj- T a l d e ,
Thiocj-anate 3Iethod s u m of COII~W
uf \-arjance
Iron ]run I.ead Iron
Degrees of Freedom
123730,.58
!I 1111
r.t.ati Inert
Squares 0.00
X lead X inert X inrrt X lead X i n e r t
8288.16 12 25 2481.09 4 17 11.08
-\lean Sciuare 61865.29 0.00 8288 16 A 13 1240 a J 4 17
Variance
Ratio
L c 3 ) -
Vcry Pignificant. T7:rri:tiice ratio esceads 1 701evr.l iiut loivt~r 0.170. Highly significant. 1-ariance ratio exceeds 0.1 levrl.
tllilll
111 studying t h c suniniarirs of the three anal~-ses, it will 1)e notccl t1i:it in c,:ti.h ('me the "iron" niain effect is rated as highl). sigiiifirant. This n-ns to I)e espected, because the esperinient was 1'1111 :it three separate ii,ori i*oni,rntrationsand there was a rcal diffi,wii(.e:imong Icvcl~.
CONCLUSION
Table J-11. \-arianee .Analysis Summary Table, Sulfide IIethod
t l i c T:iri:rni,e summary for the phenanthroline nietliod iioiie Degrees Sum of of Mean Tarianr.e the mitin rffects or interactions other than "iron" can l)i' conSorirce of Yariance Squares Freedom Square Ratio -irkred significant. It can be concluded, therefore, that riclithcr 7 Iron 83400.4 41700 2 629.7" 31fl32.1 1 Lead 31032.1 468.3" lc.atl nor iiirrt :it the (micentration used in this esperimmt offc~,id Inert 16172.1 1 16172 1 244.0" 1 :iiiy interference to the determination of iron. 1J370 3 Iron X lead 7685,2 116 . 0" 7 Iron X inert 9 31;h 1240.3 620.2 In the thiocyanate method analysis, none of the terms involviiig 17 i:j h Lead X inert 1162.0 1 1162.0 654.3 2 Iron X lead X inert 327.2 !(':id were large enough to be significant. Thus, n-e are provided 13eqidiiaI 7 9 5 , .j 1% 66.2'1 lvith n more nearly positive ansver to the iniportanime of t h e i i ~ n Total 149827 . O 23 X lead interaction which was discussed previously in a qualitative a IIiphly significant. h Very significant. \yay in t h r study of the tv-o-way tables. The inert main effcct C Significant. :rnd incrt X iron interaction were found to be highly sigiiificant. This int1io:ites not only that there was interference due to the pwsrnce of inert ninterial but also that the interference ivas not I x r m 4 n R E CITED i.ori.sistciit a t diffei~entiron levels. Reference to the iron-inert tn-o'1) 1ii~on.nlee.I- that the interference inc.reasec1 alniost S . T.. ('heiiiiml h h l i s h i i i g ( ' 0 . . 194;. i ~ 2 jFisher. K. .I,,"Desigii of Expwinients," London, ( l l i w l niitl pi,oportioiiately Kith the iron concentration. The relatioiidiip is Boyd, Ltd.. 1947. i~eutlilyconfirined 1 ) ~ . using logarithms instead of the ran- data in I S ) l:o~.tune,\T. 13.. aiitl l l c l l o i i . If. G . , ISD. Esc,. CHI:^. .Is$1.. tlir variance analysis. The inert X iron interaction term Iwroines ED., 10, DO (1WS). iiot significxit, which indic*atesthat the inert interference aniounts (4) Rosin, J o s e p h . " R e n g c u t i'heniic:rls and Standards," SCK>-o~,k, D. Van l-ostiaiid (In., 1W7. to a roughly constant peri>entageof the iron concentration. Tlir f5) Smith, G. F , "1'hcii:iiithi.oliiie aiid Substituted P h e i i a n t h r olilie question "Is there, thcn. an inert X iron interaction?" ran I J V Indicators." C'olumihns, Ohio. G . F. S m i t h Cheniical C'u., ysised, hut the point is of ininov importance provitlcd the t r u r I,('1941. 1:itionship lietiwen the two variables is clearly u i i d e i ~ ~ t o o ~ l . ( 6 ) Siiedecor. G . I\'., "Statistical lIcthods," .hies, I o x a , Ioa-a 5tate rollege PresP. 1!Mi. In the sulfide method aiialj-sis a11 the niain effect.., f m 1 - o r i I i ~ l ~ (7j Siiell and Pnell, "C'oloi.imetr ic Methods of Anal interactions, :inif the second-ordri, interaction xverr xigiiific.:iiit, p . 306. A-ew I-oi,k. D. \-ail Sost1,and C o . . 1936. :.I discussion of the m e of higher order interactions, 01' t l : c b 1)oi)Iiiig ( S ) Symposium on Statistirid N e t h o d s in Experimental and I I I ~ I I R of teriiib, i n the testing of significance is given b ~ Bron iiI(.e ( 1 ) trial Chemisti.y, .Ist i . . C H E X . .19, 943-60 (1947). :ind Siietlrcor ( 6 ) . The proredure would have little ~ i i ~ a c I i c , i ~ ! ($1) \Yernimont. G . . I h i d . , 21,115 !1940). ( I O ) M-oods. .J. T . , a11d IIelloii, 11. ( i . , ISD. ESG. C H E J I . , .ls.i.I.. E l ) , , l m r i n g on coiiclusiona made in this study, and in the iiItt'wFt of 13, 551 (1911). -implicit>- the additional calculations have been oniittetl. I The (11) 1-ouden, I\-. J., .IsAI..( ' H E M . , 19, 916 (1947). cSGtence of the second-order interaction, which XIS rlivu RECEITEDI l a y .i.1930. P r c - e n t e d hrfore t h e Division of .Innlyti
