or treatments in several periods. The

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1 ABSTRACT ANALYZING BALANCED OR UNBALANCED LATIN SQUARES AND OTHER REPEATED MEASURES DESIGNS FOR CARRYOVER EFFECTS USING THE GLM PROCEDURE D.A. Ratkowsky, Washington state University J.R. Alldredge, Washington state University J.W. cotton, university of California, Santa Barbara A general approach is presented which makes possible the analysis of any repeated-measures Latin square design, balanced or unbalanced, with a view towards obtaining estimates of carryover effects (also referred to as residual treatment effects), as well as direct treatment effects. Specifying carryover as a "classification" variable enables the adjusted sum of squares for a test for direct treatment effects and for carryover effects to be obtained using standard SAS specification statements in PRoe GLM. Differences between the means of the treatment levels and the carryover levels are readily determined using the LSMEANS statement. The approach may also be applied to many other types of repeated-measures designs including the two-treatment, two-period crossover design and ones in which each subject may receive the same treatment or treatments in several periods. The technique is illustrated using four examples. INTRODUCTION In repeated-measures Latin squares experiments, the same SUbject is used in a sequence of trials, with different treatments commonly being applied between one period of the trial and the next. Interest is often directed as to whether there are any "carryover" or "residual" effects, which can be considered to be manifestations of treatment effects in subsequent time periods of the trial. Sometimes the interval between periods is chosen to be sufficiently long to ensure that the residual effects are minimized, if not wholly eliminated, but in other experiments the question of whether there is a significant carryover effect can be the main focus of attention of the experiment. Williams (949) showed that for n x n Latin squares (n rows, n columns, and n treatments), it is possible to achieve designs that- are balanced for the estimation of carryover effects with a single square when n is an even number and with two squares when n is an odd number. Each treatment is preceded and followed by each of the other treatments an equal number of times in such squares. Wagenaar (969) presented a simple method of constructing such balanced n x n squares for all even values of n. These so-called "digram-balanced" Latin Squares possess certain optimality properties in that the separability of the estimates of the "direct" treatment effects from the residual treatment effects is greater in such designs than in those not possessing that property, but it is not necessary to have such structure to estimate carryover effects. The approach to be presented here involves the use of standard SAS specification statements and differs sharply with that of Skalland and Skalland (984), whose SAS code relied heavily on the use of SAS programming language commands. Our approach requires neither digram balance nor any other kind of balance. It is perfectly general for any Latin square or set of squares, and a similar approach can be applied to other repeated measures designs. Examples will be presented for several Latin square designs, as well as for the two-period, twotreatment IIchange-over" design studied by Grizzle (965, 974). METHODS AND EXAMPLES The key to the approach to the analysis of the designs considered here ies in the use of the CLASS statement in PROC GLM to declare variables to be "classification variables lt The approach is best illustrated by means of examples. Consider the analysis of the digrambalanced pair of Latin squares given in Cochran and Cox (957), P. 35. This is the same set of data treated by Skalland and Skalland (984). Square Cow 2 3 A 38 B 09 C 24 2 B 25 C 86 A 72 3 C 5 A 39 B 27 Square 2 Cow A 86 B 75 C 0 2 C 76 A 35 B 63 3 B 46 C 34 A 353

2 The response, milk yield, is labelled Y, and square, cow, and period, labelled SQRE, COW. and. PERIOD, respectively, are coded usng ntegers in the respective ranges -2, -6, and -3, corresponding to the appropri~te square number, cow number, and tme period at which the reading was taken. Treatment is defined as the character variable TREAT with values "A", "B" or II e" depending upon the treatment applied to that unit during that period. The character variable used for carryover effects, CARRY, is c~ded as "0" (although any other unque character would serve as well) if the reading is in period, as "A u if tl?-e reading is preceded by treatment A (n the previous time period), as "BII if the reading is preceded by tr::atme~t B and as "C" if the readng S p~eceded by treatment C. Hence, the DATA step for this example would look as follows: DATA COCHCOX; INPUT Y COW PERIOD TREAT $ CARRY $ 38 A B A 5 3 C B 09 2 B C B A C C A C B A 86 4 A C A B C B A B CA C B C A B 2 which produces the following Type I and Type II degrees of freedom, sums of squarest F-values and probabilities of exceeding F. SOURCE DF TYPE I SS F VALUE PR > F COW PERIOD (SQRE) TREAT CARRY SOURCE DF TYPE II SS F VALUE PR > F COW PERIOD (SQRE) TREAT CARRY All the sums of squares in the above table with the exception of the Type II SS for cows and for periods within squares, which are adjusted for the presence of other terms in the model, may be found in the analysis of variance table of Cochran and Cox (957), p.35. For inference on treatment effects and carryover effects, the appropriate sums of squares for significance testing is the Type II SS. This follows from the "expected mean square" table produced by SAS, which contains the following information for these effects: An important feature of the use of the PROC GLM step in this approach is the order in which the various effects are entered into the model by use of the MODEL statement. In most Latin square examples, the factors to be entered first are those which define the structural aspects of the design, that is, the "row" and "column" factors t which, in this case, are the cow effect and the period within squares effect. Then the treatment factor comes next, followed lastly by the carryover effect, which is viewed as a manifestation of a treatment effect at a subsequent time period. The SAS code for this example is PROC GLM; CLASS COW PERIOD SQRE TREAT CARRY; MODEL Y=COW PERIOD(SQRE) TREAT CARRY/SOLUTION SSl SS2; RANDOM COW; LSMEANS TREAT CARRY/PDIFF; SOURCE TREAT CARRY SOURCE TREAT CARRY TYPE I EXPECTED MEAN SQUARE VAR(ERROR) + Q(TREAT,CARRY) VAR(ERROR) + Q(CARRY) TYPE II EXPECTED MEAN SQUARE VAR(ERROR) + Q(TREAT) VAR(ERROR) + Q(CARRY) These expected mean squares tell one that the effect of treatment is appropriately tested using the F-ratio whose numerator is the mean square formed from the TyPe II sum of squares, that is, the adjusted sum af squares for treatments. Because CARRY is the last term in the model, its Type I and Type II expected mean squares are the same, as are its Type I and Type II sums of squares. Thus, either the Type I or Type II 'sums of 354

3 squares may be used in forming the F ratio for testing carryover effects. If one views a carryover effect as representing the persistence of a treatment effect from an earlier time period into a later time period, then it is reasonable to expect the carryover effect to be of a smaller magnitude than the treatment effect. It is possible, of course, for a carryover effect to exceed a treatment effect in certain circumstances where the effect of treatment is delayed by some mechanism which may trigger a late response. Furthermore, Willan and Pater (986) consider the possibility of psychological carryover as distinct from physical carryover, which may result in the magnitude of the carryover effect exceeding that of the treatment effect. If a preliminary test for the presence of carryover does not lead to rej ection of the null hypothesis, there is no need to go further and examine various components of the carryo~er effect (unless pre-planned comparsons have been specified); The analysis of variance then collapses effectively into the standard analysis of variance of a Latin square design in which the rows, columns and treatment terms are orthogonal, that is, the magnitudes of the sums of squares for those three effects do not depend upon the order in which the terms are included in the model. If, on the other hand, the estimate of carryover is significant, one would be interested in examining various contrasts among the components of the treatment and carryover effects. Unfortunately, the statement LSMEANS TREAT CARRY/PDIFF; in the FROC GLM code produces the response that the least squares means for treatment and carryover are not estimable. This relates to the fact that there is no information on carryover in the first time period, with the result that FROC GLM produces a zero estimate in the solution vector for the carryover level labelled '0', as well as a zero estimate for the last level in the alphabetical list (in this example the one labelled 'e'), in accordance with the use by that procedure of the "set-to-zero" convention. The least squares means are readily obtainable by making a simple modification to the code in the DATA step. One simply inserts the line of code IF CARRY='O' THEN CARRY='C'; prior to the CARDS statement. This modification produces the following least squares means for treatment and for carryover: LEAST SQUARES MEANS TREAT Y PROB > ITI HO: LSM(I)~LSM(J) LSMEAN I/J 2 3 A B C CARRY Y PROB > ITI HO: LSM(I)~LSM(J) LSMEAN I/J 2 3 A B C The estimates of the least squares treatment means differ from those presented in Cochran and Cox (957) by a fixed constant, reflecting the fact that the carryover effects do not sum to zero. However, the differences between least squares means are always the same, as will also be the case if, for example, the inserted line above had read IF CARRY='Q' THEN CARRY='A'i with the label for treatment level A being used in place of the label for treatment level C. A second example utilizes data from Gill (978), Exercise 8.7, pp , and consists of three 4 x 4 Latin squares which are badly unbalanced for the estimation of carryover effects. For example, treatment A is followed by treatment B four times, by treatment D four times, and by treatment C only once. Treatment C is followed by treatment B five times, by treatment D four times, and never by treatment A. Despite the imbalance, the method of using classification variables to obtain a val id overall assessment of carryover effect still applies. The coding is as follows, assuming that the days (i.e. periods) were common.to all three squares: DATA GILLX87; INPUT Y SQUARE RABBIT PERIOD TREAT$ IF CARRY='Q' THEN CARRY='D' ; 7 D C D 2 3 B C 4 A B 4 2 C D C 2 3 A D B A 355

4 3 3 B A B C A D C 4 A B A DB C D C D C A D B A D C D B C A B 2 7 A B A C B D C B A B D A C D B A B D A C D A D A C D B C 7 3 D C D B C A B C B C A B D A PROC GLM; CLASS RABBIT PERIOD TREAT CARRY; MODEL Y~RABBIT PERIOD TREAT CARRY/SOLUTION SSl SS2; RANDOM RABBIT; LSMEANS TREAT CARRY/PDIFF; The resultant analysis of variance table contains, in part, the following relevant information for assessing the importance of the various effects. SOURCE DF TYPE I SS F VALUE PR > F RABBIT PERIOD TREAT CARRY SOURCE DF TYPE II SS F VALUE PR > F RABBIT PERIOD TREAT CARRY The "omnibus ll test for carryover effects, with an adjusted sum of squares of with 3 degrees of freedom, gives no indication of a residual treatment effect. Hence, unless there are pre-planned comparisons, there is no need to examine the carryover effect in any more detail. The class variable approach exploited here may be applied to any Latin square design irrespective of whether or not the design is balanced for the estimation of carryover effects. In the above two examples, we have considered only first-order carryover effects, that is, ones in which the effect derives from the treatment in the time period preceding the current time period. The classification variable approach is readily extended to higher order carryover effects. For example, consider the data from a 6 x 6 Latin square design (Bacharach et al., 940) reproduced in Gill (978), Exercise 8.6, p. 246 I which is not balanced for the estimation of carryover effects. Order of Administration Rabbit No C D A F B E 2 E B C A D F 3 D F E C A B 4 A E F B C D 5 F C B D E A 6 B A D E F C I f one were interested in examining both first- and second-order carryover effects, the code could be written as follows, with the effects being designated by the class variables C and C2, respectively. DATA GILLX86; INPUT Y RABBIT PRESORD TREAT$ C$ IF C='Q' THEN C='F'; IF C2='O' THEN C2='F'; 7.9 C D C A D C F A D B F A E B F E B E C B E A C B DA C F D A D F D E F D C E F A C E B A C A E A F E A B F E C B F D C B 7. 5 F C F B C F D B C E D B A E D B A B D A B E D A F E D C F E PROC GLM; CLASS RABBIT PRESORD TREAT C C2; MODEL Y~RABBIT PRESORD TREAT C C2/S0LUTION SSl SS2; RANDOM RABBIT; LSMEANS TREAT Cl C2/PDIFF; 356

5 If the Type II sums of squares for C and C2 had given an indication that carryover effects were important, the differences between least squares means produced by the LSMEANS statement provide valid tests of individual contrasts 4 Of course, preplanned comparisons may be made irrespective of whether the omnibus carryover effect is significant or not. A final example of the utility ~f the class variable approach to examining data on carryover effects in repeated measures designs uses the data of Grizzle (965, 974) on a two period changeover design with two treatments. In that experiment, 6 subjects were given the sequence AB in the two periods and 8 subjects were given the sequence BA. A summary of the totals and means is presented below for the various combinations of treatment and period within each sequence: Sequence 2 Sequence 2 AB (6 subjects): Treatment Total A.8 B BA (8 subjects): Treatment Total B -9.9 A Mean Mean The sums of squares for the effects that can be obtained from these data depend upon how one views the experiment. Grizzle (965) used the term "residual effects" as a synonym for "sequence effects", that is, the sum of squares given by (5.0)2/2 + (-6.4)2/6 - (-.4)2/28 ~ "" effects may be calculated from the period totals -8. and 6.7, respectively, as (-8.)2/4 + (6.7)2/4 - (-.4)2/ "Treatmenttl effects, when calculated (see Grizzle, 974) as the interaction between period and sequence, is (.8)2/6+(322)2/6+(-9.9)2/8+(3.5)2/8- (-.4) / ~ Alternatively, treatment effects may be viewed (see Milliken and Johnson, 984, p.439) as being derived from the treatment totals 5.3 and -6.7 for treatments A and B, respectively, to give (5.3)2/4 + (-6.7)2/4 - (-.4)2/ From that viewpoint, a carryover effect is seen to be an interaction between treatment and sequence, given by [(.8)2+(3.~)2/6+[(-9.9)2+(3.5)2/8- (-.4) / ~ All of the above values may be obtained using PROC GLM by coding the original data in the manner advocated here, declaring all variables to be classification variables. Thus, the data step of the SAS program might look as follows: DATA GRIZZLE; INPUT Y SEQ SUBJ PERIOD TREAT$ 0.2 A B A A B A A B A A B A A B A.5 6 A BA B A B B A B B A B B A B B A B B A B B A B B A B By writing the PROC GLM code as follows, PRoe GLM; CLASS SEQ SUBJ PERIOD TREAT CARRY; MODEL Y~ SEQ SUBJ(SEQ) TREAT CARRY/SOLUTION SSI; one obtains , and as the Type I sums of squares for sequences, treatment, and carryover effects, respectively. These are the same values as reported in Milliken and Johnson (984, p.439). Changing the model statement to MODEL Y ~ SEQ SUBJ(SEQ) PERIOD TREAT/SOLUTION SSI; produces , and for 357

6 sequences, period and treatment effects, respectively. An equivalent generation of the five sums of squares given above uses the same DATA step as before but two new MODEL statements. One of these is MODEL Y = TREAT PERIOD CARRY SUBJ/ SOLUTION S8; yielding 5.429, and for the Type I sum of squares for what is now called treatments, periods and carryover, respectively. Following Hills and Armitage (979), this sum of squares for periods may be labeled "adjusted for treatments". Those authors refer to the carryover sum of squares as a term for the interaction between treatments and periods. The second alternative MODEL statement, MODEL Y = PERIOD TREAT CARRY SUBJ/ SOLUTION SSl; yields and as Type I sums of squares for unadjusted treatments and periods adjusted for treatments, respectively. The carryover sum of squares is still , and its interpretation is the same as before. In view of much controversy about hypothesis testing with this design (see Jones and Kenward, 989, and Freeman, 989), we do not present significance testing procedures here. REFERENCES Bacharach, A.L., Chance, M.R.A., and Middleton, T.R. (940). The biological assay of testicular diffusing factor. Biochemical Journal 34: Cochran, W.G. and Cox, G.M. (957). "Experimental Designs. John Wiley and Sons, New York. Freeman, P.R. (989). The performance of the two-stage analysis of twotreatment, two-period crossover trials. statistics in Medicine 8: Gill, J.L. (978). Design and Analysis of Experiments in the Animal and Medical Sciences. Volume 2. Iowa state University Press, Ames, Iowa. Grizzle, J.E. (965). The two-period change-over design and its use in clinical trials. Biometrics 2: Grizzle, J.E. (974). correction to lithe two-period change-over design and its use in clinical trials." Biometrics 30: 727. Hills, M. and Armitage, P. (979). The two-period cross-over clinical trial. British Journal of Clinical Pharmacology 8: Jones, B.J. and Kenward, M.G. (989). Design and Analysis of Cross-OVer Trials. Chapman and Hall, London and New York. Milliken, G.A. and Johnson, D.E. (984). Analysis of Messy Data. VolUme. Van Nostrand Reinhold, New York. Skalland, M.L. and Skalland, K.R. (984). Analysis of a balanced Latin square design using the SAS system. In proceedings of the Ninth Annual SAS Users Group International Conference, Hollywood Beach, Florida, March 8-2, 984, pp Wagenaar, W.A. (969). Note on the construction of digram-balanced Latin squares. Psychological Bulletin 72: Willan, A.R. and Pater, J.L. (986). two-period Biometrics carryover and the crossover clinical trial. 42: Williams, E.J. (949). designs balanced for of residual effects Australian 'Journal Research, Series A 2: Experimental the estimation of treatments. of Scientific The authors may be contacted at: David A. Ratkowsky, program in statistics, Washington state University, Pullman, WA, Phone: (509) J. Richard Alldredge, Program in statistics, Washington state University, Pullman, WA, Phone: (509) John W. Cotton, Departments of Education and Psychology, University of California, Santa Barbara, CA, Phone: (805)