Analysis of Covariance For Repeated Measures Design with Missing Observations

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1 Pertanika 9(), -5 (986) Analysis of Covariance For Repeated Measures Design with Missing Observations AHMAD BIN ALWI and C.J. MONLEZUN Guthrie Research Chemara, Seremban, Negeri Sembilan. Key words: Repeated measures design; subspaces; noncentrality; orthonormal basis; analysis of covariance. ABSTRAK Analisis kovarians dianggap sebagaisalah satu kaedah yang paling kurang dijahami dan diajar di antara sernua kaedah statistik gunaan. Kebanyakan buku kaedah tidak menyentuh sama sekali analisis kovarians (Guttman et. al., 982), atau dengan sepintas lalu (Brownlee, 965), dan ada yang memberi perhatian dengan banyaknya seperti Federer (955), Snedecor dan Cochran (980), Steeldan Torrie (980), dan Winer (97). Kebanyakan daripada buku-buku tersebut menumpukan kepada data yang seimbang, khususnya data yang mempunyai bilangan cerapan yang sama dalam subkelas. Apajadijika data yang diperolehi tidak seimbang dan malahan pula ada di antara cerapan yang hitangf Sudah tentu analisis kovarians menjadi lebih rumit. Penggunaan geometri dalam analisis kovarians diharap dapat memberi kejahaman dan juga meluaskan lagi jenis-jenis kaedah yang boleh dipertimbangkan bagi menyelesaikan masalah seumpama ini. ABSTRACT Analysis oj covariance might be one oj the most misunderstood and inadequately taught oj all applied statistical methods. Many methods books do not deal with it at all (Guttman et. al., 982), or sparingly (Brownlee, 965), and most of those that treat it substantially, such as Federer (955), Snedecor and Cochran (980), Steel and Torrie (980), and Winer (97), concentrate on balanced data, namely those which have equal numbers oj observations in the subclasses. What happens if the data are not balanced and moreover ij some oj the observations are missing? The missing observations complicate computations and affect what is estimable. The analysis of covariance would become more complex. The application of geometry in the analysis of covariance may offer an understanding of the analysis as well as broaden the variety of methods that can be considered. When there are no missing observations on the repeated measures factor(s), computational algorithms can be used (see Henderson and Henderson, 979). INTRODUCTION a concomitant measurement ik. If we let a 3, b - 4, n l = 3, n 2 = 2, n 3 = 4; we can tabulate a data table as in Table. The design considered here is a two-factor Repeated Measures Design. Let ijk be the measurement made on subject i ( < I < ftj) at We arbitrarily set the observations ( 3, level j ( <j <a) of factor A and level k us ), ( S2, 32 ), ( 2S, 23), ( 232, 232 ), ( < k < 6) of factor B. For every ijk, there is ( 235, 2) and ( 4, 4) be missing. Ass. Prof., Dept. of Exp. Statistics, LSU, U.S.A.

2 AHMAD BIN ALWI AND C J. MONLEZUN TABLE Data table for observations B, ] B ( B, A, u, 2, x H, x sll.x sli.,2. 2!2 32 x lls * A 3l2 V * 3 ^3 V 23 ^ 23 ** 3 m 24 34, m, x, n x JH A 2 2l ^22 ^ 22 r>2, x,., 2.,.,., V * 23 A A 223 2, x 22 A 3 23 v v,«. * 232 :»v. 432 i:i2 x m* 2» * i 2 ** 2 V 3 ^ x,,, m,,, 23,,, 4,:,, 43 *The observations assumed missing. MODEL DESCRIPTION In the model where there is no covariate, the model used by A. Ahmad and C.J. Monlezun (984) is given by ijk =U jk +S ij+ E ijk where U jkis the cell mean for the level j of factor A and the level h of factor B, S ; is the effect of subject i in the level j of factor A, and E^is the random error component. With the additional concomitant measurement ^, a model can now be written as where U..() = 0 OM + 0. ( ,) and j3 w is the intercept for group (j, k) and j3j is the common slope of all lines. Let xbe the observational vector and it can be written as illustrated in Table 2. An alternative way of writing the model is y - MVN (E ( ) GC =C0, 2 2 a E I + a s J ). Note that the covariance structure remains the same as in the ANOVA case discussed by A. Ahmad and C.J. Monlezun (984). All subspaces defined in A. Ahmad and C.J. Monlezun (984) will be used throughout this paper. HPOTHESES TESTING In the ANOVA case, we have shown that there is no exact test for testing no main effect A. A similar situation prevails in the case of analysis of covariance. Therefore we are interested in testing the following hypotheses: ( k) U.k.k = 0 (.k) H AB U jk () - U- k () = U. k - () - U jv () ~0o (ik) -0o (i k) =0o (ikl) - 3o (jv) In the analysis of variance model, all observations for the cell (j, k) have the same mean response U jk. This is not so with the covariance model, since the mean response here depends on the (j, k) combinations and also on the value of the concomitant variable ik for the experimental unit. Thus the expected response for the (j, k) cell with the covariance model is given by a regression line: where A. (AB) jjk PERTANIKA VOL. 9 NO., 986

3 ANALSIS OF COVARIANCE FOR REPEATED MEASURES DESIGN WITH MISSING OBSERVATIONS TABLE 2 Observational vector.. ij2 ii3 for ij =2,22, 3,43 ^ l2 x T A ^ ^ ^234 v = [,, 3! 3 2 " * Note that J3 o ( * k) «U.. + B for A = S (AB) = 0 If we want to measure the difference at any convenient point ijk, say ~ k =..., then (U... +B,)-(U... +B 2 ) = B,-B 2. Thus B j B 2 measures how much higher the mean response is with B x than with B 2for any value of ijk. Therefore for testing no main effect B in U jk (), the regression lines must have equal slopes and the test for main effect B is B i - B. = 0. In other words all of the B k s have to be equal. Fig. illustrates an experiment with four levels of factor B, and how these regression lines might appear. j J j J In constructing the statistics, we first need to define subspaces for the numerator space and the error space. For the error space, we need a space that is orthogonal to C x and W y Recall from ANOVA case that the smallest subspace containing both C and W s is W s [+f B ± AB, and thus the smallest subspace that contains only,c, andw s is P E (W\ B0AB). Therefore the error space that we desire must be E x = (P E (W S E]BEAB)) = 0P E PERTANKA VOL. 9 NO., 986

4 AHMAD BIN ALWI AND CJ. MONLEZUN of squares and hence derive the test statistics for testing H B and H AB. For simplicity, we let D B, AB. The sum of squares for effect D is defined as SSD = P with d degree of freedom where d = (a- l)(b- ) if D = B if D = AB Let {vj be an orthonormal basis for N D x. Then the numerator sum of squares can be d rewritten as y (y v ^ We note that Fig. : Regression lines of an experiment with four ; levels of factor B. From the ANOVA case, we have also defined the following subspaces: T = N B (W S EAB) forh B T = N AB B) forh Let M B = (W S EB AB) and M AB = (W S H B). The smallest subspace containing W s and W D x is (P ND M D ford-b,ab k=l xv kis normally distributed with mean E( x )v k (nonzero since E( ) G C x and v R G C x ) and variance o. Why is the variance equal to o E? Is any two (v., v ) independent? To A. K A m answer both the questions we look at the following calculations of variance and covariance: Var( x \) = v k Var( x )v k = v,/ [a* I + a 2 o J] v u The numerator space for testing H D has to be orthogonal to E x, W s and W D x. Therefore we can define the numerator space by + Cov( x v k) x v m )= v k [a% : I + a%j] \ v k Jv m where TJ D = N Q Q = 0 D, P E P E Since the covariance of any two xv k is zero, this would imply that they are independent random variables, and hence we can have the distribution of sum of squares for effect D as DISTRIBUTION OF SUM OF SQUARES/ TEST STATISTICS Here we are interested to determine the distribution of the numerator and the error sum SSD= [N(E( x )\,a 2 E)] 2 k=l = a 2 v 2 (d; noncentral) PERTANIKA VOL. 9 NO., 986

5 ANALSIS OF COVARIANCE FOR REPEATED MEASURES DESIGN WITH MISSING OBSERVATIONS where the noncentrality parameter is given by E( ) P N E( ) 2o\ Similarly, the distribution of the sum of squares error can be derived as t-ab-n.+a SSE = 2 [N(0,a 2 E)] 2 = o 2 E x 2 (t-ab-n.+a; central) After knowing the distribution of the sum of squares, we can now write a test statistic for testing no effect D in U jk () as SSD rd u x 2 (d; noncentral) SSE t-ab-n.+a x 2 (t-ab-n.+a; central) The test statistic above is distributed as a noncentral F distribution and when the null hypothesis is true, the test statistic becomes central F - distribution. REFERENCES A. AHMAD and C.J. MONLEZUN, (984): A Geometric Look at Repeated Measures Design with Missing Observations. PERTANIKA 7(2): 7-8. GRABILL, F.A., (976): Theory and application of the Linear Models. New ork: Holt, Rhinehart, and Winston, Inc. HENDERSON, C.R. Jr., (982): Analysis of Covariance in the mixed model: higher-level, nonhomogeneous, and random regressions. HENDERSON, C.R. Jr. and HENDERSON, C.R., (979): Analysis of Covariance in mixed models with unequal subclass numbers. Comm. in Statistics, series A, 8: NETER, J. and WASSERMAN, W., (974); Applied Linear Statistical Models. Homewood: Richard D. Irwin Inc. SCHWERTMAN, N.C., (978): A Note on the Geisser- Greenhouse correction for incomplete data splitplot analysis. Jour, Amer. Stat. Ass. 73: SEARLE, S.R., (97): Linear Models. New ork: John Wiley and Sons, Inc. SNEDECOR, G.W. andcochran, W.G., (980): Statistical Methods. The IOWA State Univ. Press, Ames, IOWA. STEEL, R.G.D. and TORRIE, J.H., (980): Principles and procedures of Statistics. New ork: McGraw- Hill Book Company, Inc. TlMM, N.H., (975): Multivariate analysis with applications in Education and Psychology. Monterey: Books/Cole publishing company. WINER, B.J., (97): Statistical principles in experimental design. New ork: McGraw-Hill Book Company, Inc. (Received 7 February 985) PERTANIKA VOL. 9 NO., 986