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combination of r periods of storage of plasma and c concentrations of adrenaline mixed with the plasma. This is a simple example of a factorial experiment,tobe discussed more generally in §9.3. The distinction between this situation and the randomized block experiment is that in the latter the `block' classification is introduced mainly to provide extra precision for treatment comparisons; differ- ences between blocks are usually of no intrinsic interest. Two-way classifications may arise also in non-experimental work, either by classifying in this way data already collected in a survey, or by arranging the data collection to fit a two-way classification. We consider first the situation in which there is just one observation at each combination of a row and a column; for the ith row and jth column the observation is y ij . To represent the possible effect of the row and column classifications on the mean value of y ij , let us consider an `additive model' by which Ey ij m a i b j 9:1 where a i and b j are constants characterizing the rows and columns. By suitable choice of m we can arrange that r i1 a i 0 and c j1 b j 0: According to (9.1) the effect of being in one row rather than another is to change the mean value by adding or subtracting a constant quantity, irrespective of which column the observation is made in. Changing from one column to another has a similar additive or subtractive effect. Any observed value y ij will, in general, vary randomly round its expectation given by (9.1). We suppose that y ij Ey ij e ij 9:2 where the e ij are independently and normally distributed with a constant vari- ance s 2 . The assumptions are, of course, not necessarily true, and we shall consider later some ways of testing their truth and of overcoming difficulties due to departures from the model. Denote the total and mean for the ith row by R i and y i: , those for the jth column by C j and y :j , and those for the whole group of N rc observations by T and y (see Table 9.1). As in the one-way analysis of variance, the total sum of squares (SSq), y ij À y 2 , will be subdivided into various parts. For any one of these deviations from the mean, y ij À y, the following is true: 9.2 Two-way analysis of variance: randomized blocks 239 Table 9.1 Notation for two-way analysis of variance data. Column 12 j c Total Mean, R i =c Row 1 y 11 y 12 y 1j y 1c R 1 y 1: 2 y 21 y 22 y 2j y 2c R 2 y 2: . . . iy i1 y i2 y ij y ic R i y i: . . . ry r1 y r2 y rj y rc R r y r: Total C1 C 2 C j C c T Mean C j =r y :1 y :2 y :j y :c y T=N y ij À y y i: À y y :j À yy ij À y i: À y :j y9:3 The three terms on the right-hand side reflect the fact that y ij differs from y partly on account of a difference characteristic of the ith row, partly because of a difference characteristic of the jth column and partly by an amount which is not explicable by either row or column differences. If (9.3) is squared and summed over all N observations, we find (the suffixes i, j being implied below each summation sign): y ij À y 2 y i: À y 2 y :j À y 2 y ij À y i: À y :j y 2 : 9:4 To show (9.4) we have to prove that all the product terms which arise from squaring the right-hand side of (9.3) are zero. For example, y i: À yy ij À y i: À y :j y0: These results can be proved by fairly simple algebra. The three terms on the right-hand side of (9.4) are called the Between- Rows SSq, the Between-Columns SSq and the Residual SSq. The first two are of exactly the same form as the Between-Groups SSq in the one-way analysis, and the usual short-cut method of calculation may be used (see (8.5)). Between rows : y i: À y 2 r i1 R 2 i =c ÀT 2 =N: 240 Experimental design Between columns : y :j À y 2 c j1 C 2 j =r ÀT 2 =N: The Total SSq is similarly calculated as y ij À y 2 y 2 ij À T 2 =N, and the Residual SSq may be obtained by subtraction: Residual SSq Total SSq À Between-Rows SSq À Between-Columns SSq: 9:5 The analysis so far is purely a consequence of algebraic identities. The relation- ships given above are true irrespective of the validity of the model. We now complete the analysis of variance by some steps which depend for their validity on that of the model. First, the degrees of freedom (DF) are allotted as shown in Table 9.2. Those for rows and columns follow from the one-way analysis; if the only classification had been into rows, for example, the first line of Table 9.2 would have been shown as Between groups and the SSq shown in Table 9.2 as Between columns and Residual would have added to form the Within-Groups SSq. With r À1 and c À 1 as degrees of freedom for rows and columns, respectively, and N À 1 for the Total SSq, the DF for Residual SSq follow by subtraction: rc À1Àr À 1Àc À 1rc Àr À c 1 r À 1c À 1: The mean squares (MSq) for rows, columns and residual are obtained in each case by the formula MSq SSq/DF, and those for rows and columns may each be tested against the Residual MSq, s 2 , as shown in Table 9.2. The test for rows, for instance, has the following justification. On the null hypothesis (which we shall call H R ) that all the row constants a i in (9.1) are equal (and therefore equal to zero, since a i 0), both s 2 R and s 2 are unbiased estimates of s 2 .IfH R is not true, so that the a i differ, s 2 R has expectation greater than s 2 whereas s 2 is still an unbiased estimate of s 2 . Hence F R tends to be greater than 1, and sufficiently high values indicate a significant departure from H R . This test is valid whatever values the b j take, since adding a constant on to all the readings in a particular column has no effect on either s 2 R or s 2 . Table 9.2 Two-way analysis of variance table. SSq DF MSq VR Between rows i R 2 i =c ÀT 2 =NrÀ 1 s 2 R F R s 2 R =s 2 Between columns j C 2 j =r ÀT 2 =NcÀ 1 s 2 C F C s 2 C =s 2 Residual By subtraction r À1c À 1 s 2 Total i, j y 2 ij À T 2 =NrcÀ 1 N À 1 9.2 Two-way analysis of variance: randomized blocks 241 Similarly, F C provides a test of the null hypothesis H C , that all the b j 0, irrespective of the values of the a i . If the additive model (9.1) is not true, the Residual SSq will be inflated by discrepancies between Ey ij and the approximations given by the best-fitting additive model, and the Residual MSq will thus be an unbiased estimate of a quantity greater than the random variance. How do we know whether this has happened? There are two main approaches, the first of which is to examine residuals. These are the individual expressions y ij À y i: À y :j y. Their sum of squares was obtained, from (9.4), by subtraction, but it could have been obtained by direct evaluation of all the N residuals and by summing their squares. These residuals add to zero along each row and down each column, like the discrepan- cies between observed and expected frequencies in a contingency table (§8.6), and (as for contingency tables) the number of DF, r À 1c À 1, is the number of values of residuals which may be independently chosen (the others being then automatically determined). Because of this lack of independence the residuals are not quite the same as the random error terms e ij of (9.2), but they have much the same distributional properties. In particular, they should not exhibit any striking patterns. Sometimes the residuals in certain parts of the two-way table seem to have predominantly the same sign; provided the ordering of the rows or columns has any meaning, this will suggest that the row-effect constants are not the same for all columns. There may be a correlation between the size of the residual and the `expected' value* y i: y :j À y: this will suggest that a change of scale would provide better agreement with the additive model. A second approach is to provide replication of observations, and this is discussed in more detail after Example 9.1. Example 9.1 Table 9.3 shows the results of a randomized block experiment to compare the effects on the clotting time of plasma of four different methods of treatment of the plasma. Samples of plasma from eight subjects (the `blocks') were assigned in random order to the four treatments. The correction term, T 2 =N, denoted here by CT, is needed for three items in the SSq column, and it is useful to calculate this at the outset. The analysis is straightforward, and the F tests show that differences between subjects and treatments are both highly sig- nificant. Differences between subjects do not interest us greatly as the main purpose of the experiment was to study differences between treatments. The standard error of the difference between two treatment means is 20Á6559=8 p 0Á405. Clearly, treatments 1, 2 and 3 do not differ significantly among themselves, but treatment 4 gives a signific- antly higher mean clotting time than the others. *This is the value expected on the basis of the average row and column effects, as may be seen from the equivalent expression y y i: À y y :j À y 242 Experimental design Table 9.3 Clotting times (min) of plasma from eight subjects, treated by four methods. Treatment Subject 1 2 3 4 Total Mean 18Á49Á49Á812Á239Á89Á95 212Á815Á212Á914Á455Á313Á82 39Á69Á111Á29Á839Á79Á92 49Á88Á89Á912Á040Á510Á12 58Á48Á28Á58Á533Á68Á40 68Á69Á99Á810Á939Á29Á80 78Á99Á09Á210Á437Á59Á38 87Á98Á18Á210Á034Á28Á55 Total 74Á477Á779Á588Á2 319Á8 Mean 9Á30 9Á71 9Á94 11Á02 (9Á99) Correction term, CT 319Á8 2 =32 3 196Á0013 Between-Subjects SSq 39Á8 2 34Á2 2 =4 ÀCT 78Á9888 Between-Treatments SSq 74Á4 2 88Á2 2 =8 ÀCT 13Á0163 Total SSq 8Á4 2 10Á0 2 À CT 105Á7788 Residual SSq 105Á7788 À 78Á9888 À 13Á0163 13Á7737 Analysis of variance SSq DF MSq VR Subjects 78Á9888 7 11Á2841 17Á20 (P < 0Á001) Treatments 13Á0163 3 4Á3388 6Á62 (P 0Á003) Residual 13Á7737 21 0Á6559 1Á00 Total 105Á7788 31 For purposes of illustration the residuals are shown below: Treatment Subject 1 2 3 4 Total 1 À0Á86 À0Á27 À0Á10 1Á22 À0Á01 2 À0Á33 1Á66 À0Á87 À0Á45 0Á01 30Á37 À0Á54 1Á33 À1Á15 0Á01 40Á37 À1Á04 À0Á17 0Á85 0Á01 50Á69 0Á08 0Á15 À0Á93 À0Á01 6 À0Á51 0Á38 0Á05 0Á07 À0Á01 70Á21 À0Á10 À0Á13 À0Á01 À0Á03 80Á04 À0Á17 À0Á30 0Á42 À0Á01 Total À0Á02 0Á00 À0Á04 0Á02 À0Á04 9.2 Two-way analysis of variance: randomized blocks 243 The sum of squares of the 32 residuals in the body of the table is 13Á7744, in agreement with the value found by subtraction in Table 9.3 apart from rounding errors. (These errors also account for the fact that the residuals as shown do not add exactly to zero along the rows and columns.) No particular pattern emerges from the table of residuals, nor does the distribution appear to be grossly non-normal. There are 16 negative values and 16 positive values; the highest three in absolute value are positive (1Á66, 1Á33 and 1Á22), which suggests mildly that the random error distribution may have slight positive skewness. If the linear model (9.1) is wrong, there is said to be an interaction between the row and column effects. In the absence of an interaction the expected differences between observations in different columns are the same for all rows (and the statement is true if we interchange the words `columns' and `rows'). If there is an interaction, the expected column differences vary from row to row (and, similarly, expected row differences vary from column to column). With one observation in each row/column cell, the effect of an interaction is inextricably mixed with the residual variation. Suppose, however, that we have more than one observation per cell. The variation between observations within the same cell provides direct evidence about the random variance s 2 , and may therefore be used as a basis of comparison for the between-cells residual. This is illustrated in the next example. Example 9.2 In Table 9.4 we show some hypothetical data related to the data of Table 9.3. There are three subjects and three treatments, and for each subject±treatment combination three rep- licate observations are made. The mean of each group of three replicates will be seen to agree with the value shown in Table 9.3 for the same subject and treatment. Under each group of replicates is shown the total T ij and the sum of squares, S ij (as indicated for T 11 and S 11 ). The Subjects and Treatments SSq are obtained straightforwardly, using the divisor 9 for the sums of squares of row (or column) totals, since there are nine observations in each row (or column), and using a divisor 27 in the correction term. The Interaction SSq is obtained in a similar way to the Residual in Table 9.3, but using the totals T ij as the basis of calculation. Thus, Interaction SSq = SSq for differences between the nine subject/treatment cells ± Subjects SSq ± Treatment SSq, and the degrees of freedom are, correspondingly, 8 À 2 À2 4. The Total SSq is obtained in the usual way and the Residual SSq follows by subtraction. The Residual SSq could have been obtained directly as the sum over the nine cells of the sum of squares about the mean of each triplet, i.e. as S 11 À T 2 11 =3S 12 À T 2 12 =3 S 33 À T 2 33 =3: The F tests show the effects of subjects and treatments to be highly significant. The interaction term is not significant at the 5% level, but the variance ratio (VR) is never- theless rather high. It is due mainly to the mean value for subject 8 and treatment 4 being higher than expected. 244 Experimental design Table 9.4 Clotting time (min) of plasma from three subjects, three methods of treatment and three replications of each subject±treatment combination. Treatment Subject 2 3 4 Total 69Á89Á911Á3 10Á19Á510Á7 9Á810Á010Á7 T 11 29Á729Á432Á7 R 1 91Á8 S 11 294Á09 288Á26 356Á67 79Á29Á110Á3 8Á69Á110Á7 9Á29Á410Á2 27Á027Á631Á2 R 2 85Á8 243Á24 253Á98 324Á62 88Á48Á69Á8 7Á98Á010Á1 8Á08Á010Á1 24Á324Á630Á0 R 3 78Á9 196Á97 201Á96 300Á06 Total C 1 81Á0 C 2 81Á6 C 3 93Á9 T 256Á5 y 2 2459Á85 CT T 2 =27 2436Á75 Subjects SSq 91Á8 2 78Á9 2 =9 ÀCT 9Á2600 Treatments SSq 81Á0 2 93Á9 2 =9 ÀCT 11Á7800 Interaction SSq 29Á7 2 30Á0 2 =3 ÀCT À Subj: SSq À Treat:SSq 0Á7400 Total SSq 9Á8 2 10Á1 2 À CT 2459Á85 À CT 23Á1000 Residual SSq Total À Subjects À Treatments À Interaction 1Á3200 Analysis of variance SSq DF MSq VR Subjects 9Á2600 2 4Á6300 63Á1 Treatments 11Á7800 2 5Á8900 80Á3 Interaction 0Á7400 4 0Á1850 2Á52 (P 0Á077) Residual 1Á3200 18 0Á0733 1Á00 Total 23Á1000 26 The interpretation of significant interactions and the interpretation of the tests for the `main effects' (subjects and treatments in Examples 9.1 and 9.2) when interactions are present will be discussed in the next section. 9.2 Two-way analysis of variance: randomized blocks 245 If, in a two-way classification without replication, c 2, the situation is the same as that for which the paired t test was used in §4.3. There is a close analogy here with the relationship between the one-way analysis of variance and the two-sample t test noted in §8.1. In the two-way case the F test provided by the analysis of variance is equivalent to the paired t test in that: (i) F is numerically equal to t 2 ; (ii) the F statistic has 1 and r À1 DF while t has r À 1 DF, and, as noted in §5.1, the distributions of t 2 and F are identical. The Residual MSq in the analysis of variance is half the corresponding s 2 in the t test, since the latter is an estimate of the variance of the difference between the two readings. In Example 9.2 the number of replications at each row±column combination was constant. This is not a necessary requirement. The number of observations at the ith row and jth column, n ij , may vary, but the method of analysis indicated in Example 9.2 is valid only if the n ij are proportional to the total row and column frequencies; that is, denoting the latter by n i: and n :j , n ij n i: n :j N : 9:6 In Example 9.2 all the n i: and n :j were equal to 9, N was 27, and n ij 81=27 3, for all i and j. If (9.6) is not true, an attempt to follow the standard method of analysis may lead to negative sums of squares for the interaction or residual, which is, of course, an impossible situation. Condition (9.6) raises a more general issue, namely that many of the rela- tively straightforward forms of analysis, not only for the two-way layout but also for many of the other arrangements in this chapter, are only possible if the numbers of outcomes in different parts of the experiment satisfy certain quite strict conditions, such as (9.6). Data which fail to satisfy such conditions are said to lack balance. In medical applications difficulties with recruitment or with- drawal will readily lead to unbalanced data. In these cases it may be necessary to use more general methods of analysis, such as those discussed in Chapters 11 and 12. If the data are unbalanced because of the absence of just a very small proportion of the data, then one approach is to impute the missing values on the basis of the available data and the fitted model. Details can be found in Cochran and Cox (1957). However, when addressing problems of missing data the issues of why the data are missing can be more important than how to cope with the resulting imbalance: see §12.6. 9.3 Factorial designs In §9.2 an example was described of a design for a factorial experiment in which the variable to be analysed was blood-clotting time and the effects of two factors were to be measured: r periods of storage and c concentrations of adrenaline. Observations were made at each combination of storage periods 246 Experimental design and adrenaline concentrations. There are two factors here, one at r levels and the other at c levels, and the design is called an r Âc factorial. This design contravenes what used to be regarded as a good principle of experimentation, namely that only one factor should be changed at a time. The advantages of factorial experimentation over the one-factor-at-a-time approach were pointed out by Fisher. If we make one observation at each of the rc combinations, we can make comparisons of the mean effects of different periods of storage on the basis of c observations at each period. To get the same precision with a non-factorial design we would have to choose one particular concentration of adrenaline and make c observations for each storage period: rc in all. This would give us no information about the effect of varying the concentration of adrenaline. An experiment to throw light on this factor with the same precision as the factorial design would need a further rc observations, all with the same storage period. Twice as many observations as in the factorial design would therefore be needed. Moreover, the factorial design permits a comparison of the effect of one factor at different levels of the other: it permits the detection of an interaction between the two factors. This cannot be done without the factorial approach. The two-factor design considered in §9.2 can clearly be generalized to allow the simultaneous study of three or more factors. Strictly, the term `factorial design' should be reserved for situations in which the factors are all controllable experimental treatments and in which all the combinations of levels are ran- domly allocated to the experimental units. The analysis is, however, essentially the same in the slightly different situation in which one or more of the factors represents a form of blockingÐa source of known or suspected variation which can usefully be eliminated in comparing the real treatments. We shall therefore include this extended form of factorial design in the present discussion. Notation becomes troublesome if we aim at complete generality, so we shall discuss in detail a three-factor design. The directions of generalization should be clear. Suppose there are three factors: A at I levels, B at J levels and C at K levels. As in §9.2, we consider a linear model whereby the mean response at the ith level of A, the jth level of B and the kth level of C is Ey ijk m a i b j g k ab ij ag ik bg jk abg ijk , 9:7 with i a i i ab ij i abg ijk 0, etc: Here, the terms like ab ij are to be read as single constants, the notation being chosen to indicate the interpretation of each term as an interaction between two or more factors. The constants a i measure the effects of the different levels of factor A averaged over the various levels of the other factors; these are called the main effects of A. The constant ab ij indicates the extent to which the mean 9.3 Factorial designs 247 Table 9.5 Structure of analysis of three-factor design with replication. SSq DF MSq VR ( MSq/s 2 ) Main effects A S A I À 1 s 2 A F A B S B J À1 s 2 B F B C S C K À 1 s 2 C F C Two-factor interactions AB S AB I À 1J À1 s 2 AB F AB AC S AC I À 1K À1 s 2 AC F AC BC S BC J À1K À1 s 2 BC F BC Three-factor interaction ABC S ABC I À 1J À1K À 1 s 2 ABC F ABC Residual S R IJKn À 1 s 2 1 Total SNÀ 1 response at level i of A and level j of B, averaged over all levels of C, is not determined purely by a i and b j , and it thus measures one aspect of the interaction of A and B. It is called a first-order interaction term or two-factor interaction term. Similarly, the constant abg ijk indicates how the mean response at the triple combination of A, B and C is not determined purely by main effects and first- order interaction terms. It is called a second-order or three-factor interaction term. To complete the model, suppose that y ijk is distributed about Ey ijk with a constant variance s 2 . Suppose now that we make n observations at each combination of A, B and C. The total number of observations is nIJK N, say. The structure of the analysis of variance is shown in Table 9.5. The DF for the main effects and two-factor interactions follow directly from the results for two-way analyses. That for the three-factor interaction is a natural extension. The residual DF are IJKn À1 because there are n À 1 DF between replicates at each of the IJK factor combinations. The SSq terms are calculated as follows. 1 Main effects. As for a one-way analysis, remembering that the divisor for the square of a group total is the total number of observations in that group. Thus, if the total for ith level of A is T i:: , and the grand total is T, the SSq for A is S A i T 2 i:: =nJK À T 2 =N: 9:8 2 Two-factor interactions. Form a two-way table of totals, calculate the appro- priate corrected sum of squares between these totals and subtract the SSq for the two relevant main effects. For AB, for instance, suppose T ij: is the total for levels i of A and j of B. Then 248 Experimental design [...]... iv 7 4 vi 7Á7 v 6Á0 iii 6Á8 i 7Á3 ii 7Á3 i 7 4 v 7Á1 vi 6 4 ii 7Á7 iii 6 4 iv 5Á8 vi 7Á1 iii 8Á1 ii 6Á2 iv 8Á5 v 6 4 i 6 4 ii 8Á2 i 5Á9 iv 7Á5 v 8Á5 vi 7Á3 iii 7Á7 Total Mean 42 4 7Á067 51Á7 8Á617 42 Á5 7Á083 40 Á8 6Á800 42 Á7 7Á117 45 Á1 7Á517 Order Total Mean i 43 Á0 7Á167 ii 44 Á3 7Á383 iii 45 Á0 7Á500 iv 45 Á2 7Á533 v 44 Á0 7Á333 vi 43 Á7 7Á283 a b c d e f Total Mean 46 Á7 7Á783 43 Á1 7Á183 41 Á7 6Á950 46 Á9... Father's strain Mother's strain 1 2 3 4 Total 1 2 3 4 4Á15 4 62 6Á39 4 32 4 74 4Á92 6Á05 5Á 34 5Á19 6Á27 4 64 5Á80 5Á06 5Á52 4 93 4 25 19Á 14 21Á33 22Á01 19Á71 Total 19 48 21Á05 21Á90 19Á76 82Á19 Strains 2 and 3 give relatively high readings for M and F, suggesting a systematic effect which has not achieved significance for either parent separately The interaction is due partly to the high reading for (M3,... are within families, vary from one degree of crowding to another; it is therefore based entirely on within-families contrasts Table 9.11 Analysis of variance for data in Table 9.10 VR against: SSq Between families Crowding Residual Within families Status Status  crowding Residual 3122Á80 Total 42 68Á89 DF 1 146 Á09 MSq a 2 15 235Á 24 45Á04b 1Á78 4 8 60 383 42 9Á05 25Á28a b 15Á17** 0Á36 1Á00 17 47 0 49 675Á60... shown in the DF column of Table 9.11 The total DF are 89, since there are 90 observations These are split (as in a one-way analysis of variance) into 17
18 À 1 between families and 72
18 Â 4 within families The between-families DF are split (again as in a one-way analysis) into 2
3 À 1 for degrees of crowding and 15
3 Â 5 for residual variation within crowding categories The within-families... 1Á11 1Á85 0 43 13Á39 6Á32 19Á71 1Á38 0Á59 13Á57 5Á91 19 48 14 39 6Á66 21Á05 14 60 7Á30 21Á90 13Á97 5Á79 56Á53 25Á66 19Á76 82Á19 1 Total CT 105Á 549 9 Analysis of variance Mother's strain, M Father's strain, F Sex of animal, S MF MS FS MFS Residual DF 3 3 1 9 3 3 9 32 SSq 0Á3396 0Á 240 1 14 8900 1Á2988 0Á3 945 0Á0 245 0Á2612 1Á2 647 Total 63 4 18Á71 34 Differences , À < MSq 0Á1132 0Á0800 14 8900 0Á 144 3 0Á1315... AD, AB against ABD The justification for this follows by interpreting the interaction terms involving D in the model like (9.7) as independent observations on random variables with zero mean The concept of a random interaction is reasonable; if, for example, D is a blocking system, any linear contrast representing part of a main effect or interaction of the other factors can be regarded as varying randomly... 12 5 11 3 3 10 9 41 Crowded 1 2 3 4 5 6 Child 33 45 95 86 298 6 9 2 0 3 6 3 6 2 2 2 2 5 6 6 10 0 4 7 14 15 16 3 7 3 10 8 21 14 20 24 45 33 49 22 39 26 Total 17 31 62 76 212 106 105 145 258 276 890 268 Experimental design categories of family status, 8
4 Â 2 for the interaction between the two main effects, and 60 for within-families residual variation The latter number can be obtained by subtraction... mother's strain, for example, is
19Á 14 2
19Á712 =16 À CT, where the correction term, CT, is
82Á192 = 64 105Á 549 9 The two-factor interaction, MF, is obtained as 4 152 4 252 =4 À CT À SM À SF , where 4 15 is the sum of the responses in the first cell (0Á93 1Á70 0Á69 0Á83), and SM and SF are the SSq for the two main effects Similarly, the three-factor interaction is obtained... described in this section have found little use in medical research, examples of their application being drawn usually from industrial and agricultural research This contrast is perhaps partly due to inadequate appreciation of the less familiar designs by medical research workers, but it is likely also that the organizational problems of experimentation are more severe in medical research than in many... is again convenient to redefine the interaction as ABC=4n The results are summarized in Table 9.7 Note that the positive and negative signs for the two-factor interactions are easily obtained by multiplying together the coefficients for the corresponding main effects; and those for the three-factor interaction by multiplying the coefficients for A and BC, B and AC, or C and AB The final . strain Mother's strain 1 2 3 4 Total 14 15 4 74 5Á 19 5Á06 19Á 14 24 62 4 92 6Á 27 5Á52 21Á33 36Á39 6Á05 4 64 4Á93 22Á01 44 Á32 5Á 34 5Á 80 4 25 19Á71 Total 19 48 21Á05 21Á90 19Á76 82Á19 Strains. term, CT, is 82Á19 2 = 64 105Á 549 9. The two-factor interaction, MF, is obtained as 4 15 2 4 25 2 =4 À CT ÀS M À S F , where 4 15 is the sum of the responses in the first cell (0Á93. differ- ences between blocks are usually of no intrinsic interest. Two-way classifications may arise also in non-experimental work, either by classifying in this way data already collected in a
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