JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS TABLE III TRANSFORMED DATA Soil •2 Operator X ¾ Z X Z Total , Date 6a 9p 2a 8p 3p 7a 6p 9a 2p 8a 3a 7p 1 g/litre .77 8 1 8 -7 t -7 1 1 -7 1 1 1 2 2g/litre 1.00 0 11 0 0 4 -5 -5 0 -5 0 0 0 0 4 g/litre 1.15 11 5 5 -7 0 -4 5 0 0 -7 0 3 11 6 g/litre 1.25 5 -2 5 -2 1 3 -2 1 -5 -2 -2 1 1 Total 24 15 18 -16 I 6 -13 -1 2 -17 -8 -1 5 14 We now proceed to carry out an analysis of variance. An introduction to the technique is given by Middleton', but the present example is an incomplete four factor experiment dealt with fully in Brownlee4b, to which the reader is referred. Our analysis gives us the following table ß . : TABLE IV '• :'•.• ANALYSIS OF VARIANCE , Source of Variance d.f. S.S. M.S. -- Between samples 3 6-4 Between soils 1 60.8 60.8 Between operators 2 129. $ 64.8 Samples X soils 3 12.4 4.1 Samples x operators 6 54'0 9.0 'Soils X operators 2 82-6 41.3 Samples x soils X operators 6 39-2 6.5 Replications 24 531.0 23.1 Total 47 915.9 The experiment was designed with each condition of test reveated ifi•:j• order to provide a good estimate of the random errors. Another e•timate error is given by the highest interaction, that of samples x soils x operat )peratoL•!• and, as is not infrequent, this is smaller than that of the replications, altho the F test shows no significance at the 0.05 level. Nevertheless, the pu will be reconsidered later to determine if some factor had changed the replications. For the time being, we pool the S.S. contributions of the replica• variance and aH smaller effects giving a mean estimate of error 636.6/39 = 16.3. The F test is used to determine if the interaction vari a•i•

DEVELOPMENT OF A. DETERGENT TEST--I due to soil x operators is significant, and the value found, 2-54, does not reach the tabulated figure at the 0.05 level. Therefore, the effect is considered not significant and the variance pooled with the error variance to give a new estimate of 719-2/41 = 17.6 for the error variance. The value of F for the operators effect is 3-69 which exceeds the value of 3.2 for N1 •-2, N2 =41, p = 0.05, and the factor is therefore deemed "probably significant." The soils difference is found to be not significant. On returning to the replications variance, a closer examination of its sources revealed a trend pointing to a totally unexpected new factor--the age of the soil. The last row of Table III shows that with soil Q2 each operator obtained a smaller result on replication, whilst with Q3 the results were, if anything, higher. The effects are of different magnitude with the different operators, but it is noticed that the changes are in the same order as the intervals between replications. This is brought out in the table below. Soil Q2 Soil Q3 Operator X (3 days later) 9 units lower 3 units higher Operator Y (6 days later) 34 units lower 9 units higher Operator Z (4 days later) 19 units lower 6 units higher To test the significance of the data we postulate a null hypothesis that a coefficient b representing the mean daily change is not significantly different from zero. The best value of b is that which reduces the residual S.S. to a minimum, and for soil Q2 calculated thus: Original S.S. due to age -- (9 • q- 34 •- q- 192)/8 •- 199.7 Residual S.S. ---- ((9 -- 3b)' q- (34 -- 6b)' q- (19 -- 4b)')/8 When the residual is a minimum, its derivative with respect to b is zero, and thus: 3(9 -- 3b) q- 6(34 -- 6b) q- 4 (19-- 4b) = 0 whence b ----- 5.03. Substitution of this value in the previous equation gives 6.6 for the residual S.S., hence S.S. due to age is, by difference, 193.1. A similar calcula- tion for soil Q3 gives a coefficient of 1.42 (representing the daily increase in results) and an S.S. due to age of 15.5. These terms are deducted from the replications term of Table IV leaving 322.4 with 22 d.f. The three interactions of M.S. less than 14 give a combined S.S. of 105.6 based on 15 d.f., which is added to the foregoing to give a more precise estimate of the error. The new analysis is: 103