374 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



Note that now each of the variables associated with Row, Column, 

 Latin letter, Greek letter has the property that each element of the four 

 categories contains one and only one element of the remaining three 

 categories. For example, consider the five Latin-Greek letter combina- 

 tions, or sample cells, containing e: 



Row 1 Co\ 5, E; Row 2 Col 3, D; Row 3 Col 1, C; Row 4 Col 4, B; Row 



5 Col 2, .4. 



Then the sum of these five cells will contain the contribution of the 5 

 Rows, 5 Columns, and 5 Latin letters. 



It should })e noted that the rows and columns can be permuted with- 

 out affecting the properties of the scjuare. Lideed to protect against 

 systematic effects which may be detrimental, it is usual to assign at 

 random the row and column number, as well as the Latin and Greek 

 letters to the values of variables represented by them. A notable excep- 

 tion to randomization occurs when time is a variable and those measure- 

 ments made under essentially the same conditions within the same unit 

 of time become the experimental unit. In the first experiment, all meas- 

 urements made on one Run become the experimental unit with respect 

 to time. 



The Factorial Design serves a different purpose. In this discussion only 

 two independent variables A", Z will be predicated but the extension to 

 more variables follows directly. The XZ plane is the plane of the in- 

 dependent variables and we seek the point (Xo , Zq) which gives a value 

 of y (the dependent variable), Y = ^(A', Z), which is optimum in some 

 sense. That is, min Y, max Y may be sought, or the surface f(X, Y^, Z) 

 shown to be a plane. 



Generally only the region in the neighborhood of y (optimum) is of 

 interest to the experimenter. Hence it is imperative (1) to bracket this 

 point with respect to each independent variable and (2) to have a method 

 of estimating // (optimum). If a factorial experiment has U different 

 values of X and Iz different values of Z, then each replication of the 

 experiment will recjuire U • h units or points (A', Z). The first repetition of 

 an experiment is called the second replication in the same way that the 

 first overtone in music is called the second harmonic by engineers. 

 Since the number of units available for test is usually limited, this places 

 a practical ceiling on the magnitude of 4 and h . As a practical limit in 

 general / should be 7 or less and the values 2, 3, or 4 are far more common. 

 It is generally better to use the smaller values of I and repeat the expei'i- 

 ment, than to conduct an experiment involving only a single replication. 

 In addition to evaluating one variable averaged ovei- the second, we 



