Notation for Two-Factor Experiments

QR Code Reader In NoneUsing Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications.

QR Code JIS X 0510 Generation In NoneUsing Barcode generator for Software Control to generate, create QR Code ISO/IEC18004 image in Software applications.

Assuming that we have a treatments and b blocks, we construct Table 9-4, where it is supposed that there is one experimental value (for example, yield per acre) corresponding to each treatment and block. For treatment j and block k we denote this value by xjk. The mean of the entries in the jth row is denoted by xj., where j 1, . . . , a, #

QR Code 2d Barcode Decoder In NoneUsing Barcode decoder for Software Control to read, scan read, scan image in Software applications.

QR Creator In C#.NETUsing Barcode generation for .NET framework Control to generate, create QR Code image in Visual Studio .NET applications.

CHAPTER 9 Analysis of Variance

Quick Response Code Printer In VS .NETUsing Barcode generator for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications.

Denso QR Bar Code Printer In Visual Studio .NETUsing Barcode encoder for VS .NET Control to generate, create Quick Response Code image in .NET framework applications.

while the mean of the entries in the kth column is denoted by x.k, where k # is denoted by x. In symbols, # xj. # 1 x , b ka jk 1

QR Code JIS X 0510 Generator In Visual Basic .NETUsing Barcode creator for VS .NET Control to generate, create QR Code image in .NET applications.

Generating Barcode In NoneUsing Barcode generator for Software Control to generate, create barcode image in Software applications.

1, . . . , b. The overall, or grand, mean

Making GTIN - 128 In NoneUsing Barcode maker for Software Control to generate, create GS1 128 image in Software applications.

Painting Barcode In NoneUsing Barcode maker for Software Control to generate, create barcode image in Software applications.

x.k #

Creating DataMatrix In NoneUsing Barcode creator for Software Control to generate, create ECC200 image in Software applications.

Draw UPC-A In NoneUsing Barcode generator for Software Control to generate, create UCC - 12 image in Software applications.

1 a a xjk,

Identcode Drawer In NoneUsing Barcode maker for Software Control to generate, create Identcode image in Software applications.

Barcode Drawer In JavaUsing Barcode generation for Eclipse BIRT Control to generate, create barcode image in BIRT applications.

1 x ab a jk j,k

Reading Barcode In JavaUsing Barcode recognizer for Java Control to read, scan read, scan image in Java applications.

Scanning Barcode In NoneUsing Barcode reader for Software Control to read, scan read, scan image in Software applications.

(27)

Code-39 Decoder In NoneUsing Barcode recognizer for Software Control to read, scan read, scan image in Software applications.

Recognizing ANSI/AIM Code 39 In VS .NETUsing Barcode scanner for VS .NET Control to read, scan read, scan image in .NET framework applications.

Table 9-4 Blocks 1 Treatments 1 2 ( a x11 x21 ( xa1 x.1 # 2 x12 x22 ( xa2 x.2 # c c c ( ( ( c c b x1b x2b xab x.b # x1. # x2. # xa. #

Create Barcode In JavaUsing Barcode creation for Android Control to generate, create bar code image in Android applications.

Code 128 Code Set C Maker In Visual C#.NETUsing Barcode maker for .NET Control to generate, create Code 128C image in .NET applications.

Variations for Two-Factor Experiments

As in the case of one-factor experiments, we can define variations for two-factor experiments. We first define the total variation, as in (3), to be v By writing the identity xjk x # (xjk xj. # v where ve variation due to error or chance vr vc x.k # ve a (xjk

a (xjk

x)2 #

(28)

x) # vr xj. #

(xj. # vc x.k #

x) #

(x.k #

x) #

(29)

and then squaring and summing over j and k, we can show that (30) x)2 #

variation between rows (treatments)

b a (xj. #

j 1 b

x)2 #

variation between columns (blocks)

a a (x.k #

x)2 #

The variation due to error or chance is also known as the residual variation. The following are short formulas for computation, analogous to (10), (11), and (12). v vr vc ve a xjk

t2 ab t2 ab t2 ab vc

(31)

1 t2 b ja j. 1 1 2 a a t.k

k 1 b

(32)

(33) (34)

where tj. is the total of entries of the jth row, tk is the total of entries in the kth column, and t is the total of all . entries.

CHAPTER 9 Analysis of Variance

Analysis of Variance for Two-Factor Experiments

For the mathematical model of two-factor experiments, let us assume that the random variables Xjk whose values are the xjk can be written as Xjk m aj bk

(35)

Here m is the population grand mean, aj is that part of Xjk due to the different treatments (sometimes called the treatment effects), bk is that part of Xjk due to the different blocks (sometimes called the block effects), and jk is that part of Xjk due to chance or error. As before, we can take the jk as independent normally distributed random variables with mean zero and variance s2, so that the Xjk are also independent normally distributed variables with variance s2. Under suitable assumptions on the means of the Xjk, we have a aj

a bk

(36)

which makes m 1 E(Xjk) ab a j,k

Corresponding to the results (16) through (18), we can prove that E(Vr) E(Vc) E(Ve) E(V) (a (b (a (ab 1)s2 1)s2 1)(b 1)s2 b a a2 j

(37) (38) (39) a a b2 k

a a b2 k

1)s2 b a a2 j

(40)

There are two null hypotheses that we would want to test: H(1): All treatment (row) means are equal, i.e., aj 0 H(2): 0 All block (column) means are equal, i.e., bk 0, j 0, k 1, . . . , a 1, . . . , b

We see from (39) that, without regard to H(1) or H(2), a best (unbiased) estimate of s2 is provided by 0 0 S2 e

Ve 1)(b

i.e.,

E(S 2) e

(41)

Also, if the hypotheses H(1) and H(2) are true, then 0 0 S2 r

Vr a 1

, S2 c

Vc b 1

V ab 1

(42)

will be unbiased estimates of s2. If H(1) and H(2) are not true, however, we have from (37) and (38), respectively, 0 0 E(S 2) r

b a a b

1a j 1a k

a2 j

(43)

E(S 2) c

b2 k

(44)

The following theorems are similar to Theorems 9-1 and 9-2. Theorem 9-4 Ve >s2 is chi-square distributed with (a or H(2). 0 1)(b 1) degrees of freedom, without regard to H(1) 0