Analysis of Variance

BIOMETRICS

COMPARING TWO MEANS: Xbar₁ – Xbar₂ / S_x1 – S_x2 ~ t 

can’t define numerator for >2 means

ANALYSIS OF VARIANCE: Comparing more than two means. When μ₁ not = μ₂, an estimate of variance using means will include a contribution attributable to the difference between population means. This ‘treatment’ effect will lead to a biased estimate of variance and large F.

F = σ ²+ trt / σ = S²_b  / S²_w ~ F(.)

If means are different, then  S²_b  > S²_w  , then F will be large. A large significant F value allows you to reject the null hypothesis that the means are equal. 

Linear Additive Model: Y_ij = U + T_i + E_ij

Or in matrix form: Y = Xb + e  where  Y = a matrix of individual observations. X = a ‘design matrix of 1’s and 0’s, & b =  a matrix of co-efficients of the population mean and treatment effects ‘t’.  or ( U + T_i) = Xb

SS Total = Y’Y – Y’CY’C(1/n)

SS Between/Model=b’X’Y - Y’CY’C(1/n)

SS Within/Error  =Y’Y’ – B’X’Y

Where ‘C’ is a matrix of 1’s

RCB:

                                    ( Blocks)

(treatments)               A            B            C             D             E

1            x             x            x            x            x

2            x            x            x            x            x

3            x            x            x            x            x

4            x            x            x            x            x

1.  If treatments or blocks differ, look at individual differences in means via LSD.
2.  Contrasts: Q = CiYi  ( orthogonal)   MS(Q) = Q² / r sum(Ci²), a subset of a linear function, a way of comparing different means.

LATIN SQUARE:

A	D	C	B
B	C	A	D
D	A	B	C
C	B	D	A

Each row is a comparable block with treatments A-D occurring only once in each block.

FACTORIAL EXPERIMENTS:

1. # treatments consists of several categories, levels
2. May be used to determine optimal levels
3. Interaction: difference in response levels-non additive,  measures homogeneity of  a response.

Polynomial Responses:  partitioning SS into linear and quadratic components- response surfaces

SPLIT PLOT DESIGNS

Whole plots divided into sub-plots where levels of factors are applied.

Block #1

A4B2 A4B1

A1B2 A1B1

A2B1 A2B2

A3B1 A3B2

A4B2 A4B1

Incomplete block re ‘A’ , complete block re ‘B’

SPLIT BLOCK EXPERIMENTS

Split plot in time vs. space ex:  time or cutting represents an abstract split plot ‘a’ cultivars, ‘b’ cuttings.

ANALYSIS OF COVARIANCE

1. Observed variation in Y is partly due to variation in X.
2. Use regression to eliminate effects that cannot be controlled by experimental design.

Ex: animals in a block with varying initial weights, use covariance to remove effect from experimental error.

Y_ij = μ + T_i + p_j + B(X_ij – Xbar) + e_ij

B = Cov(X,Y) / Var(X)

Carry out AOV on values adjusted for regression on an independent variable.

References:

Principles and Procedures of Statistics: A Biometrical Approach 3rd Edition. Robert George Douglas Steel, James Hiram Torrie, David A. Dickey