Sunday, January 16, 2011
Analysis of Variance
BIOMETRICS
COMPARING TWO MEANS: Xbar1 – Xbar2 / Sx1 – Sx2 ~ 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 = S2b / S2w ~ F(.)
If means are different, then S2b > S2w , 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: Yij = U + Ti + Eij
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 + Ti) = 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
- If treatments or blocks differ, look at individual differences in means via LSD.
- 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:
- # treatments consists of several categories, levels
- May be used to determine optimal levels
- 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
- Observed variation in Y is partly due to variation in X.
- 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.
Yij = μ + Ti + pj + B(Xij – Xbar) + eij
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
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