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 = S2/ 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


  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.

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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