# BIOMETRICS

__COMPARING TWO MEANS__: Xbar

_{1}– Xbar

_{2}/ S

_{x1}– S

_{x2}~ t

can’t define numerator for >2 means

__ANALYSIS OF VARIANCE:__Comparing more than two means. When μ

_{1}not = μ

_{2}, 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 = S^{2}_{b }/ S^{2}_{w}~ F(.)If means are different, then S

^{2}_{b }> S^{2}_{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

- If treatments or blocks differ, look at individual differences in means via LSD.
- Contrasts: Q = CiYi ( orthogonal) MS(Q) = Q
^{2}/ r sum(Ci^{2}), 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.

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

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