So far... Until we looked at factorial interactions, we were

looking at differences and their significance - or the probability that an observed difference was due to chance

But we had not learned anything about how two (or more) variables are related

What is regression? The way one variable is related to another.

As you change one, how are others affected?

Yield

Protein %

Types of Variables in Crop Experiments:

Treatments such as fertilizer rates, varieties, and weed control methods which are the primary focus of the experiment

Environmental factors, such as rainfall and solar radiation which are not within the researcher’s control

Responses which represent the biological and physical features of the experimental units that are expected to be affected by the treatments being tested.

Usual associations within ANOVA...

Association between response and treatment– when treatments are quantitative - such as

fertilizer levels - it is possible to describe the association between treatment and response

– the response could then be specified for not only the treatment levels actually tested but for all other intermediate points within the range of the treatments tested

Partitioning SST into Regression Components

Agronomic experiments frequently consist of different levels of one or more quantitative variables:– Varying amounts of fertilizer– Several different row spacings– Two or more depths of seeding

Would be useful to develop an equation to describe the relationship between plant response and treatment level

Fitting the Model

WheatYield(Y)

Applied N Level

X1 X2 X3 X4

Y3

Y1

Y2

Y4

Y = 0 + 1X +

where:Y =wheat yieldX =nitrogen level0=yield with no

nitrogen1=change in yield

per unit of applied N

=random error

Partitioning SST

Sums of Squares for Treatments (SST) contains:– SSLIN = Sum of squares associated with the

linear regression of Y on X

– SSLOF = Sum of squares for the failure of the regression model to describe the relationship between Y and X (lack of fit)

One way: Find a set of coefficients that define a linear

contrast– use the deviations of the treatment levels from

the mean level of all treatments– so that j jk X X

Therefore

The sum of the coefficients will be zero, satisfying the definition of a contrast

jLIN j jL (X X)Y

Computing SSLIN

SSLOF (sum of squares for lack of fit) is computed by subtraction

SSLOF = SST - SSLIN (df is df for treatments - 1)

Not to be confused with SSE which is still the SS for pure error (experimental error)

_ SSLIN = r*LLIN

2/[j (Xj - X)2]

really no different from any other contrast - df is always 1

F Ratios and their meaning All F ratios have MSE as a denominator

FT = MST/MSE tests– significance of differences among the treatment means

FLIN = MSLIN/MSE tests– H0: no linear relationship between X and Y (1 = 0)– Ha: there is a linear relationship between X and Y ( 1 0)

FLOF = MSLOF/MSE tests

– H0: the simple linear regression model describes the data

E(Y) = 0 + 1X

– Ha: there is significant deviation from a linear relationship between X and Y

E(Y) 0 + 1X

The linear relationship

The expected value of Y given X is described by the equation:

where:– = grand mean of Y

– Xj = value of X (treatment level) at which Y is estimated

–

j 1 jY Y b (X X)

Y

jLIN j jL (X X)Y

LIN1 2

j j

Lb

(X X)

2LIN

LIN 2j j

r *LSS

(X X)

Sources of Variation in Regression

WheatYield

(y)

Applied N Level

x1 x2 x3 x4

Y3

Y1

Y2

Y4

Y

Y

V

Y Y b(X X)

iˆY Y

Orthogonal Polynomials If the relationship is not linear, we can simplify

curve fitting within the ANOVA with the use of orthogonal polynomial coefficients under these conditions:– equal replication– the levels of the treatment variable must be equally

spaced• e.g., 20, 40, 60, 80, 100 kg of fertilizer per plot

Curve fitting Model: E(Y) = 0 + 1X + 2X2 + 3X3 +…

Determine the coefficients for 2nd order and higher polynomials from a table

Use the F ratio to test the significance of each contrast.

Unless there is prior reason to believe that the equation is of a particular order, it is customary to fit the terms sequentially

Include all terms in the equation up to and including the term at which lack of fit first becomes nonsignificant

Table of coefficients

Where do linear contrast coefficients come from? (revisited)

Assume 5 Nitrogen levels: 30, 60, 90, 120, 150

x = 90

k1 = (-60, -30, 0, 30, 60)

If we code the treatments as 1, 2, 3, 4, 5

x = 3

k1 = (-2, -1, 0, 1, 2)

b1 = LLIN / [r j (xj - x)2], but must be decoded back to original scale

_

_

_

jLIN j jL (X X)Y

1 1

X Xk

d

Consider an experiment Five levels of N (10, 30, 50, 70, 90) with four

replications 2LIN

LIN 2j j

r *LSS

(X X)

LIN 1 2 3 4 5L ( 2)Y ( 1)Y (0)Y (1)Y (2)Y

Linear contrast–

– SSLIN = 4* LLIN2

/ 10

QUAD 1 2 3 4 5L (2)Y ( 1)Y ( 2)Y ( 1)Y (2)Y

Quadratic–

– SSQUAD = 4*LQUAD2

/ 14

LOF still significant? Keep going… Cubic

–

– SSCUB = 4*LCUB2

/ 10CUB 1 2 3 4 5L ( 1)Y (2)Y (0)Y ( 2)Y (1)Y

QUAR 1 2 3 4 5L (1)Y ( 4)Y (6)Y ( 4)Y (1)Y

Quartic–

– SSQUAR = 4*LQUAR2

/ 70

Each contrast has 1 degree of freedom

Each F has MSE in denominator

Numerical Example An experiment to determine the effect of nitrogen on the

yield of sugarbeet roots:– RBD– three blocks– 5 levels of N (0, 35, 70, 105, and 140) kg/ha

Meets the criteria– N is a quantitative variable– levels are equally spaced– equally replicated

Significant SST so we go to contrasts

Orthogonal Partition of SST

N level (kg/ha)

0 35 70 105 140

Order Mean 28.4 66.8 87.0 92.0 85.7 Li j kj2 SS(L)i

Linear -2 -1 0 +1 +2 46.60 10 651.4780

Quadratic +2 -1 -2 -1 +2 -34.87 14 260.5038

Cubic -1 +2 0 -2 +1 2.30 10 1.5870

Quartic +1 -4 +6 -4 +1 0.30 70 .0039

Sequential Test of Nitrogen Effects

Source df SS MS F

(1)Nitrogen 4 913.5627 228.3907 64.41**

(2)Linear 1 651.4680 651.4680 183.73**

Dev (LOF) 3 262.0947 87.3649 24.64**

(3)Quadratic 1 260.5038 260.5038 73.47**

Dev (LOF) 2 1.5909 .7955 0.22ns

Choose a quadratic model– First point at which the LOF is not significant– Implies that a cubic term would not be significant

Regression Equation

bi = LREG / j kj2 Coefficient b0 b1 b2

23.99 4.66 -2.49

2Y 9.69 0.418X 0.002X

To scale to original X values

j 1j 2 j

1

for example, at 0 k

Y Y 4.66k 2.49k

Y 23.99 0.418( 2) 0.002(2) 9.69

g N/ha

1 1

X Xk

d

2 2

2 2

X X t 1k

d 12

Common misuse of regression... Broad Generalization

– Extrapolating the result of a regression line outside the range of X values tested

– Don’t go beyond the highest nitrogen rate tested, for example

– Or don’t generalize over all varieties when you have just tested one

Do not over interpret higher order polynomials– with t-1 df, they will explain all of the variation among

treatments, whether there is any meaningful pattern to the data or not

Class vs nonclass variables General linear model in matrix notation

Y = Xß + X is the design matrix

– Assume CRD with 3 fertilizer treatments, 2 replications

1 1 0 0

1 1 0 0

1 0 1 0

1 0 1 0

1 0 0 1

1 0 0 1

1 -1 1

1 -1 1

1 0 -2

1 0 -2

1 1 1

1 1 1

1 30 900

1 30 900

1 60 3600

1 60 3600

1 90 8100

1 90 8100

x1 x2 x3 L1 L2 b0 x x2

ANOVA(class variables)

Orthogonalpolynomials

Regression(continuous variables)