So far...

We have been estimating differences caused by application of various treatments, and determining the probability that an observed difference was due to chance

The presence of interactions may indicate that two or more treatment factors have a joint effect on a response variable

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

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

What is Regression?

The way that one variable is related to another.

As you change one, how are others affected?

Yield

Grain Protein %

May want to– Develop and test a model for a biological system– Predict the values of one variable from another

Usual associations within ANOVA... 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– the response could then be specified for not only the

treatment levels actually tested but for all other intermediate points within the range of those treatments

Simplest form of response is a straight line

Fitting the Linear Regression Model

WheatYield(Y)

Applied N Level

X1 X2 X3 X4

Y3

Y1

Y2

Y4Y = 0 + 1X +

where:Y = wheat yieldX = nitrogen level0 = yield with no N 1 = change in yield per unit of applied N = random error

Choose a line that minimizes deviation of observed values from the line (predicted values)

Types of regression models Model I

– Values of the independent variable X are controlled by the experimenter

– Assumed to be measured without error– We measure response of the independent variable Y to

changes in X

Model II– Both the X and the Y variables are measured and subject to

error (e.g., in an observational study)– Either variable could be considered as the independent

variable; choice depends on the context of the experiment– Often interested in correlations between variables– May be descriptive, but might not be reliable for prediction

Sums of Squares due to Regression

2

j j j

2j j

(X X)(Y Y)SSR

(X X)

j j j XY XY2 2

j j X

(X X)(Y Y) SCPb

(X X) SSX

0 1Y X Y a bX

Y Y b X X Y a bX a Y bX

Because the line passes through X,Y

Partitioning SST

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

linear regression of Y on X (with 1 df)– SSLOF = Sum of squares for the failure of the

regression model to describe the relationship between Y and X (lack of fit) (with t-2 df)

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/[Sj (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)

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 S 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 2

LINLIN 2

j 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* LLIN

2 / 10

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

Quadratic– – SSQUAD = 4*LQUAD

2 / 14

LOF still significant? Keep going…

Cubic– – SSCUB = 4*LCUB

2 / 10

CUB 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*LQUAR

2 / 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 / Sj 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

Easier way 1) use contrasts to find the best model and estimate pure error

2) get the equation from a graph or from regression analysis

Useful for prediction

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 a 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)

This column is dropped - it provides no additional information