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So far...

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

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So far...

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  1. 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

  2. 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

  3. Yield Grain Protein % What is Regression? • The way that one variable is related to another. • As you change one, how are others affected? • May want to • Develop and test a model for a biological system • Predict the values of one variable from another

  4. 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

  5. Fitting the Linear Regression Model Y = 0 + 1X +  Y4 where: Y = wheat yield X = nitrogen level 0 = yield with no N 1 = change in yield per unit of applied N = random error Y2 Wheat Yield (Y) Y3 Y1 X1 X2 X3 X4 Applied N Level • Choose a line that minimizes deviation of observed values from the line (predicted values)

  6. Sums of Squares due to Regression

  7. 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)

  8. 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 • Therefore • The sum of the coefficients will be zero, satisfying the definition of a contrast One way:

  9. _ • SSLIN = r*LLIN2/[Sj (Xj - X)2] really no different from any other contrast - df is always 1 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)

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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 Sj (xj - x)2], but must be decoded back to original scale _ _ _

  15. Linear contrast • SSLIN = 4* LLIN2/ 10 • Quadratic • SSQUAD = 4*LQUAD2/ 14 Consider an experiment • Five levels of N (10, 30, 50, 70, 90) with four replications

  16. Quartic • SSQUAR = 4*LQUAR2/ 70 • Each contrast has 1 degree of freedom • Each F has MSE in denominator LOF still significant? Keep going… • Cubic • SSCUB = 4*LCUB2/ 10

  17. 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

  18. N level (kg/ha) 0 35 70 105 140 Order Mean 28.4 66.8 87.0 92.0 85.7 Lij kj2SS(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 Orthogonal Partition of SST

  19. 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

  20. Regression Equation bi = LREG / Sj kj2 Coefficient b0 b1 b2 23.99 4.66 -2.49 To scale to original X values

  21. 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

  22.  x1 x2 x3  L1 L2 b0 x x2 ANOVA (class variables) Orthogonal polynomials Regression (continuous variables) 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

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