Making a curved line straight Data Transformation & Regression

Last Class  Predicting the dependant variable and standard errors of predicted values.  Outliers.  Need to visually inspect data in graphic form.  Making a curved line straight.  Transformation.

Early Growth Pattern of Plants

y = Ln(y)

Early Growth Pattern of Plants y =  y

Homogeneity of Error Variance

y =Ln(y)

Growth Curve Y = e x

Growth Curve Y = Log(x)

Sigmoid Growth Curve 

 Accululative Normal Distribution

Sigmoid Growth Curve  Accululative Normal Distribution  T T-- T T-- ƒ (  d  d  T

Sigmoid Growth Curve  Accululative Normal Distribution  T T-- T T-- ƒ (  d  d  T

Probit Analysis Group of plants/insects exposed to different concentrations of a specific stimulant (i.e. insecticide). Group of plants/insects exposed to different concentrations of a specific stimulant (i.e. insecticide). Data are counts (or proportions), say number killed. Data are counts (or proportions), say number killed. Usually concerned or interested in concentration which causes specific event (i.e. LD 50%). Usually concerned or interested in concentration which causes specific event (i.e. LD 50%).

Probit Analysis ~ Example

Estimating the Mean  y = 50% Killed  x ~ 2.8

Estimating the Standard Deviation 2.8

2.8 2222

2.8 2222 95% values Estimating the Standard Deviation

2.8 2222 95% values  = 1.2 Estimating the Standard Deviation

Probit Analysis

Probit (  ) =  + . Log 10 (concentration)  =  = Log 10 (conc) to kill 50% (LD-50) is probit 0.5 = 0 0 = x LD-50 LD-50 = = 2.65%

Problems  Obtaining “good estimates” of the mean and standard deviation of the data.  Make a calculated guess, use iteration to get “better fit” to observed data.

Where Straight Lines Meet

Optimal Assent

Y 1 =a 1 +b 1 x

Optimal Assent Y 1 =a 1 +b 1 x Y 2 =a 2 +b 2 x

Optimal Assent Y 1 =a 1 +b 1 x Y 2 =a 2 +b 2 x t =[b 1 -b 2 ]/se(b) = ns = ns

Optimal Assent Y 1 =a 1 +b 1 x Y 3 =a 3 +b 3 x

Optimal Assent Y 1 =a 1 +b 1 x Y 3 =a 3 +b 3 x t =[b 1 -b 3 ]/se(b) = *** = ***

Optimal Assent Y 1 =a 1 +b 1 x Y 3 =a 3 +b 3 x

Optimal Assent Y 1 =a 1 +b 1 x Y 3 =a 3 +b 3 x t =[b 1 -b n ]/se(b) = *** = *** Y n =a n +b n x

Optimal Assent Y 1 =a 1 +b 1 x Y 3 =a 3 +b 3 x

Yield and Nitrogen

What application of nitrogen will result in the optimum yield response?

Intersecting Lines

Y = 2.81x Y = 9.01x

Intersecting Lines t = [b 11 - b 21 ]/average se(b) 6.2/0.593 = *, With 3 df Intersect = same value of y b 10 + b 11 x = y = b 20 + b 21 x x = [b 20 - b 10 ]/[b 11 - b 21 ] = lb N/acre with lb/acre seed yield

Intersecting Lines Y = 2.81x Y = 9.01x lb N/acre lb/acre

Linear Y = b 0 + b 1 x Quadratic Y = b 0 + b 1 x + b 2 x 2 Cubic Y = b 0 + b 1 x + b 2 x 2 + b 3 x 3 Bi-variate Distribution Correlation