1. More Regression

2. Last Class Association between characters.
Simple linear regression model.
Estimation of parameters.
Analysis of variance of regression.
Testing regression parameters (t tests).

3. Predicting the dependant variable

4. Predicting the Dependant Variable
Y = x
At x = Rads
y = x 2.5 =
How accurate is this estimation? -
se(yp) = {2 [1+1/n+(xp-x)2/SS(x)]}

5. Linear Regression Example
Predicting at x = 2.5
se(yp) = {0.234 [1+1/5+( )2/12.800]}
se(yp) = 0.531
yp =

6. Linear Regression Example
Predicting at x = 4.5
se(yp) = {0.234 [1+1/5+( )2/12.800]}
se(yp) = 0.663
yp =

7. Mutation Frequency in Drosophila

8. Outliers and Regression

9. Outliers and Regression

10. Outliers in Regression
Y = x ?

11. Outliers and Regression

12. Outliers and Regression

13. Outliers in Regression
Y = x

14. Use your Eyes

15. Value of Scatter Diagrams

16. Value of Scatter Diagrams Set A
Y = x

17. Value of Scatter Diagrams

18. Value of Scatter Diagrams
Source Df
MSq
Set A
Set B
Set C
Set D
Regression 1
27.5***
Residual 9
1.5
Slope
0.5
Intersept
3.0

19. Value of Scatter Diagrams Set A
Y = x

20. Value of Scatter Diagrams Set B
Y = x

21. Value of Scatter Diagrams Set C
Y = x

22. Value of Scatter Diagrams Set D
Y = x

23. Value of Scatter Diagrams

24. Making a curved line straight

25. Curved Relationships

26. Curved Relationships Why make it straight?
Makes analyses and interpretation simpler, or possible

27. Transformation How do researchers know what transformation to use?
From experience.
From knowledge of the biological character being investigated.
Transformation should therefore have some biological meaning in addition to making the line straight.

28. Early Growth Pattern of Plants

29. Early Growth Pattern of Plants
y =Ln(y)

30. Early Growth Pattern of Plants

31. Early Growth Pattern of Plants
y =Ln(y)

32. Something Fishy

33. Something Fishy Experiment

34. Something Fishy Experiment
x = 1/x

35. Growth Curve
Y = ex

36. Growth Curve
Y = Log(x)

37. Sigmoid Growth Curve 

38. Accululative Normal Distribution
Sigmoid Growth Curve
Accululative Normal Distribution 

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

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

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

42. Probit Analysis ~ Example

43. Probit Analysis ~ Example

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

45. Estimating the Standard Deviation
2.8

46. Estimating the Standard Deviation2
2.8

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

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

49. Probit Analysis

50. Probit Analysis

51. Probit Analysis Probit () =  +  . Log10(concentration)
 =
 =
Log10 (conc) to kill 50% (LD-50) is probit 0.5 = 0
0 = x LD-50
LD-50 = 0.423
= 2.65%

52. 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.

53. Where Straight Lines Meet

54. Optimal Assent

55. Optimal Assent
Y1=a1+b1x

56. Optimal Assent
Y2=a2+b2x
Y1=a1+b1x

57. Optimal Assent
t =[b1-b2]/se(b)
= ns
Y2=a2+b2x
Y1=a1+b1x

58. Optimal Assent
Y3=a3+b3x
Y1=a1+b1x

59. Optimal Assent
Y3=a3+b3x
t =[b1-b3]/se(b)
= ***
Y1=a1+b1x

60. Optimal Assent
Y3=a3+b3x
Y1=a1+b1x

61. Optimal Assent
t =[b1-bn]/se(b)
= ***
Yn=an+bnx
Y3=a3+b3x
Y1=a1+b1x

62. Optimal Assent
Y3=a3+b3x
Y3=a3+b3x
Y1=a1+b1x

63. Yield and Nitrogen

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

65. Intersecting Lines

66. Intersecting Lines
Y = 9.01x
Y = 2.81x

67. Intersecting Lines t = [b11 - b21]/average se(b)
6.2/0.593 = * , With 3 df
Intersect = same value of y
b10 + b11x = y = b20 + b21x
x = [b20 - b10]/[b11 - b21]
= lb N/acre
with lb/acre seed yield

68. Intersecting Lines 1321.83 lb/acre 94.92 lb N/acre Y = 9.01x + 466.60

69. Bi-variate Distribution
Linear
Y = b0 + b1x
Quadratic
Y = b0 + b1x + b2 x2
Cubic
Y = b0 + b1x + b2 x2 + b3 x3
Bi-variate Distribution
Correlation