Growth of bean plants in four different media

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Growth of bean plants in four different media
Completely randomized design (one-way anova)
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data MyExample Infile 'h\Forsøgsplanlægning\2005
\forelæsninger\Lecture 2.prn' firstobs 2 input
Treat Rep Temp Biomass proc
print run proc glm title 'One-way
anova' class Treat Rep model Biomass
Treat/solution run
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One-way anova 1158 Tuesday, February 15,
2005 19
The GLM Procedure Dependent Variable Biomass
Sum of
Source DF
Squares Mean Square F Value Pr gt F
Model 3 153.2820000
51.0940000 25.31 lt.0001 Error
16 32.2960000
2.0185000 Corrected Total 19
185.5780000
R-Square Coeff Var Root MSE
Biomass Mean 0.825971
2.306769 1.420739 61.59000
Source DF Type I SS
Mean Square F Value Pr gt F Treat
3 153.2820000
51.0940000 25.31 lt.0001
Source DF Type III SS
Mean Square F Value Pr gt F Treat
3 153.2820000
51.0940000 25.31 lt.0001

Standard Parameter
Estimate Error t Value
Pr gt t Intercept
59.68000000 B 0.63537391 93.93
lt.0001 Treat Control
4.62000000 B 0.89855439 5.14
lt.0001 Treat Cu
-1.60000000 B 0.89855439 -1.78
0.0940 Treat Mn
4.62000000 B 0.89855439 5.14
lt.0001 Treat Zn
0.00000000 B . . .
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Block design with replicate as blocks (two-way
anova)
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proc glm title 'Two-way anova' class Treat
Rep model Biomass Treat Rep / solution run
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Two-way anova 1158 Tuesday, February 15,
2005 21
The GLM Procedure Dependent Variable Biomass
Sum of
Source DF
Squares Mean Square F Value Pr gt F
Model 7 161.2250000
23.0321429 11.35 0.0002 Error
12 24.3530000
2.0294167 Corrected Total 19
185.5780000 R-Square
Coeff Var Root MSE Biomass Mean
0.868772 2.312999 1.424576
61.59000 Source
DF Type I SS Mean Square F Value
Pr gt F Treat 3
153.2820000 51.0940000 25.18 lt.0001
Rep 4
7.9430000 1.9857500 0.98 0.4551
Source DF Type III
SS Mean Square F Value Pr gt F
Treat 3 153.2820000
51.0940000 25.18 lt.0001 Rep
4 7.9430000
1.9857500 0.98 0.4551

Standard Parameter
Estimate Error t Value
Pr gt t Intercept
59.51500000 B 0.90098095 66.06
lt.0001 Treat Control
4.62000000 B 0.90098095 5.13
0.0003 Treat Cu
-1.60000000 B 0.90098095 -1.78
0.1011 Treat Mn
4.62000000 B 0.90098095 5.13
0.0003 Treat Zn
0.00000000 B . . .
Rep 1 -0.60000000 B
1.00732732 -0.60 0.5625 Rep
2 1.12500000 B 1.00732732
1.12 0.2859 Rep 3
0.62500000 B 1.00732732 0.62
0.5466 Rep 4
-0.32500000 B 1.00732732 -0.32
0.7525 Rep 5
0.00000000 B . . .
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Analysis of covariance (ANCOVA) with Temp. as
covariate
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Treat Rep Temp Biomass Zn 1
24.8 61.7 Zn 2 25.2 59.4 Zn
3 25.4 60.5 Zn 4 24.9
59.2 Zn 5 24.6 57.6 Cu 1
24.8 57 Cu 2 25.2 58.4
Cu 3 25.4 57.3 Cu 4
24.9 57.8 Cu 5 24.6 59.9 Mn
1 24.8 62.3 Mn 2 25.2
66.2 Mn 3 25.4 65.2 Mn 4
24.9 63.7 Mn 5 24.6
64.1 Control 1 24.8 62.3 Control 2
25.2 66.2 Control 3 25.4
65.2 Control 4 24.9 63.7 Control 5
24.6 64.1
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proc glm title 'Ancova' class Treat Rep model
Biomass Treat Temp / solution run
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Ancova 1158 Tuesday, February 15, 2005
23 The GLM
Procedure Dependent Variable Biomass
Sum of
Source DF Squares
Mean Square F Value Pr gt F Model
4 157.3480392
39.3370098 20.90 lt.0001 Error
15 28.2299608
1.8819974 Corrected Total 19
185.5780000
R-Square Coeff Var Root MSE
Biomass Mean 0.847881
2.227406 1.371859 61.59000
Source DF Type I SS
Mean Square F Value Pr gt F Treat
3 153.2820000
51.0940000 27.15 lt.0001 Temp
1 4.0660392
4.0660392 2.16 0.1623 Source
DF Type III SS Mean
Square F Value Pr gt F Treat
3 153.2820000 51.0940000
27.15 lt.0001 Temp
1 4.0660392 4.0660392 2.16
0.1623
Standard
Parameter Estimate
Error t Value Pr gt t
Intercept 20.25078431 B
26.83214670 0.75 0.4621
Treat Control 4.62000000 B
0.86763988 5.32 lt.0001 Treat
Cu -1.60000000 B 0.86763988
-1.84 0.0850 Treat Mn
4.62000000 B 0.86763988 5.32
lt.0001 Treat Zn
0.00000000 B . . .
Temp 1.57843137
1.07386436 1.47 0.1623
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Matrix Notation

• Of particular interest to us is the fact that not
  even in regression analysis was much use made of
  matrix algebra. In fact one of us, as a
  statistics graduate student at Cambridge
  University in the early 1950s, had lectures on
  multiple regression that were couched in scalar
  notation!
• This absence of matrices and vectors is surely
  surprising when one thinks of A.C. Aitken. His
  two books, Matrices and Determinants and
  Statistical Mathematics were both first published
  in 1939, had fourth and fifth editions,
  respectively, in 1947 and 1948, and are still in
  print. Yet, very surprisingly, the latter makes
  no use of matrices and vectors which are so
  thoroughly dealt with in the former.
• There were exceptions, of course, as have already
  been noted, such as Kempthorne (1952) and his
  co-workers, e.g. Wilk and Kempthorne (1955, 1956)
  and others, too. Even with matrix expressions
  available, arithmetic was a real problem. A
  regression analysis in the New Zealand Department
  of Agriculture in the mid-1950s involved 40
  regressors. Using electromechanical calculators,
  two calculators (people) using row echelon
  methods needed six weeks to invert the 40 x 40
  matrix. One person could do a row, then the other
  checked it (to a maximum capacity of 8 to 10
  digits, hoping for 4- or 5-digit accuracy in the
  final result). That person did the next row and
  passed it to the first person for checking and
  so on. This was the impasse matrix algebra was
  appropriate and not really difficult. But the
  arithmetic stemming therefrom could be a
  nightmare.
• (From Linear Models 1945-1995 by Shayle R. Searle
  and Charles E. McCulloch in Advances in Biometry
  (eds. Peter Armitage and Herbert A. David), John
  Wiley Sons, 1996)