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An introduction to gexp package
Ivan Bezerra Allaman José Cláudio Faria
2023-06-20
The gexp package was elaborated inspired by the personal
functions of Professor José Cláudio Faria, whose objective is to
generate one or p random variables according to the
type of experiment, whose effects of the treatments are indicated by the
user.
Most packages like agricolae, for example, have functions that allow
the user to generate only designs, thus helping the researcher who wants
to set up an experiment. However, if the user has the interest in
generating random variables according to the type of experiment and
still, being able to manipulate which treatments will be different, then
the gexp package has this differential.
The experiments included in the package are: a completely randomized
design (CRD), randomized complete block design (RCBD), latin square
design (LSD), factorial schemes considering CRD, DBC and DQL designs and
a schema in split-plots also considering the CRD, RCBD and LSD designs
(Aquino 1992) .
The arguments of the function are based on a linear model with the
following matrix form (Ferreira (2008) ,
Rencher and Schaalje (2007) ):
\[Y = X\beta + E \hspace{5cm}
(1)\]
then:
Y is a n\(\times\) p matrix, with
n observation of p random
variables;
X is a n\(\times\) m matrix, where
m is the number of parameters that influence the
response variable (\(\mu, \alpha's,
\beta's, etc.\) );
\(\beta\) is a m\(\times\) p matrix of factors effects;
\(E\) is a n\(\times\) p matrix of the errors with
p-variate normal distribution with mean vectors 0 and
\(\Sigma\)
In the case of the split-plot design, the package works considering
the following linear mixed model (Naes, Aastveit,
and Sahni 2007) :
\[Y = X\beta + Zu + E \hspace{5cm}
(2)\]
then:
Z is a n\(\times\) q matrix, where
q is the number of parameters of the interaction that
influence the plot error (CRD - repetition:\(\alpha\) , RCBD - block:\(\alpha\) , LSD - row:column:\(\alpha\) );
u is a q\(\times\) p matrix of the
plot errors with p-variate normal distribution with mean vectors
0 and \(\Sigma_{plot}\)
\(E\) is a n\(\times\) p matrix of the errors with
p-variate normal distribution with mean vectors 0 and
\(\Sigma_{sub-plot}\)
The other components of the equation 2 are analogous to equation
1.
The symbology adopted was as follows:
Sources of Variation
Symbol
Treatments
\(\alpha,\tau\)
Block
\(\beta\)
Interaction
\(\gamma\)
Row block
\(\eta\)
Column block
\(\delta\)
The multivariate case , will only be addressed in the
CRD, because for the other designs the reasoning is the same.
Simulating an Completely Randomized Design (CRD)
Univariate case
Let us assume that we want to generate a single
random variable according to a CRD with a factor and two levels being 3
replicates. Then we can denote \(Y_{ik}\) as being the random variable
observed in the k-th experimental unit (k = 1, 2, 3) that received the
i-th level of factor X1 (i = 1,2). Also, assuming that this variable is
under the effect of a constant (\(\mu =
15\) ) and each level of the factor under study influence \(\alpha_1 = 1\) and \(\alpha_2 = -2\) the response variable
respectively, we will have the following matrix configuration:
\[
\begin{eqnarray}
\begin{bmatrix}
Y_{11} \\
Y_{21} \\
Y_{12} \\
Y_{22} \\
Y_{13} \\
Y_{23}
\end{bmatrix}
&=&
\begin{array}{p{5cm}}
\begin{matrix}
\mu & \!\scriptsize{x11}\! & \scriptsize{x12}
\end{matrix}\\
\left[\begin{array}{c|cc}
1\hphantom{0} & 1\hphantom{0} & 0 \\
1\hphantom{0} & 0\hphantom{0} & 1 \\
1\hphantom{0} & 1\hphantom{0} & 0 \\
1\hphantom{0} & 0\hphantom{0} & 1 \\
1\hphantom{0} & 1\hphantom{0} & 0 \\
1\hphantom{0} & 0\hphantom{0} & 1
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
\mu \\
\alpha_{1} \\
\alpha_{2} \\
\end{bmatrix}
+
\begin{bmatrix}
e_{11} \\
e_{21} \\
e_{12} \\
e_{22} \\
e_{13} \\
e_{23}
\end{bmatrix}
\\
&=&
\begin{array}{c}
\begin{matrix}
\mu & \!\scriptsize{x11}\! & \scriptsize{x12}
\end{matrix}\\
\left[\begin{array}{c|cc}
1\hphantom{0} & 1\hphantom{0} & 0 \\
1\hphantom{0} & 0\hphantom{0} & 1 \\
1\hphantom{0} & 1\hphantom{0} & 0 \\
1\hphantom{0} & 0\hphantom{0} & 1 \\
1\hphantom{0} & 1\hphantom{0} & 0 \\
1\hphantom{0} & 0\hphantom{0} & 1
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
15 \\
1 \\
-2 \\
\end{bmatrix}
+
\begin{bmatrix}
e_{11} \\
e_{21} \\
e_{12} \\
e_{22} \\
e_{13} \\
e_{23}
\end{bmatrix}
\\
&=&
\begin{bmatrix}
16 \\
13 \\
16 \\
13 \\
16 \\
13
\end{bmatrix}
+
\begin{bmatrix}
e_{11} \\
e_{21} \\
e_{12} \\
e_{22} \\
e_{13} \\
e_{23}
\end{bmatrix}
\end{eqnarray}
\]
If we consider an error with mean 0 and variance 0 then we have the
following simulated random variable:
\[
\begin{bmatrix}
Y_{11} \\
Y_{21} \\
Y_{12} \\
Y_{22} \\
Y_{13} \\
Y_{23}
\end{bmatrix}
=
\begin{bmatrix}
16 \\
13 \\
16 \\
13 \\
16 \\
13
\end{bmatrix}
\]
With the gexp package we have:
library (gexp)<- gexp (mu = 15 ,err = matrix (rep (0 , 6 ),nrow = 6 ),r = 3 ,fe = list (alpha = c (1 , - 2 )))summary (crd)#> Database #> r X1 Y1 #> 1 1 x11 16 #> 2 2 x11 16 #> 3 3 x11 16 #> 4 1 x12 13 #> 5 2 x12 13 #> 6 3 x12 13
Multivariate Case
Let us now assume that we want to generate two
random variables by considering the same number of factors and the same
number of repetitions of the previous case. Then we can denote \(Y_{ikl}\) as being the lth random variable
(l = 1,2) observed in the k-th experimental unit (k = 1,2,3) that
received the i-th factor level X1 (i = 1,2). Also, assuming that this
variable is under the effect of a constant (\(\mu_1 = 15\) and \(\mu_2 = 6\) ) and each level of the factor
under study influence \(\alpha_{11} =
1\) and \(\alpha_{21} = -2\) the
variable 1 and, \(\alpha_{12} = 2\) and
\(\alpha_{22} = 3\) the variable 2, we
will have the following matrix configuration:
\[
\begin{eqnarray}
\begin{bmatrix}
Y_{111} & Y_{112} \\
Y_{211} & Y_{212} \\
Y_{121} & Y_{122} \\
Y_{221} & Y_{222} \\
Y_{131} & Y_{132} \\
Y_{231} & Y_{232}
\end{bmatrix}
&=&
\begin{array}{c}
\begin{matrix}
\mu & \!\scriptsize{x11}\! & \scriptsize{x12}
\end{matrix}\\
\left[\begin{array}{c|cc}
1\hphantom{0} & 1\hphantom{0} & 0 \\
1\hphantom{0} & 0\hphantom{0} & 1 \\
1\hphantom{0} & 1\hphantom{0} & 0 \\
1\hphantom{0} & 0\hphantom{0} & 1 \\
1\hphantom{0} & 1\hphantom{0} & 0 \\
1\hphantom{0} & 0\hphantom{0} & 1
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
\mu_1 & \mu_2 \\
\alpha_{11} & \alpha_{12} \\
\alpha_{21} & \alpha_{22} \\
\end{bmatrix}
+
\begin{bmatrix}
e_{111} & e_{112} \\
e_{211} & e_{212} \\
e_{121} & e_{122} \\
e_{221} & e_{222} \\
e_{131} & e_{132} \\
e_{231} & e_{232}
\end{bmatrix}
\\
&=&
\begin{array}{c}
\begin{matrix}
\mu & \!\scriptsize{x11}\! & \scriptsize{x12}
\end{matrix}\\
\left[\begin{array}{c|cc}
1\hphantom{0} & 1\hphantom{0} & 0 \\
1\hphantom{0} & 0\hphantom{0} & 1 \\
1\hphantom{0} & 1\hphantom{0} & 0 \\
1\hphantom{0} & 0\hphantom{0} & 1 \\
1\hphantom{0} & 1\hphantom{0} & 0 \\
1\hphantom{0} & 0\hphantom{0} & 1
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
15 & 6 \\
1 & 2 \\
-2 & 3 \\
\end{bmatrix}
+
\begin{bmatrix}
e_{111} & e_{112} \\
e_{211} & e_{212} \\
e_{121} & e_{122} \\
e_{221} & e_{222} \\
e_{131} & e_{132} \\
e_{231} & e_{232}
\end{bmatrix}
\\
&=&
\begin{bmatrix}
16 & 8 \\
13 & 9 \\
16 & 8 \\
13 & 9 \\
16 & 8 \\
13 & 9
\end{bmatrix}
+
\begin{bmatrix}
e_{111} & e_{112} \\
e_{211} & e_{212} \\
e_{121} & e_{122} \\
e_{221} & e_{222} \\
e_{131} & e_{132} \\
e_{231} & e_{232}
\end{bmatrix}
\end{eqnarray}
\]
If we consider an error with averages 0 and a matrix of
variance-covariance 0 then we will have the following simulated random
variables:
\[
\begin{bmatrix}
Y_{111} & Y_{112} \\
Y_{211} & Y_{212} \\
Y_{121} & Y_{122} \\
Y_{221} & Y_{222} \\
Y_{131} & Y_{132} \\
Y_{231} & Y_{232}
\end{bmatrix}
=
\begin{bmatrix}
16 & 8 \\
13 & 9 \\
16 & 8 \\
13 & 9 \\
16 & 8 \\
13 & 9
\end{bmatrix}
\]
With the gexp package we have:
<- gexp (mu = c (15 , 6 ),err = matrix (c (0 , 0 ,0 , 0 ,0 , 0 ,0 , 0 ,0 , 0 ,0 , 0 ), ncol = 2 ),r = 3L,fe = list (alpha = matrix (c (1 , - 2 ,2 , 3 ), ncol = 2 )))summary (crd_mv)#> Database #> r X1 Y1 Y2 #> 1 1 x11 16 8 #> 2 2 x11 16 8 #> 3 3 x11 16 8 #> 4 1 x12 13 9 #> 5 2 x12 13 9 #> 6 3 x12 13 9
Simulating a Randomized Completely Block Design (RCBD)
Assuming we want to generate a single random
variable according to a RCBD with a factor and two levels being a block
with three levels. Then we can denote \(Y_
{ij}\) as the random variable observed in the j-th block (j = 1,
2, 3) that received the i-th level of factor X1 (i = 1, 2). Also,
assuming that this variable is under the effect of a constant (\(\mu = 15\) ) and each level of the factor
under study influence \(\alpha_1 = 1\)
and \(\alpha_2 = -2\) the variable
response respectively, and each block influences in \(\beta_1 = 2, \beta_2 = 4\) and \(\beta_3 = 6\) respectively, we will have
the following matrix configuration:
\[
\begin{eqnarray}
\begin{bmatrix}
Y_{11} \\
Y_{21} \\
Y_{12} \\
Y_{22} \\
Y_{13} \\
Y_{23}
\end{bmatrix}
&=&
\begin{array}{c}
\begin{matrix}
\mu &
\!\scriptsize{x11}\!&\scriptsize{x12}\hphantom{0}&\scriptsize{b1}&\scriptsize{b2}&\scriptsize{b3}
\end{matrix}\\
\left[\begin{array}{c|cc|ccc}
1 & 1\hphantom{0}& 0\hphantom{0}& 1\hphantom{0}&
0\hphantom{0}& 0\\
1 & 0\hphantom{0}& 1\hphantom{0}& 1\hphantom{0}&
0\hphantom{0}& 0\\
1 & 1\hphantom{0}& 0\hphantom{0}& 0\hphantom{0}&
1\hphantom{0}& 0\\
1 & 0\hphantom{0}& 1\hphantom{0}& 0\hphantom{0}&
1\hphantom{0}& 0\\
1 & 1\hphantom{0}& 0\hphantom{0}& 0\hphantom{0}&
0\hphantom{0}& 1\\
1 & 0\hphantom{0}& 1\hphantom{0}& 0\hphantom{0}&
0\hphantom{0}& 1\\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
\mu \\
\alpha_1 \\
\alpha_2 \\
\beta_1 \\
\beta_2 \\
\beta_3 \\
\end{bmatrix}
+
\begin{bmatrix}
e_{11} \\
e_{21} \\
e_{12} \\
e_{22} \\
e_{13} \\
e_{23}
\end{bmatrix}
\\
&=&
\begin{array}{c}
\begin{matrix}
\mu &
\!\scriptsize{x11}\!&\scriptsize{x12}\hphantom{0}&\scriptsize{b1}&\scriptsize{b2}&\scriptsize{b3}
\end{matrix}\\
\left[\begin{array}{c|cc|ccc}
1 & 1\hphantom{0}& 0\hphantom{0}& 1\hphantom{0}&
0\hphantom{0}& 0\\
1 & 0\hphantom{0}& 1\hphantom{0}& 1\hphantom{0}&
0\hphantom{0}& 0\\
1 & 1\hphantom{0}& 0\hphantom{0}& 0\hphantom{0}&
1\hphantom{0}& 0\\
1 & 0\hphantom{0}& 1\hphantom{0}& 0\hphantom{0}&
1\hphantom{0}& 0\\
1 & 1\hphantom{0}& 0\hphantom{0}& 0\hphantom{0}&
0\hphantom{0}& 1\\
1 & 0\hphantom{0}& 1\hphantom{0}& 0\hphantom{0}&
0\hphantom{0}& 1\\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
15 \\
1 \\
-2 \\
2 \\
4 \\
6
\end{bmatrix}
+
\begin{bmatrix}
e_{11} \\
e_{21} \\
e_{12} \\
e_{22} \\
e_{13} \\
e_{23}
\end{bmatrix}
\\
&=&
\begin{bmatrix}
18 \\
15 \\
20 \\
17 \\
22 \\
19
\end{bmatrix}
+
\begin{bmatrix}
e_{11} \\
e_{21} \\
e_{12} \\
e_{22} \\
e_{13} \\
e_{23}
\end{bmatrix}
\end{eqnarray}
\]
Assuming an error with mean 0 and variance 0, we have:
\[
\begin{bmatrix}
Y_{11} \\
Y_{21} \\
Y_{12} \\
Y_{22} \\
Y_{13} \\
Y_{23}
\end{bmatrix}
=
\begin{bmatrix}
18 \\
15 \\
20 \\
17 \\
22 \\
19
\end{bmatrix}
\]
With the gexp package we have:
<- gexp (mu = 15 ,r = 1 ,err = matrix (rep (0 , 6 ),nrow = 6 ),fe = list (alpha = c (1 , - 2 )),blke = c (2 , 4 , 6 ),design = 'RCBD' )summary (rcbd)#> Database #> r Block X1 Y1 #> 1 1 1 x11 18 #> 2 1 2 x11 20 #> 3 1 3 x11 22 #> 4 1 1 x12 15 #> 5 1 2 x12 17 #> 6 1 3 x12 19
Simulating a Latin Square Design (LSD)
Assuming we want to generate a single random
variable according to a LSD with a factor and three levels, with a
blocking factor in the row sense with three levels and another blocking
factor in the column sense with three levels. Then we can denote \(Y_{rci}\) as being the random variable
observed in the c-th column (c = 1, 2, 3) and r-th row (r = 1, 2, 3)
that received the i-th level of factor X1 (i = 1, 2, 3). Also, assuming
this variable is under the effect of a constant (\(\mu = 15\) ) and each level of the factor
under study influence \(\alpha_1 = 1, \alpha_2
= -2\) e \(\alpha_3 = 3\) the
response variable respectively, each row influencing in \(\eta_1 = 2,\eta_2 = 3\) and \(\eta_3 = 4\) respectively and each column
influencing in \(\delta_1 = 6, \delta_2 =
7\) and \(\delta_3 = 8\)
respectively we will have the following matrix configuration:
\[
\begin{eqnarray}
\begin{bmatrix}
Y_{111} \\
Y_{122} \\
Y_{133} \\
Y_{212} \\
Y_{223} \\
Y_{231} \\
Y_{313} \\
Y_{321} \\
Y_{332}
\end{bmatrix}
&=&
\begin{array}{c}
\begin{matrix}
\mu &
\!\scriptsize{r1}\!&\scriptsize{r2}&\scriptsize{r3}&\scriptsize{c1}&\scriptsize{c2}&\scriptsize{c3}&\scriptsize{x11}&\scriptsize{x12}&\scriptsize{x13}
\end{matrix}\\
\left[\begin{array}{c|ccc|ccc|ccc}
1&1&0&0\hphantom{0}&1&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\\
1&1&0&0\hphantom{0}&0&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\\
1&1&0&0\hphantom{0}&0&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\\
1&0&1&0\hphantom{0}&1&0\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\\
1&0&1&0\hphantom{0}&0&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\\
1&0&1&0\hphantom{0}&0&0\hphantom{0}&1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\\
1&0&0&1\hphantom{0}&1&0\hphantom{0}&0\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\\
1&0&0&1\hphantom{0}&0&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\\
1&0&0&1\hphantom{0}&0&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
\mu \\
\eta_1 \\
\eta_2 \\
\eta_3 \\
\delta_1 \\
\delta_2 \\
\delta_3 \\
\alpha_1 \\
\alpha_2 \\
\alpha_3 \\
\end{bmatrix}
+
\begin{bmatrix}
e_{111} \\
e_{122} \\
e_{133} \\
e_{212} \\
e_{223} \\
e_{231} \\
e_{313} \\
e_{321} \\
e_{332}
\end{bmatrix}
\\
&=&
\begin{array}{c}
\begin{matrix}
\mu &
\!\scriptsize{r1}\!&\scriptsize{r2}&\scriptsize{r3}&\scriptsize{c1}&\scriptsize{c2}&\scriptsize{c3}&\scriptsize{x11}&\scriptsize{x12}&\scriptsize{x13}
\end{matrix}\\
\left[\begin{array}{c|ccc|ccc|ccc}
1&1&0&0\hphantom{0}&1&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\\
1&1&0&0\hphantom{0}&0&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\\
1&1&0&0\hphantom{0}&0&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\\
1&0&1&0\hphantom{0}&1&0\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\\
1&0&1&0\hphantom{0}&0&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\\
1&0&1&0\hphantom{0}&0&0\hphantom{0}&1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\\
1&0&0&1\hphantom{0}&1&0\hphantom{0}&0\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\\
1&0&0&1\hphantom{0}&0&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\\
1&0&0&1\hphantom{0}&0&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
15 \\
2 \\
3 \\
4 \\
6 \\
7 \\
8 \\
1 \\
-2 \\
3 \\
\end{bmatrix}
+
\begin{bmatrix}
e_{111} \\
e_{122} \\
e_{133} \\
e_{212} \\
e_{223} \\
e_{231} \\
e_{313} \\
e_{321} \\
e_{332}
\end{bmatrix}
\\
&=&
\begin{bmatrix}
24 \\
22 \\
28 \\
22 \\
28 \\
27 \\
28 \\
27 \\
25 \\
\end{bmatrix}
+
\begin{bmatrix}
e_{111} \\
e_{122} \\
e_{133} \\
e_{212} \\
e_{223} \\
e_{231} \\
e_{313} \\
e_{321} \\
e_{332}
\end{bmatrix}
\end{eqnarray}
\]
Considering that the error has mean 0 and variance 0 we have:
\[
\begin{bmatrix}
Y_{111} \\
Y_{122} \\
Y_{133} \\
Y_{212} \\
Y_{223} \\
Y_{231} \\
Y_{313} \\
Y_{321} \\
Y_{332}
\end{bmatrix}
=
\begin{bmatrix}
24 \\
22 \\
28 \\
22 \\
28 \\
27 \\
28 \\
27 \\
25 \\
\end{bmatrix}
\]
Using the gexp function we have:
<- gexp (mu = 15 ,r = 1 ,err = matrix (rep (0 , 9 ), nrow = 9 ),fe = list (alpha = c (1 , - 2 , 3 )),rowe = c (2 , 3 , 4 ),cole = c (6 , 7 , 8 ),design = 'LSD' )summary (lsd)#> Database #> Row Column X1 Y1 #> 1 1 1 x11 24 #> 2 2 1 x12 22 #> 3 3 1 x13 28 #> 4 1 2 x12 22 #> 5 2 2 x13 28 #> 6 3 2 x11 27 #> 7 1 3 x13 28 #> 8 2 3 x11 27 #> 9 3 3 x12 25
Factorial Experiments
CRD
Considering that we want to generate a single random
variable according to a CRD in a factorial scheme of type 2 x 3 (two
levels of factor X1 and three levels of factor X2) with 2 replicates.
Then we can denote \(Y_{ijk}\) as being
the random variable observed in the k-th experimental unit (k = 1, 2)
that received the i-th level of factor X1 (i = 1,2) and jth level of
factor X2 (j = 1,2,3). Also, assuming this variable is under the effect
of a constant (\(\mu = 15\) ), and each
level of factor X1 influences \(\alpha_1 =
1\) and \(\alpha_2 = -2\) , each
level of factor X2 influences in \(\tau_1 = 1,
\tau_2 = -1\) and \(\tau_3 = 1\)
and the effect of the interaction is \(\gamma_{11} = 3,\gamma_{21} = 1,\gamma_{12} =
1,\gamma_{22} = -5,\gamma_{13} = 1\) And \(\gamma_{23} = 1\) respectively, we shall
have following matrix configuration:
\[
\begin{eqnarray}
\begin{bmatrix}
Y_{111} \\
Y_{211} \\
Y_{121} \\
Y_{221} \\
Y_{131} \\
Y_{231} \\
Y_{112} \\
Y_{212} \\
Y_{122} \\
Y_{222} \\
Y_{132} \\
Y_{232}
\end{bmatrix}
&=&
\begin{array}{c}
\begin{matrix}
\hspace{0.2cm}\mu&\scriptsize{x11}&\scriptsize{x12}\hspace{0.1cm}&\scriptsize{x21}&\scriptsize{x22}&\scriptsize{x23}&\scriptsize{x11x21}&\scriptsize{x12x21}&\scriptsize{x11x22}&\scriptsize{x12x22}&\scriptsize{x11x23}&\scriptsize{x12x23}\hspace{0.6cm}
\end{matrix}\\
\left[\begin{array}{c|cc|ccc|cccccc}
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&0\hphantom{000}&1\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&1\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&1\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
1\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&1\hphantom{000}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&0\hphantom{000}&1\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&1\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&1\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
1\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&1\hphantom{000}\\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
\mu \\
\alpha_1\\
\alpha_2\\
\tau_1\\
\tau_2\\
\tau_3\\
\gamma_{11}\\
\gamma_{21}\\
\gamma_{12}\\
\gamma_{22}\\
\gamma_{13}\\
\gamma_{23}
\end{bmatrix}
+
\begin{bmatrix}
e_{111} \\
e_{211} \\
e_{121} \\
e_{221} \\
e_{131} \\
e_{231} \\
e_{112} \\
e_{212} \\
e_{122} \\
e_{222} \\
e_{132} \\
e_{232}
\end{bmatrix}
\\
&=&
\begin{array}{c}
\begin{matrix}
\hspace{0.2cm}\mu&\scriptsize{x11}&\scriptsize{x12}\hspace{0.1cm}&\scriptsize{x21}&\scriptsize{x22}&\scriptsize{x23}&\scriptsize{x11x21}&\scriptsize{x12x21}&\scriptsize{x11x22}&\scriptsize{x12x22}&\scriptsize{x11x23}&\scriptsize{x12x23}\hspace{0.6cm}
\end{matrix}\\
\left[\begin{array}{c|cc|ccc|cccccc}
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&0\hphantom{000}&1\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&1\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&1\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
1\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&1\hphantom{000}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&0\hphantom{000}&1\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&1\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&1\hphantom{0000}&
0\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
1\hphantom{0000}&0\hphantom{000}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{000}&0\hphantom{0000}&0\hphantom{0000}&0\hphantom{0000}&
0\hphantom{0000}&1\hphantom{000}\\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
15 \\
1\\
-2\\
1\\
-1\\
1\\
3\\
1\\
1\\
-5\\
1\\
1
\end{bmatrix}
+
\begin{bmatrix}
e_{111} \\
e_{211} \\
e_{121} \\
e_{221} \\
e_{131} \\
e_{231} \\
e_{112} \\
e_{212} \\
e_{122} \\
e_{222} \\
e_{132} \\
e_{232}
\end{bmatrix}
\\
&=&
\begin{bmatrix}
20 \\
15\\
16\\
7\\
18\\
15\\
20\\
15\\
16\\
7\\
18\\
15
\end{bmatrix}
+
\begin{bmatrix}
e_{111} \\
e_{211} \\
e_{121} \\
e_{221} \\
e_{131} \\
e_{231} \\
e_{112} \\
e_{212} \\
e_{122} \\
e_{222} \\
e_{132} \\
e_{232}
\end{bmatrix}
\end{eqnarray}
\]
Considering that the error has mean 0 and variance 0 we have:
\[
\begin{bmatrix}
Y_{111} \\
Y_{211} \\
Y_{121} \\
Y_{221} \\
Y_{131} \\
Y_{231} \\
Y_{112} \\
Y_{212} \\
Y_{122} \\
Y_{222} \\
Y_{132} \\
Y_{232}
\end{bmatrix}
=
\begin{bmatrix}
20 \\
15\\
16\\
7\\
18\\
15\\
20\\
15\\
16\\
7\\
18\\
15
\end{bmatrix}
\]
Using the gexp function we have:
<- gexp (mu = 15 ,r = 2 , err = matrix (rep (0 , 12 ),nrow = 12 ),fe = list (alpha = c (1 , - 2 ),tau = c (1 , - 1 , 1 )),inte = c (3 , 1 , 1 , - 5 , 1 , 1 ),type = 'FE' )summary (FE_crd)#> Database #> r X1 X2 Y1 #> 1 1 x11 x21 20 #> 2 2 x11 x21 20 #> 3 1 x12 x21 15 #> 4 2 x12 x21 15 #> 5 1 x11 x22 16 #> 6 2 x11 x22 16 #> 7 1 x12 x22 7 #> 8 2 x12 x22 7 #> 9 1 x11 x23 18 #> 10 2 x11 x23 18 #> 11 1 x12 x23 15 #> 12 2 x12 x23 15
Split-plot experiments
CRD
Supondo que queiramos gerar uma variável aleatória segundo um DIC em
parcelas subdivididas com dois níveis do fator X1 na parcela e dois
níveis do fator X2 na subparcela com 3 repetições. Então, podemos
denominar \(Y_{ijk}\) como sendo a
variável aleatória observada na subparcela na k-ésima unidade
experimental (k = 1, 2 e 3) que recebeu o i-ésimo nível do fator X1
(i=1,2) e j-ésimo nível do fator X2 (j=1,2). Ainda, supondo que esta
variável esteja sob o efeito de uma constante (\(\mu=15\) ), e cada nível do fator X1
influenciem em \(\alpha_{1}=1\) e \(\alpha_{2}=-2\) , cada nível do fator X2
influenciam em \(\tau_{1}=1\) e \(\tau_{2}=-1\) e que o efeito da interação
seja de \(\gamma_{11}=3, \gamma_{21}=1,
\gamma_{12}=1\) e \(\gamma_{22}=-5\) respectivamente, teremos a
seguinte configuração matricial conforme a equação 2.
Portanto, tem-se a seguinte configuração matricial:
\[
\begin{eqnarray}
\begin{bmatrix}
Y_{111} \\
Y_{211} \\
Y_{121} \\
Y_{221} \\
Y_{112} \\
Y_{212} \\
Y_{122} \\
Y_{222} \\
Y_{113} \\
Y_{213} \\
Y_{123} \\
Y_{223}
\end{bmatrix}
&=&
\begin{array}{c}
\begin{matrix}
\mu&\scriptsize{x11}&\scriptsize{x12}&\scriptsize{x21}&\scriptsize{x22}&\scriptsize{x11x21}&\scriptsize{x12x21}&\scriptsize{x11x22}&\scriptsize{x12x22}
\end{matrix}\\
\left[\begin{array}{c|cc|cc|cccc}
1\hphantom{0}& 1\hphantom{0} & 0\hphantom{0} & 1\hphantom{0}
& 0\hphantom{00} & 1\hphantom{000} & 0\hphantom{000} &
0\hphantom{000} & 0\hphantom{00}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&1\hphantom{0}&0\hphantom{00}&0\hphantom{000}&1\hphantom{000}&0\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{00}&0\hphantom{000}&0\hphantom{000}&1\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{00}&0\hphantom{000}&0\hphantom{000}&0\hphantom{000}&1\hphantom{00}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{00}&1\hphantom{000}&0\hphantom{000}&0\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&1\hphantom{0}&0\hphantom{00}&0\hphantom{000}&1\hphantom{000}&0\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{00}&0\hphantom{000}&0\hphantom{000}&1\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{00}&0\hphantom{000}&0\hphantom{000}&0\hphantom{000}&1\hphantom{00}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{00}&1\hphantom{000}&0\hphantom{000}&0\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&1\hphantom{0}&0\hphantom{00}&0\hphantom{000}&1\hphantom{000}&0\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{00}&0\hphantom{000}&0\hphantom{000}&1\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{00}&0\hphantom{000}&0\hphantom{000}&0\hphantom{000}&1\hphantom{00}\\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
\mu \\
\alpha_1\\
\alpha_2\\
\tau_1\\
\tau_2\\
\gamma_{11}\\
\gamma_{21}\\
\gamma_{12}\\
\gamma_{22}
\end{bmatrix}
+
\begin{array}{c}
\begin{matrix}
\scriptsize{x11r1}&\scriptsize{x12r1}\hspace{0.1cm}&\scriptsize{x11r2}&\scriptsize{x12r2}&\scriptsize{x11r3}&\scriptsize{x12r3}\hspace{0.6cm}
\end{matrix}\\
\left[\begin{array}{cccccc}
1\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 1\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
1\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 1\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 1\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
1\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 1\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
1\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 1\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 1\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 1\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 1\hphantom{0} \\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
u_{11} \\
u_{21} \\
u_{12} \\
u_{22} \\
u_{13} \\
u_{23}
\end{bmatrix}
+
\begin{bmatrix}
e_{111} \\
e_{211} \\
e_{121} \\
e_{221} \\
e_{112} \\
e_{212} \\
e_{122} \\
e_{222} \\
e_{113} \\
e_{213} \\
e_{123} \\
e_{223}
\end{bmatrix}
\\
&=&
\begin{array}{c}
\begin{matrix}
\mu&\scriptsize{x11}&\scriptsize{x12}&\scriptsize{x21}&\scriptsize{x22}&\scriptsize{x11x21}&\scriptsize{x12x21}&\scriptsize{x11x22}&\scriptsize{x12x22}
\end{matrix}\\
\left[\begin{array}{c|cc|cc|cccc}
1\hphantom{0}& 1\hphantom{0} & 0\hphantom{0} & 1\hphantom{0}
& 0\hphantom{00} & 1\hphantom{000} & 0\hphantom{000} &
0\hphantom{000} & 0\hphantom{00}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&1\hphantom{0}&0\hphantom{00}&0\hphantom{000}&1\hphantom{000}&0\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{00}&0\hphantom{000}&0\hphantom{000}&1\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{00}&0\hphantom{000}&0\hphantom{000}&0\hphantom{000}&1\hphantom{00}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{00}&1\hphantom{000}&0\hphantom{000}&0\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&1\hphantom{0}&0\hphantom{00}&0\hphantom{000}&1\hphantom{000}&0\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{00}&0\hphantom{000}&0\hphantom{000}&1\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{00}&0\hphantom{000}&0\hphantom{000}&0\hphantom{000}&1\hphantom{00}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{00}&1\hphantom{000}&0\hphantom{000}&0\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&1\hphantom{0}&0\hphantom{00}&0\hphantom{000}&1\hphantom{000}&0\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&1\hphantom{0}&0\hphantom{0}&0\hphantom{0}&1\hphantom{00}&0\hphantom{000}&0\hphantom{000}&1\hphantom{000}&0\hphantom{00}\\
1\hphantom{0}&0\hphantom{0}&1\hphantom{0}&0\hphantom{0}&1\hphantom{00}&0\hphantom{000}&0\hphantom{000}&0\hphantom{000}&1\hphantom{00}\\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
15 \\
1\\
-2\\
1\\
-1\\
3\\
1\\
1\\
-5
\end{bmatrix}
+
\begin{array}{c}
\begin{matrix}
\scriptsize{x11r1}&\scriptsize{x12r1}\hspace{0.1cm}&\scriptsize{x11r2}&\scriptsize{x12r2}&\scriptsize{x11r3}&\scriptsize{x12r3}\hspace{0.6cm}
\end{matrix}\\
\left[\begin{array}{cccccc}
1\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 1\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
1\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 1\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 1\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
1\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 1\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
1\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 1\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 1\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 1\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 1\hphantom{0} \\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
u_{11} \\
u_{21} \\
u_{12} \\
u_{22} \\
u_{13} \\
u_{23}
\end{bmatrix}
+
\begin{bmatrix}
e_{111} \\
e_{211} \\
e_{121} \\
e_{221} \\
e_{112} \\
e_{212} \\
e_{122} \\
e_{222} \\
e_{113} \\
e_{213} \\
e_{123} \\
e_{223}
\end{bmatrix}
\\
&=&
\begin{bmatrix}
20 \\
15 \\
16 \\
7 \\
20 \\
15 \\
16 \\
7 \\
20 \\
15 \\
16 \\
7
\end{bmatrix}
+
\begin{array}{c}
\begin{matrix}
\scriptsize{x11r1}&\scriptsize{x12r1}\hspace{0.1cm}&\scriptsize{x11r2}&\scriptsize{x12r2}&\scriptsize{x11r3}&\scriptsize{x12r3}\hspace{0.6cm}
\end{matrix}\\
\left[\begin{array}{cccccc}
1\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 1\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
1\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 1\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 1\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
1\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 1\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
1\hphantom{000}& 0\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 1\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 1\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 1\hphantom{00}& 0\hphantom{0} \\
0\hphantom{000}& 0\hphantom{000}& 0\hphantom{000}&
0\hphantom{000}& 0\hphantom{00}& 1\hphantom{0} \\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
u_{11} \\
u_{21} \\
u_{12} \\
u_{22} \\
u_{13} \\
u_{23}
\end{bmatrix}
+
\begin{bmatrix}
e_{111} \\
e_{211} \\
e_{121} \\
e_{221} \\
e_{112} \\
e_{212} \\
e_{122} \\
e_{222} \\
e_{113} \\
e_{213} \\
e_{123} \\
e_{223}
\end{bmatrix}
\end{eqnarray}
\]
Considering that the error of plot and subplot has mean 0 and
variance 0 we have:
\[
\begin{bmatrix}
Y_{111} \\
Y_{211} \\
Y_{121} \\
Y_{221} \\
Y_{112} \\
Y_{212} \\
Y_{122} \\
Y_{222} \\
Y_{113} \\
Y_{213} \\
Y_{123} \\
Y_{223}
\end{bmatrix}
=
\begin{bmatrix}
20 \\
15 \\
16 \\
7 \\
20 \\
15 \\
16 \\
7 \\
20 \\
15 \\
16 \\
7
\end{bmatrix}
\]
Using the gexp package we have:
<- gexp (mu = 15 ,err = matrix (rep (0 , 12 ),nrow = 12 ),errp = matrix (rep (0 , 6 ),nrow = 6 ),r = 3 ,fe = list (alpha = c (1 , - 2 ),tau = c (1 , - 1 )),inte = c (3 , 1 , 1 , - 5 ),type = 'SPE' )summary (SPE_crd)#> Database #> r X1 X2 Y1 #> 1 1 x11 x21 20 #> 2 2 x11 x21 20 #> 3 3 x11 x21 20 #> 4 1 x12 x21 15 #> 5 2 x12 x21 15 #> 6 3 x12 x21 15 #> 7 1 x11 x22 16 #> 8 2 x11 x22 16 #> 9 3 x11 x22 16 #> 10 1 x12 x22 7 #> 11 2 x12 x22 7 #> 12 3 x12 x22 7
Simulating a CRD with a quantitative factor
Non-Orthogonal Contrast
Linear Effect
Let us suppose that we want to generate a single
random variable according to a CRD with one factor and four levels (0,
5, 10, 15) being 3 repetitions. Then we can denote \(Y_ {ik}\) as being the random variable
observed in the k-th experimental unit (k = 1, 2, 3) that received the
i-th level of factor X1 (i = 1,2,3,4). In this case, it is only possible
to adjust a regression up to the third degree. As the treatment is
quantitative, instead of indicating the effects of treatments as
deviations from the general mean, we will indicate the value we want for
each regression coefficient. So, instead of having \(\mu, \alpha_1, \alpha_2, \alpha_3\) and
\(\alpha_4\) , we have \(\beta_0\) (the intercept), \(\beta_1\) (the angular coefficient), \(\beta_2\) the quadratic coefficient) and
\(\beta_3\) (the cubic coefficient).
Since the effect we want is linear, then we can provide the following
values for betas: \(\beta_0 = 2, \beta_1 = 3,
\beta_2 = 0\) and \(\beta_3 =
0\) . Therefore, we will have the following matrix
configuration:
\[
\begin{eqnarray}
\begin{bmatrix}
Y_{11} \\
Y_{21} \\
Y_{31} \\
Y_{41} \\
Y_{12} \\
Y_{22} \\
Y_{32} \\
Y_{42} \\
Y_{13} \\
Y_{23} \\
Y_{33} \\
Y_{43}
\end{bmatrix}
&=&
\begin{array}{p{5cm}}
\begin{matrix}
\beta_0 & \!\scriptsize{x11}\! & \scriptsize{x12} &
\scriptsize{x13}
\end{matrix}\\
\left[\begin{array}{c|cc}
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 \\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
\beta_{0} \\
\beta_{1} \\
\beta_{2} \\
\beta_{3}
\end{bmatrix}
+
\begin{bmatrix}
e_{11} \\
e_{21} \\
e_{31} \\
e_{41} \\
e_{12} \\
e_{22} \\
e_{23} \\
e_{24} \\
e_{13} \\
e_{23} \\
e_{33} \\
e_{43}
\end{bmatrix}
\\
&=&
\begin{array}{c}
\begin{matrix}
\beta_0 & \!\scriptsize{x11}\! & \scriptsize{x12} &
\scriptsize{x13}
\end{matrix}\\
\left[\begin{array}{c|cc}
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 \\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
2 \\
3 \\
0 \\
0 \\
\end{bmatrix}
+
\begin{bmatrix}
e_{11} \\
e_{21} \\
e_{31} \\
e_{41} \\
e_{12} \\
e_{22} \\
e_{23} \\
e_{24} \\
e_{13} \\
e_{23} \\
e_{33} \\
e_{43}
\end{bmatrix}
\\
&=&
\begin{bmatrix}
2 \\
17 \\
32 \\
47 \\
2 \\
17 \\
32 \\
47 \\
2 \\
17 \\
32 \\
47
\end{bmatrix}
+
\begin{bmatrix}
e_{11} \\
e_{21} \\
e_{31} \\
e_{41} \\
e_{12} \\
e_{22} \\
e_{23} \\
e_{24} \\
e_{13} \\
e_{23} \\
e_{33} \\
e_{43}
\end{bmatrix}
\end{eqnarray}
\]
If we consider an error with mean 0 and variance 0 then we have the
following simulated random variable:
\[
\begin{bmatrix}
Y_{11} \\
Y_{21} \\
Y_{31} \\
Y_{41} \\
Y_{12} \\
Y_{22} \\
Y_{32} \\
Y_{42} \\
Y_{13} \\
Y_{23} \\
Y_{33} \\
Y_{43}
\end{bmatrix}
=
\begin{bmatrix}
2 \\
17 \\
32 \\
47 \\
2 \\
17 \\
32 \\
47 \\
2 \\
17 \\
32 \\
47
\end{bmatrix}
\]
Using the gexp package we have:
<- seq (0 , 15 , 5 )<- matrix (c (level,^ 2 ,^ 3 ),ncol = 3 )<- gexp (mu = 2 ,r = 3 ,err = matrix (rep (0 ,12 ),nrow = 12 ),fe = list (f1 = c (3 , 0 , 0 )),fl = list (dose = level),contrasts = list (dose = cont_crd))summary (crd_l)#> Database #> r dose Y1 #> 1 1 0 2 #> 2 2 0 2 #> 3 3 0 2 #> 4 1 5 17 #> 5 2 5 17 #> 6 3 5 17 #> 7 1 10 32 #> 8 2 10 32 #> 9 3 10 32 #> 10 1 15 47 #> 11 2 15 47 #> 12 3 15 47 Below the simulation chart.
Quadratic effect
Taking advantage of the same previous example, adding only the
quadratic effect to the data, that is, \(\beta_2 = 4\) . Therefore, we will have the
following matrix configuration:
\[
\begin{eqnarray}
\begin{bmatrix}
Y_{11} \\
Y_{21} \\
Y_{31} \\
Y_{41} \\
Y_{12} \\
Y_{22} \\
Y_{32} \\
Y_{42} \\
Y_{13} \\
Y_{23} \\
Y_{33} \\
Y_{43}
\end{bmatrix}
&=&
\begin{array}{p{5cm}}
\begin{matrix}
\beta_0 & \!\scriptsize{x11}\! & \scriptsize{x12} &
\scriptsize{x13}
\end{matrix}\\
\left[\begin{array}{c|cc}
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 \\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
\beta_{0} \\
\beta_{1} \\
\beta_{2} \\
\beta_{3}
\end{bmatrix}
+
\begin{bmatrix}
e_{11} \\
e_{21} \\
e_{31} \\
e_{41} \\
e_{12} \\
e_{22} \\
e_{23} \\
e_{24} \\
e_{13} \\
e_{23} \\
e_{33} \\
e_{43}
\end{bmatrix}
\\
&=&
\begin{array}{c}
\begin{matrix}
\beta_0 & \!\scriptsize{x11}\! & \scriptsize{x12} &
\scriptsize{x13}
\end{matrix}\\
\left[\begin{array}{c|cc}
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 \\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
2 \\
3 \\
4 \\
0 \\
\end{bmatrix}
+
\begin{bmatrix}
e_{11} \\
e_{21} \\
e_{31} \\
e_{41} \\
e_{12} \\
e_{22} \\
e_{23} \\
e_{24} \\
e_{13} \\
e_{23} \\
e_{33} \\
e_{43}
\end{bmatrix}
\\
&=&
\begin{bmatrix}
2 \\
117 \\
432 \\
947 \\
2 \\
117 \\
432 \\
947 \\
2 \\
117 \\
432 \\
947
\end{bmatrix}
+
\begin{bmatrix}
e_{11} \\
e_{21} \\
e_{31} \\
e_{41} \\
e_{12} \\
e_{22} \\
e_{23} \\
e_{24} \\
e_{13} \\
e_{23} \\
e_{33} \\
e_{43}
\end{bmatrix}
\end{eqnarray}
\]
Using the gexp package we have:
<- gexp (mu = 2 ,r = 3 ,err = matrix (rep (0 ,12 ),nrow = 12 ),fe = list (f1 = c (3 , 4 , 0 )),fl = list (dose = level),contrasts = list (dose = cont_crd))summary (crd_q)#> Database #> r dose Y1 #> 1 1 0 2 #> 2 2 0 2 #> 3 3 0 2 #> 4 1 5 117 #> 5 2 5 117 #> 6 3 5 117 #> 7 1 10 432 #> 8 2 10 432 #> 9 3 10 432 #> 10 1 15 947 #> 11 2 15 947 #> 12 3 15 947 Below the simulation chart.
Simulating a CRD in a factorial experiments with two quantitative
factors
Linear Response Surface
Let us suppose that we want to generate a single
random variable according to a CRD with two quantitative factors with
levels 0, 5, 10, 15 and 2, 4, 6, 8 respectively, being 2 replicates.
Then we can denote \(Y_{ijk}\) as being
the random variable observed in the k-th experimental unit (k = 1, 2)
that received the i-th level of factor X1 (i = 1,2,3,4) and j-th level
of factor X2 (j = 1,2,3,4). In these cases, we should note that there is
only one intercept. Therefore, such an intercept must appear in the
first factor to be declared. In the others, the value of the intercept
must be declared as 0 (only to compute the length of the vector
correctly). Therefore, consider the following values for the first
factor: \(\beta_0 = 1, \beta_1 = 3, \beta_2 =
0, \beta_3 = 0\) . For the second factor we have: \(\beta_4 = 2, \beta_5 = 0, \beta_6 = 0\) .
Matrically we have:
\[
\begin{eqnarray}
\begin{bmatrix}
Y_{111} \\
Y_{211} \\
Y_{311} \\
Y_{411} \\
Y_{121} \\
Y_{221} \\
Y_{321} \\
Y_{421} \\
Y_{131} \\
Y_{231} \\
Y_{331} \\
Y_{431} \\
Y_{141} \\
Y_{241} \\
Y_{341} \\
Y_{441} \\
Y_{112} \\
Y_{212} \\
Y_{312} \\
Y_{412} \\
Y_{122} \\
Y_{222} \\
Y_{322} \\
Y_{422} \\
Y_{132} \\
Y_{232} \\
Y_{332} \\
Y_{432} \\
Y_{142} \\
Y_{242} \\
Y_{342} \\
Y_{442} \\
\end{bmatrix}
&=&
\begin{array}{p{5cm}}
\begin{matrix}
\beta_0 & \!\scriptsize{x11}\! & \scriptsize{x12} &
\scriptsize{x13} & \!\scriptsize{x21}\! & \scriptsize{x22} &
\scriptsize{x23}
\end{matrix}\\
\left[\begin{array}{c|cc}
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
8\hphantom{0} & 8^2 & 8^3 \\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
\beta_{0} \\
\beta_{1} \\
\beta_{2} \\
\beta_{3} \\
\beta_{4} \\
\beta_{5} \\
\beta_{6}
\end{bmatrix}
+
\begin{bmatrix}
e_{111} \\
e_{211} \\
e_{311} \\
e_{411} \\
e_{121} \\
e_{221} \\
e_{321} \\
e_{421} \\
e_{131} \\
e_{231} \\
e_{331} \\
e_{431} \\
e_{141} \\
e_{241} \\
e_{341} \\
e_{441} \\
e_{112} \\
e_{212} \\
e_{312} \\
e_{412} \\
e_{122} \\
e_{222} \\
e_{322} \\
e_{422} \\
e_{132} \\
e_{232} \\
e_{332} \\
e_{432} \\
e_{142} \\
e_{242} \\
e_{342} \\
e_{442} \\
\end{bmatrix}
\\
&=&
\begin{array}{c}
\begin{matrix}
\beta_0 & \!\scriptsize{x11}\! & \scriptsize{x12} &
\scriptsize{x13} & \!\scriptsize{x21}\! & \scriptsize{x22} &
\scriptsize{x23}
\end{matrix}\\
\left[\begin{array}{c|cc}
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
2\hphantom{0} & 2^2 & 2^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
4\hphantom{0} & 4^2 & 4^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
6\hphantom{0} & 6^2 & 6^3 \\
1\hphantom{0} & 0\hphantom{0} & 0^2 & 0^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 5\hphantom{0} & 5^2 & 5^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 10\hphantom{0} & 10^2 & 10^3 &
8\hphantom{0} & 8^2 & 8^3 \\
1\hphantom{0} & 15\hphantom{0} & 15^2 & 15^3 &
8\hphantom{0} & 8^2 & 8^3 \\
\end{array}\right]
\end{array}
\cdot
\begin{bmatrix}
1 \\
3 \\
0 \\
0 \\
2 \\
0 \\
0
\end{bmatrix}
+
\begin{bmatrix}
e_{111} \\
e_{211} \\
e_{311} \\
e_{411} \\
e_{121} \\
e_{221} \\
e_{321} \\
e_{421} \\
e_{131} \\
e_{231} \\
e_{331} \\
e_{431} \\
e_{141} \\
e_{241} \\
e_{341} \\
e_{441} \\
e_{112} \\
e_{212} \\
e_{312} \\
e_{412} \\
e_{122} \\
e_{222} \\
e_{322} \\
e_{422} \\
e_{132} \\
e_{232} \\
e_{332} \\
e_{432} \\
e_{142} \\
e_{242} \\
e_{342} \\
e_{442} \\
\end{bmatrix}
\\
&=&
\begin{bmatrix}
5 \\
20 \\
35 \\
50 \\
9 \\
24 \\
39 \\
54 \\
13 \\
28 \\
43 \\
58 \\
17 \\
32 \\
47 \\
62 \\
5 \\
20 \\
35 \\
50 \\
9 \\
24 \\
39 \\
54 \\
13 \\
28 \\
43 \\
58 \\
17 \\
32 \\
47 \\
62
\end{bmatrix}
+
\begin{bmatrix}
e_{111} \\
e_{211} \\
e_{311} \\
e_{411} \\
e_{121} \\
e_{221} \\
e_{321} \\
e_{421} \\
e_{131} \\
e_{231} \\
e_{331} \\
e_{431} \\
e_{141} \\
e_{241} \\
e_{341} \\
e_{441} \\
e_{112} \\
e_{212} \\
e_{312} \\
e_{412} \\
e_{122} \\
e_{222} \\
e_{322} \\
e_{422} \\
e_{132} \\
e_{232} \\
e_{332} \\
e_{432} \\
e_{142} \\
e_{242} \\
e_{342} \\
e_{442}
\end{bmatrix}
\end{eqnarray}
\]
Using the gexp package we have:
<- seq (2 ,8 ,2 )<- matrix (c (level2,level2^ 2 ,level2^ 3 ),ncol = 3 )<- gexp (mu = 1 ,r = 2 ,err = matrix (rep (0 ,32 ),nrow = 32 ),fe = list (f1 = c (3 , 0 , 0 ),f2 = c (2 , 0 , 0 )),fl = list (N = level,P = level2),inte = c (6 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ),contrasts = list (N = cont_crd,P = cont_crd2),type = 'FE' )summary (crd_hb)#> Database #> r N P Y1 #> 1 1 0 2 5 #> 2 2 0 2 5 #> 3 1 5 2 80 #> 4 2 5 2 80 #> 5 1 10 2 155 #> 6 2 10 2 155 #> 7 1 15 2 230 #> 8 2 15 2 230 #> 9 1 0 4 9 #> 10 2 0 4 9 #> 11 1 5 4 144 #> 12 2 5 4 144 #> 13 1 10 4 279 #> 14 2 10 4 279 #> 15 1 15 4 414 #> 16 2 15 4 414 #> 17 1 0 6 13 #> 18 2 0 6 13 #> 19 1 5 6 208 #> 20 2 5 6 208 #> 21 1 10 6 403 #> 22 2 10 6 403 #> 23 1 15 6 598 #> 24 2 15 6 598 #> 25 1 0 8 17 #> 26 2 0 8 17 #> 27 1 5 8 272 #> 28 2 5 8 272 #> 29 1 10 8 527 #> 30 2 10 8 527 #> 31 1 15 8 782 #> 32 2 15 8 782 Below the simulation chart.
Quadratic response surface
Let’s omit matrix algebra as it is analogous to the one presented
above. Considering now \(\beta_2 = 2\)
and \(\beta_5 = 3\) , we have the
following result using the gexp package.
<- gexp (mu = 1 ,r = 2 ,err = matrix (rep (0 ,32 ),nrow = 32 ),fe = list (f1 = c (3 , 2 , 0 ),f2 = c (2 , 3 , 0 )),fl = list (N = level,P = level2),inte = c (2 ,1 ,0 ,1 ,8 ,0 ,0 ,0 ,0 ), contrasts = list (N = cont_crd,P = cont_crd2),type = 'FE' )summary (crd_hb2)#> Database #> r N P Y1 #> 1 1 0 2 17 #> 2 2 0 2 17 #> 3 1 5 2 972 #> 4 2 5 2 972 #> 5 1 10 2 3727 #> 6 2 10 2 3727 #> 7 1 15 2 8282 #> 8 2 15 2 8282 #> 9 1 0 4 57 #> 10 2 0 4 57 #> 11 1 5 4 3542 #> 12 2 5 4 3542 #> 13 1 10 4 13727 #> 14 2 10 4 13727 #> 15 1 15 4 30612 #> 16 2 15 4 30612 #> 17 1 0 6 121 #> 18 2 0 6 121 #> 19 1 5 6 7776 #> 20 2 5 6 7776 #> 21 1 10 6 30231 #> 22 2 10 6 30231 #> 23 1 15 6 67486 #> 24 2 15 6 67486 #> 25 1 0 8 209 #> 26 2 0 8 209 #> 27 1 5 8 13674 #> 28 2 5 8 13674 #> 29 1 10 8 53239 #> 30 2 10 8 53239 #> 31 1 15 8 118904 #> 32 2 15 8 118904 Below the simulation chart.
Doing experiment planning
Not interactive
CRD
Let’s suppose that we are interested in planning a DIC with three
treatments (default) and five repetitions (default). In this case, as we
are not interested in simulating data, we can give any value to the
effects of the treatments, because what matters is the arrangement of
the treatments in the experimental units.
So, using the gexp package we have:
<- gexp ()plot (crd_design)
RCBD without repetition
Let us consider an RCBD in which the block is the very repetition of
the experiment. Consider also three treatments (default) and three
blocks (default).
With the gexp package we have:
<- gexp (r = 1 , design = 'RCBD' )Graphically, we have:
RCBD with repetition
Considering the same experiment previous, alternating the number of
repetitions inside blocks. Let us now consider two replicates per
block.
With the gexp package we have:
<- gexp (r = 2 , design = 'RCBD' )Graphically, we have:
LSD
Consider a LSD with three treatments (default), three rows and three
columns. In this case we have the following planning.
<- gexp (r = 1 ,design = 'LSD' )Graphically, we have:
Factorial - CRD
Let us assume that we are interested in planning a 2x3 factorial
(default) design with three replicates.
<- gexp (r = 3 , type = 'FE' )Using the graph we have:
Split-plot - CRD
Assuming that we are interested in planning a CRD in split-plot, with
two treatments in the plot (default) and three treatments in the subplot
(default), with three replications.
<- gexp (r = 3 , type = 'SPE' )summary (split_design)#> Database #> r X1 X2 Y1 #> 1 1 x11 x21 28.2 #> 2 2 x11 x21 27.0 #> 3 3 x11 x21 28.1 #> 4 1 x12 x21 30.6 #> 5 2 x12 x21 29.5 #> 6 3 x12 x21 28.3 #> 7 1 x13 x21 29.4 #> 8 2 x13 x21 30.2 #> 9 3 x13 x21 28.2 #> 10 1 x11 x22 30.5 #> 11 2 x11 x22 28.3 #> 12 3 x11 x22 28.4 #> 13 1 x12 x22 29.8 #> 14 2 x12 x22 31.1 #> 15 3 x12 x22 27.6 #> 16 1 x13 x22 30.0 #> 17 2 x13 x22 30.5 #> 18 3 x13 x22 29.3 Using the graph we have:
Interactive
CRD
If the researcher has a sketch of the experimental area in the PNG,
JPG or JPEG formats, it is possible to import such a file with the
functions of the gexp package and to randomly distribute the treatments
in the experimental units. Suppose we have a sketch with six fish tanks,
and we want to distribute two treatments with three replicates in the
image. Then we proceed from the following form:
<- gexp (r = 3 , fe = list (f1 = rep (1 ,2 )))plot (crd_plot,dynamic = TRUE )
References
Aquino, Luiz Henrique de. 1992. “Técnica Experimental Com Animais
i.”
Ferreira, Daniel Furtado. 2008. Estatística Multivariada .
Edited by UFLA. 1st ed.
Naes, T., A. H. Aastveit, and N. S. Sahni. 2007. “Analysis of
Split-Plot Designs: An Overview and Comparison of Methods.”
Qual. Reliab. Engng. Int. 23: 801–20.
Rencher, Alvin C., and Bruce G. Schaalje. 2007. Linear Models in
Statistics . Edited by John Wiley & Sons. 2nd ed.
These binaries (installable software) and packages are in development.
