Two level factorial design

We will work with yield experiment (BHH - 308)
Let's set up data first:

> ntt<-factor(rep(c(160,180),8))

> ncc<-factor(rep(c(20,20,40,40),4))

> nkk<-rep(rep(c("A","B"),c(4,4)),2)

> yield<-c(60,72,54,68,52,83,45,80,59,74,50,69,50,81,46,79)

> options(contrasts = c("contr.helmert", "contr.poly"))

>

Now let's fit model:

> fit<-lm(yield~ntt*ncc*nkk)

> model.matrix(fit)

(Intercept) ntt ncc nkk ntt:ncc ntt:nkk ncc:nkk ntt:ncc:nkk

1 1 -1 -1 -1 1 1 1 -1

2 1 1 -1 -1 -1 -1 1 1

3 1 -1 1 -1 -1 1 -1 1

4 1 1 1 -1 1 -1 -1 -1

5 1 -1 -1 1 1 -1 -1 1

6 1 1 -1 1 -1 1 -1 -1

7 1 -1 1 1 -1 -1 1 -1

8 1 1 1 1 1 1 1 1

9 1 -1 -1 -1 1 1 1 -1

10 1 1 -1 -1 -1 -1 1 1

11 1 -1 1 -1 -1 1 -1 1

12 1 1 1 -1 1 -1 -1 -1

13 1 -1 -1 1 1 -1 -1 1

14 1 1 -1 1 -1 1 -1 -1

15 1 -1 1 1 -1 -1 1 -1

16 1 1 1 1 1 1 1 1

> anova(fit)

Analysis of Variance Table

Response: yield

Terms added sequentially (first to last)
            Df Sum of Sq Mean Sq F Value     Pr(F)
ntt          1   2256.25 2256.25 1128.125 0.0000000
ncc          1    100.00 100.00   50.000 0.0001050
nkk          1      6.25    6.25    3.125 0.1150771
ntt:ncc      1      9.00    9.00    4.500 0.0666880
ntt:nkk      1    306.25 306.25 153.125 0.0000017
ncc:nkk      1      4.00    4.00    2.000 0.1950155
ntt:ncc:nkk 1      0.00    0.00    0.000 1.0000000
Residuals    8     16.00    2.00
> summary(fit)

Call: lm(formula = yield ~ ntt * ncc * nkk)
Residuals:
Min 1Q Median 3Q Max
-2 -0.625 -2.22e-16 0.625 2

Coefficients:
                Value Std. Error   t value Pr(>|t|)
(Intercept)   63.8750    0.3536   180.6658    0.0000
        ntt   11.8750    0.3536    33.5876    0.0000
        ncc   -2.5000    0.3536    -7.0711    0.0001
        nkk    0.6250    0.3536     1.7678    0.1151
    ntt:ncc    0.7500    0.3536     2.1213    0.0667
    ntt:nkk    4.3750    0.3536    12.3744    0.0000
    ncc:nkk    0.5000    0.3536     1.4142    0.1950
ntt:ncc:nkk    0.0000    0.3536     0.0000    1.0000

Residual standard error: 1.414 on 8 degrees of freedom
Multiple R-Squared: 0.9941
F-statistic: 191.6 on 7 and 8 degrees of freedom, the p-value is 2.869e-08

Correlation of Coefficients:
            (Intercept) ntt ncc nkk ntt:ncc ntt:nkk ncc:nkk
        ntt 0
        ncc 0           0
        nkk 0           0   0
    ntt:ncc 0           0   0   0
    ntt:nkk 0           0   0   0   0
    ncc:nkk 0           0   0   0   0       0
ntt:ncc:nkk 0           0   0   0   0       0       0

Let's make a few plots:

> X11()

> qqnorm.aov(fit,label=5)

Another model:

> fit<-lm(yield~(ntt+ncc+nkk)^2)

> model.matrix(fit)

(Intercept) ntt ncc nkk ntt:ncc ntt:nkk ncc:nkk

1 1 -1 -1 -1 1 1 1

2 1 1 -1 -1 -1 -1 1

3 1 -1 1 -1 -1 1 -1

4 1 1 1 -1 1 -1 -1

5 1 -1 -1 1 1 -1 -1

6 1 1 -1 1 -1 1 -1

7 1 -1 1 1 -1 -1 1

8 1 1 1 1 1 1 1

9 1 -1 -1 -1 1 1 1

10 1 1 -1 -1 -1 -1 1

11 1 -1 1 -1 -1 1 -1

12 1 1 1 -1 1 -1 -1

13 1 -1 -1 1 1 -1 -1

14 1 1 -1 1 -1 1 -1

15 1 -1 1 1 -1 -1 1

16 1 1 1 1 1 1 1

> anova(fit)

Analysis of Variance Table

Response: yield

Terms added sequentially (first to last)
          Df Sum of Sq Mean Sq F Value     Pr(F)
ntt        1   2256.25 2256.250 1269.141 0.0000000
ncc        1    100.00 100.000   56.250 0.0000369
nkk        1      6.25    6.250    3.516 0.0935497
ntt:ncc    1      9.00    9.000    5.062 0.0510033
ntt:nkk    1    306.25 306.250 172.266 0.0000004
ncc:nkk    1      4.00    4.000    2.250 0.1678507
Residuals 9     16.00    1.778
> summary(fit)

Call: lm(formula = yield ~ (ntt + ncc + nkk)^2)
Residuals:
Min 1Q Median 3Q Max
-2 -0.625 2.498e-16 0.625 2

Coefficients:
                Value Std. Error   t value Pr(>|t|)
(Intercept)   63.8750    0.3333   191.6250    0.0000
        ntt   11.8750    0.3333    35.6250    0.0000
        ncc   -2.5000    0.3333    -7.5000    0.0000
        nkk    0.6250    0.3333     1.8750    0.0935
    ntt:ncc    0.7500    0.3333     2.2500    0.0510
    ntt:nkk    4.3750    0.3333    13.1250    0.0000
    ncc:nkk    0.5000    0.3333     1.5000    0.1679

Residual standard error: 1.333 on 9 degrees of freedom
Multiple R-Squared: 0.9941
F-statistic: 251.4 on 6 and 9 degrees of freedom, the p-value is 1.687e-09

Correlation of Coefficients:
        (Intercept) ntt ncc nkk ntt:ncc ntt:nkk
    ntt 0
    ncc 0           0
    nkk 0           0   0
ntt:ncc 0           0   0   0
ntt:nkk 0           0   0   0   0
ncc:nkk 0           0   0   0   0       0
> qqnorm.aov(fit,label=5)
> par(mfrow=c(2,3))
> plot(fit)
> par(mfrow=c(1,1))

Now let's look what would happend if we don't have replicates. I will use the same dataset, but only first 8 replications.

> yield1<-yield[1:8]

Error: Object "yeild" not found

> yield1<-yield[1:8]

> nkk1<-factor(nkk[1:8])

> ncc1<-factor(ncc[1:8])

> ntt1<-factor(ntt[1:8])

> fit<-lm(yield1~nkk1*ncc1*ntt1)

> model.matrix(fit)

(Intercept) nkk1 ncc1 ntt1 nkk1:ncc1 nkk1:ntt1 ncc1:ntt1 nkk1:ncc1:ntt1

1 1 -1 -1 -1 1 1 1 -1

2 1 -1 -1 1 1 -1 -1 1

3 1 -1 1 -1 -1 1 -1 1

4 1 -1 1 1 -1 -1 1 -1

5 1 1 -1 -1 -1 -1 1 1

6 1 1 -1 1 -1 1 -1 -1

7 1 1 1 -1 1 -1 -1 -1

8 1 1 1 1 1 1 1 1

> anova(fit)

Analysis of Variance Table

Response: yield1

Terms added sequentially (first to last)
               Df Sum of Sq Mean Sq
nkk1            1       4.5     4.5
ncc1            1      50.0    50.0
ntt1            1    1058.0 1058.0
nkk1:ncc1       1       0.0     0.0
nkk1:ntt1       1     200.0   200.0
ncc1:ntt1       1       4.5     4.5
nkk1:ncc1:ntt1 1       0.5     0.5
> summary(fit)

Call: lm(formula = yield1 ~ nkk1 * ncc1 * ntt1)

Coefficients:
                Value Std. Error t value Pr(>|t|)
   (Intercept) 64.25     NA         NA      NA
          nkk1   0.75     NA         NA      NA
          ncc1 -2.50     NA         NA      NA
          ntt1 11.50     NA         NA      NA
     nkk1:ncc1   0.00     NA         NA      NA
     nkk1:ntt1   5.00     NA         NA      NA
     ncc1:ntt1   0.75     NA         NA      NA
nkk1:ncc1:ntt1   0.25     NA         NA      NA

Residual standard error: NA on 0 degrees of freedom
Multiple R-Squared: 1
F-statistic: NA on 7 and 0 degrees of freedom, the p-value is NA

Correlation of Coefficients:
               (Intercept) nkk1 ncc1 ntt1 nkk1:ncc1 nkk1:ntt1 ncc1:ntt1
          nkk1 0
          ncc1 0           0
          ntt1 0           0    0
     nkk1:ncc1 0           0    0    0
     nkk1:ntt1 0           0    0    0    0
     ncc1:ntt1 0           0    0    0    0         0
nkk1:ncc1:ntt1 0           0    0    0    0         0         0
Warning messages:
One or more nonpositive parameters in: pf(x$fstatistic[1], x$fstatistic[2],
x$fstatistic[3])
> qqnorm.aov(fit,label=5)
> par(mfrow=c(2,3))
> plot(fit)
Error in .Fortran(if(!cmplx) "dbksl" else "zbksl",: subroutine dbksl: 24 missing
value(s) in argument 4
Dumped
> par(mfrow=c(1,1))
> fit<-lm(yield1~(nkk1+ncc1+ntt1)^2)
> model.matrix(fit)
(Intercept) nkk1 ncc1 ntt1 nkk1:ncc1 nkk1:ntt1 ncc1:ntt1
1           1   -1   -1   -1         1         1         1
2           1   -1   -1    1         1        -1        -1
3           1   -1    1   -1        -1         1        -1
4           1   -1    1    1        -1        -1         1
5           1    1   -1   -1        -1        -1         1
6           1    1   -1    1        -1         1        -1
7           1    1    1   -1         1        -1        -1
8           1    1    1    1         1         1         1
> summary(fit)

Call: lm(formula = yield1 ~ (nkk1 + ncc1 + ntt1)^2)
Residuals:
1 2 3 4 5 6 7 8
-0.25 0.25 0.25 -0.25 0.25 -0.25 -0.25 0.25

Coefficients:
                Value Std. Error   t value Pr(>|t|)
(Intercept)   64.2500    0.2500   257.0000    0.0025
       nkk1    0.7500    0.2500     3.0000    0.2048
       ncc1   -2.5000    0.2500   -10.0000    0.0635
       ntt1   11.5000    0.2500    46.0000    0.0138
nkk1:ncc1    0.0000    0.2500     0.0000    1.0000
nkk1:ntt1    5.0000    0.2500    20.0000    0.0318
ncc1:ntt1    0.7500    0.2500     3.0000    0.2048

Residual standard error: 0.7071 on 1 degrees of freedom
Multiple R-Squared: 0.9996
F-statistic: 439 on 6 and 1 degrees of freedom, the p-value is 0.03652

Correlation of Coefficients:
          (Intercept) nkk1 ncc1 ntt1 nkk1:ncc1 nkk1:ntt1
     nkk1 0
     ncc1 0           0
     ntt1 0           0    0
nkk1:ncc1 0           0    0    0
nkk1:ntt1 0           0    0    0    0
ncc1:ntt1 0           0    0    0    0         0
> anova(fit)
Analysis of Variance Table

Response: yield1

Terms added sequentially (first to last)
          Df Sum of Sq Mean Sq F Value     Pr(F)
nkk1       1       4.5     4.5       9 0.2048328
ncc1       1      50.0    50.0     100 0.0634510
ntt1       1    1058.0 1058.0    2116 0.0138374
nkk1:ncc1 1       0.0     0.0       0 1.0000000
nkk1:ntt1 1     200.0   200.0     400 0.0318045
ncc1:ntt1 1       4.5     4.5       9 0.2048328
Residuals 1       0.5     0.5
>