Aircraft primer paints are applied to aluminum surfaces by two methods; dipping and
spraying. A factorial experiment was performed to investigate the effect of paint primer
type and application method on paint adhesion. Adhesion force was measured, with three
different primers and two application methods.
> paint <- data.frame(adhf =
c(4.0,4.5,4.3,5.6,4.9,5.4,3.8,3.7,4.0,5.4,4.9,5.6,5.8,6.1,6.3,5.5,5.0,5
.0),
primer = factor(rep(rep(1:3,rep(3,3)),2)),applic =
factor(rep(c("D","S"),c(9,9))))
> paint
adhf primer applic
1
4.0
1
D
2
4.5
1
D
3
4.3
1
D
4
5.6
2
D
5
4.9
2
D
6
5.4
2
D
7
3.8
3
D
8
3.7
3
D
9
4.0
3
D
10 5.4
1
S
11 4.9
1
S
12 5.6
1
S
13 5.8
2
S
14 6.1
2
S
15 6.3
2
S
16 5.5
3
S
17 5.0
3
S
18 5.0
3
S
This is a one-way anova for primer, ignoring application method; lm() fits the linear
model and anova() displayes the results in an anova table.
> anova(lm(adhf~primer, data=paint))
Analysis of Variance Table
Response: adhf
Df Sum Sq Mean Sq F value Pr(>F)
primer
2 4.5811 2.2906 5.5989 0.01527 *
Residuals 15 6.1367 0.4091
--Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
This is a one-way anova for application method, ignoring primer.
> anova(lm(adhf~applic, data=paint))
Analysis of Variance Table
Response: adhf
Df Sum Sq Mean Sq F value
Pr(>F)
applic
1 4.9089 4.9089 13.521 0.002039 **
Residuals 16 5.8089 0.3631
---
Signif. codes:
0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
This is a two-way anova for application method, primer and their interaction. Note that
the sums of square for primer and for applic are the same as computed in the respective
one-way analyses.
> anova(lm(adhf~primer*applic, data=paint))
Analysis of Variance Table
Response: adhf
Df Sum Sq Mean Sq F value
Pr(>F)
primer
2 4.5811 2.2906 27.8581 3.097e-05 ***
applic
1 4.9089 4.9089 59.7027 5.357e-06 ***
primer:applic 2 0.2411 0.1206 1.4662
0.2693
Residuals
12 0.9867 0.0822
--Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
The interaction is not significant (P = 0.27) so we can test the main effects, both of which
are highly significant. We conclude that both the choice of primer and the choice of
application method affect the adhesive force of the paint, and the differences between the
three primers are the same with both application methods, and the difference between the
two application methods is the same with each primer.
The summary shows the fitted coefficients and their t-tests.
> summary(lm(adhf~primer*applic, data=paint))
Call:
lm(formula = adhf ~ primer * applic, data = paint)
Residuals:
Min
1Q
-0.40000 -0.16667
Median
0.03333
3Q
0.21667
Max
0.33333
Coefficients:
Estimate Std. Error t value
(Intercept)
4.2667
0.1656 25.772
primer2
1.0333
0.2341
4.414
primer3
-0.4333
0.2341 -1.851
applicS
1.0333
0.2341
4.414
primer2:applicS -0.2667
0.3311 -0.805
primer3:applicS
0.3000
0.3311
0.906
--Signif. codes: 0 `***' 0.001 `**' 0.01 `*'
Pr(>|t|)
7.1e-12
0.000845
0.088949
0.000845
0.436265
0.382736
***
***
.
***
0.05 `.' 0.1 ` ' 1
Residual standard error: 0.2867 on 12 degrees of freedom
Multiple R-Squared: 0.9079,
Adjusted R-squared: 0.8696
F-statistic: 23.67 on 5 and 12 DF, p-value: 7.89e-06
The next commands show how to compute means for the six different primer:applic
combinations and how to arrange the means in a matrix to give the interaction plots.
> split(paint$adhf,paint$applic:paint$primer)
$"D:1"
[1] 4.0 4.5 4.3
$"D:2"
[1] 5.6 4.9 5.4
$"D:3"
[1] 3.8 3.7 4.0
$"S:1"
[1] 5.4 4.9 5.6
$"S:2"
[1] 5.8 6.1 6.3
$"S:3"
[1] 5.5 5.0 5.0
> sapply(split(paint$adhf,paint$applic:paint$primer),mean)
D:1
D:2
D:3
S:1
S:2
S:3
4.266667 5.300000 3.833333 5.300000 6.066667 5.166667
>
matrix(sapply(split(paint$adhf,paint$applic:paint$primer),mean),ncol=2)
[,1]
[,2]
[1,] 4.266667 5.300000
[2,] 5.300000 6.066667
[3,] 3.833333 5.166667
matplot() plots each column of the matrix on the same graph.
>
matplot(matrix(sapply(split(paint$adhf,paint$applic:paint$primer),mean)
,ncol=2),
type="l", xlab="Primer",ylab="Adhesion Force")
The two lines are close to parallel, confirming the conclusion of no interaction. That is,
the differences between primers are the same for each application method.
>
matplot(matrix(sapply(split(paint$adhf,paint$primer:paint$applic),mean)
,ncol=3),
type="l", xlab="Application",ylab="Adhesion Force")
You could also plot adhesion force against application method, with a different line for
each primer. Again, the lines are close to parallel.