# Unreplicated Two-Way ANOVA Model

+     pressure=rep(seq(from=120,to=150,by=10),3),

+     temperature=rep(seq(250,270,10),c(4,4,4)))

1: 9.60 9.69 8.43 9.98 11.28 10.10 11.01 10.44

9: 9.00 9.57 9.03 9.80

13:

```Read 12 items
```

The session below 'simulate' hand calculations. It's used only for demonstration.

```        temperature
pressure   250   260   270
120  9.60 11.28  9.00
130  9.69 10.10  9.57
140  8.43 11.01  9.03
150  9.98 10.44  9.80
```

```pressure
120   130   140   150
29.88 29.36 28.47 30.22
```

```temperature
250   260   270
37.70 42.83 37.40
```

```[1] 117.93
```

```[1] 1166.349
```

```[1] 3478.613
```

```[1] 4654.459
```

> SST

```[1] 7.392225
```

> SSA <− sum(margin.table(adhesive.xtabs, 1)^2)/3 -

> SSA

```[1] 0.5806917
```

> SSB

```[1] 4.65765
```

> SSAB <− SST - SSA - SSB

> SSAB

```[1] 2.153883
```

> print( Abar <− rowMeans(adhesive.xtabs) ) # ybar-i.

```      120       130       140       150
9.960000  9.786667  9.490000 10.073333
```

> print( Bbar <− colMeans(adhesive.xtabs) ) # ybar-.j

```    250     260     270
9.4250 10.7075  9.3500
```

> print( ylim <− range(pretty(range(Abar,Bbar,ybar))) )

```[1]  9.2 10.8
```

> windows(width=6,height=3,pointsize=10)

> par(mfrow=c(1,2), mar=c(4,4,.5,.5)+.1, oma=c(2,0,3,0))

> plot(seq(120,150,10),Abar,ylim=ylim,

+      xlab=bquote(plain("pressure (")*italic("lb/in")^2*plain(")")),

+      ylab="mean strength",xaxt="n",type="l",lwd=2)

> axis(1, at=seq(120,150,10))

> abline(h=ybar, lty=3, col="darkorange4")

> box(which="figure",col="grey75")

> plot(seq(250,270,10),Bbar,ylim=ylim,

+      xlab=bquote(plain("temperature (")*degree*italic("F")*plain(")")),

+      ylab="mean strength",xaxt="n",type="l",lwd=2)

> axis(1, at=seq(250,270,10))

> abline(h=ybar, lty=3, col="darkorange4")

> box(which="figure",col="grey75")

> mtext(bquote(bar(y)[..]==.(ybar)), side=1,line=0.5,outer=T)

> title(main="Main Effect Plots",outer=T)

> box(which="outer", col="grey50")

The main effect plots were produced by above codes.

> Alpha <− Abar - ( ybar <− mean(adhesive.xtabs) )

> Alpha # A effects

```        120         130         140         150
0.13250000 -0.04083333 -0.33750000  0.24583333
```

> Beta <− Bbar - ybar

> Beta # B effects

```    250     260     270
-0.4025  0.8800 -0.4775
```

> num <− sum( (Alpha %o% Beta) * adhesive.xtabs )

> num # numerator of gamma.hat

```[1] -0.3321467
```

> denom <− sum(Alpha^2) * sum(Beta^2)

> denom # denominator of gamma.hat

```[1] 0.2253882
```

> num / denom # gamma.hat

```[1] -1.473665
```

> print( SSgamma <− num^2 / denom ) # SSγ

```[1] 0.489473
```

> print( SSrem <− SSAB - SSgamma ) # SSrem

```[1] 1.664410
```

> print( MSrem <− SSrem / 5 ) # MSrem

```[1] 0.3328821
```

> print( Fgamma <− SSgamma / MSrem ) # Fγ

```[1] 1.470409
```

> pf(Fgamma, 1, 5, lower=F) # Pvalγ

```[1] 0.2794440
```

> cols <− paste("dark",c("magenta","green","blue","orange4"),sep="")

> windows(width=6,height=3,pointsize=10)

> par(mfrow=c(1,2),mar=c(4,4,.5,3)+.1,oma=c(2,0,3,0),bty="l")

+   interaction.plot(pressure,temperature,strength,type="b",

+     col=cols[1:3], pch=1:3)

+   box(which="figure",col="grey80")

+   interaction.plot(temperature,pressure,strength,type="b",

+     col=cols[1:4], pch=1:4)

+   box(which="figure",col="grey80")

+ } )

> title(main="Interaction Plots", outer=T)

> box(which="outer",col="grey50")

The above codes generate the interaction plots below:

```                    Df Sum Sq Mean Sq F value  Pr(>F)
factor(pressure)     3 0.5807  0.1936  0.5392 0.67270
factor(temperature)  2 4.6576  2.3288  6.4873 0.03162 *
Residuals            6 2.1539  0.3590
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

> require(multcomp)

```Loading required package: multcomp
[1] TRUE
```

+     linfct=mcp("factor(temperature)"="Tukey"))

```         Simultaneous Tests for General Linear Hypotheses

Multiple Comparisons of Means: Tukey Contrasts

Fit: aov(formula = strength ~ factor(pressure) + factor(temperature),

Linear Hypotheses:
Estimate Std. Error t value p value
260 - 250 == 0   1.2825     0.4237   3.027  0.0526 .
270 - 250 == 0  -0.0750     0.4237  -0.177  0.9829
270 - 260 == 0  -1.3575     0.4237  -3.204  0.0423 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

```         Simultaneous Confidence Intervals for General Linear Hypotheses

Multiple Comparisons of Means: Tukey Contrasts

Fit: aov(formula = strength ~ factor(pressure) + factor(temperature),

Estimated Quantile = 2.5165

Linear Hypotheses:
Estimate lwr     upr
260 - 250 == 0  1.2825   0.2164  2.3486
270 - 250 == 0 -0.0750  -1.1411  0.9911
270 - 260 == 0 -1.3575  -2.4236 -0.2914

90% family-wise confidence level
```

```         Simultaneous Confidence Intervals for General Linear Hypotheses

Multiple Comparisons of Means: Tukey Contrasts

Fit: aov(formula = strength ~ factor(pressure) + factor(temperature),

Estimated Quantile = 3.068

Linear Hypotheses:
Estimate lwr      upr
260 - 250 == 0  1.28250 -0.01731  2.58231
270 - 250 == 0 -0.07500 -1.37481  1.22481
270 - 260 == 0 -1.35750 -2.65731 -0.05769

95% family-wise confidence level
```

> windows(width=5,height=3,pointsize=10)

> par(mar=c(5,5,4,1)+.1)

+     ylim=c(.5,3.5))

> mtext("Tukey's Method for Factor temperature",side=3,line=0.5)

The codes above generate a plot of Tukey's 95% confidence intervals.

> summary(lm(strength~factor(pressure)+poly(temperature,2),

```Call:
lm(formula = strength ~ factor(pressure) + poly(temperature,

Residuals:
Min      1Q  Median      3Q     Max
-0.6575 -0.4902  0.1233  0.3067  0.6400

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)             9.9600     0.3459  28.793 1.16e-07 ***
factor(pressure)130    -0.1733     0.4892  -0.354   0.7352
factor(pressure)140    -0.4700     0.4892  -0.961   0.3738
factor(pressure)150     0.1133     0.4892   0.232   0.8245
poly(temperature, 2)1  -0.1061     0.5991  -0.177   0.8653
poly(temperature, 2)2  -2.1556     0.5991  -3.598   0.0114 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.5991 on 6 degrees of freedom
Multiple R-Squared: 0.7086,     Adjusted R-squared: 0.4658
F-statistic: 2.918 on 5 and 6 DF,  p-value: 0.1124
```