Laboratory Exercise 4: Factorial Design (Two-Factor Factorial Design)
JORIELYN S. MANLA
2023-12-05
Given Data
PROBLEM: An experiment was conducted to determine if either firing temperature or furnace position affects the baked density of a carbon anode. The data are shown below.
| Position | 800°C | 825°C | 850°C |
|---|---|---|---|
| 1 | 570 | 1063 | 565 |
| 1 | 565 | 1080 | 510 |
| 1 | 583 | 1043 | 590 |
| 2 | 528 | 988 | 526 |
| 2 | 547 | 1026 | 538 |
| 2 | 521 | 1004 | 532 |
1. Write your experimental question. Use \(\alpha = 0.05\).
Is there a significant effect between firing temperature, furnace position and their interaction in the baked density of a carbon anode, using \(\alpha = 0.05\)?
2. Construct your null and alternative hypotheses.
The hypothesis being tested are the following:
For firing temperature:
\(H_o:\) There is no significant
main effect of firing temperature in the baked density of a carbon
anode.
\(H_a:\) There is a significant main
effect of firing temperature in the baked density of a carbon
anode.
For furnace position:
\(H_o:\) There is no significant
main effect of furnace position in the baked density of a carbon
anode.
\(H_a:\) There is a significant main
effect of furnace position in the baked density of a carbon anode.
For firing temperature and furnace position interaction:
\(H_o:\) There is no significant
interaction effect between firing temperature and furnace position in
the baked density of a carbon anode.
\(H_a:\) There is a significant
interaction effect between firing temperature and furnace position in
the baked density of a carbon anode.
3. Suppose we assume that no interaction exists.
(a) Write down the statistical model.
The model for the two-factor, no interaction model is \(Y_{ijk} =\mu + \tau_i +\beta_i +
\epsilon_{ijk}\)
where
- \(\mu\) - overall mean effect
- \(\tau_i\) - effect of the \(i\)th level of the row factor furnace
position \((A)\)
- \(\beta_i\) - effect of the \(j\)th level of the column factor firing
temperature \((B)\)
- \(\epsilon_{ijk}\) - random error
component.
(b) Conduct the analysis of variance assuming both treatments to be fixed effects.
Analysis of Variance Table
Response: obs
Df Sum Sq Mean Sq F value Pr(>F)
pos 1 7160 7160 15.998 0.001762 **
temp 2 945342 472671 1056.117 3.25e-14 ***
pos:temp 2 818 409 0.914 0.427110
Residual 12 5371 448
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1(c) What conclusions/interpretations can be drawn?
Firing Temperature:
- A two-way ANOVA was conducted to examine the effect of firing
temperature on the baked density of carbon anodes. Using \(\alpha=0.05\), the results revealed a
significant main effect of firing temperature in which \(F_{0.05, 2,12}=3.89\) \(<\) the F-value \((1056.117)\). Therefore, we reject the null
hypothesis and this suggests that the effect of temperature is
statistically significant. This means that the baked density of carbon
anodes was significantly affected by firing temperature.
Furnace Position:
- A two-way ANOVA was conducted to examine the effect of furnace
position on the baked density of carbon anodes. Using \(\alpha=0.05\), the results revealed a
significant main effect of furnace position in which \(F_{0.05, 1,12}=4.75\) \(<\) the F-value \((15.998)\). Thus, we will reject the null
hypothesis, indicating that the effect of position is statistically
significant. This means that the baked density of carbon anodes was
significantly affected by furnace position.
Position:Temperature Interaction:
- A two-way ANOVA was conducted to examine the interaction effect
between firing temperature and furnace position on the baked density of
carbon anodes. The results revealed no significant interaction effect
between firing temperature and furnace position, since the critical
value \(F_{0.05, 2,12}=3.89 >\) the
F-value \((0.914)\), this means that we
failed to reject the null hypothesis. This suggests that the effect of
firing temperature on the baked density of carbon anodes was consistent
across all furnace positions.
Conclusion:
- The results show that both firing temperature and furnace position have significant effects on the baked density of carbon anodes. Furthermore, no significant interaction effect between firing temperature and furnace position was found, suggesting that the effect of firing temperature on the baked density of carbon anodes was consistent across all furnace positions.
(d) Derive the expected mean squares.
The expected mean squares \((EMS)\) provide insights into the expected variability for each factor influencing the observed outcome. In a case without interaction effects, there are two main effects, one associated with each factor, and an error term. Consequently, the expected mean square for furnace position is 7160, while the expected mean square for firing temperature is 472671, and the expected mean square for the error term is 448.
(e) Comment on model adequacy.
- First the residual mean square (448) is relatively small compared to
the mean squares for the main effects, suggesting that the model fits
the data well.
- The model appears to adequately explain the variability in the
response variable (observations) based on the factors considered
(Position and Temperature).
- The main effects of furnace position and firing temperature are both
statistically significant, indicating that they contribute to explaining
the variability in baked density, also indicating that the model as a
whole is a good fit for the data. The lack of significance for the
interaction term suggests that the effect of one factor is not dependent
on the level of the other factor.
- Therefore, based on the ANOVA output, the model appears to be adequate for explaining the observed variation in the data. The significant main effects of both position and temperature, along with the non-significant interaction effect, suggest that the model captures the essential relationships between the independent variables and the dependent variable.
The data is generally normally distributed with a few outliers. We
should consider discounting observations 6 and 9.
The data sets for each treatment generally have uniform variance. The
model is adequate without transformation or non-parametric tests.
The same plot is shown below:
4. Perform a post-hoc test using Tukey multiple comparisons of means. Use \(\alpha = 0.05\) . Provide interpretations of your findings.
Since there is no significant interaction between furnace position
and firing temperature, then we only compare the means between each
factor.
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = obs ~ pos + temp + pos * temp)
$pos
diff lwr upr p adj
2-1 -39.88889 -61.61776 -18.16002 0.0017624The table above shows and suggests that the two furnace position
differs by \(-39.88889\) with the
corresponding p-value \((0.0017624)\)
which is less than to our significance level \(\alpha=0.05\), hence this difference is
statistically significant.
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = obs ~ pos + temp + pos * temp)
$temp
diff lwr upr p adj
825-800 481.666667 449.08101 514.25232 0.0000000
850-800 -8.833333 -41.41899 23.75232 0.7547952
850-825 -490.500000 -523.08566 -457.91434 0.0000000Here, we observe that all pairs of means differ. Meanwhile, the pair
of temperatures \(850\) and \(800\) is the only one with a
non-significant difference. The pairs \(825−800\) and \(850−825\) exhibit statistically significant
differences.
5. Give details on the syntax used to produce your answer.
First, export the data from excel using library(readxl).
setwd("E:/Experimental Design")dats<-read_xlsx("E:/Experimental Design//Problem Set 4.xlsx")To customize the table of the data, we use library(knitr) and
library(kableExtra) and manipulate the design of the table.
kable(dats, format = "html") %>%
kable_styling(full_width = FALSE) %>%
row_spec(0, bold = TRUE, color = "black", background = "skyblue") %>%
row_spec(1:4, background = "white")We use the package GAD which contains functions for the analysis of any complex ANOVA models with any combination of orthogonal/nested and fixed/random factors. We input our data into a variable and specify it as fixed using as.fixed. Then we use the function aov() and store it in the variable model to compute the estimation of model parameters. To create the ANOVA table, we use the function gad(model).
obs<- c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,1004,532)
temp<- c(800,825,850)
pos<-c(rep(1,9),rep(2,9))
temp<-c(rep(temp,6))
pos <- as.fixed(pos)
temp <- as.fixed(temp)
model <- aov(obs~pos+temp+pos*temp)
GAD::gad(model)To plot the model, we use the function plot(), in order to check on model adequacy.
plot(model,1)
plot(model,2)check_model_adequacy <- function(model) {
# Diagnostic plots
par(mfrow = c(2, 2))
plot(model)
plot(model$fitted.values, model$residuals, main = "Residuals vs Fitted",
xlab = "Fitted Values", ylab = "Residuals")
qqnorm(model$residuals, main = "Normal Q-Q Plot")
qqline(model$residuals)
plot(sqrt(abs(model$residuals)), model$residuals, main = "Scale-Location Plot",
xlab = "Fitted Values", ylab = "sqrt(|Residuals|)")
}
model <- aov(obs ~ pos + temp + pos*temp)
check_model_adequacy(model)To compute for Post-hoc test using Tukey multiple comparisons of
means with \(\alpha= 0.05\), we use the
function TukeyHSD() and store in the variable tukey_result.
tukey_res= TukeyHSD(model, "pos")
tukey_restukey_res= TukeyHSD(model, "temp")
tukey_res