Question

3. Using the R data set called warpbreaks (See ?warpbreaks for more info), we want to compare the mean breaks across both the different types of wool and the different levels of tension. In this problem, use ?? = 0.10.

a. Make a boxplot to compare breaks across both wool and tension. Color-code the three different tension levels for easier visibility. Within wool A, describe the relationship between tension and breaks. Within wool B, describe the relationship between tension and breaks.

b. Run a two-way ANOVA to compare breaks across both wool and tension, including their interaction. Is the interaction significant? Which main effects are significant?

c. Regardless of your answer to part b, make an interaction plot (color coding might help, but is not required) and interpret it.

d. Run a one-way ANOVA (or a two-sample ??-test) to compare breaks across just wool. What is the result of the test, and how do you reconcile that with our previous results?

Answers should be in the form of R code on how to accomplish each part and include the correct statistical explanation for those that require it in the question. Please be as thorough as possible. Thank you so much!!!

Answer 1

R-commands and outputs:

d=warpbreaks
head(d)
breaks wool tension
1 26 A L
2 30 A L
3 54 A L
4 25 A L
5 70 A L
6 52 A L

brk=d$breaks
wool=d$wool
tension=d$tension

# a)
?boxplot
boxplot(brk~wool+tension, col=c(3,3,7,7,2,2))

# There is high variability between most pairs of wool and tension, especially wool A and tension L.
# This variability may be inherent in the data itself OR it is due to small sample size n=9 for each pair. This question should be thought here.
# Within wool A, the relationship between tension and breaks is MORE VARIED as compared to that witin wool B. This is indicated by larger length of rectangle in the plot as well the whiskers.

# b)
## Two-way ANOVA to compare breaks across both wool and tension, including their interaction
ANOVA=aov(breaks~wool+tension+wool*tension,data=d)
ANOVA
Call:
aov(formula = breaks ~ wool + tension + wool * tension, data = d)
Terms:
wool tension wool:tension Residuals
Sum of Squares 450.667 2034.259 1002.778 5745.111
Deg. of Freedom 1 2 2 48
Residual standard error: 10.94028
Estimated effects may be unbalanced

s=summary(ANOVA)
s
Df Sum Sq Mean Sq F value Pr(>F)
wool 1 451 450.7 3.765 0.058213 .
tension 2 2034 1017.1 8.498 0.000693 ***
wool:tension 2 1003 501.4 4.189 0.021044 *
Residuals 48 5745 119.7   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

## alpha= 0.10
## H0I: Interaction effect is insignificant.
## H0A: main effect A (wool) is insignificant.
## H0B: main effect B (wool) is insignificant.
## Clearly, for interaction (wool:tension) effect, p-value is 0.021044, which is less than alpha. Therefore, we Reject H0I.
## Conclude that Interaction is significant.
## Also,for main effect A (wool), p-value is 0.058213, which is less than alpha(0.10). Thus, we Reject H0A.
## For main effect B (tension), p-value is 0.000693, which is less than alpha(0.10). Thus, we Reject H0B.
## Conclude that both main effects are significant.

# c)
## An interaction between factors occurs when the change in response from the low level to the high level of one factor is not the same as the change in response at the same two levels of a second factor. That is, the effect of one factor is dependent upon a second factor. You can use interaction plots to compare the relative strength of the effects across factors.
interaction.plot(tension,wool,brk) #Response=brk

## From Interaction plot, we can see that it appears that wool A has a decrease in breaks between low(L) and medium(M) tension, while wool B has a decrease in breaks between medium(M) and high(H).

# d)
oneANOVA=aov(breaks~wool,data=d)
oneANOVA
Call:
aov(formula = breaks ~ wool, data = d)
Terms:
wool Residuals
Sum of Squares 450.667 8782.148
Deg. of Freedom 1 52
Residual standard error: 12.99567
Estimated effects may be unbalanced

ones=summary(oneANOVA)
ones
Df Sum Sq Mean Sq F value Pr(>F)
wool 1 451 450.7 2.668 0.108
Residuals 52 8782 168.9   

# From the output of summary command, p-value for wool is 0.108 which is greater (though slightly) than alpha=0.1. We fail to Reject H0.
## Conclusion: 'wool' is not significant effect.
## This result is different from the result of two-way anova.