# 3.10: Practice problems

- Page ID
- 33237

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## 3.10 Practice problems

3.1. **Cholesterol Analysis** For the first practice problems, you will work with the cholesterol data set from the `multcomp`

package that was used to generate the Tukey’s HSD results. To load the data set and learn more about the study, use the following code:

```
library(multcomp)
data(cholesterol)
library(tibble)
cholesterol <- as_tibble(cholesterol)
help(cholesterol)
```

3.1.1. Graphically explore the differences in the changes in Cholesterol levels for the five levels using pirate-plots.

3.1.2. Is the design balanced? Generate R output to support this assessment.

3.1.3. Complete all 6+ steps of the hypothesis test using the parametric \(F\)-test, reporting the ANOVA table and the distribution of the test statistic under the null. When you discuss the scope of inference, make sure you note that the treatment levels were randomly assigned to volunteers in the study.

3.1.4. Generate the permutation distribution and find the p-value. Compare the parametric p-value to the permutation test results.

3.1.5. Perform Tukey’s HSD on the data set. Discuss the results – which pairs were detected as different and which were not? Bigger reductions in cholesterol are good, so are there any levels you would recommend or that might provide similar reductions?

3.1.6. Find and interpret the CLD and compare that to your interpretation of results from 3.1.5.

3.2. **Sting Location Analysis** These data come from (Smith 2014) where the author experimented on himself by daily stinging himself five times on randomly selected body locations over the course of months. You can read more about this fascinating (and cringe inducing) study at https://peerj.com/articles/338/. The following code gets the data prepared for analysis by removing the observations he took each day on how painful it was to sting himself on his forearm before and after the other three observations that were of interest each day of the study. This is done with a negation (using “!” of the `%in%`

which identifies rows related to the two daily forearm locations (`Forearm`

and `Forearm1`

) to leave all the rows in the data set for any levels of `Body_Location`

that were not in these two levels. This is easier than trying to list all 24 other levels, then `Body_Location`

variable is re-factored to clean out its unused levels, and finally the `reorder`

function is used to order the levels based on the sample mean pain rating – and the results of these steps are stored in the `sd_fixedR`

tibble.

```
library(readr)
sd_fixed <- read_csv("http://www.math.montana.edu/courses/s217/documents/stingdata_fixed.csv")
sd_fixedR <- sd_fixed %>%
filter(!(Body_Location %in% c("Forearm", "Forearm1"))) %>%
mutate(Body_Location = factor(Body_Location),
Body_Location = reorder(Body_Location, Rating, FUN = mean)
)
```

3.2.1. Graphically explore the differences in the pain ratings (`Rating`

) across the different `Body_Location`

levels using boxplots and pirate-plots. How are boxplots misleading for representing these data? **Hint**: look for discreteness in the responses.

3.2.2. Is the design balanced?

3.2.3. How does taking 3 measurements that are of interest each day lead to a violation of the independence assumption here?

3.2.4. Complete all 6+ steps of the hypothesis test using the parametric \(F\)-test, reporting the ANOVA table and the distribution of the test statistic under the null. For the scope of inference use the information that the sting locations were randomly assigned but only one person (the researcher) participated in the study.

3.2.5. Generate the permutation distribution and find the p-value. Compare the parametric p-value to the permutation test results.

3.2.6. Generate an effects plot (use something like `plot(allEffects(lm_model), rotx = 45)`

to rotate the x-axis text 45 degrees so you can read it!). Which of the locations did he find most painful on average?

3.2.7. Generated our standard panel of diagnostic plots. In the QQ-Plot, you should see a stair-step pattern that presents a violation of the normality assumption that we have not see before. Look at your answer to 3.2.1 and try to explain why this pattern is present.

3.2.8. Often we might consider Tukey’s pairwise comparisons given the initial result here. How many levels are there in `Body_Location`

in the filtered data set? How many pairs would be compared if we tried Tukey’s – calculate this using the `choose`

function?

### References

*The Journal of Nutrition*33 (5): 491–504. http://jn.nutrition.org/content/33/5/491.full.pdf.

*Effects: Effect Displays for Linear, Generalized Linear, and Other Models*. https://CRAN.R-project.org/package=effects.

*Biometrical Journal*50 (3): 346–63.

*Multcomp: Simultaneous Inference in General Parametric Models*. https://CRAN.R-project.org/package=multcomp.

*Journal of Computational and Graphical Statistics*13 (2): 456–66.

*PeerJ*2 (April): e338. https://doi.org/10.7717/peerj.338.

- In Chapter 4, methods are discussed for when there are two categorical explanatory variables that is called the Two-Way ANOVA and related ANOVA tests are used in Chapter 8 for working with extensions of these models.↩︎
- In Chapter 2, we used
`lm`

to get these estimates and focused on the estimate of the difference between the second group and the baseline – that was and still is the difference in the sample means. Now there are potentially more than two groups and we need to formalize notation to handle this more complex situation.↩︎ - If you look closely in the code for the rest of the book, any model for a quantitative response will use this function, suggesting a common thread in the most commonly used statistical models.↩︎
- We can and will select the order of the levels of categorical variables as it can make plots easier to interpret.↩︎
- Suppose we were doing environmental monitoring and were studying asbestos levels in soils. We might be hoping that the mean-only model were reasonable to use if the groups being compared were in remediated areas and in areas known to have never been contaminated.↩︎
- Make sure you can work from left to right and top down to fill in the ANOVA table given just the necessary information to determine the other components or from a study description to complete the
*DF*part of the table – there are always questions like these on exams…↩︎ - Any further claimed precision is an exaggeration and eventually we might see p-values that approach the precision of the computer at 2.2e-16 and anything below 0.0001 should just be reported as being below 0.0001. Also note the way that R represents small or extremely large numbers using scientific notation such as
`3e-4`

which is \(3 \cdot 10^{-4} = 0.0003\).↩︎ - This would be another type of publication bias – where researchers search across groups and only report their biggest differences and fail to report the other pairs that they compared. As discussed before, this biases the results to detecting results more than they should be and then when other researchers try to repeat the same studies and compare just, say, two groups, they likely will fail to find similar results unless they also search across many different possible comparisons and only report the most extreme. The better approach is to do the ANOVA \(F\)-test first and then Tukey’s comparisons and report all these results, as discussed below.↩︎
- You need to use this command for linear model diagnostics or you won’t get the plots we want from the model. And you really just need
`plot(lm2)`

but the`pch = 16`

option makes it easier to see some of the points in the plots.↩︎ - Along with multiple names, there is variation of what is plotted on the x and y axes, the scaling of the values plotted, and even the way the line is chosen to represent the 1-1 relationship, increasing the challenge of interpreting QQ-plots. We are consistent about the x and y axis choices throughout this book and how the line is drawn but different versions of these plots do vary in what is presented, so be careful with using QQ-plots.↩︎
- Here this means re-scaled so that they should have similar scaling to a standard normal with mean 0 and standard deviation 1. This does not change the shape of the distribution but can make outlier identification simpler – having a standardized residual more extreme than 5 or -5 would suggest a deviation from normality since we rarely see values that many standard deviations from the mean in a normal distribution. But mainly focus on the pattern in points in the QQ-plot and whether it matches the 1-1 line that is being plotted.↩︎
- A resistant procedure is one that is not severely impacted by a particular violation of an assumption. For example, the median is resistant to the impact of an outlier. But the mean is not a resistant measure as changing the value of a single point changes the mean.↩︎
- A violation of the independence assumption could have easily been created if they measured cells in two locations on each guinea pig or took measurements over time on each subject.↩︎
- Note that to see all the group labels in the plot when making the figure, you have to widen the plot window before copying the figure out of R. You can resize the plot window using the small vertical and horizontal “=” signs in the grey bars that separate the different panels in RStudio.↩︎
- In working with researchers on hundreds of projects, my experience has been that many conversations are often required to discover all the potential sources of issues in data sets, especially related to assessing independence of the observations. Discussing how the data were collected is sometimes the only way to understand whether violations of independence are present or not.↩︎
- When this procedure is used with unequal group sizes it is also sometimes called Tukey-Kramer’s method.↩︎
- We often use “spurious” to describe falsely rejected null hypotheses, but they are also called false detections.↩︎
- In more complex models, this code can be used to create pair-wise comparisons on one of many explanatory variables.↩︎
- The plot of results usually contains all the labels of groups but if the labels are long or there many groups, sometimes the row labels are hard to see even with re-sizing the plot to make it taller in RStudio. The numerical output is useful as a guide to help you read the plot in those situations.↩︎
- Note that this method is implemented slightly differently than explained here in some software packages so if you see this in a journal article, read the discussion carefully.↩︎
- There is a warning message produced by the default Tukey’s code here related to the algorithms used to generate approximate p-values and then the CLD, but the results seem reasonable and just a few p-values seem to vary in the second or third decimal points.↩︎