4.9: Practice problems
- Page ID
- 33247
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)4.1. Mathematics Usage Test Scores Analysis To practice the Two-Way ANOVA, consider a data set on \(N = 861\) ACT Mathematics Usage Test scores from 1987. The test was given to a sample of high school seniors who met one of three profiles of high school mathematics course work: (a) Algebra I only; (b) two Algebra courses and Geometry; and (c) two Algebra courses, Geometry, Trigonometry, Advanced Mathematics, and Beginning Calculus. These data were generated from summary statistics for one particular form of the test as reported by Doolittle and Welch (1989). The source of this version of the data set is Ramsey and Schafer (2012) and the Sleuth3 package (F. L. Ramsey et al. 2019). First install and then load that package.
library(Sleuth3)
library(mosaic)
library(tibble)
math <- as_tibble(ex1320)
math
names(math)
favstats(Score ~ Sex + Background, data = math)4.1.1. Use the favstats summary to discuss whether the design was balanced or not.
4.1.2. Make a pirate-plot and interaction plot array of the results and discuss the relationship between Sex, Background, and ACT Score.
4.1.3. Write out the interaction model in terms of the Greek letters, making sure to define all the terms and don’t forget the error terms in the model.
4.1.4. Fit the interaction plot and find the ANOVA table. For the test you should consider first (the interaction), write out the hypotheses, report the test statistic, p-value, distribution of the test statistic under the null, and write a conclusion related to the results of this test.
4.1.5. Re-fit the model as an additive model (why is this reasonable here?) and use Anova to find the Type II sums of squares ANOVA. Write out the hypothesis for the Background variable, report the test statistic, p-value, distribution of the test statistic under the null, and write a conclusion related to the results of this test. Make sure to discuss the scope of inference for this result.
4.1.6. Use the effects package to make a term-plot from the additive model from 4.5 and discuss the results. Specifically, discuss what you can conclude about the average relationship across both sexes, between Background and average ACT score?
4.1.7. Add partial residuals to the term-plot and make our standard diagnostic plots and assess the assumptions using these plots. Can you assess independence using these plots? Discuss this assumption in this situation.
4.1.8. Use the term-plot and the estimated model coefficients to determine which of the combinations of levels provides the highest estimated average score.
4.2. Sleep Quality Analysis As a second example, consider data based on Figure 3 from Puhan et al. (2006), which is available at http://www.bmj.com/content/332/7536/266. In this study, the researchers were interested in whether didgeridoo playing might impact sleep quality (and therefore daytime sleepiness). They obtained volunteers and they randomized the subjects to either get a lesson or be placed on a waiting list for lessons. They constrained the randomization based on the high/low apnoea and high/low on the Epworth scale of the subjects in their initial observations to make sure they balanced the types of subjects going into the treatment and control groups. They measured the subjects’ Epworth value (daytime sleepiness, higher is more sleepy) initially and after four months, where only the treated subjects (those who took lessons) had any intervention. We are interested in whether the mean Epworth scale values changed differently over the four months in the group that got didgeridoo lessons than it did in the control group (that got no lessons). Each subject was measured twice in the data set provided that is available at http://www.math.montana.edu/courses/s217/documents/epworthdata.csv.
library(readr)
epworthdata <- read_csv("http://www.math.montana.edu/courses/s217/documents/epworthdata.csv")
epworthdata <- epworthdata %>% mutate(Time = factor(Time),
Group = factor(Group)
)
levels(epworthdata$Time) <- c("Pre" , "Post")
levels(epworthdata$Group) <- c("Control" , "Didgeridoo")4.2.1. Make a pirate-plot and an interaction plot array to graphically explore the potential interaction of Time and Group on the Epworth responses.
4.2.2. Fit the interaction model and find the ANOVA table. For the test you should consider first (the interaction), write out the hypotheses, report the test statistic, p-value, distribution of the test statistic under the null, and write a conclusion related to the results of this test.
4.2.3. Discuss the independence assumption for the previous model. The researchers used an analysis based on matched pairs. Discuss how using ideas from matched pairs might be applicable to the scenario discussed here.
4.2.4. Refine the model based on the previous test result and continue refining the model as the results might suggest. This should lead to retaining just a single variable. Make term-plot plot for this model and discuss this result related to the intent of the original research. If you read the original paper, they did find evidence of an effect of learning to play the didgeridoo (that there was a different change over time in the treated control when compared to the control group) – why might they have gotten a different result (hint: think about the previous question).
Note that the didgeridoo example is revisited in the case-studies in Chapter 9 with some information on an even better way to analyze these data.
References
- We would not suggest throwing away observations to get balanced designs. Plan in advance to try to have a balanced design but analyze the responses you get.↩︎
- Github.com is a version control system used for software development and collaborative work, which we used to allow us to make changes to it and track the modifications. This book is also written using github to allow the same connection for writing and editing it, and one location where the digital version is hosted: https://greenwood-stat.github.io/GreenwoodBookHTML/.↩︎
- Copy and include this code in the first code chunk in any document where you want to use the
intplotorinplotarrayfunctions.↩︎ - We will use “main effects” to refer to the two explanatory variables in the additive model even if they are not randomly assigned to contrast the terminology with having those variables involved in an interaction term in the model. It is the one place in the book where we use “effects” without worrying about the causal connotation of that word.↩︎
- In the standard ANOVA table, \(\text{SS}_A + \text{SS}_B + \text{SS}_{AB} + \text{SS}_E = \text{SS}_{\text{Total}}\). However, to get the tests we really desire when our designs are not balanced, a slight modification of the SS is used, using what are called Type II sums of squares and this result doesn’t hold in the output you will see for additive models. This is discussed further below.↩︎
- This does not mean that there is truly no interaction in the population but does mean that we are going to proceed assuming it is not present since we couldn’t prove the null was wrong.↩︎
- The
anovaresults are not wrong, just not what we want in all situations.↩︎ - Actually, the tests are only conditional on other main effects if Type II Sums of Squares are used for an interaction model, but we rarely focus on the main effect tests when the interaction is present.↩︎
- In Multiple Linear Regression models in Chapter 8, the reasons for this wording will (hopefully) become clearer.↩︎
- This goes beyond our considerations with character variables that have text levels but are not declared as factors in the first chapters. Those often will be modeled correctly in linear models whether they are characters or factors – but numerical variables will be modeled in a way that you did not intend for these predictors that we will discuss in Chapters 7 and 8.↩︎
- Just so you don’t think that perfect R code should occur on the first try, I have made similarly serious coding mistakes even after accumulating more than decade of experience with R. It is finding those mistakes (in time) that matters.↩︎
- To get
dosefon the x-axis in the plot, thex.var = "dosef"option was employed to force the Dose to be the variable on the x-axis.↩︎ - We can also use select to only retain these three variables and then
drop_na()to get the same result for these three variables.↩︎ - Correctly accounting for these missing data is a complex topic and you should not always engage
drop_na(), but the first step to handling missing data issues is to find out (1) if you have an issue, (2) how prevalent it is, and (3) whether it is systematic in any way – in other words (and to date myself), “knowing is half the battle” with missing data. Consult a statistician or take more advanced statistics courses to explore this challenging topic further.↩︎ - We switched back to the
anovafunction here as theAnovafunction only reportsError in Anova.lm(lm(responses ~ dropsf * brand, data = ptR)) : residual df = 0, which is fine but not as useful for understanding this issue as whatanovaprovides.↩︎ - This package is not on the “CRAN” repository and from time to time involves more complex installation requirements to install it from its “github” repository and some packages it depends on. In order to install this package, we usually can use the following code after installing the
remotes(Csárdi et al. 2021) package in the regular way:library(remotes); remotes::install_github("mtennekes/tabplot")↩︎ - In larger data sets, multiple subjects are displayed in each row as proportions of the rows in each category.↩︎
- Quantitative variables are displayed with boxplot-like bounds to describe the variability in the variable for that row of responses for larger data sets.↩︎