Informal ANOVA Example

The Yarn Breaks Data

To demonstrate informal ANOVA, we will use the yarn breaks data set, called warpbreaks. (To roll with the “knitting” in R theme! Lol). Convieniently, this data is already loaded into R. It exists in the background. To look at it, simply run the following command.

View(warpbreaks)

There are two structural factors: wool type (2 levels: A and B), and tension (3 levels: L, M, and H). L, M, and H, correspond to “low”, “medium”, and “high”, respectively. These two factors are crossed, creating 6 conditions, or cells. There are 9 looms for each of the 6 conditions, for a total of 54 looms tested. The response variable is called breaks. This is the number of breaks for each loom.

You can read more about the data set using the following command:

?warpbreaks

What kind of design is this? A two-way basic factorial design, BF[2]. The same design you are using to MP1.

If you would like to get it into your enviroment, use the data function. It will first show up as <Promise>, but as soon as you start using it, it will appear as a data frame.

data(warpbreaks)

Paralell Dot Graph

Paralell Dot Graphs are used to get a sense of the variability within and bewteen the six conditions. We will use the ggplot2 package to make our graph.

ggplot(warpbreaks, aes(x = tension, y = breaks, color = wool)) +
  geom_point()

Alternatively, you can make a side-by-side boxplot to view the within and between variability. Each type of visualization has its pros and cons.

ggplot(warpbreaks, aes(x = tension, y = breaks, color = wool)) +
  geom_boxplot()

As can be seen in both figures, the most breaks occur for the looms using low tension with wool type A. However, it also looks like there is more variability for this condition. This is a potential violation of the Fisher assumtion of the Same standard deviations (S). Further, the largest variation goes with the group that has the largest median.

Let’s compute some summary statistics to get a better idea of spreads for each of the six conditions.

 sum_stats <- warpbreaks %>%
  group_by(tension, wool) %>%
  summarise(mean = mean(breaks),
            sd = sd(breaks),
            min = min(breaks),
            max = max(breaks),
            range = max - min) %>%
  arrange(range)

 sum_stats

Next, we can check how many times bigger the largest spread is than the smallest spread. We can use the range, and standard deviation as our measures of spread with the following two lines of code, respectively.

sum_stats[6,7]/sum_stats[1,7]

sum_stats[6,4]/sum_stats[1,4]

The range is 3 times as big and the standard deviation is 3.70 times as big.

Homework 4 Problems

For homework 4, complete the following exercises. When you are done, please knit your homework to an HTML file and submit the HTML file on Moodle.

  1. List the structural factors for the data set in Fig. 2.12 (above). No need to draw the factor diagram.

  2. Create a parallel dot graph, and a side-by-side boxplot for the data in Fig 2.12 (run the code provided below to get the data into R).

hamster <- rep(c(4, 1, 7, 2, 3, 8, 5, 6), 2)
day_length <- c(rep("long", 4), rep("short", 4), rep("long", 4), rep("short", 4))
organ <- c(rep("heart", 8), rep("brain", 8))
enzyme <- c(246, 169, 186, 183, 468, 390, 314, 508, 394, 297, 216, 373, 216, 198, 194, 192)

hamsters <- data.frame(hamster, day_length, organ, enzyme)

  1. Compute the average, standard deviation (SD), and range using the same four groups as in Section 2.3 (see Table 2.2 on pg. 49) and comment on what you find. Are the four ranges roughly equal? Are the four SD’s roughly equal? Does the range and SD seem to be related to the average by a pattern?

  2. I find it helps me remember the Fisher assumptions if I remember the nonsense word CA-SINZ, California SINZ. Each letter stands for an assumption: Constant effects, Additivity, Same standard deviation, Independence, Normality, and Zero average for errors. List the four letters for assumptions about residuals.

  3. Explain how the assumption of constant effects (C) is part of the justification for computing averages.

This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was adapted by Randi Garcia from labs originally written by Mark Hansen, further adapted for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel.

Resources for learning R and working in RStudio

That was a short introduction to R and RStudio, but we will provide you with more functions and a more complete sense of the language as the course progresses.

In this course we will be using R packages called dplyr for data wrangling and ggplot2 for data visualization. If you are googling for R code, make sure to also include these package names in your search query. For example, instead of googling “scatterplot in R”, google “scatterplot in R with ggplot2”.

These cheatsheets may come in handy throughout the semester:

Note that some of the code on these cheatsheets may be too advanced for this course, however majority of it will become useful throughout the semester.