Frequently asked questions

No attempt is made to fully explain the following concepts, but hopefully this gets you started. The Internet has plenty of resources on these topics if you would like to learn more.

What is a trimmed mean?

The trimmed mean involves calculating the sample mean after removing a proportion of values from each tail of the distribution. In symbols the trimmed mean is expressed as follows:

$$\bar{X}_t = \frac{X_{(g+1)}\,+,...,+\,X_{(n-g)}}{n-2g}$$

where $X_1, \,X_2,\,...\,,X_n$ is a random sample and $X_{(1)}, \le X_{(2)}\,,...,\,\le X_{(n)}$ are the observations in ascending order. The proportion to trim is $\gamma\,(0\lt \gamma \lt.5)$ and $g = [ \gamma n ]$ rounded down to the nearest integer.

What is bootstrapping?

In the context of hypothesis testing and generally speaking, bootstrapping involves taking many random samples (with replacement) from the data at hand in order to estimate a sampling distribution of interest. This is in contrast to traditional methods which assume the shape of the particular sampling distribution under study. Once we have an emprically derived sampling distribution, obtaining CIs and p values is relatively straightforward.

What is a contrast matrix?

First, it is helpful to imagine your design arranged into a JxK matrix.

$$A=\begin{bmatrix} a_{1,1} & a_{1,2} & ... & a_{1,K} \\ a_{2,1} & a_{2,2} & ... & a_{2,K} \\ a_{J,1} & a_{J,2} & ... & a_{J,K} \end{bmatrix}$$

A contrast matrix specifies which cells (or elements) in the above design are to be compared. The rows in a contrast matrix correspond to the cells in your design. The columns correspond to the contrasts that you wish to make.

Examples of contrast matrices for different designs

Matrix notation is used to explain which cells are being compared, followed by the corresponding contrast matrix.

design with 2 groups

${a_{1,1} - a_{1,2}}$

contrast 1
1
-1

design with 3 groups

1. $\Large{a_{1,1} - a_{1,2}}$
2. $\Large{a_{1,1} - a_{1,3}}$
3. $\Large{a_{1,2} - a_{1,3}}$

contrast 1	contrast 2	contrast 3
1	1	0
-1	0	1
0	-1	-1

2x2 design

Factor A

$\Large{(a_{1,1} + a_{1,2})-(a_{2,1} + a_{2,2})}$

contrast 1
1
1
-1
-1

Factor B

$\Large{(a_{1,1} + a_{2,1})-(a_{1,2} + a_{2,2})}$

contrast 1
1
-1
1
-1

Interaction

$\Large{(a_{1,1} + a_{2,2})-(a_{1,2} + a_{2,1})}$

That is, the difference of the differences

contrast 1
1
-1
-1
1

2x3 design

Factor A

$\Large{(a_{1,1} + a_{1,2} + a_{1,3})-(a_{2,1} + a_{2,2} + a_{2,3})}$

contrast 1
1
1
1
-1
-1
-1

Factor B

1. $\Large{(a_{1,1} + a_{2,1})-(a_{1,2} + a_{2,2})}$
2. $\Large{(a_{1,1} + a_{2,1})-(a_{1,3} + a_{2,3})}$
3. $\Large{(a_{1,2} + a_{2,2})-(a_{1,3} + a_{2,3})}$

contrast 1	contrast 2	contrast 3
1	1	0
-1	0	1
0	-1	-1
1	1	0
-1	0	1
0	-1	-1

Interactions

1. $\Large{(a_{1,1} + a_{2,2})-(a_{1,2} + a_{2,1})}$
2. $\Large{(a_{1,1} + a_{2,3})-(a_{1,3} + a_{2,1})}$
3. $\Large{(a_{1,2} + a_{2,3})-(a_{1,3} + a_{2,2})}$

contrast 1	contrast 2	contrast 3
1	1	0
-1	0	1
0	-1	-1
-1	-1	0
1	0	-1
0	1	1

Not a fan of contrast matrices?

Don't worry, Hypothesize can generate all linear contrasts automatically (see functions con1Way and con2way). However, it is useful to understand this concept so that you know which comparisons are being made and how to specify your own if necessary.