Section author: Danielle J. Navarro and David R. Foxcroft

Different ways to specify contrasts

In the previous section, I showed you a method for converting a factor into a collection of contrasts. In the method I showed you we specify a set of binary variables in which we defined a table like this one:

drug druganxifree drugjoyzepam
placebo 0 0
anxifree 1 0
joyzepam 0 1

Each row in the table corresponds to one of the factor levels, and each column corresponds to one of the contrasts. This table, which always has one more row than columns, has a special name. It is called a contrast matrix. However, there are lots of different ways to specify a contrast matrix. In this section I discuss a few of the standard contrast matrices that statisticians use and how you can use them in jamovi. If you’re planning to read the section on unbalanced ANOVA later on (section Factorial ANOVA 3: unbalanced designs), it’s worth reading this section carefully. If not, you can get away with skimming it, because the choice of contrasts doesn’t matter much for balanced designs.

Treatment contrasts

In the particular kind of contrasts that I’ve described above, one level of the factor is special, and acts as a kind of “baseline” category (i.e., placebo in our example), against which the other two are defined. The name for these kinds of contrasts is treatment contrasts, also known as “dummy coding”. In this contrast each level of the factor is compared to a base reference level, and the base reference level is the value of the intercept.

The name reflects the fact that these contrasts are quite natural and sensible when one of the categories in your factor really is special because it actually does represent a baseline. That makes sense in our clinicaltrial data. The placebo condition corresponds to the situation where you don’t give people any real drugs, and so it’s special. The other two conditions are defined in relation to the placebo. In one case you replace the placebo with Anxifree, and in the other case your replace it with Joyzepam.

The table shown above is a matrix of treatment contrasts for a factor that has 3 levels. But suppose I want a matrix of treatment contrasts for a factor with 5 levels? You would set this out like this:

Level   2 3 4 5
  1     0 0 0 0
  2     1 0 0 0
  3     0 1 0 0
  4     0 0 1 0
  5     0 0 0 1

In this example, the first contrast is level 2 compared with level 1, the second contrast is level 3 compared with level 1, and so on. Notice that, by default, the first level of the factor is always treated as the baseline category (i.e., it’s the one that has all zeros and doesn’t have an explicit contrast associated with it). In jamovi you can change which category is the first level of the factor by manipulating the order of the levels of the variable shown in the Data Variable window (double click on the name of the variable in the spreadsheet column to bring up the Data Variable view).

Helmert contrasts

Treatment contrasts are useful for a lot of situations. However, they make most sense in the situation when there really is a baseline category, and you want to assess all the other groups in relation to that one. In other situations, however, no such baseline category exists, and it may make more sense to compare each group to the mean of the other groups. This is where we meet Helmert contrasts, generated by the Helmert option in the jamovi ANOVA - Contrasts selection box. The idea behind Helmert contrasts is to compare each group to the mean of the “previous” ones. That is, the first contrast represents the difference between group 2 and group 1, the second contrast represents the difference between group 3 and the mean of groups 1 and 2, and so on. This translates to a contrast matrix that looks like this for a factor with five levels:

1   -1   -1   -1   -1
2    1   -1   -1   -1
3    0    2   -1   -1
4    0    0    3   -1
5    0    0    0    4

One useful thing about Helmert contrasts is that every contrast sums to zero (i.e., all the columns sum to zero). This has the consequence that, when we interpret the ANOVA as a regression, the intercept term corresponds to the grand mean µ.. if we are using Helmert contrasts. Compare this to treatment contrasts, in which the intercept term corresponds to the group mean for the baseline category. This property can be very useful in some situations. It doesn’t matter very much if you have a balanced design, which we’ve been assuming so far, but it will turn out to be important later when we consider unbalanced designs in section Factorial ANOVA 3: unbalanced designs. In fact, the main reason why I’ve even bothered to include this section is that contrasts become important if you want to understand unbalanced ANOVA.

Sum to zero contrasts

The third option that I should briefly mention are “sum to zero” contrasts, called “Simple” contrasts in jamovi, which are used to construct pairwise comparisons between groups. Specifically, each contrast encodes the difference between one of the groups and a baseline category, which in this case corresponds to the first group:

1   -1   -1   -1   -1
2    1    0    0    0
3    0    1    0    0
4    0    0    1    0
5    0    0    0    1

Much like Helmert contrasts, we see that each column sums to zero, which means that the intercept term corresponds to the grand mean when ANOVA is treated as a regression model. When interpreting these contrasts, the thing to recognise is that each of these contrasts is a pairwise comparison between group 1 and one of the other four groups. Specifically, contrast 1 corresponds to a “group 2 minus group 1” comparison, contrast 2 corresponds to a “group 3 minus group 1” comparison, and so on.[1]

Optional contrasts in jamovi

jamovi also comes with a variety of options that can generate different kinds of contrasts in ANOVA. These can be found in the Contrasts option in the main ANOVA analysis window, where the following contrast types are listed:

Contrast type What does the contrast compare?
Deviation Compares the mean of each level (except a reference category) to the mean of all of the levels (grand mean).
Simple Like the treatment contrasts, the simple contrast compares the mean of each level to the mean of a specified level. This type of contrast is useful when there is a control group. By default the first category is the reference. However, with a simple contrast the intercept is the grand mean of all the levels of the factors.
Difference Compares the mean of each level (except the first) to the mean of previous levels. (Sometimes called reverse Helmert contrasts).
Helmert Compares the mean of each level of the factor (except the last) to the mean of subsequent levels.
Repeated Compares the mean of each level (except the last) to the mean of the subsequent level.
Polynomial Compares the linear effect and quadratic effect. The first degree of freedom contains the linear effect across all categories; the second degree of freedom, the quadratic effect. These contrasts are often used to estimate polynomial trends.
[1]What’s the difference between treatment and simple contrasts, I hear you ask? Well, as a basic example consider a gender main effect, with m = 0 and f = 1. The coefficient corresponding to the treatment contrast will measure the difference in mean between females and males, and the intercept would be the mean of the males. However, with a simple contrast, i.e., m = -1 and f = 1, the intercept is the average of the means and the main effect is the difference of each group mean from the intercept.