The "constrasts" set in your R environment determine how categorical variables are handled in your models. The most common scheme in regression is called "treatment contrasts": with treatment contrasts, the first level of the categorical variable is assigned the value 0, and then other levels measure the change from the first level.

Why do we need this? Because with k categories, we really need only k-1 pieces of information to represent any of the values. If the values are "Red," "White," and "Blue," for example, we might have a column named "Red," containing 1's for Red items and 0 for non-Red items, and a column named "White," containing 1's for White items and 0 for non-White items. That's all we need -- if we see an item with 0's for both Red and White, it must be blue. So we need one constraint on our contrasts, or, to put it another way, we need k-1 columns to represent a categorical variable with k levels.

As an example, consider the "Gun" data set in the MEMSS library. First, look at the levels associated with each of the categorical variables.

> sapply (Gun, levels) # see ?Gun for more info $rounds NULL $Method [1] "M1" "M2" $Team [1] "T1A" "T1H" "T1S" "T2A" "T2H" "T2S" "T3A" "T3H" "T3S" $Physique [1] "Slight" "Average" "Heavy"There are two levels of Method, three of Physique and nine of Team. rounds, being continuous, has no levels. When you have treatment contrasts set, a simple linear model, ignoring Team, looks like this. (Since the response variable is an integer, perhaps this model isn't 100% appropriate, but hey -- this is just an example, right?)

> options (contrasts = rep("contr.treatment", 2)) # Set contrasts -- see below > lm (rounds ~ Physique + Method, data = Gun) Call: lm(formula = rounds ~ Physique + Method, data = Gun) Coefficients: (Intercept) PhysiqueAverage PhysiqueHeavy MethodM2 24.3806 -0.7417 -1.6333 -8.5111

The interpretation is clear here. Physique Small is the baseline; you can think of it as having a coefficient of 0. Physique Average has a predicted average number of "rounds" that is .74 less than Physique Small, and Physique Heavy has a predicted number that is 1.63 lower than Physique Small. Both coefficients compare levels to the baseline, and the baseline is the first level on the list of levels.

Similarly Method M1 is set to be the baseline, and the difference -- the contrast -- between Method M2 and M1 is estimated as -8.51. The coefficient labels consist of the column name and the level name, pasted together; the baseline level isn't listed as all.

The intercept here is 24.38. This is the estimated average rounds when Physique = Small and Method = M1 (that is, when each variable is at its basline). Now we have enough information to get the estimated average for every combination of Physique and Method.

> options(contrasts = rep("contr.sum", 2)) # Set contrasts -- see below > contr.sum (3) # This shows what the Physique contrasts measure, by column [,1] [,2] 1 1 0 2 0 1 3 -1 -1 > contr.sum (2) # This is what the Method contrast measures. [,1] 1 1 2 -1 > lm (rounds ~ Physique + Method, data = gun) Call: lm(formula = rounds ~ Physique + Method, data = Gun) Coefficients: (Intercept) Physique1 Physique2 Method1 19.3333 0.7917 0.0500 4.2556This looks different, but it really isn't -- it's just a question of scoring. All the residuals from this model are identical to those from the other. So are the predictions. For example, the first column of the contrast matrix (above) is (1, 0, -1), meaning it measures the difference between physiques Small and Heavy. That effect is estimated as .791667. So if an observation has Physique Small, it gets +.791667, and if it has Heavy, it gets -.791667. The estimated difference between A and H is .05. If an observation has Physique Average, it gets +.05, and if it has Heavy, it gets -.05. Both of these contrasts compare the baseline, but unlike the case treatment contrasts, the baseline is the

> contr.poly (3) .L .Q [1,] -7.071068e-01 0.4082483 [2,] -7.850462e-17 -0.8164966 [3,] 7.071068e-01 0.4082483As you can see, the first column computes (1/sqrt(2)) * (3rd level) - (1/sqrt(2)) * 1st level. This contrast measures the rate of change of the set of coefficients; the 1/sqrt(2) part just makes sure that the vector of contrasts has length 1. If the contrasts weren't ordered this wouldn't seem like a great move. The second column measures how quadratic the effects are, by comparing the middle one to the average of #1 and #3. Again, the specfic values are chosen so that the middle is -2 * the others and the whole vector has length 1. Polynomial contrasts are orthogonal, which is good, but to me they're harder to interpret than treatment contrasts.

options(contrasts = rep ("contr.treatment", 2))Notice that we need to enter two contrast settings. The first handles unordered categorical variables, the second, ordered. The default settings are "contr.treatment" for unordered variables, and "contr.poly" for ordered ones. Notice also that we pass the contrast settings to options() as the

> options(contrasts = rep ("contr.treatment", 2)) # Set default contrasts > Gun <- Gun # Make local copy of data > attr (Gun$Physique, "contrasts") <- contr.poly (3) # Attach contrasts to "Physique" by creating an > lm (rounds ~ Physique + Method, data = Gun) # attribute; the 3 tells S-Plus there are 3 levels Call: lm(formula = rounds ~ Physique + Method, data = Gun) Coefficients: (Intercept) Physique.L Physique.Q MethodM2 23.58889 -1.15494 -0.06124 -8.51111Here the "Physique" variable is using polynomial contrasts but the "Method" variable is using treatment contrasts.