# What is a contrast matrix?

What exactly is contrast matrix (a term, pertaining to an analysis with categorical predictors) and how exactly is contrast matrix specified? I.e. what are columns, what are rows, what are the constraints on that matrix and what does number in column j and row i mean? I tried to look into the docs and web but it seems that everyone uses it yet there's no defition anywhere. I could backward-engineer the available pre-defined contrasts, but I think the definition should be available without that.

> contr.treatment(4)
2 3 4
1 0 0 0
2 1 0 0
3 0 1 0
4 0 0 1
> contr.sum(4)
[,1] [,2] [,3]
1    1    0    0
2    0    1    0
3    0    0    1
4   -1   -1   -1
> contr.helmert(4)
[,1] [,2] [,3]
1   -1   -1   -1
2    1   -1   -1
3    0    2   -1
4    0    0    3
> contr.SAS(4)
1 2 3
1 1 0 0
2 0 1 0
3 0 0 1
4 0 0 0

• "Contrast matrix" is used to represent categorical IVs (factors) in modeling. In particularly, it is used to recode a factor into a set of "contrast variables" (dummy variables being just an example). Each type of contrast variables has its own corresponding contrast matrix. See for example my own related question, not answered yet. – ttnphns Dec 3 '13 at 7:21
• @ttnphns Sorry but you keep doing what all the docs and webs do: you explain what are contrast matrices used for, without addressing the question what the contrast matrix is. This is the purpose of a definition. – Curious Dec 3 '13 at 10:39
• Of course it is related, but deriving "what it is" from "what it is needed for" is a detective's job, which shouldn't be needed. That's reverse engineering. Things should be documented. – Curious Dec 3 '13 at 11:00
• ats.ucla.edu/stat/r/library/contrast_coding.htm is a good R-oriented resource on coding methods. – whuber Jul 4 '16 at 17:44
• @Curious, just to let you know: I awarded 100 bounty to ttnphns, but I will start another bounty (or ask somebody else to do it) in order to award Gus_est as well. I have also written my own answer, just in case you prefer to have a shorter one :-) – amoeba Jul 8 '16 at 23:46

In their nice answer, @Gus_est, undertook a mathematical explanation of the essence of the contrast coefficient matrix L (notated there a C). $\bf Lb=k$ is the fundamental formula for testing hypotheses in univariate general linear modeling (where $\bf b$ are parameters and $\bf k$ are estimable function representing a null hypothesis), and that answer shows some necessary formulas used in modern ANOVA programs.

My answer is styled very differently. It is for a data analyst who sees himself rather an "engineer" than a "mathematician", so the answer will be a (superficial) "practical" or "didactic" account and will focus to answer just topics (1) what do the contrast coefficients mean and (2) how can they help to perform ANOVA via linear regression program.

ANOVA as regression with dummy variables: introducing contrasts.

Let us imagine ANOVA with dependent variable Y and categorical factor A having 3 levels (groups). Let us glance at the ANOVA from the linear regression point of view, that is - via turning the factor into the set of dummy (aka indicator aka treatment aka one-hot) binary variables. This is our independent set X. (Probably everybody has heard that it is possible to do ANOVA this way - as linear regression with dummy predictors.)

Since one of the three groups is redundant, only two dummy variables will enter the linear model. Let's appoint Group3 to be redundant, or reference. The dummy predictors constituting X are an example of contrast variables, i.e. elementary variables representing categories of a factor. X itself is often called design matrix. We can now input the dataset in a multiple linear regression program which will center the data and find the regression coefficients (parameters) $\bf b= (X'X)^{-1}X'y=X^+y$, where "+" designates pseudoinverse.

Equivalent pass will be not to do the centering but rather add constant term of the model as the first column of 1s in X, then estimate the coefficients same way as above $\bf b= (X'X)^{-1}X'y=X^+y$. So far so good.

Let us define matrix C to be the aggregation (summarization) of the independent variables design matrix X. It simply shows us the coding scheme observed there, - the contrast coding matrix (= basis matrix): $\bf C= {\it{aggr}} X$.

C
Const  A1    A2
Gr1 (A=1)       1     1     0
Gr2 (A=2)       1     0     1
Gr3 (A=3,ref)   1     0     0


The colums are the variables (columns) of X - the elementary contrast variables A1 A2, dummy in this instance, and the rows are all the groups/levels of the factor. So was our coding matrix C for indicator or dummy contrast coding scheme.

Now, $\bf C^+=L$ is called the contrast coefficient matrix, or L-matrix. Since C is square, $\bf L=C^+=C^{-1}$. The contrast matrix, corresponding to our C - that is for indicator contrasts of our example - is therefore:

L
Gr1   Gr2   Gr3
(A=1) (A=2) (A=3)
Const      0     0     1            => Const = Mean_Gr3
A1         1     0    -1            => Param1 = Mean_Gr1-Mean_Gr3
A2         0     1    -1            => Param2 = Mean_Gr2-Mean_Gr3


L-matrix is the matrix showing contrast coefficients. Note that sum of contrast coefficients in every row (except row Constant) is $0$. Every such row is called a contrast. Rows correspond to the contrast variables and columns correspond to the groups, factor levels.

The significance of contrast coefficients is that they help understand what each effect (each parameter b estimated in the regression with our X, coded as it is) represent in the sense of the difference (the group comparison). We immediately see, following the coefficients, that the estimated Constant will equal the Y mean in the reference group; that parameter b1 (i.e. of dummy variable A1) will equal the difference: Y mean in group1 minus Y mean in group3; and parameter b2 is the difference: mean in group2 minus mean in group3.

Note: Saying "mean" right above (and further below) we mean estimated (predicted by the model) mean for a group, not the observed mean in a group.

An instructive remark: When we do a regression by binary predictor variables, the parameter of such a variable says about the difference in Y between variable=1 and variable=0 groups. However, in the situation when the binary variables are the set of k-1 dummy variables representing a k-level factor, the meaning of the parameter gets narrower: it shows the difference in Y between variable=1 and (not just variable=0 but even) reference_variable=1 groups.

Like $\bf X^+$ (after multiplied by $\bf y$) brings us values of b, similarly $\bf(\it{aggr} \bf X)^+$ brings in meanings of b.

OK, we've given the definition of contrast coefficient matrix L. Since $\bf L=C^+=C^{-1}$, symmetrically $\bf C=L^+=L^{-1}$, which means that if you were given or have constructed a contrast matrix L based on categorical factor(s) - to test that L in your analysis, then you have clue for how to code correctly your contrast predictor variables X in order to test the L via an ordinary regression software (i.e. the one processing just "continuous" variables the standard OLS way, and not recognizing categorical factors at all). In our present example the coding was - indicator (dummy) type variables.

ANOVA as regression: other contrast types.

Let us briefly observe other contrast types (= coding schemes, = parameterization styles) for a categorical factor A.

Deviation or effect contrasts. C and L matrices and parameter meaning:

C
Const  A1    A2
Gr1 (A=1)       1     1     0
Gr2 (A=2)       1     0     1
Gr3 (A=3,ref)   1    -1    -1

L
Gr1   Gr2   Gr3
(A=1) (A=2) (A=3)
Const     1/3   1/3   1/3      => Const = 1/3Mean_Gr3+1/3Mean_Gr2+1/3Mean_Gr3 = Mean_GU
A1        2/3  -1/3  -1/3      => Param1 = 2/3Mean_Gr1-1/3(Mean_Gr2+Mean_Gr3) = Mean_Gr1-Mean_GU
A2       -1/3   2/3  -1/3      => Param2 = 2/3Mean_Gr2-1/3(Mean_Gr1+Mean_Gr3) = Mean_Gr2-Mean_GU

Parameter for the reference group3 = -(Param1+Param2) = Mean_Gr3-Mean_GU

Mean_GU is grand unweighted mean = 1/3(Mean_Gr1+Mean_Gr2+Mean_Gr3)


By deviation coding, each group of the factor is being compared with the unweighted grand mean, while Constant is that grand mean. This is what you get in regression with contrast predictors X coded in deviation or effect "manner".

Simple contrasts. This contrasts/coding scheme is a hybrid of indicator and deviation types, it gives the meaning of Constant as in deviation type and the meaning of the other parameters as in indicator type:

C
Const  A1    A2
Gr1 (A=1)       1   2/3  -1/3
Gr2 (A=2)       1  -1/3   2/3
Gr3 (A=3,ref)   1  -1/3  -1/3

L
Gr1   Gr2   Gr3
(A=1) (A=2) (A=3)
Const     1/3   1/3   1/3        => Const = as in Deviation
A1         1     0    -1         => Param1 = as in Indicator
A2         0     1    -1         => Param2 = as in Indicator


Helmert contrasts. Compares each group (except reference) with the unweighted mean of the subsequent groups, and Constant is the unweighted grand mean. C and L matrces:

C
Const  A1    A2
Gr1 (A=1)       1   2/3    0
Gr2 (A=2)       1  -1/3   1/2
Gr3 (A=3,ref)   1  -1/3  -1/2

L
Gr1   Gr2   Gr3
(A=1) (A=2) (A=3)
Const     1/3   1/3   1/3        => Const = Mean_GU
A1         1   -1/2  -1/2        => Param1 = Mean_Gr1-1/2(Mean_Gr2+Mean_Gr3)
A2         0     1    -1         => Param2 = Mean_Gr2-Mean_Gr3


Difference or reverse Helmert contrasts. Compares each group (except reference) with the unweighted mean of the previous groups, and Constant is the unweighted grand mean.

C
Const  A1    A2
Gr1 (A=1)       1  -1/2  -1/3
Gr2 (A=2)       1   1/2  -1/3
Gr3 (A=3,ref)   1    0    2/3

L
Gr1   Gr2   Gr3
(A=1) (A=2) (A=3)
Const     1/3   1/3   1/3        => Const = Mean_GU
A1        -1     1     0         => Param1 = Mean_Gr2-Mean_Gr1
A2       -1/2  -1/2    1         => Param2 = Mean_Gr3-1/2(Mean_Gr2+Mean_Gr1)


Repeated contrasts. Compares each group (except reference) with the next group, and Constant is the unweighted grand mean.

C
Const  A1    A2
Gr1 (A=1)       1   2/3   1/3
Gr2 (A=2)       1  -1/3   1/3
Gr3 (A=3,ref)   1  -1/3  -2/3

L
Gr1   Gr2   Gr3
(A=1) (A=2) (A=3)
Const     1/3   1/3   1/3        => Const = Mean_GU
A1         1    -1     0         => Param1 = Mean_Gr1-Mean_Gr2
A2         0     1    -1         => Param2 = Mean_Gr2-Mean_Gr3


The Question asks: how exactly is contrast matrix specified? Looking at the types of contrasts outlined so far it is possible to grasp how. Each type has its logic how to "fill in" the values in L. The logic reflects what each parameter means - what are the two combinations of groups it is planned to compare.

Polynomial contrasts. These are a bit special, nonlinear. The first effect is a linear one, the second is quadratic, next is cubic. I'm leaving here unaccounted the question how their C and L matrices are to be constructed and if they are the inverse of each other. Please consult with profound @Antoni Parellada's explanations of this type of contrast: 1, 2.

In balanced designs, Helmert, reverse Helmert, and polynomial contrasts are always orthogonal contrasts. Other types considered above are not orthogonal contrasts. Orthogonal (under balancedness) is the contrast where in contrast matrix L sum in each row (except Const) is zero and sum of products of the corresponding elements of each pair of rows is zero.

Here is the angle similarity measures (cosine and Pearson correlation) under different contrast types, except polynomial which I didn't test. Let us have single factor A with k levels, and it was then recoded into the set of k-1 contrast variables of a specific type. What are the values in the correlation or cosine matrix between these contrast variables?

                     Balanced (equal size) groups     Unbalanced groups
Contrast type             cos        corr              cos        corr

INDICATOR                  0       -1/(k-1)             0         varied
DEVIATION                 .5          .5              varied      varied
SIMPLE                 -1/(k-1)    -1/(k-1)           varied      varied
HELMERT, REVHELMERT        0           0              varied      varied
REPEATED                varied   =  varied            varied      varied

"=" means the two matrices are same while elements in matrix vary


I'm giving the table for information and leaving it uncommented. It is of some importance for a deeper glance into general linear modeling.

User-defined contrasts. This is what we compose to test a custom comparison hypothesis. Normally sum in every but the first row of L should be 0 which means that two groups or two compositions of groups are being compared in that row (i.e. by that parameter).

Where are the model parameters after all?

Are they the rows or the columns of L? Throughout the text above I was saying that parameters correspond to the rows of L, as the rows represent contrast-variables, the predictors. While the columns are levels of a factor, the groups. That may appear to fall in contradiction with such, for example, theoretical block from @Gus_est answer, where clearly the columns correspond to the parameters:

$H_0: \begin{bmatrix} 0 & 1 & -1 & \phantom{-}0 & \phantom{-}0 \\ 0 & 0 & \phantom{-}1 & -1 & \phantom{-}0 \\ 0 & 0 & \phantom{-}0 & \phantom{-}1 & -1 \end{bmatrix} \begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3 \\ \beta_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$

Actually, there is no contradiction and the answer to the "problem" is: both rows and columns of the contrast coefficient matrix correspond to the parameters! Just recall that contrasts (contrast variables), the rows, were initially created to represent nothing else than the factor levels: they are the levels except the omitted reference one. Compare please these two equivalent spelling of the L-matrix for the simple contrast:

L
Gr1   Gr2   Gr3
A=1   A=2   A=3(reference)
Const     1/3   1/3   1/3
A1         1     0    -1
A2         0     1    -1

L
b0    b1    b2    b3(redundant)
Const  A=1   A=2   A=3(reference)
b0  Const   1    1/3   1/3   1/3
b1  A1      0     1     0    -1
b2  A2      0     0     1    -1


The first one is what I've shown before, the second is more "theoretical" (for general linear model algebra) layout. Simply, a column corresponding to Constant term was added. Parameter coefficients b label the rows and columns. Parameter b3, as redundant, will be set to zero. You may pseudoinverse the second layout to get the coding matrix C, where inside in the bottom-right part you will find still the correct codes for contrast variables A1 and A2. That will be so for any contrast type described (except for indicator type - where the pseudoinverse of such rectangular layout won't give correct result; this is probably why simple contrast type was invented for convenience: contrast coefficients identical to indicator type, but for row Constant).

Contrast type and ANOVA table results.

ANOVA table shows effects as combined (aggregated) - for example main effect of factor A, whereas contrasts correspond to elementary effects, of contrast variables - A1, A2, and (omitted, reference) A3. The parameter estimates for the elementary terms depend on the type of the contrast selected, but the combined result - its mean square and significance level - is the same, whatever the type is. Omnibus ANOVA (say, one-way) null hypothesis that all the three means of A are equal may be put out in a number of equivalent statements, and each will correspond to a specific contrast type: $(\mu_1=\mu_2, \mu_2=\mu_3)$ = repeated type; $(\mu_1=\mu_{23}, \mu_2=\mu_3)$ = Helmert type; $(\mu_1=\mu_{123}, \mu_2=\mu_{123})$ = Deviation type; $(\mu_1=\mu_3, \mu_2=\mu_3)$ = indicator or simple types.

ANOVA programs implemented via general linear model paradigm can display both ANOVA table (combined effects: main, interactions) and parameter estimates table (elementary effects b). Some programs may output the latter table correspondent to the contrast type as bid by the user, but most will output always the parameters correspondent to one type - often, indicator type, because ANOVA programs based on general linear model parameterize specifically dummy variables (most convenient to do) and then switch over for contrasts by special "linking" formulae interpreting the fixed dummy input to a (arbitrary) contrast.

Whereas in my answer - showing ANOVA as regression - the "link" is realized as early as at the level of the input X, which called to introduce the notion of the appropriarte coding schema for the data.

A few examples showing testing of ANOVA contrasts via usual regression.

Showing in SPSS the request a contrast type in ANOVA and getting the same result via linear regression. We have some dataset with Y and factors A (3 levels, reference=last) and B (4 levels, reference=last); find the data below later on.

Deviation contrasts example under full factorial model (A, B, A*B). Deviation type requested for both A and B (we might choose to demand different type for each factor, for your information).

Contrast coefficient matrix L for A and for B:

            A=1      A=2      A=3
Const     .3333    .3333    .3333
dev_a1    .6667   -.3333   -.3333
dev_a2   -.3333    .6667   -.3333

B=1      B=2      B=3      B=4
Const     .2500    .2500    .2500    .2500
dev_b1    .7500   -.2500   -.2500   -.2500
dev_b2   -.2500    .7500   -.2500   -.2500
dev_b3   -.2500   -.2500    .7500   -.2500


Request ANOVA program (GLM in SPSS) to do analysis of variance and to output explicit results for deviation contrasts:  Deviation contrast type compared A=1 vs Grand unweighted Mean and A=2 with that same Mean. Red ellipses ink the difference estimates and their p-values. The combined effect over the factor A is inked by red rectangle. For factor B, everyting is analogously inked in blue. Displaying also the ANOVA table. Note there that the combined contrast effects equal the main effects in it. Let us now create physically contrast variables dev_a1, dev_a2, dev_b1, dev_b2, dev_b3 and run regression. Invert the L-matrices to obtain the coding C matrices:

      dev_a1   dev_a2
A=1   1.0000    .0000
A=2    .0000   1.0000
A=3  -1.0000  -1.0000

dev_b1   dev_b2   dev_b3
B=1   1.0000    .0000    .0000
B=2    .0000   1.0000    .0000
B=3    .0000    .0000   1.0000
B=4  -1.0000  -1.0000  -1.0000


The column of ones (Constant) is omitted: because we'll use regular regression program (which internally centers variables, and is also intolerant to singularity) variable Constant won't be needed. Now create data X: actually no manual recoding of the factors into these values is needed, the one-stroke solution is $\bf X=DC$, where $\bf D$ is the indicator (dummy) variables, all k columns (k is the number of levels in a factor).

Having created the contrast variables, multiply among those from different factors to get variables to represent interactions (our ANOVA model was full factorial): dev_a1b1, dev_a1b2, dev_a1b3, dev_a2b1, dev_a2b2, dev_a2b3. Then run multiple linear regression with all the predictors. As expected, dev_a1 is the same as effect as was the contrast "Level 1 vs Mean"; dev_a2 is the same as was "Level 2 vs Mean", etc etc, - compare the inked parts with the ANOVA contrast analysis above.

Note that if we were not using interaction variables dev_a1b1, dev_a1b2... in regression the results will coincide with results of main-effects-only ANOVA contrast analysis.

Simple contrasts example under the same full factorial model (A, B, A*B).

Contrast coefficient matrix L for A and for B:

            A=1      A=2      A=3
Const     .3333    .3333    .3333
sim_a1   1.0000    .0000  -1.0000
sim_a2    .0000   1.0000  -1.0000

B=1      B=2      B=3      B=4
Const     .2500    .2500    .2500    .2500
sim_b1   1.0000    .0000    .0000  -1.0000
sim_b2    .0000   1.0000    .0000  -1.0000
sim_b3    .0000    .0000   1.0000  -1.0000


ANOVA results for simple contrasts:  The overall results (ANOVA table) is the same as with deviation contrasts (not displaying now).

Create physically contrast variables sim_a1, sim_a2, sim_b1, sim_b2, sim_b3. The coding matrices by inverting of the L-matrices are (w/o Const column):

      sim_a1   sim_a2
A=1    .6667   -.3333
A=2   -.3333    .6667
A=3   -.3333   -.3333

sim_b1   sim_b2   sim_b3
B=1    .7500   -.2500   -.2500
B=2   -.2500    .7500   -.2500
B=3   -.2500   -.2500    .7500
B=4   -.2500   -.2500   -.2500


Create the data $\bf X=DC$ and add there the interaction contrast variables sim_a1b1, sim_a1b2, ... etc, as the products of the main effects contrast variables. Perform the regression. As before, we see that the results of regression and ANOVA match. A regression parameter of a simple contrast variable is the difference (and significance test of it) between that level of the factor and the reference (the last, in our example) level of it.

The two-factor data used in the examples:

     Y      A      B
.2260      1      1
.6836      1      1
-1.772      1      1
-.5085      1      1
1.1836      1      2
.5633      1      2
.8709      1      2
.2858      1      2
.4057      1      2
-1.156      1      3
1.5199      1      3
-.1388      1      3
.4865      1      3
-.7653      1      3
.3418      1      4
-1.273      1      4
1.4042      1      4
-.1622      2      1
.3347      2      1
-.4576      2      1
.7585      2      1
.4084      2      2
1.4165      2      2
-.5138      2      2
.9725      2      2
.2373      2      2
-1.562      2      2
1.3985      2      3
.0397      2      3
-.4689      2      3
-1.499      2      3
-.7654      2      3
.1442      2      3
-1.404      2      3
-.2201      2      4
-1.166      2      4
.7282      2      4
.9524      2      4
-1.462      2      4
-.3478      3      1
.5679      3      1
.5608      3      2
1.0338      3      2
-1.161      3      2
-.1037      3      3
2.0470      3      3
2.3613      3      3
.1222      3      4


User defined contrast example. Let us have single factor F with 5 levels. I will create and test a set of custom orthogonal contrasts, in ANOVA and in regression. The picture shows the process (one of possible) of combining/splitting among the 5 groups to obtain 4 orthogonal contrasts, and the L matrix of contrast coefficints resultant from that process is on the right. All the contrasts are orthogonal to each other: $\bf LL'$ is diagonal. (This example schema was years ago copied from D. Howell's book on Statistics for psychologist.)

Let us submit the matrix to SPSS' ANOVA procedure to test the contrasts. Well, we might submit even any one row (contrast) from the matrix, but we'll submit the whole matrix because - as in previous examples - we'll want to receive the same results via regression, and regression program will need the complete set of contrast variables (to be aware that they belong together to one factor!). We'll add the constant row to L, just as we did before, although if we don't need to test for the intercept we may safely omit it.

UNIANOVA Y BY F
/METHOD=SSTYPE(3)
/INTERCEPT=INCLUDE
/CONTRAST (F)= special
(.2 .2 .2 .2 .2
3  3 -2 -2 -2
1 -1  0  0  0
0  0  2 -1 -1
0  0  0  1 -1)
/DESIGN=F.

Equivalently, we might also use this syntax (with a more flexible /LMATRIX subcommand)
if we omit the Constant row from the matrix.
UNIANOVA Y BY F
/METHOD=SSTYPE(3)
/INTERCEPT=INCLUDE
/LMATRIX= "User contrasts"
F  3  3 -2 -2 -2;
F  1 -1  0  0  0;
F  0  0  2 -1 -1;
F  0  0  0  1 -1
/DESIGN=F. The overall contrasts effect (in the bottom of the pic) is not the same as the expected overall ANOVA effect: but it is simply the artefact of our inserting Constant term into the L matrix. For, SPSS already implies Constant when user-defined contrasts are specified. Remove the constant row from L and we'll get the same contrasts results (matrix K on the pic above) except that L0 contrast won't be displayed. And the overall contrast effect will match the overall ANOVA: OK, now create the contrast variables physically and submit them to regression. $\bf C=L^+$, $\bf X=DC$.

C
use_f1   use_f2   use_f3   use_f4
F=1    .1000    .5000    .0000    .0000
F=2    .1000   -.5000    .0000    .0000
F=3   -.0667    .0000    .3333    .0000
F=4   -.0667    .0000   -.1667    .5000
F=5   -.0667    .0000   -.1667   -.5000 Observe the identity of results. The data used in this example:

     Y      F
.2260      1
.6836      1
-1.772      1
-.5085      1
1.1836      1
.5633      1
.8709      1
.2858      1
.4057      1
-1.156      1
1.5199      2
-.1388      2
.4865      2
-.7653      2
.3418      2
-1.273      2
1.4042      2
-.1622      3
.3347      3
-.4576      3
.7585      3
.4084      3
1.4165      3
-.5138      3
.9725      3
.2373      3
-1.562      3
1.3985      3
.0397      4
-.4689      4
-1.499      4
-.7654      4
.1442      4
-1.404      4
-.2201      4
-1.166      4
.7282      4
.9524      5
-1.462      5
-.3478      5
.5679      5
.5608      5
1.0338      5
-1.161      5
-.1037      5
2.0470      5
2.3613      5
.1222      5


Contrasts in other than (M)ANOVA analyses.

Wherever nominal predictors appear, the question of contrast (which contrast type to select for which predictor) arise. Some programs solve it behind the scene internally when the overall, omnibus results won't depend on the type selected. If you want a specific type to see more "elementary" results, you have to select. You select (or, rather, compose) a contrast also when you are testing a custom comparison hypothesis.

(M)ANOVA and Loglinear analysis, Mixed and sometimes Generalized linear modeling include options to treat predictors via different types of contrasts. But as I've tried to show, it is possible to create contrasts as contrast variables explicitly and by hand. Then, if you don't have ANOVA package at hand, you might do it - in many respects with as good luck - with multiple regression.

• please do not restrict this answer just to anova if possible. The [anova] tag was added by @amoeba by the time when you answered my question, but I don't want the answer to be restricted just to anova. – Curious Jul 6 '16 at 10:07
• Thanks for writing and updating this answer! I have several questions, here is the first one. In your answer you introduced "contrast coding matrix" ($C$) and "contrast coefficient matrix" ($L$). (By the way, are these standard terms? When I google "contrast coding matrix", I get only 5 hits, two of which lead to this very page). The OP, however, asks about "contrast matrix" and also gives several examples of those as used in R (see also this manual). Am I right in understanding that this "contrast matrix" is your $C$ (not $L$)? – amoeba Jul 6 '16 at 22:37
• @amoeba, I'm not familiar with "contrast matrix" and almost sure it stands for "contrast coefficient matrix" or L-matrix, which is an official or at least wide spread term in (M)ANOVA/GLM. "Contrast coding matrix" term is much less mentioned as it is simply the aggrigated view of the design matrix X; I've seen "basis matrix" word used in papers of one SPSS's senior statistician Dave Nichols. Absolutely, L (official label) and C (arbitrary label?) matrices are so closely related that one can hardly discuss one w/o the other. I suppose that "contrast matrix" should be considered as this pair. – ttnphns Jul 6 '16 at 23:57
• Yes, I agree. By now I am convinced that "contrast matrix" is a term that is only used in the R community and refers to the coding scheme. I checked the textbook that Gus_est refers to and they never use the term "contrast matrix", they only talk about "contrasts" (see my last comment under his answer). The OP clearly was asking about the "contrast matrix" in the R sense. – amoeba Jul 7 '16 at 11:32
• That L will determine what are you going to test, you aren't free anymore to choose what to test: No, I disagree with that. As far as I understand, one can "manually" perform a test that is not tied to the coding scheme. The formulas for that are provided in the Gus'es answer. I am not saying it's convenient in practice, I am just saying that it's possible. I think what you are saying is that C-matrix determines the meaning of each beta coefficient and the corresponding p-values will be for $\beta_i=0$. This is clear. But one can still "manually" test e.g. if $\beta_1-\beta_2/2-\beta_3/2=0$. – amoeba Jul 7 '16 at 14:21

I'll use lower-case letters for vectors and upper-case letters for matrices.

In case of a linear model of the form:

$$\mathbf{y}=\mathbf{X} \boldsymbol{\beta} + \boldsymbol{\varepsilon}$$

where $$\bf{X}$$ is a $$n \times (k+1)$$ matrix of rank $$k+1 \leq n$$, and we assume $$\boldsymbol{\varepsilon} \sim \mathcal N(0,\sigma^2)$$.

We can estimate $$\hat{\boldsymbol{\beta}}$$ by $$(\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{X}^\top \mathbf{y}$$, since the inverse of $$\mathbf{X}^\top \mathbf{X}$$ exists.

Now, take an ANOVA case in which $$\mathbf{X}$$ is not full-rank anymore. The implication of this is that we don't have $$(\mathbf{X}^\top\mathbf{X})^{-1}$$ and we have to settle for the generalized inverse $$(\mathbf{X}^\top\mathbf{X})^{-}$$.

One of the problems of using this generalized inverse is that it's not unique. Another problem is that we cannot find an unbiased estimator for $$\boldsymbol{\beta}$$, since $$\hat{\boldsymbol{\beta}}=(\mathbf{X}^\top\mathbf{X})^{-}\mathbf{X}^\top\mathbf{y} \implies E(\hat{\boldsymbol{\beta}})=(\mathbf{X}^\top\mathbf{X})^{-}\mathbf{X}^\top\mathbf{X}\boldsymbol{\beta}.$$

So, we cannot estimate an unique and unbiased $$\boldsymbol{\beta}$$. There are various approaches to work around the lack of uniqueness of the parameters in an ANOVA case with non-full-rank $$\mathbf{X}$$. One of them is to work with the overparameterized model and define linear combinations of the $$\boldsymbol{\beta}$$'s that are unique and can be estimated.

We have that a linear combination of the $$\boldsymbol{\beta}$$'s, say $$\mathbf{g}^\top \boldsymbol{\beta}$$, is estimable if there exists a vector $$\mathbf{a}$$ such that $$E(\mathbf{a}^\top \mathbf{y})=\mathbf{g}^\top \boldsymbol{\beta}$$.

The contrasts are a special case of estimable functions in which the sum of the coefficients of $$\mathbf{g}$$ is equal to zero.

And, contrasts come up in the context of categorical predictors in a linear model. (if you check the manual linked by @amoeba, you see that all their contrast coding are related to categorical variables). Then, answering @Curious and @amoeba, we see that they arise in ANOVA, but not in a "pure" regression model with only continuous predictors (we can also talk about contrasts in ANCOVA, since we have some categorical variables in it).

Now, in the model $$\mathbf{y}=\mathbf{X} \boldsymbol{\beta} + \boldsymbol{\varepsilon}$$ where $$\mathbf{X}$$ is not full-rank, and $$E(\mathbf{y})=\mathbf{X}^\top \boldsymbol{\beta}$$, the linear function $$\mathbf{g}^\top \boldsymbol{\beta}$$ is estimable iff there exists a vector $$\mathbf{a}$$ such that $$\mathbf{a}^\top \mathbf{X}=\mathbf{g}^\top$$. That is, $$\mathbf{g}^\top$$ is a linear combination of the rows of $$\mathbf{X}$$. Also, there are many choices of the vector $$\mathbf{a}$$, such that $$\mathbf{a}^\top \mathbf{X}=\mathbf{g}^\top$$, as we can see in the example below.

## Example 1

Consider the one-way model: $$y_{ij}=\mu + \alpha_i + \varepsilon_{ij}, \quad i=1,2 \, , j=1,2,3.$$

\begin{align} \mathbf{X} = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \end{bmatrix} \, , \quad \boldsymbol{\beta}=\begin{bmatrix} \mu \\ \tau_1 \\ \tau_2 \end{bmatrix} \end{align}

And suppose $$\mathbf{g}^\top = [0, 1, -1]$$, so we want to estimate $$[0, 1, -1] \boldsymbol{\beta}=\tau_1-\tau_2$$.

We can see that there are different choices of the vector $$\mathbf{a}$$ that yield $$\mathbf{a}^\top \mathbf{X}=\mathbf{g}^\top$$: take $$\mathbf{a}^\top=[0 , 0,1,-1,0,0]$$; or $$\mathbf{a}^\top = [1,0,0,0,0,-1]$$; or $$\mathbf{a}^\top = [2,-1,0,0,1,-2]$$.

## Example 2

Take the two-way model: $$y_{ij}=\mu+\alpha_i+\beta_j+\varepsilon_{ij}, \, i=1,2, \, j=1,2$$.

\begin{align} \mathbf{X} = \begin{bmatrix} 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1\\ 1 & 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 \end{bmatrix} \, , \quad \boldsymbol{\beta}=\begin{bmatrix} \mu \\ \alpha_1 \\ \alpha_2 \\ \beta_1 \\ \beta_2 \end{bmatrix} \end{align}

We can define the estimable functions by taking linear combinations of the rows of $$\mathbf{X}$$.

Subtracting Row 1 from Rows 2, 3, and 4 (of $$\mathbf{X}$$): $$\begin{bmatrix} 1 & \phantom{-}1 & 0 & \phantom{-}1 & 0 \\ 0 & 0 & 0 & -1 & 1\\ 0 & -1 & 1 & \phantom{-}0 & 0 \\ 0 & -1 & 1 & -1 & 1 \end{bmatrix}$$

And taking Rows 2 and 3 from the fourth row: $$\begin{bmatrix} 1 & \phantom{-}1 & 0 & \phantom{-}1 & 0 \\ 0 & 0 & 0 & -1 & 1\\ 0 & -1 & 1 & \phantom{-}0 & 0 \\ 0 & \phantom{-}0 & 0 & \phantom{-}0 & 0 \end{bmatrix}$$

Multiplying this by $$\boldsymbol{\beta}$$ yields: \begin{align} \mathbf{g}_1^\top \boldsymbol{\beta} &= \mu + \alpha_1 + \beta_1 \\ \mathbf{g}_2^\top \boldsymbol{\beta} &= \beta_2 - \beta_1 \\ \mathbf{g}_3^\top \boldsymbol{\beta} &= \alpha_2 - \alpha_1 \end{align}

So, we have three linearly independent estimable functions. Now, only $$\mathbf{g}_2^\top \boldsymbol{\beta}$$ and $$\mathbf{g}_3^\top \boldsymbol{\beta}$$ can be considered contrasts, since the sum of its coefficients (or, the row sum of the respective vector $$\mathbf{g}$$) is equal to zero.

Going back to a one-way balanced model $$y_{ij}=\mu + \alpha_i + \varepsilon_{ij}, \quad i=1,2, \ldots, k \, , j=1,2,\ldots,n.$$

And suppose we want to test the hypothesis $$H_0: \alpha_1 = \ldots = \alpha_k$$.

In this setting the matrix $$\mathbf{X}$$ is not full-rank, so $$\boldsymbol{\beta}=(\mu,\alpha_1,\ldots,\alpha_k)^\top$$ is not unique and not estimable. To make it estimable we can multiply $$\boldsymbol{\beta}$$ by $$\mathbf{g}^\top$$, as long as $$\sum_{i} g_i = 0$$. In other words, $$\sum_{i} g_i \alpha_i$$ is estimable iff $$\sum_{i} g_i = 0$$.

Why this is true?

We know that $$\mathbf{g}^\top \boldsymbol{\beta}=(0,g_1,\ldots,g_k) \boldsymbol{\beta} = \sum_{i} g_i \alpha_i$$ is estimable iff there exists a vector $$\mathbf{a}$$ such that $$\mathbf{g}^\top = \mathbf{a}^\top \mathbf{X}$$. Taking the distinct rows of $$\mathbf{X}$$ and $$\mathbf{a}^\top=[a_1,\ldots,a_k]$$, then: $$[0,g_1,\ldots,g_k]=\mathbf{g}^\top=\mathbf{a}^\top \mathbf{X} = \left(\sum_i a_i,a_1,\ldots,a_k \right)$$

And the result follows.

If we would like to test a specific contrast, our hypothesis is $$H_0: \sum g_i \alpha_i = 0$$. For instance: $$H_0: 2 \alpha_1 = \alpha_2 + \alpha_3$$, which can be written as $$H_0: \alpha_1 = \frac{\alpha_2+\alpha_3}{2}$$, so we are comparing $$\alpha_1$$ to the average of $$\alpha_2$$ and $$\alpha_3$$.

This hypothesis can be expressed as $$H_0: \mathbf{g}^\top \boldsymbol{\beta}=0$$, where $${\mathbf{g}}^\top = (0,g_1,g_2,\ldots,g_k)$$. In this case, $$q=1$$ and we test this hypothesis with the following statistic: $$F=\cfrac{\left[\mathbf{g}^\top \hat{\boldsymbol{\beta}}\right]^\top \left[\mathbf{g}^\top(\mathbf{X}^\top\mathbf{X})^{-}\mathbf{g} \right]^{-1}\mathbf{g}^\top \hat{\boldsymbol{\beta}}}{SSE/k(n-1)}.$$

If $$H_0: \alpha_1 = \alpha_2 = \ldots = \alpha_k$$ is expressed as $$\mathbf{G}\boldsymbol{\beta}=\boldsymbol{0}$$ where the rows of the matrix $$\mathbf{G} = \begin{bmatrix} \mathbf{g}_1^\top \\ \mathbf{g}_2^\top \\ \vdots \\ \mathbf{g}_k^\top \end{bmatrix}$$ are mutually orthogonal contrasts ($${\mathbf{g}_i^\top\mathbf{g}}_j = 0$$), then we can test $$H_0: \mathbf{G}\boldsymbol{\beta}=\boldsymbol{0}$$ using the statistic $$F=\cfrac{\frac{\mbox{SSH}}{\mbox{rank}(\mathbf{G})}}{\frac{\mbox{SSE}}{k(n-1)}}$$, where $$\mbox{SSH}=\left[\mathbf{G}\hat{\boldsymbol{\beta}}\right]^\top \left[\mathbf{G}(\mathbf{X}^\top\mathbf{X})^{-1} \mathbf{G}^\top \right]^{-1}\mathbf{G}\hat{\boldsymbol{\beta}}$$.

## Example 3

To understand this better, let's use $$k=4$$, and suppose we want to test $$H_0: \alpha_1 = \alpha_2 = \alpha_3 = \alpha_4,$$ which can be expressed as $$H_0: \begin{bmatrix} \alpha_1 - \alpha_2 \\ \alpha_1 - \alpha_3 \\ \alpha_1 - \alpha_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$

Or, as $$H_0: \mathbf{G}\boldsymbol{\beta}=\boldsymbol{0}$$: $$H_0: \underbrace{\begin{bmatrix} 0 & 1 & -1 & \phantom{-}0 & \phantom{-}0 \\ 0 & 1 & \phantom{-}0 & -1 & \phantom{-}0 \\ 0 & 1 & \phantom{-}0 & \phantom{-}1 & -1 \end{bmatrix}}_{{\mathbf{G}}, \mbox{our contrast matrix}} \begin{bmatrix} \mu \\ \alpha_1 \\ \alpha_2 \\ \alpha_3 \\ \alpha_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$

So, we see that the three rows of our contrast matrix are defined by the coefficients of the contrasts of interest. And each column gives the factor level that we are using in our comparison.

Pretty much all I've written was taken\copied (shamelessly) from Rencher & Schaalje, "Linear Models in Statistics", chapters 8 and 13 (examples, wording of theorems, some interpretations), but other things like the term "contrast matrix" (which, indeed, doesn't appear in this book) and its definition given here were my own.

## Relating OP's contrast matrix to my answer

One of OP's matrix (which can also be found in this manual) is the following:

> contr.treatment(4)
2 3 4
1 0 0 0
2 1 0 0
3 0 1 0
4 0 0 1


In this case, our factor has 4 levels, and we can write the model as follows: This can be written in matrix form as: \begin{align} \begin{bmatrix} y_{11} \\ y_{21} \\ y_{31} \\ y_{41} \end{bmatrix} = \begin{bmatrix} \mu \\ \mu \\ \mu \\ \mu \end{bmatrix} + \begin{bmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{bmatrix} + \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{21} \\ \varepsilon_{31} \\ \varepsilon_{41} \end{bmatrix} \end{align}

Or \begin{align} \begin{bmatrix} y_{11} \\ y_{21} \\ y_{31} \\ y_{41} \end{bmatrix} = \underbrace{\begin{bmatrix} 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0 & 1\\ \end{bmatrix}}_{\mathbf{X}} \underbrace{\begin{bmatrix} \mu \\ a_1 \\ a_2 \\ a_3 \\ a_4 \end{bmatrix}}_{\boldsymbol{\beta}} + \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{21} \\ \varepsilon_{31} \\ \varepsilon_{41} \end{bmatrix} \end{align}

Now, for the dummy coding example on the same manual, they use $$a_1$$ as the reference group. Thus, we subtract Row 1 from every other row in matrix $$\mathbf{X}$$, which yields the $$\widetilde{\mathbf{X}}$$:

\begin{align} \begin{bmatrix} 1 & \phantom{-}1 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0\\ 0 & -1 & 0 & 1 & 0\\ 0 & -1 & 0 & 0 & 1 \end{bmatrix} \end{align}

If you observe the numeration of the rows and columns in the contr.treatment(4) matrix, you'll see that they consider all rows and only the columns related to the factors 2, 3, and 4. If we do the same in the above matrix yields: \begin{align} \begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \end{align}

This way, the contr.treatment(4) matrix is telling us that they are comparing factors 2, 3 and 4 to factor 1, and comparing factor 1 to the constant (this is my understanding of the above).

And, defining $$\mathbf{G}$$ (i.e. taking only the rows that sum to 0 in the above matrix): \begin{align} \begin{bmatrix} 0 & -1 & 1 & 0 & 0\\ 0 & -1 & 0 & 1 & 0\\ 0 & -1 & 0 & 0 & 1 \end{bmatrix} \end{align}

We can test $$H_0: \mathbf{G}\boldsymbol{\beta}=0$$ and find the estimates of the contrasts.

hsb2 = read.table('http://www.ats.ucla.edu/stat/data/hsb2.csv', header=T, sep=",")

y<-hsb2$write dummies <- model.matrix(~factor(hsb2$race)+0)
X<-cbind(1,dummies)

# Defining G, what I call contrast matrix
G<-matrix(0,3,5)
G[1,]<-c(0,-1,1,0,0)
G[2,]<-c(0,-1,0,1,0)
G[3,]<-c(0,-1,0,0,1)
G
[,1] [,2] [,3] [,4] [,5]
[1,]    0   -1    1    0    0
[2,]    0   -1    0    1    0
[3,]    0   -1    0    0    1

# Estimating Beta

X.X<-t(X)%*%X
X.y<-t(X)%*%y

library(MASS)
Betas<-ginv(X.X)%*%X.y

# Final estimators:
G%*%Betas
[,1]
[1,] 11.541667
[2,]  1.741667
[3,]  7.596839


And the estimates are the same.

## Relating @ttnphns' answer to mine.

On their first example, the setup has a categorical factor A having three levels. We can write this as the model (suppose, for simplicity, that $$j=1$$): $$y_{ij}=\mu+a_i+\varepsilon_{ij}\, , \quad \mbox{for } i=1,2,3$$

And suppose we want to test $$H_0: a_1 = a_2 = a_3$$, or $$H_0: a_1 - a_3 = a_2 - a_3=0$$, with $$a_3$$ as our reference group/factor.

This can be written in matrix form as: \begin{align} \begin{bmatrix} y_{11} \\ y_{21} \\ y_{31} \end{bmatrix} = \begin{bmatrix} \mu \\ \mu \\ \mu \end{bmatrix} + \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} + \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{21} \\ \varepsilon_{31} \end{bmatrix} \end{align}

Or \begin{align} \begin{bmatrix} y_{11} \\ y_{21} \\ y_{31} \end{bmatrix} = \underbrace{\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ \end{bmatrix}}_{\mathbf{X}} \underbrace{\begin{bmatrix} \mu \\ a_1 \\ a_2 \\ a_3 \end{bmatrix}}_{\boldsymbol{\beta}} + \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{21} \\ \varepsilon_{31} \end{bmatrix} \end{align}

Now, if we subtract Row 3 from Row 1 and Row 2, we have that $$\mathbf{X}$$ becomes (I will call it $$\widetilde{\mathbf{X}}$$:

\begin{align} \widetilde{\mathbf{X}} =\begin{bmatrix} 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 1 & 0 & 0 & \phantom{-}1 \\ \end{bmatrix} \end{align}

Compare the last 3 columns of the above matrix with @ttnphns' matrix $$\mathbf{L}$$. Despite of the order, they are quite similar. Indeed, if multiply $$\widetilde{\mathbf{X}} \boldsymbol{\beta}$$, we get:

\begin{align} \begin{bmatrix} 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 1 & 0 & 0 & \phantom{-}1 \\ \end{bmatrix} \begin{bmatrix} \mu \\ a_1 \\ a_2 \\ a_3 \end{bmatrix} = \begin{bmatrix} a_1 - a_3 \\ a_2 - a_3 \\ \mu + a_3 \end{bmatrix} \end{align}

So, we have the estimable functions: $$\mathbf{c}_1^\top \boldsymbol{\beta} = a_1-a_3$$; $$\mathbf{c}_2^\top \boldsymbol{\beta} = a_2-a_3$$; $$\mathbf{c}_3^\top \boldsymbol{\beta} = \mu + a_3$$.

Since $$H_0: \mathbf{c}_i^\top \boldsymbol{\beta} = 0$$, we see from the above that we are comparing our constant to the coefficient for the reference group (a_3); the coefficient of group1 to the coefficient of group3; and the coefficient of group2 to the group3. Or, as @ttnphns said: "We immediately see, following the coefficients, that the estimated Constant will equal the Y mean in the reference group; that parameter b1 (i.e. of dummy variable A1) will equal the difference: Y mean in group1 minus Y mean in group3; and parameter b2 is the difference: mean in group2 minus mean in group3."

Moreover, observe that (following the definition of contrast: estimable function+row sum =0), that the vectors $$\mathbf{c}_1$$ and $$\mathbf{c}_2$$ are contrasts. And, if we create a matrix $$\mathbf{G}$$ of constrasts, we have:

\begin{align} \mathbf{G} = \begin{bmatrix} 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \end{bmatrix} \end{align}

Our contrast matrix to test $$H_0: \mathbf{G}\boldsymbol{\beta}=0$$

## Example

We will use the same data as @ttnphns' "User defined contrast example" (I'd like to mention that the theory that I've written here requires a few modifications to consider models with interactions, that's why I chose this example. However, the definitions of contrasts and - what I call - contrast matrix remain the same).

Y<-c(0.226,0.6836,-1.772,-0.5085,1.1836,0.5633,0.8709,0.2858,0.4057,-1.156,1.5199,
-0.1388,0.4865,-0.7653,0.3418,-1.273,1.4042,-0.1622,0.3347,-0.4576,0.7585,0.4084,
1.4165,-0.5138,0.9725,0.2373,-1.562,1.3985,0.0397,-0.4689,-1.499,-0.7654,0.1442,
-1.404,-0.2201,-1.166,0.7282,0.9524,-1.462,-0.3478,0.5679,0.5608,1.0338,-1.161,
-0.1037,2.047,2.3613,0.1222)

F_<-c(1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,
5,5,5,5,5,5,5,5,5,5,5)

dummies.F<-model.matrix(~as.factor(F_)+0)

X_F<-cbind(1,dummies.F)

G_F<-matrix(0,4,6)
G_F[1,]<-c(0,3,3,-2,-2,-2)
G_F[2,]<-c(0,1,-1,0,0,0)
G_F[3,]<-c(0,0,0,2,-1,-1)
G_F[4,]<-c(0,0,0,0,1,-1)

G
[,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    3    3   -2   -2   -2
[2,]    0    1   -1    0    0    0
[3,]    0    0    0    2   -1   -1
[4,]    0    0    0    0    1   -1

# Estimating Beta

X_F.X_F<-t(X_F)%*%X_F
X_F.Y<-t(X_F)%*%Y

Betas_F<-ginv(X_F.X_F)%*%X_F.Y

# Final estimators:
G_F%*%Betas_F
[,1]
[1,]  0.5888183
[2,] -0.1468029
[3,]  0.6115212
[4,] -0.9279030


So, we have the same results.

## Conclusion

It seems to me that there isn't one defining concept of what a contrast matrix is.

If you take the definition of contrast, given by Scheffe ("The Analysis of Variance", page 66), you'll see that it's an estimable function whose coefficients sum to zero. So, if we wish to test different linear combinations of the coefficients of our categorical variables, we use the matrix $$\mathbf{G}$$. This is a matrix in which the rows sum to zero, that we use to multiply our matrix of coefficients by in order to make those coefficients estimable. Its rows indicate the different linear combinations of contrasts that we are testing and its columns indicate which factors (coefficients) are being compared.

As the matrix $$\mathbf{G}$$ above is constructed in a way that each of its rows is composed by a contrast vector (which sum to 0), for me it makes sense to call $$\mathbf{G}$$ a "contrast matrix" (Monahan - "A primer on linear models" - also uses this terminology).

However, as beautifully explained by @ttnphns, softwares are calling something else as "contrast matrix", and I couldn't find a direct relationship between the matrix $$\mathbf{G}$$ and the built-in commands/matrices from SPSS (@ttnphns) or R (OP's question), only similarities. But I believe that the nice discussion/colaboration presented here will help clarify such concepts and definitions.

• please do not restrict this answer just to anova if possible. The [anova] tag was added by @amoeba by the time when you answered my question, but I don't want the answer to be restricted just to anova. – Curious Jul 6 '16 at 10:07
• Thanks a lot for such a big update. I removed some of my comments above that were obsolete by now (you can remove some of yours, e.g. the first one). However, by now it is clear to me that "contrast matrix" in your (and Monahan's) sense is something entirely different from "contrast matrix" in the sense it's used in this R manual and also in the original question here (what ttnphns calls C-matrix). I think it would make sense if you make a note somewhere in your answer about this difference. – amoeba Jul 8 '16 at 0:17
• I'm having troubles with understanding starting right from Example1. What is $i$ an $j$ in your notation $y_{ij}$? What is $a_i$ and what do the columns od $X$ represent? Is that Constant term (column of ones) and the two dummy variables? – ttnphns Jul 8 '16 at 9:58
• @ttnphns: $i$ is indexing group (there are two groups in Example 1), $j$ is indexing data point inside each group. $\mu$ is a constant and $\alpha_i$ are constants for each group such that $\mu+\alpha_i$ are group means (so $\mu$ can be total mean and $\alpha_i$ can be deviation of the group means from the total mean). Columns of $X$ are constant term and two dummies, yes. – amoeba Jul 8 '16 at 14:23
• @fabiob technically, ANOVA is not GLM. But, I think I understand your question. When you have an overparameterized model you can: i) reparameterize the model to get a full-rank matrix; ii) define side conditions (which gives you a full-rank matrix); or iii) work with the model as is and define lin. comb. of the parameters that are unique and can be estimated (these are called contrasts). Since the question is about a contrast matrix, I used iii). However, we also have ANOVA cases where X is full-rank. I updated my answer to clarify that I'm talking about the special non-full-rank case. – Gus_est Feb 3 at 21:05

"Contrast matrix" is not a standard term in the statistical literature. It can have [at least] two related by distinct meanings:

1. A matrix specifying a particular null hypothesis in an ANOVA regression (unrelated to the coding scheme), where each row is a contrast. This is not a standard usage of the term. I used full text search in Christensen Plane Answers to Complex Questions, Rutherford Introducing ANOVA and ANCOVA; GLM Approach, and Rencher & Schaalje Linear Models in Statistics. They all talk a lot about "contrasts" but never ever mention the term "contrast matrix". However, as @Gus_est found, this term is used in Monahan's A Primer on Linear Models.

2. A matrix specifying the coding scheme for the design matrix in an ANOVA regression. This is how the term "contrast matrix" is used in the R community (see e.g. this manual or this help page).

The answer by @Gus_est explores the first meaning. The answer by @ttnphns explores the second meaning (he calls it "contrast coding matrix" and also discusses "contrast coefficient matrix" which is a standard term in SPSS literature).

My understanding is that you were asking about meaning #2, so here goes the definition:

"Contrast matrix" in the R sense is $k\times k$ matrix $\mathbf C$ where $k$ is the number of groups, specifying how group membership is encoded in the design matrix $\mathbf X$. Specifically, if a $m$-th observation belongs to the group $i$ then $X_{mj}=C_{ij}$.

Note: usually the first column of $\mathbf C$ is the column of all ones (corresponding to the intercept column in the design matrix). When you call R commands like contr.treatment(4), you get matrix $\mathbf C$ without this first column.

I am planning to extend this answer to make an extended comment on how the answers by @ttnphns and @Gus_est fit together.

• The answer by @Gus_est explores the first meaning. The answer by @ttnphns explores the second meaning. I protest. (And am surprised to hear - after we both had a long conversation on the definitions in the comments to mty answer.) I invited two terms: contrast coefficient matrix (where rows are the contrasts, linear combibnation of means) aka L-matrix, and contrast coding schema matrix, aka C matrix. Both are related, I discussed both. – ttnphns Jul 8 '16 at 16:58
• (cont.) Contrast coefficent L matrix is a standard term in ANOVA / General linear model, used in texts and in SPSS docs, for example. The coding schemes see here. – ttnphns Jul 8 '16 at 16:59
• You were asking about meaning #2 We actually are not sure what meaning of the term the OP implied. The OP displayed some examples of contrast coding schemes, - it doesn't necessarily mean s/he wasn't interested in L matrices. – ttnphns Jul 8 '16 at 17:11
• I'm happy that we kinda speak the same language now. It seems so, at least. It would be great for everybody, especially a visitor reader, if you accomplish your answer, showing how Gus' and ttnphns' reports convert to the same result. If you want to accomplish. – ttnphns Jul 9 '16 at 12:04
• (cont.) Of course the L matrix in both "approaches " is the same (and no mysterious G matrix is needed). Show that two equivalent paths (L is arbitrary, X is dummies): L -> XC -> regression -> result and X -> [regression -> adjusting to test for L] -> result leave the same result. The 2nd path is how an ANOVA program will do (the bracketed part []); the 1st path is a didactic demonstration how contrasts are solvable via only regression program. – ttnphns Jul 9 '16 at 12:12

A contrast compares two groups by comparing their difference with zero. In a contrast matrix the rows are the contrasts and must add to zero, the columns are the groups. For example:

Let's say you have 4 groups A,B,C,D that you want to compare, then the contrast matrix would be:

Group: A B C D
A vs B: 1 -1 0 0
C vs D: 0 0 -1 1
A,B vs D,C: 1 1 -1 -1

Paraphrasing from Understanding Industrial Experimentation:

If there's a group of k objects to be compared, with k subgroups averages, a contrast is defined on this set of k objects by any set of k coefficients, [c1, c2, c3, ... cj, ..., ck] that sum to zero.

Let C be a contrast then,

$$C = c_{1}\mu_{1} + c_{2}\mu_{2} + ... c_{j}\mu_{j} + ... c_{k}\mu_{k}$$

$$C = \sum_{j=1}^{k} c_{j}\mu{j}$$

with the constraint $$\sum_{j=1}^{k} c_{j} = 0$$

Those subgroups that are assigned a coefficient of zero will be excluded from the comparison.(*)

It is the signs of the coefficients that actually define the comparison, not the values chosen. The absolute values of the coefficients can be anything as long as the sum of the coefficients is zero.

(*)Each statistical software has a different way of indicating which subgroups will be excluded/included.