Relationship Between Formula and Design Matrices

Formula

In general, a formula for model specification is a character vector or string scalar of the form 'y ~ terms'. For the linear mixed-effects models, this formula is in the form 'y ~ fixed + (random1|grouping1) + ... + (randomR|groupingR)', where fixed and random contain the fixed-effects and the random-effects terms.

Suppose a table tbl contains the following:

A response variable, y
Predictor variables, X_j, which can be continuous or grouping variables
Grouping variables, g₁, g₂, ..., g_R,

where the grouping variables in X_j and g_r can be categorical, logical, character arrays, string arrays, or cell arrays of character vectors.

Then, in a formula of the form, 'y ~ fixed + (random₁|g₁) + ... + (random_R|g_R)', the term fixed corresponds to a specification of the fixed-effects design matrix X, random₁ is a specification of the random-effects design matrix Z₁ corresponding to grouping variable g₁, and similarly random_R is a specification of the random-effects design matrix Z_R corresponding to grouping variable g_R. You can express the fixed and random terms using Wilkinson notation.

Wilkinson notation describes the factors present in models. The notation relates to factors present in models, not to the multipliers (coefficients) of those factors.

Wilkinson Notation	Factors in Standard Notation
`1`	Constant (intercept) term
`X^k`, where `k` is a positive integer	`X`, `X²`, ..., `X^k`
`X1 + X2`	`X1`, `X2`
`X1*X2`	`X1`, `X2`, `X1.*X2 (elementwise multiplication of X1 and X2)`
`X1:X2`	`X1.*X2` only
`- X2`	Do not include `X2`
`X1*X2 + X3`	`X1`, `X2`, `X3`, `X1*X2`
`X1 + X2 + X3 + X1:X2`	`X1`, `X2`, `X3`, `X1*X2`
`X1X2X3 - X1:X2:X3`	`X1`, `X2`, `X3`, `X1X2`, `X1X3`, `X2*X3`
`X1*(X2 + X3)`	`X1`, `X2`, `X3`, `X1X2`, `X1X3`

Statistics and Machine Learning Toolbox™ notation always includes a constant term unless you explicitly remove the term using -1. Here are some examples for linear mixed-effects model specification.

Examples:

Formula	Description
`'y ~ X1 + X2'`	Fixed effects for the intercept, `X1` and `X2`. This is equivalent to `'y ~ 1 + X1 + X2'`.
`'y ~ -1 + X1 + X2'`	No intercept and fixed effects for `X1` and `X2`. The implicit intercept term is suppressed by including `-1`.
`'y ~ 1 + (1 \| g1)'`	Fixed effects for the intercept plus random effect for the intercept for each level of the grouping variable `g1`.
`'y ~ X1 + (1 \| g1)'`	Random intercept model with a fixed slope.
`'y ~ X1 + (X1 \| g1)'`	Random intercept and slope, with possible correlation between them. This is equivalent to `'y ~ 1 + X1 + (1 + X1\|g1)'`.
`'y ~ X1 + (1 \| g1) + (-1 + X1 \| g1)'`	Independent random effects terms for intercept and slope.
`'y ~ 1 + (1 \| g1) + (1 \| g2) + (1 \| g1:g2)'`	Random intercept model with independent main effects for `g1` and `g2`, plus an independent interaction effect.

Design Matrices for Fixed and Random Effects

fitlme converts the expressions in the fixed and random parts (not grouping variables) of a formula into design matrices as follows:

Each term in a formula adds one or more columns to the corresponding design matrix.
A term containing a single continuous variable adds one column to the design matrix.
A fixed term containing a categorical variable X with k levels adds (k – 1) dummy variables to the design matrix.
For example, if the variable Supplier represents three different suppliers a manufacturer receives parts from, i.e. a categorical variable with three levels, and out of six batches of parts, the first two batches come from supplier 1 (level 1), the second two batches come from supplier 2 (level 2), and the last two batches come from supplier 3 (level 3), such as
```
Supplier =

     1
     1
     2
     2
     3
     3 
```
Then, adding Supplier to the formula as a fixed-effects or random-effects term adds the following two dummy variables to the corresponding design matrix, using the 'reference' contrast:
```
 0     0
 0     0
 1     0
 1     0
 0     1
 0     1
```
For more details on dummy variables, see Dummy Variables. For other contrast options, see the 'DummyVarCoding' name-value pair argument of fitlme.
If X1 and X2 are continuous variables, the product term X1:X2 adds one column obtained by elementwise multiplication of X1 and X2 to the design matrix.
If X1 is continuous and X2 is categorical with k levels, the product term X1:X2 multiplies elementwise X1 with the (k – 1) dummy variables representing X2, and adds these (k – 1) columns to the design matrix.
For example, if Drug is the amount of a drug given to patients, a continuous treatment, and Time is three distinct points in time when the health measures are taken, a categorical variable with three levels, and out of nine observations, the first three are observed at time point 1, the second three are observed at time point 2, and the last three are observed at time point 3 so that
```
[Drug Time] =

    0.1000    1.0000
    0.2000    1.0000
    0.5000    2.0000
    0.6000    2.0000
    0.3000    3.0000
    0.8000    3.0000
```
Then, the product term Drug:Time adds the following two variables to the design matrix:
```
     0         0
     0         0
0.5000         0
0.6000         0
     0    0.3000
     0    0.8000
```
If X1 and X2 are categorical variables with k and m levels respectively, the product term X1:X2 adds (k – 1)*(m – 1) dummy variables to the design matrix formed by taking the elementwise product of each dummy variable representing X1 with each dummy variable representing X2.
For example, in an experiment to determine the impact of the type of corn and the popping method on the yield, suppose there are three types of Corn and two types of Method as follows:
```
    1    oil
    1    oil
    1    air
    1    air
    2    oil
    2    oil
    2    air
    2    air
    3    oil
    3    oil
    3    air
    3    air
```
Then, the interaction term Corn:Method adds the following to the design matrix:
```
     0     0
     0     0
     0     0
     0     0
     1     0
     1     0
     0     0
     0     0
     0     1
     0     1
     0     0
     0     0
```
The term X1*X2 adds the necessary number of columns for X1, X2, and X1:X2 to the design matrix.
The term X1^2 adds the necessary number of columns for X1 and X1:X1 to the design matrix.
The symbol 1 (one) in the formula stands for a column of all 1s. By default a column of 1s is included in the design matrix. To exclude a column of ones from the design matrix, you must explicitly specify –1 as a term in the expression.

Grouping Variables

fitlme handles the grouping variables in the (.|group) part of a formula as follows:

If a grouping variable has k levels, then k dummy variables represent this grouping.
For example, suppose District is a categorical grouping variable with three levels, showing the three types of districts, and out of six schools, the first two are in district 1, the second two are in district 2, and the last two are in district 3, so that
```
District =

     1
     1
     2
     2
     3
     3
```
Then, the dummy variables that represent this grouping are:
```
  1     0     0
  1     0     0
  0     1     0
  0     1     0
  0     0     1
  0     0     1
```
If X1 is a continuous random-effects variable and X2 is a grouping variable with k levels, then the random term (X1 – 1|X2) multiplies elementwise X1 with the k dummy variables representing X2 and adds these k columns to the random-effects design matrix.
For example, suppose Score is a continuous variable showing the scores of students from a math exam in a school, and Class is a categorical variable with three levels, showing the three different classes in a school. Also, suppose out of nine observations first three correspond to the scores of students in the first class, the second three correspond to scores of students in the second class, and the last three correspond to the scores of students in the third class, such as
```
[Score Class] =

    78.0000    1.0000
    68.0000    1.0000
    81.0000    2.0000
    53.0000    2.0000
    85.0000    3.0000
    72.0000    3.0000
```
Then, the random term (Score – 1|Class) adds the following three columns to the random-effects design matrix:
```
 78.0000          0          0
 68.0000          0          0
       0    81.0000          0
       0    53.0000          0
       0          0    85.0000
       0          0    72.0000    
```
If X1 is a continuous predictor variable and X2 and X3 are grouping variables with k and m levels respectively, the term (X1|X2:X3) represents this grouping of X1 with k*m dummy variables formed by taking the elementwise product of each dummy variable representing X2 with each dummy variable representing X3.
For example, suppose Treatment is a continuous predictor variable, and there are three levels of Block and two levels of Plot nested within the block as follows:
```
   0.1000    1    a
   0.2000    1    b 
   0.5000    2    a  
   0.6000    2    b 
   0.3000    3    a 
   0.8000    3    b  
```
Then, the random term (Treatment – 1|Block:Plot) adds the following to the random-effects design matrix:
```
 0.1000         0         0         0         0         0
      0    0.2000         0         0         0         0
      0         0    0.5000         0         0         0
      0         0         0    0.6000         0         0
      0         0         0         0    0.3000         0
      0         0         0         0         0    0.8000
```