# Defining Contrast Matrix

#### Cynderella

##### New Member
Suppose I have a binary Covariate $$X$$ which is defined as

$$X = \begin{cases} 1, & \text{if treatment group} \\ 0, & \text{if placebo group} \end{cases}$$

The model is $$\mathbb E[Y] = \beta_0 + \beta_1X,$$
where $$Y$$ is a continuous random variable.

If $$X=0$$, then $$\mathbb E[Y] = \beta_0\ldots (1)$$,

if $$X=1$$, then $$\mathbb E[Y] = \beta_0 + \beta_1\ldots (2)$$.

Now I want to calculate the Wald Statistics $$W^2 = (\mathbf L\mathbf\beta)'\{\mathbf L\hat{\text{Cov}(\hat{\mathbf\beta})}\mathbf L'\}(\mathbf L\mathbf\beta),$$

where $$\mathbf L$$ is a contrast matrix.

But I can't write the contrast matrix for both $$(1)$$ and $$(2)$$.

The hypothesis is $$\mathbf L\mathbf\beta = 0,$$

where $$\mathbf\beta = \begin{pmatrix} \beta_0\\ \beta_1\\ \end{pmatrix}.$$

For $$(1)$$, I tried to write the contrast matrix $$\mathbf L=(1, 0)$$ so that

$$\mathbf L\mathbf\beta = 0$$
$$\Rightarrow (1, 0)\begin{pmatrix} \beta_0\\ \beta_1\\ \end{pmatrix}=0$$
$$\Rightarrow \beta_0 = 0,$$

but necessarily the contrast matrix is incorrect as the row sum of a contrast matrix is equal to $$0$$. How can I define the contrast matrix?