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