Scalar Projections and Linear Contrasts

Why Scalar Projections Matter

A multivariate normal vector describes several random quantities jointly, but many questions concern just one numerical summary of them: a weighted score, a difference between measurements, or a treatment contrast. Such a summary is a scalar projection. Its distribution can be calculated directly from the vector's mean and covariance, without first finding the full joint density.

The key calculation is compact, but the covariance terms matter. In particular, the coordinates of a Gaussian vector need not be independent, and omitting their covariances can give a wrong variance and therefore a wrong probability. A second important case occurs when the projection has zero variance: then it is deterministic, not an ordinary normal variable that can be standardized.

Distribution of a Scalar Projection

We use \(N(m,v)\) for a univariate normal distribution with mean \(m\) and variance \(v\). When \(v=0\), this notation denotes the point mass at \(m\). The standard normal cumulative distribution function is \(\Phi(z)=P(Z\leq z)\) for \(Z\sim N(0,1)\).

The statement applies to nonsingular and singular Gaussian laws alike. To verify it, recall the characteristic function of a Gaussian vector: \[ \varphi_X(t)=E[\exp(i t^\mathsf{T}X)] =\exp\!\left(i t^\mathsf{T}\mu-\tfrac12 t^\mathsf{T}\Sigma t\right), \qquad t\in\mathbb{R}^p. \] For the scalar \(Y=a^\mathsf{T}X\), its characteristic function at \(s\in\mathbb{R}\) is \[ \varphi_Y(s)=E[\exp(i s a^\mathsf{T}X)] =\varphi_X(sa) =\exp\!\left(i s a^\mathsf{T}\mu-\tfrac12 s^2 a^\mathsf{T}\Sigma a\right). \] This is the characteristic function of a univariate normal variable with the stated mean and variance when \(a^\mathsf{T}\Sigma a>0\). Uniqueness of characteristic functions identifies the distribution. Also, \(a^\mathsf{T}\Sigma a=\operatorname{Var}(a^\mathsf{T}X)\geq 0\). If it equals zero, then \[ E\!\left[\{a^\mathsf{T}X-a^\mathsf{T}\mu\}^2\right] =\operatorname{Var}(a^\mathsf{T}X)=0. \] A nonnegative random variable with expectation zero is zero with probability one, so \(a^\mathsf{T}X=a^\mathsf{T}\mu\) almost surely. Its distribution is the point mass specified above. \(\square\)

The formula also follows at the level of means and variances from linearity and the covariance definition: \[ E[a^\mathsf{T}X]=a^\mathsf{T}\mu,\qquad \operatorname{Var}(a^\mathsf{T}X) =E\!\left[a^\mathsf{T}(X-\mu)(X-\mu)^\mathsf{T}a\right] =a^\mathsf{T}\Sigma a. \] These calculations identify the parameters; Gaussianity is what makes those parameters determine the entire univariate distribution.

Worked Scalar Projections

For any threshold \(t\), the same calculation gives a direct probability rule. If \(v=a^\mathsf{T}\Sigma a>0\) and \(m=a^\mathsf{T}\mu\), then \[ P(a^\mathsf{T}X\leq t)=\Phi\!\left(\frac{t-m}{\sqrt{v}}\right), \qquad P(a^\mathsf{T}X>t)=1-\Phi\!\left(\frac{t-m}{\sqrt{v}}\right). \] A positive-variance normal distribution is continuous, so replacing \(>\) by \(\geq\), or \(\leq\) by \(<\), does not change the corresponding probability. The zero-variance case must instead be handled as a point mass: if \(m=a^\mathsf{T}\mu\), then \(P(a^\mathsf{T}X\leq t)\) is \(1\) when \(m\leq t\) and \(0\) when \(m>t\).

Linear Contrasts and Common Shifts

A contrast is a projection whose coefficients sum to zero. For example, a difference \(X_1-X_2\) has coefficients \(1\) and \(-1\); a comparison of three coordinates might use weights \(1,-\tfrac12,-\tfrac12\). A contrast has a useful invariance: adding the same constant to every coordinate does not change its value. Indeed, if \(\mathbf{1}\) is the vector of \(p\) ones and \(a^\mathsf{T}\mathbf{1}=0\), then for any constant \(c\), \[ a^\mathsf{T}(X+c\mathbf{1}) =a^\mathsf{T}X+c\,a^\mathsf{T}\mathbf{1} =a^\mathsf{T}X. \] The variance formula for a contrast is still \(a^\mathsf{T}\Sigma a\); the zero-sum condition changes the interpretation of the projection, not the distribution rule.

A Reliable Calculation and Common Pitfalls

A common mistake is to calculate the variance of a weighted sum using only the marginal variances. In coordinates, \[ a^\mathsf{T}\Sigma a =\sum_{i=1}^{p}a_i^2\operatorname{Var}(X_i) +2\sum_{1\leq i<j\leq p}a_i a_j\operatorname{Cov}(X_i,X_j). \] The second sum is generally present, and its sign depends on both the weights and the covariances. Another mistake is to apply the positive-variance standardization formula when \(v=0\); division by \(\sqrt{v}\) is then undefined, and the correct answer comes from the deterministic value of the projection.

Scalar projections are useful because they convert joint Gaussian questions into univariate ones whenever the quantity of interest is a linear combination. This is central to interpreting differences, weighted scores, and other linear summaries: the mean describes their center, while the covariance matrix determines how variable they are. A correct threshold probability follows once those two quantities are calculated and the degenerate case is checked.