Joint mean test for setosa

Section of Estimation and inference for the multivariate normal, part of Statistical learning

Type	Section
Taught in	Estimation and inference for the multivariate normal
Part	Statistical learning

The one-sample test is applied here to the two sepal measurements of Iris setosa, with illustrative null mean (5.0, 3.4) cm. That vector is a chosen comparison point. It is not a biological standard. The probability from the F calibration is exact under independent bivariate normal sampling, with n>p and nonsingular S.

x <- as.matrix(subset(iris, Species == "setosa")[, 1:2])
n <- nrow(x); p <- ncol(x)
d <- colMeans(x) - c(5.0, 3.4)
T2 <- n * drop(t(d) %*% solve(cov(x), d))
Fstat <- (n - p) * T2 / (p * (n - 1))
c(T_squared = T2, F = Fstat,
  df1 = p, df2 = n - p,
  p_value = pf(Fstat, p, n - p, lower.tail = FALSE))

T_squared         F       df1       df2   p_value
    0.4323    0.2117    2.0000   48.0000    0.8099

The computed p-value is 0.81. At the 5\% level this joint null is not rejected. A failure to reject does not establish that the true mean equals (5.0, 3.4). The reference distribution F_{p,n-p} is the exact null law only under the independent normal sampling used to derive the F calibration. The p-value does not check that sampling model.

Key idea

Hotelling’s T^2=n(\bar{\mathbf{x}}-\boldsymbol{\mu}_0)^\top S^{-1}(\bar{\mathbf{x}}-\boldsymbol{\mu}_0) is n times the squared Mahalanobis distance of the sample mean from the hypothesised mean, in the metric of the sample covariance. It is a strictly decreasing function of the likelihood ratio. Under independent normal sampling with n>p,

\frac{n-p}{p(n-1)}T^2\sim F_{p,\,n-p}

when the null mean is true. When p=1 the statistic is the square of the ordinary t statistic. The two-sample version uses the same geometry with the pooled covariance and the coefficient n_1 n_2/(n_1+n_2).

Joint mean test for setosa

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See also