# Chapter 21 Multivariate Methods

$$y_1,...,y_p$$ are possibly correlated random variables with means $$\mu_1,...,\mu_p$$

$\mathbf{y} = \left( \begin{array} {c} y_1 \\ . \\ y_p \\ \end{array} \right)$

$E(\mathbf{y}) = \left( \begin{array} {c} \mu_1 \\ . \\ \mu_p \\ \end{array} \right)$

Let $$\sigma_{ij} = cov(y_i, y_j)$$ for $$i,j = 1,…,p$$

$\mathbf{\Sigma} = (\sigma_{ij}) = \left( \begin{array} {cccc} \sigma_{11} & \sigma_{22} & ... & \sigma_{1p} \\ \sigma_{21} & \sigma_{22} & ... & \sigma_{2p} \\ . & . & . & . \\ \sigma_{p1} & \sigma_{p2} & ... & \sigma_{pp} \end{array} \right)$

where $$\mathbf{\Sigma}$$ (symmetric) is the variance-covariance or dispersion matrix

Let $$\mathbf{u}_{p \times 1}$$ and $$\mathbf{v}_{q \times 1}$$ be random vectors with means $$\mu_u$$ and $$\mu_v$$ . Then

$\mathbf{\Sigma}_{uv} = cov(\mathbf{u,v}) = E[(\mathbf{u} - \mu_u)(\mathbf{v} - \mu_v)']$

in which $$\mathbf{\Sigma}_{uv} \neq \mathbf{\Sigma}_{vu}$$ and $$\mathbf{\Sigma}_{uv} = \mathbf{\Sigma}_{vu}'$$

Properties of Covariance Matrices

1. Symmetric $$\mathbf{\Sigma}' = \mathbf{\Sigma}$$
2. Non-negative definite $$\mathbf{a'\Sigma a} \ge 0$$ for any $$\mathbf{a} \in R^p$$, which is equivalent to eigenvalues of $$\mathbf{\Sigma}$$, $$\lambda_1 \ge \lambda_2 \ge ... \ge \lambda_p \ge 0$$
3. $$|\mathbf{\Sigma}| = \lambda_1 \lambda_2 ... \lambda_p \ge 0$$ (generalized variance) (the bigger this number is, the more variation there is
4. $$trace(\mathbf{\Sigma}) = tr(\mathbf{\Sigma}) = \lambda_1 + ... + \lambda_p = \sigma_{11} + ... + \sigma_{pp} =$$ sum of variance (total variance)

Note:

• $$\mathbf{\Sigma}$$ is typically required to be positive definite, which means all eigenvalues are positive, and $$\mathbf{\Sigma}$$ has an inverse $$\mathbf{\Sigma}^{-1}$$ such that $$\mathbf{\Sigma}^{-1}\mathbf{\Sigma} = \mathbf{I}_{p \times p} = \mathbf{\Sigma \Sigma}^{-1}$$

Correlation Matrices

$\rho_{ij} = \frac{\sigma_{ij}}{\sqrt{\sigma_{ii} \sigma_{jj}}}$

$\mathbf{R} = \left( \begin{array} {cccc} \rho_{11} & \rho_{12} & ... & \rho_{1p} \\ \rho_{21} & \rho_{22} & ... & \rho_{2p} \\ . & . & . &. \\ \rho_{p1} & \rho_{p2} & ... & \rho_{pp} \\ \end{array} \right)$

where $$\rho_{ij}$$ is the correlation, and $$\rho_{ii} = 1$$ for all i

Alternatively,

$\mathbf{R} = [diag(\mathbf{\Sigma})]^{-1/2}\mathbf{\Sigma}[diag(\mathbf{\Sigma})]^{-1/2}$

where $$diag(\mathbf{\Sigma})$$ is the matrix which has the $$\sigma_{ii}$$’s on the diagonal and 0’s elsewhere

and $$\mathbf{A}^{1/2}$$ (the square root of a symmetric matrix) is a symmetric matrix such as $$\mathbf{A} = \mathbf{A}^{1/2}\mathbf{A}^{1/2}$$

Equalities

Let

• $$\mathbf{x}$$ and $$\mathbf{y}$$ be random vectors with means $$\mu_x$$ and $$\mu_y$$ and variance -variance matrices $$\mathbf{\Sigma}_x$$ and $$\mathbf{\Sigma}_y$$.

• $$\mathbf{A}$$ and $$\mathbf{B}$$ be matrices of constants and $$\mathbf{c}$$ and $$\mathbf{d}$$ be vectors of constants

Then

• $$E(\mathbf{Ay + c} ) = \mathbf{A} \mu_y + c$$

• $$var(\mathbf{Ay + c}) = \mathbf{A} var(\mathbf{y})\mathbf{A}' = \mathbf{A \Sigma_y A}'$$

• $$cov(\mathbf{Ay + c, By+ d}) = \mathbf{A\Sigma_y B}'$$

• $$E(\mathbf{Ay + Bx + c}) = \mathbf{A \mu_y + B \mu_x + c}$$

• $$var(\mathbf{Ay + Bx + c}) = \mathbf{A \Sigma_y A' + B \Sigma_x B' + A \Sigma_{yx}B' + B\Sigma'_{yx}A'}$$

Multivariate Normal Distribution

Let $$\mathbf{y}$$ be a multivariate normal (MVN) random variable with mean $$\mu$$ and variance $$\mathbf{\Sigma}$$. Then the density of $$\mathbf{y}$$ is

$f(\mathbf{y}) = \frac{1}{(2\pi)^{p/2}|\mathbf{\Sigma}|^{1/2}} \exp(-\frac{1}{2} \mathbf{(y-\mu)'\Sigma^{-1}(y-\mu)} )$

$$\mathbf{y} \sim N_p(\mu, \mathbf{\Sigma})$$

### 21.0.1 Properties of MVN

• Let $$\mathbf{A}_{r \times p}$$ be a fixed matrix. Then $$\mathbf{Ay} \sim N_r (\mathbf{A \mu, A \Sigma A'})$$ . $$r \le p$$ and all rows of $$\mathbf{A}$$ must be linearly independent to guarantee that $$\mathbf{A \Sigma A}'$$ is non-singular.

• Let $$\mathbf{G}$$ be a matrix such that $$\mathbf{\Sigma}^{-1} = \mathbf{GG}'$$. Then $$\mathbf{G'y} \sim N_p(\mathbf{G' \mu, I})$$ and $$\mathbf{G'(y-\mu)} \sim N_p (0,\mathbf{I})$$

• Any fixed linear combination of $$y_1,...,y_p$$ (say $$\mathbf{c'y}$$) follows $$\mathbf{c'y} \sim N_1 (\mathbf{c' \mu, c' \Sigma c})$$

• Define a partition, $$[\mathbf{y}'_1,\mathbf{y}_2']'$$ where

• $$\mathbf{y}_1$$ is $$p_1 \times 1$$

• $$\mathbf{y}_2$$ is $$p_2 \times 1$$,

• $$p_1 + p_2 = p$$

• $$p_1,p_2 \ge 1$$ Then

$\left( \begin{array} {c} \mathbf{y}_1 \\ \mathbf{y}_2 \\ \end{array} \right) \sim N \left( \left( \begin{array} {c} \mu_1 \\ \mu_2 \\ \end{array} \right), \left( \begin{array} {cc} \mathbf{\Sigma}_{11} & \mathbf{\Sigma}_{12} \\ \mathbf{\Sigma}_{21} & \mathbf{\Sigma}_{22}\\ \end{array} \right) \right)$

• The marginal distributions of $$\mathbf{y}_1$$ and $$\mathbf{y}_2$$ are $$\mathbf{y}_1 \sim N_{p1}(\mathbf{\mu_1, \Sigma_{11}})$$ and $$\mathbf{y}_2 \sim N_{p2}(\mathbf{\mu_2, \Sigma_{22}})$$

• Individual components $$y_1,...,y_p$$ are all normally distributed $$y_i \sim N_1(\mu_i, \sigma_{ii})$$

• The conditional distribution of $$\mathbf{y}_1$$ and $$\mathbf{y}_2$$ is normal

• $$\mathbf{y}_1 | \mathbf{y}_2 \sim N_{p1}(\mathbf{\mu_1 + \Sigma_{12} \Sigma_{22}^{-1}(y_2 - \mu_2),\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \sigma_{21}})$$

• In this formula, we see if we know (have info about) $$\mathbf{y}_2$$, we can re-weight $$\mathbf{y}_1$$ ’s mean, and the variance is reduced because we know more about $$\mathbf{y}_1$$ because we know $$\mathbf{y}_2$$
• which is analogous to $$\mathbf{y}_2 | \mathbf{y}_1$$. And $$\mathbf{y}_1$$ and $$\mathbf{y}_2$$ are independently distrusted only if $$\mathbf{\Sigma}_{12} = 0$$

• If $$\mathbf{y} \sim N(\mathbf{\mu, \Sigma})$$ and $$\mathbf{\Sigma}$$ is positive definite, then $$\mathbf{(y-\mu)' \Sigma^{-1} (y - \mu)} \sim \chi^2_{(p)}$$

• If $$\mathbf{y}_i$$ are independent $$N_p (\mathbf{\mu}_i , \mathbf{\Sigma}_i)$$ random variables, then for fixed matrices $$\mathbf{A}_{i(m \times p)}$$, $$\sum_{i=1}^k \mathbf{A}_i \mathbf{y}_i \sim N_m (\sum_{i=1}^{k} \mathbf{A}_i \mathbf{\mu}_i, \sum_{i=1}^k \mathbf{A}_i \mathbf{\Sigma}_i \mathbf{A}_i)$$

Multiple Regression

$\left( \begin{array} {c} Y \\ \mathbf{x} \end{array} \right) \sim N_{p+1} \left( \left[ \begin{array} {c} \mu_y \\ \mathbf{\mu}_x \end{array} \right] , \left[ \begin{array} {cc} \sigma^2_Y & \mathbf{\Sigma}_{yx} \\ \mathbf{\Sigma}_{yx} & \mathbf{\Sigma}_{xx} \end{array} \right] \right)$

The conditional distribution of Y given x follows a univariate normal distribution with

\begin{aligned} E(Y| \mathbf{x}) &= \mu_y + \mathbf{\Sigma}_{yx} \Sigma_{xx}^{-1} (\mathbf{x}- \mu_x) \\ &= \mu_y - \Sigma_{yx} \Sigma_{xx}^{-1}\mu_x + \Sigma_{yx} \Sigma_{xx}^{-1}\mathbf{x} \\ &= \beta_0 + \mathbf{\beta'x} \end{aligned}

where $$\beta = (\beta_1,...,\beta_p)' = \mathbf{\Sigma}_{xx}^{-1} \mathbf{\Sigma}_{yx}'$$ (e.g., analogous to $$\mathbf{(x'x)^{-1}x'y}$$ but not the same if we consider $$Y_i$$ and $$\mathbf{x}_i$$, $$i = 1,..,n$$ and use the empirical covariance formula: $$var(Y|\mathbf{x}) = \sigma^2_Y - \mathbf{\Sigma_{yx}\Sigma^{-1}_{xx} \Sigma'_{yx}}$$)

Samples from Multivariate Normal Populations

A random sample of size n, $$\mathbf{y}_1,.., \mathbf{y}_n$$ from $$N_p (\mathbf{\mu}, \mathbf{\Sigma})$$. Then

• Since $$\mathbf{y}_1,..., \mathbf{y}_n$$ are iid, their sample mean, $$\bar{\mathbf{y}} = \sum_{i=1}^n \mathbf{y}_i/n \sim N_p (\mathbf{\mu}, \mathbf{\Sigma}/n)$$. that is, $$\bar{\mathbf{y}}$$ is an unbiased estimator of $$\mathbf{\mu}$$

• The $$p \times p$$ sample variance-covariance matrix, $$\mathbf{S}$$ is $$\mathbf{S} = \frac{1}{n-1}\sum_{i=1}^n (\mathbf{y}_i - \bar{\mathbf{y}})(\mathbf{y}_i - \bar{\mathbf{y}})' = \frac{1}{n-1} (\sum_{i=1}^n \mathbf{y}_i \mathbf{y}_i' - n \bar{\mathbf{y}}\bar{\mathbf{y}}')$$

• where $$\mathbf{S}$$ is symmetric, unbiased estimator of $$\mathbf{\Sigma}$$ and has $$p(p+1)/2$$ random variables.
• $$(n-1)\mathbf{S} \sim W_p (n-1, \mathbf{\Sigma})$$ is a Wishart distribution with n-1 degrees of freedom and expectation $$(n-1) \mathbf{\Sigma}$$. The Wishart distribution is a multivariate extension of the Chi-squared distribution.

• $$\bar{\mathbf{y}}$$ and $$\mathbf{S}$$ are independent

• $$\bar{\mathbf{y}}$$ and $$\mathbf{S}$$ are sufficient statistics. (All of the info in the data about $$\mathbf{\mu}$$ and $$\mathbf{\Sigma}$$ is contained in $$\bar{\mathbf{y}}$$ and $$\mathbf{S}$$ , regardless of sample size).

Large Sample Properties

$$\mathbf{y}_1,..., \mathbf{y}_n$$ are a random sample from some population with mean $$\mathbf{\mu}$$ and variance-covariance matrix $$\mathbf{\Sigma}$$

• $$\bar{\mathbf{y}}$$ is a consistent estimator for $$\mu$$

• $$\mathbf{S}$$ is a consistent estimator for $$\mathbf{\Sigma}$$

• Multivariate Central Limit Theorem: Similar to the univariate case, $$\sqrt{n}(\bar{\mathbf{y}} - \mu) \dot{\sim} N_p (\mathbf{0,\Sigma})$$ where n is large relative to p ($$n \ge 25p$$), which is equivalent to $$\bar{\mathbf{y}} \dot{\sim} N_p (\mu, \mathbf{\Sigma}/n)$$

• Wald’s Theorem: $$n(\bar{\mathbf{y}} - \mu)' \mathbf{S}^{-1} (\bar{\mathbf{y}} - \mu)$$ when n is large relative to p.

Maximum Likelihood Estimation for MVN

Suppose iid $$\mathbf{y}_1 ,... \mathbf{y}_n \sim N_p (\mu, \mathbf{\Sigma})$$, the likelihood function for the data is

\begin{aligned} L(\mu, \mathbf{\Sigma}) &= \prod_{j=1}^n (\frac{1}{(2\pi)^{p/2}|\mathbf{\Sigma}|^{1/2}} \exp(-\frac{1}{2}(\mathbf{y}_j -\mu)'\mathbf{\Sigma}^{-1})(\mathbf{y}_j -\mu)) \\ &= \frac{1}{(2\pi)^{np/2}|\mathbf{\Sigma}|^{n/2}} \exp(-\frac{1}{2} \sum_{j=1}^n(\mathbf{y}_j -\mu)'\mathbf{\Sigma}^{-1})(\mathbf{y}_j -\mu) \end{aligned}

Then, the MLEs are

$\hat{\mu} = \bar{\mathbf{y}}$

$\hat{\mathbf{\Sigma}} = \frac{n-1}{n} \mathbf{S}$

using derivatives of the log of the likelihood function with respect to $$\mu$$ and $$\mathbf{\Sigma}$$

Properties of MLEs

• Invariance: If $$\hat{\theta}$$ is the MLE of $$\theta$$, then the MLE of $$h(\theta)$$ is $$h(\hat{\theta})$$ for any function h(.)

• Consistency: MLEs are consistent estimators, but they are usually biased

• Efficiency: MLEs are efficient estimators (no other estimator has a smaller variance for large samples)

• Asymptotic normality: Suppose that $$\hat{\theta}_n$$ is the MLE for $$\theta$$ based upon n independent observations. Then $$\hat{\theta}_n \dot{\sim} N(\theta, \mathbf{H}^{-1})$$

• $$\mathbf{H}$$ is the Fisher Information Matrix, which contains the expected values of the second partial derivatives fo the log-likelihood function. the (i,j)th element of $$\mathbf{H}$$ is $$-E(\frac{\partial^2 l(\mathbf{\theta})}{\partial \theta_i \partial \theta_j})$$

• we can estimate $$\mathbf{H}$$ by finding the form determined above, and evaluate it at $$\theta = \hat{\theta}_n$$

• Likelihood ratio testing: for some null hypothesis, $$H_0$$ we can form a likelihood ratio test

• The statistic is: $$\Lambda = \frac{\max_{H_0}l(\mathbf{\mu}, \mathbf{\Sigma|Y})}{\max l(\mu, \mathbf{\Sigma | Y})}$$

• For large n, $$-2 \log \Lambda \sim \chi^2_{(v)}$$ where v is the number of parameters in the unrestricted space minus the number of parameters under $$H_0$$

Test of Multivariate Normality

• Check univariate normality for each trait (X) separately

• Can check Normality Assessment

• The good thing is that if any of the univariate trait is not normal, then the joint distribution is not normal (see again Properties of MVN). If a joint multivariate distribution is normal, then the marginal distribution has to be normal.

• However, marginal normality of all traits does not imply joint MVN

• Easily rule out multivariate normality, but not easy to prove it

• Mardia’s tests for multivariate normality

• Multivariate skewness is$\beta_{1,p} = E[(\mathbf{y}- \mathbf{\mu})' \mathbf{\Sigma}^{-1} (\mathbf{x} - \mathbf{\mu})]^3$

• where $$\mathbf{x}$$ and $$\mathbf{y}$$ are independent, but have the same distribution (note: $$\beta$$ here is not regression coefficient)

• Multivariate kurtosis is defined as

• $\beta_{2,p} - E[(\mathbf{y}- \mathbf{\mu})' \mathbf{\Sigma}^{-1} (\mathbf{x} - \mathbf{\mu})]^2$

• For the MVN distribution, we have $$\beta_{1,p} = 0$$ and $$\beta_{2,p} = p(p+2)$$

• For a sample of size n, we can estimate

$\hat{\beta}_{1,p} = \frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n g^2_{ij}$

$\hat{\beta}_{2,p} = \frac{1}{n} \sum_{i=1}^n g^2_{ii}$

• where $$g_{ij} = (\mathbf{y}_i - \bar{\mathbf{y}})' \mathbf{S}^{-1} (\mathbf{y}_j - \bar{\mathbf{y}})$$. Note: $$g_{ii} = d^2_i$$ where $$d^2_i$$ is the Mahalanobis distance
• shows for large n

$\kappa_1 = \frac{n \hat{\beta}_{1,p}}{6} \dot{\sim} \chi^2_{p(p+1)(p+2)/6}$

$\kappa_2 = \frac{\hat{\beta}_{2,p} - p(p+2)}{\sqrt{8p(p+2)/n}} \sim N(0,1)$

• Hence, we can use $$\kappa_1$$ and $$\kappa_2$$ to test the null hypothesis of MVN.

• When the data are non-normal, normal theory tests on the mean are sensitive to $$\beta_{1,p}$$ , while tests on the covariance are sensitive to $$\beta_{2,p}$$

• Chi-square Q-Q plot

• Let $$\mathbf{y}_i, i = 1,...,n$$ be a random sample sample from $$N_p(\mathbf{\mu}, \mathbf{\Sigma})$$

• Then $$\mathbf{z}_i = \mathbf{\Sigma}^{-1/2}(\mathbf{y}_i - \mathbf{\mu}), i = 1,...,n$$ are iid $$N_p (\mathbf{0}, \mathbf{I})$$. Thus, $$d_i^2 = \mathbf{z}_i' \mathbf{z}_i \sim \chi^2_p , i = 1,...,n$$

• plot the ordered $$d_i^2$$ values against the qualities of the $$\chi^2_p$$ distribution. When normality holds, the plot should approximately resemble a straight lien passing through the origin at a 45 degree

• it requires large sample size (i.e., sensitive to sample size). Even if we generate data from a MVN, the tail of the Chi-square Q-Q plot can still be out of line.

• If the data are not normal, we can

• ignore it

• use nonparametric methods

• use models based upon an approximate distirubiton (e.g., GLMM)

• try performing a transformation

library(heplots)
## Warning: package 'heplots' was built under R version 4.0.5
library(ICSNP)
## Warning: package 'ICSNP' was built under R version 4.0.5
## Warning: package 'ICS' was built under R version 4.0.5
library(MVN)
## Warning: package 'MVN' was built under R version 4.0.5
library(tidyverse)

names(trees) <- c("Nitrogen","Phosphorous","Potassium","Ash","Height")
str(trees)
## 'data.frame':    26 obs. of  5 variables:
##  $Nitrogen : num 2.2 2.1 1.52 2.88 2.18 1.87 1.52 2.37 2.06 1.84 ... ##$ Phosphorous: num  0.417 0.354 0.208 0.335 0.314 0.271 0.164 0.302 0.373 0.265 ...
##  $Potassium : num 1.35 0.9 0.71 0.9 1.26 1.15 0.83 0.89 0.79 0.72 ... ##$ Ash        : num  1.79 1.08 0.47 1.48 1.09 0.99 0.85 0.94 0.8 0.77 ...
##  $Height : int 351 249 171 373 321 191 225 291 284 213 ... summary(trees) ## Nitrogen Phosphorous Potassium Ash ## Min. :1.130 Min. :0.1570 Min. :0.3800 Min. :0.4500 ## 1st Qu.:1.532 1st Qu.:0.1963 1st Qu.:0.6050 1st Qu.:0.6375 ## Median :1.855 Median :0.2250 Median :0.7150 Median :0.9300 ## Mean :1.896 Mean :0.2506 Mean :0.7619 Mean :0.8873 ## 3rd Qu.:2.160 3rd Qu.:0.2975 3rd Qu.:0.8975 3rd Qu.:0.9825 ## Max. :2.880 Max. :0.4170 Max. :1.3500 Max. :1.7900 ## Height ## Min. : 65.0 ## 1st Qu.:122.5 ## Median :181.0 ## Mean :196.6 ## 3rd Qu.:276.0 ## Max. :373.0 cor(trees, method = "pearson") # correlation matrix ## Nitrogen Phosphorous Potassium Ash Height ## Nitrogen 1.0000000 0.6023902 0.5462456 0.6509771 0.8181641 ## Phosphorous 0.6023902 1.0000000 0.7037469 0.6707871 0.7739656 ## Potassium 0.5462456 0.7037469 1.0000000 0.6710548 0.7915683 ## Ash 0.6509771 0.6707871 0.6710548 1.0000000 0.7676771 ## Height 0.8181641 0.7739656 0.7915683 0.7676771 1.0000000 # qq-plot gg <- trees %>% pivot_longer(everything(), names_to = "Var", values_to = "Value") %>% ggplot(aes(sample = Value)) + geom_qq() + geom_qq_line() + facet_wrap("Var", scales = "free") gg # Univariate normality sw_tests <- apply(trees, MARGIN = 2, FUN = shapiro.test) sw_tests ##$Nitrogen
##
##  Shapiro-Wilk normality test
##
## data:  newX[, i]
## W = 0.96829, p-value = 0.5794
##
##
## $Phosphorous ## ## Shapiro-Wilk normality test ## ## data: newX[, i] ## W = 0.93644, p-value = 0.1104 ## ## ##$Potassium
##
##  Shapiro-Wilk normality test
##
## data:  newX[, i]
## W = 0.95709, p-value = 0.3375
##
##
## $Ash ## ## Shapiro-Wilk normality test ## ## data: newX[, i] ## W = 0.92071, p-value = 0.04671 ## ## ##$Height
##
##  Shapiro-Wilk normality test
##
## data:  newX[, i]
## W = 0.94107, p-value = 0.1424
# Kolmogorov-Smirnov test
ks_tests <- map(trees, ~ ks.test(scale(.x),"pnorm"))
## Warning in ks.test(scale(.x), "pnorm"): ties should not be present for the
## Kolmogorov-Smirnov test
## Warning in ks.test(scale(.x), "pnorm"): ties should not be present for the
## Kolmogorov-Smirnov test

## Warning in ks.test(scale(.x), "pnorm"): ties should not be present for the
## Kolmogorov-Smirnov test

## Warning in ks.test(scale(.x), "pnorm"): ties should not be present for the
## Kolmogorov-Smirnov test

## Warning in ks.test(scale(.x), "pnorm"): ties should not be present for the
## Kolmogorov-Smirnov test
ks_tests
## $Nitrogen ## ## One-sample Kolmogorov-Smirnov test ## ## data: scale(.x) ## D = 0.12182, p-value = 0.8351 ## alternative hypothesis: two-sided ## ## ##$Phosphorous
##
##  One-sample Kolmogorov-Smirnov test
##
## data:  scale(.x)
## D = 0.17627, p-value = 0.3944
## alternative hypothesis: two-sided
##
##
## $Potassium ## ## One-sample Kolmogorov-Smirnov test ## ## data: scale(.x) ## D = 0.10542, p-value = 0.9348 ## alternative hypothesis: two-sided ## ## ##$Ash
##
##  One-sample Kolmogorov-Smirnov test
##
## data:  scale(.x)
## D = 0.14503, p-value = 0.6449
## alternative hypothesis: two-sided
##
##
## $Height ## ## One-sample Kolmogorov-Smirnov test ## ## data: scale(.x) ## D = 0.1107, p-value = 0.9076 ## alternative hypothesis: two-sided # Mardia's test, need large sample size for power mardia_test <- mvn( trees, mvnTest = "mardia", covariance = FALSE, multivariatePlot = "qq" ) mardia_test$multivariateNormality
##              Test         Statistic            p value Result
## 1 Mardia Skewness  29.7248528871795   0.72054426745778    YES
## 2 Mardia Kurtosis -1.67743173185383 0.0934580886477281    YES
## 3             MVN              <NA>               <NA>    YES

### 21.0.2 Mean Vector Inference

In the univariate normal distribution, we test $$H_0: \mu =\mu_0$$ by using

$T = \frac{\bar{y}- \mu_0}{s/\sqrt{n}} \sim t_{n-1}$

under the null hypothesis. And reject the null if $$|T|$$ is large relative to $$t_{(1-\alpha/2,n-1)}$$ because it means that seeing a value as large as what we observed is rare if the null is true

Equivalently,

$T^2 = \frac{(\bar{y}- \mu_0)^2}{s^2/n} = n(\bar{y}- \mu_0)(s^2)^{-1}(\bar{y}- \mu_0) \sim f_{(1,n-1)}$

#### 21.0.2.1Natural Multivariate Generalization

$H_0: \mathbf{\mu} = \mathbf{\mu}_0 \\ H_a: \mathbf{\mu} \neq \mathbf{\mu}_0$

Define Hotelling’s $$T^2$$ by

$T^2 = n(\bar{\mathbf{y}} - \mathbf{\mu}_0)'\mathbf{S}^{-1}(\bar{\mathbf{y}} - \mathbf{\mu}_0)$

which can be viewed as a generalized distance between $$\bar{\mathbf{y}}$$ and $$\mathbf{\mu}_0$$

Under the assumption of normality,

$F = \frac{n-p}{(n-1)p} T^2 \sim f_{(p,n-p)}$

and reject the null hypothesis when $$F > f_{(1-\alpha, p, n-p)}$$

• The $$T^2$$ test is invariant to changes in measurement units.

• If $$\mathbf{z = Cy + d}$$ where $$\mathbf{C}$$ and $$\mathbf{d}$$ do not depend on $$\mathbf{y}$$, then $$T^2(\mathbf{z}) - T^2(\mathbf{y})$$
• The $$T^2$$ test can be derived as a likelihood ratio test of $$H_0: \mu = \mu_0$$

#### 21.0.2.2 Confidence Intervals

##### 21.0.2.2.1 Confidence Region

An “exact” $$100(1-\alpha)\%$$ confidence region for $$\mathbf{\mu}$$ is the set of all vectors, $$\mathbf{v}$$, which are “close enough” to the observed mean vector, $$\bar{\mathbf{y}}$$ to satisfy

$n(\bar{\mathbf{y}} - \mathbf{\mu}_0)'\mathbf{S}^{-1}(\bar{\mathbf{y}} - \mathbf{\mu}_0) \le \frac{(n-1)p}{n-p} f_{(1-\alpha, p, n-p)}$

• $$\mathbf{v}$$ are just the mean vectors that are not rejected by the $$T^2$$ test when $$\mathbf{\bar{y}}$$ is observed.

In case that you have 2 parameters, the confidence region is a “hyper-ellipsoid.”

In this region, it consists of all $$\mathbf{\mu}_0$$ vectors for which the $$T^2$$ test would not reject $$H_0$$ at significance level $$\alpha$$

Even though the confidence region better assesses the joint knowledge concerning plausible values of $$\mathbf{\mu}$$ , people typically include confidence statement about the individual component means. We’d like all of the separate confidence statements to hold simultaneously with a specified high probability. Simultaneous confidence intervals: intervals against any statement being incorrect

###### 21.0.2.2.1.1 Simultaneous Confidence Statements
• Intervals based on a rectangular confidence region by projecting the previous region onto the coordinate axes:

$\bar{y}_{i} \pm \sqrt{\frac{(n-1)p}{n-p}f_{(1-\alpha, p,n-p)}\frac{s_{ii}}{n}}$

for all $$i = 1,..,p$$

which implied confidence region is conservative; it has at least $$100(1- \alpha)\%$$

Generally, simultaneous $$100(1-\alpha) \%$$ confidence intervals for all linear combinations , $$\mathbf{a}$$ of the elements of the mean vector are given by

$\mathbf{a'\bar{y}} \pm \sqrt{\frac{(n-1)p}{n-p}f_{(1-\alpha, p,n-p)}\frac{\mathbf{a'Sa}}{n}}$

• works for any arbitrary linear combination $$\mathbf{a'\mu} = a_1 \mu_1 + ... + a_p \mu_p$$, which is a projection onto the axis in the direction of $$\mathbf{a}$$

• These intervals have the property that the probability that at least one such interval does not contain the appropriate $$\mathbf{a' \mu}$$ is no more than $$\alpha$$

• These types of intervals can be used for “data snooping” (like Scheffe)

###### 21.0.2.2.1.2 One $$\mu$$ at a time
• One at a time confidence intervals:

$\bar{y}_i \pm t_{(1 - \alpha/2, n-1} \sqrt{\frac{s_{ii}}{n}}$

• Each of these intervals has a probability of $$1-\alpha$$ of covering the appropriate $$\mu_i$$

• But they ignore the covariance structure of the $$p$$ variables

• If we only care about $$k$$ simultaneous intervals, we can use “one at a time” method with the Bonferroni correction.

• This method gets more conservative as the number of intervals $$k$$ increases.

### 21.0.3 General Hypothesis Testing

#### 21.0.3.1 One-sample Tests

$H_0: \mathbf{C \mu= 0}$

where

• $$\mathbf{C}$$ is a $$c \times p$$ matrix of rank c where $$c \le p$$

We can test this hypothesis using the following statistic

$F = \frac{n - c}{(n-1)c} T^2$

where $$T^2 = n(\mathbf{C\bar{y}})' (\mathbf{CSC'})^{-1} (\mathbf{C\bar{y}})$$

Example:

$H_0: \mu_1 = \mu_2 = ... = \mu_p$

Equivalently,

$\mu_1 - \mu_2 = 0 \\ \vdots \\ \mu_{p-1} - \mu_p = 0$

a total of $$p-1$$ tests. Hence, we have $$\mathbf{C}$$ as the $$p - 1 \times p$$ matrix

$\mathbf{C} = \left( \begin{array} {ccccc} 1 & -1 & 0 & \ldots & 0 \\ 0 & 1 & -1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & 1 & -1 \end{array} \right)$

number of rows = $$c = p -1$$

Equivalently, we can also compare all of the other means to the first mean. Then, we test $$\mu_1 - \mu_2 = 0, \mu_1 - \mu_3 = 0,..., \mu_1 - \mu_p = 0$$, the $$(p-1) \times p$$ matrix $$\mathbf{C}$$ is

$\mathbf{C} = \left( \begin{array} {ccccc} -1 & 1 & 0 & \ldots & 0 \\ -1 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -1 & 0 & \ldots & 0 & 1 \end{array} \right)$

The value of $$T^2$$ is invariant to these equivalent choices of $$\mathbf{C}$$

This is often used for repeated measures designs, where each subject receives each treatment once over successive periods of time (all treatments are administered to each unit).

Example:

Let $$y_{ij}$$ be the response from subject i at time j for $$i = 1,..,n, j = 1,...,T$$. In this case, $$\mathbf{y}_i = (y_{i1}, ..., y_{iT})', i = 1,...,n$$ are a random sample from $$N_T (\mathbf{\mu}, \mathbf{\Sigma})$$

Let $$n=8$$ subjects, $$T = 6$$. We are interested in $$\mu_1, .., \mu_6$$

$H_0: \mu_1 = \mu_2 = ... = \mu_6$

Equivalently,

$\mu_1 - \mu_2 = 0 \\ \mu_2 - \mu_3 = 0 \\ ... \\ \mu_5 - \mu_6 = 0$

We can test orthogonal polynomials for 4 equally spaced time points. To test for example the null hypothesis that quadratic and cubic effects are jointly equal to 0, we would define $$\mathbf{C}$$

$\mathbf{C} = \left( \begin{array} {cccc} 1 & -1 & -1 & 1 \\ -1 & 3 & -3 & 1 \end{array} \right)$

#### 21.0.3.2 Two-Sample Tests

Consider the analogous two sample multivariate tests.

Example: we have data on two independent random samples, one sample from each of two populations

$\mathbf{y}_{1i} \sim N_p (\mathbf{\mu_1, \Sigma}) \\ \mathbf{y}_{2j} \sim N_p (\mathbf{\mu_2, \Sigma})$

We assume

• normality

• equal variance-covariance matrices

• independent random samples

We can summarize our data using the sufficient statistics $$\mathbf{\bar{y}}_1, \mathbf{S}_1, \mathbf{\bar{y}}_2, \mathbf{S}_2$$ with respective sample sizes, $$n_1,n_2$$

Since we assume that $$\mathbf{\Sigma}_1 = \mathbf{\Sigma}_2 = \mathbf{\Sigma}$$, compute a pooled estimate of the variance-covariance matrix on $$n_1 + n_2 - 2$$ df

$\mathbf{S} = \frac{(n_1 - 1)\mathbf{S}_1 + (n_2-1) \mathbf{S}_2}{(n_1 -1) + (n_2 - 1)}$

$H_0: \mathbf{\mu}_1 = \mathbf{\mu}_2 \\ H_a: \mathbf{\mu}_1 \neq \mathbf{\mu}_2$

At least one element of the mean vectors is different

We use

• $$\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2$$ to estimate $$\mu_1 - \mu_2$$

• $$\mathbf{S}$$ to estimate $$\mathbf{\Sigma}$$

Note: because we assume the two populations are independent, there is no covariance

$$cov(\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2) = var(\mathbf{\bar{y}}_1) + var(\mathbf{\bar{y}}_2) = \frac{\mathbf{\Sigma_1}}{n_1} + \frac{\mathbf{\Sigma_2}}{n_2} = \mathbf{\Sigma}(\frac{1}{n_1} + \frac{1}{n_2})$$

Reject $$H_0$$ if

\begin{aligned} T^2 &= (\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2)'\{ \mathbf{S} (\frac{1}{n_1} + \frac{1}{n_2})\}^{-1} (\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2)\\ &= \frac{n_1 n_2}{n_1 +n_2} (\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2)'\{ \mathbf{S} \}^{-1} (\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2)\\ & \ge \frac{(n_1 + n_2 -2)p}{n_1 + n_2 - p - 1} f_{(1- \alpha,n_1 + n_2 - p -1)} \end{aligned}

or equivalently, if

$F = \frac{n_1 + n_2 - p -1}{(n_1 + n_2 -2)p} T^2 \ge f_{(1- \alpha, p , n_1 + n_2 -p -1)}$

A $$100(1-\alpha) \%$$ confidence region for $$\mu_1 - \mu_2$$ consists of all vector $$\delta$$ which satisfy

$\frac{n_1 n_2}{n_1 + n_2} (\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2 - \mathbf{\delta})' \mathbf{S}^{-1}(\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2 - \mathbf{\delta}) \le \frac{(n_1 + n_2 - 2)p}{n_1 + n_2 -p - 1}f_{(1-\alpha, p , n_1 + n_2 - p -1)}$

The simultaneous confidence intervals for all linear combinations of $$\mu_1 - \mu_2$$ have the form

$\mathbf{a'}(\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2) \pm \sqrt{\frac{(n_1 + n_2 -2)p}{n_1 + n_2 - p -1}}f_{(1-\alpha, p, n_1 + n_2 -p -1)} \times \sqrt{\mathbf{a'Sa}(\frac{1}{n_1} + \frac{1}{n_2})}$

Bonferroni intervals, for k combinations

$(\bar{y}_{1i} - \bar{y}_{2i}) \pm t_{(1-\alpha/2k, n_1 + n_2 - 2)}\sqrt{(\frac{1}{n_1} + \frac{1}{n_2})s_{ii}}$

#### 21.0.3.3 Model Assumptions

If model assumption are not met

• Unequal Covariance Matrices

• If $$n_1 = n_2$$ (large samples) there is little effect on the Type I error rate and power fo the two sample test

• If $$n_1 > n_2$$ and the eigenvalues of $$\mathbf{\Sigma}_1 \mathbf{\Sigma}^{-1}_2$$ are less than 1, the Type I error level is inflated

• If $$n_1 > n_2$$ and some eigenvalues of $$\mathbf{\Sigma}_1 \mathbf{\Sigma}_2^{-1}$$ are greater than 1, the Type I error rate is too small, leading to a reduction in power

• Sample Not Normal

• Type I error level of the two sample $$T^2$$ test isn’t much affect by moderate departures from normality if the two populations being sampled have similar distributions

• One sample $$T^2$$ test is much more sensitive to lack of normality, especially when the distribution is skewed.

• Intuitively, you can think that in one sample your distribution will be sensitive, but the distribution of the difference between two similar distributions will not be as sensitive.

• Solutions:

• Transform to make the data more normal

• Large large samples, use the $$\chi^2$$ (Wald) test, in which populations don’t need to be normal, or equal sample sizes, or equal variance-covariance matrices

• $$H_0: \mu_1 - \mu_2 =0$$ use $$(\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2)'( \frac{1}{n_1} \mathbf{S}_1 + \frac{1}{n_2}\mathbf{S}_2)^{-1}(\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2) \dot{\sim} \chi^2_{(p)}$$

##### 21.0.3.3.1 Equal Covariance Matrices Tests

With independent random samples from k populations of p-dimensional vectors. We compute the sample covariance matrix for each, $$\mathbf{S}_i$$, where $$i = 1,...,k$$

$H_0: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2 = \ldots = \mathbf{\Sigma}_k = \mathbf{\Sigma} \\ H_a: \text{at least 2 are different}$

Assume $$H_0$$ is true, we would use a pooled estimate of the common covariance matrix, $$\mathbf{\Sigma}$$

$\mathbf{S} = \frac{\sum_{i=1}^k (n_i -1)\mathbf{S}_i}{\sum_{i=1}^k (n_i - 1)}$

with $$\sum_{i=1}^k (n_i -1)$$

###### 21.0.3.3.1.1 Bartlett’s Test

(a modification of the likelihood ratio test). Define

$N = \sum_{i=1}^k n_i$

and (note: $$| |$$ are determinants here, not absolute value)

$M = (N - k) \log|\mathbf{S}| - \sum_{i=1}^k (n_i - 1) \log|\mathbf{S}_i|$

$C^{-1} = 1 - \frac{2p^2 + 3p - 1}{6(p+1)(k-1)} \{\sum_{i=1}^k (\frac{1}{n_i - 1}) - \frac{1}{N-k} \}$

• Reject $$H_0$$ when $$MC^{-1} > \chi^2_{1- \alpha, (k-1)p(p+1)/2}$$

• If not all samples are from normal populations, $$MC^{-1}$$ has a distribution which is often shifted to the right of the nominal $$\chi^2$$ distribution, which means $$H_0$$ is often rejected even when it is true (the Type I error level is inflated). Hence, it is better to test individual normality first, or then multivariate normality before you do Bartlett’s test.

#### 21.0.3.4 Two-Sample Repeated Measurements

• Define $$\mathbf{y}_{hi} = (y_{hi1}, ..., y_{hit})'$$ to be the observations from the i-th subject in the h-th group for times 1 through T

• Assume that $$\mathbf{y}_{11}, ..., \mathbf{y}_{1n_1}$$ are iid $$N_t(\mathbf{\mu}_1, \mathbf{\Sigma})$$ and that $$\mathbf{y}_{21},...,\mathbf{y}_{2n_2}$$ are iid $$N_t(\mathbf{\mu}_2, \mathbf{\Sigma})$$

• $$H_0: \mathbf{C}(\mathbf{\mu}_1 - \mathbf{\mu}_2) = \mathbf{0}_c$$ where $$\mathbf{C}$$ is a $$c \times t$$ matrix of rank $$c$$ where $$c \le t$$

• The test statistic has the form

$T^2 = \frac{n_1 n_2}{n_1 + n_2} (\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2)' \mathbf{C}'(\mathbf{CSC}')^{-1}\mathbf{C} (\mathbf{\bar{y}}_1 - \mathbf{\bar{y}}_2)$

where $$\mathbf{S}$$ is the pooled covariance estimate. Then,

$F = \frac{n_1 + n_2 - c -1}{(n_1 + n_2-2)c} T^2 \sim f_{(c, n_1 + n_2 - c-1)}$

when $$H_0$$ is true

If the null hypothesis $$H_0: \mu_1 = \mu_2$$ is rejected. A weaker hypothesis is that the profiles for the two groups are parallel.

$\mu_{11} - \mu_{21} = \mu_{12} - \mu_{22} \\ \vdots \\ \mu_{1t-1} - \mu_{2t-1} = \mu_{1t} - \mu_{2t}$

The null hypothesis matrix term is then

$$H_0: \mathbf{C}(\mu_1 - \mu_2) = \mathbf{0}_c$$ , where $$c = t - 1$$ and

$\mathbf{C} = \left( \begin{array} {ccccc} 1 & -1 & 0 & \ldots & 0 \\ 0 & 1 & -1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & -1 \end{array} \right)_{(t-1) \times t}$

# One-sample Hotelling's T^2 test
#  Create data frame
plants <- data.frame(
y1 = c(2.11, 2.36, 2.13, 2.78, 2.17),
y2 = c(10.1, 35.0, 2.0, 6.0, 2.0),
y3 = c(3.4, 4.1, 1.9, 3.8, 1.7)
)

# Center the data with the hypothesized means and make a matrix
plants_ctr <- plants %>%
transmute(y1_ctr = y1 - 2.85,
y2_ctr = y2 - 15.0,
y3_ctr = y3 - 6.0) %>%
as.matrix()

# Use anova.mlm to calculate Wilks' lambda
onesamp_fit <- anova(lm(plants_ctr ~ 1), test = "Wilks")
onesamp_fit # can't reject the null of hypothesized vector of means
## Analysis of Variance Table
##
##             Df    Wilks approx F num Df den Df  Pr(>F)
## (Intercept)  1 0.054219   11.629      3      2 0.08022 .
## Residuals    4
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Paired-Sample Hotelling's T^2 test
library(ICSNP)

#  Create data frame
waste <- data.frame(
case = 1:11,
com_y1 = c(6, 6, 18, 8, 11, 34, 28, 71, 43, 33, 20),
com_y2 = c(27, 23, 64, 44, 30, 75, 26, 124, 54, 30, 14),
state_y1 = c(25, 28, 36, 35, 15, 44, 42, 54, 34, 29, 39),
state_y2 = c(15, 13, 22, 29, 31, 64, 30, 64, 56, 20, 21)
)

# Calculate the difference between commercial and state labs
waste_diff <- waste %>%
transmute(y1_diff = com_y1 - state_y1,
y2_diff = com_y2 - state_y2)
# Run the test
paired_fit <- HotellingsT2(waste_diff)
paired_fit # value T.2 in the output corresponds to the approximate F-value in the output from anova.mlm
##
##  Hotelling's one sample T2-test
##
## data:  waste_diff
## T.2 = 6.1377, df1 = 2, df2 = 9, p-value = 0.02083
## alternative hypothesis: true location is not equal to c(0,0)
# reject the null that the two labs' measurements are equal
# Independent-Sample Hotelling's T^2 test with Bartlett's test

names(steel) <- c("Temp", "Yield", "Strength")
str(steel)
## 'data.frame':    12 obs. of  3 variables:
##  $Temp : int 1 1 1 1 1 2 2 2 2 2 ... ##$ Yield   : int  33 36 35 38 40 35 36 38 39 41 ...
##  $Strength: int 60 61 64 63 65 57 59 59 61 63 ... # Plot the data ggplot(steel, aes(x = Yield, y = Strength)) + geom_text(aes(label = Temp), size = 5) + geom_segment(aes( x = 33, y = 57.5, xend = 42, yend = 65 ), col = "red") # Bartlett's test for equality of covariance matrices # same thing as Box's M test in the multivariate setting bart_test <- boxM(steel[, -1], steel$Temp)
bart_test # fail to reject the null of equal covariances 
##
##  Box's M-test for Homogeneity of Covariance Matrices
##
## data:  steel[, -1]
## Chi-Sq (approx.) = 0.38077, df = 3, p-value = 0.9442
# anova.mlm
twosamp_fit <-
anova(lm(cbind(Yield, Strength) ~ factor(Temp), data = steel), test = "Wilks")
twosamp_fit
## Analysis of Variance Table
##
##              Df    Wilks approx F num Df den Df    Pr(>F)
## (Intercept)   1 0.001177   3818.1      2      9 6.589e-14 ***
## factor(Temp)  1 0.294883     10.8      2      9  0.004106 **
## Residuals    10
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# ICSNP package
twosamp_fit2 <-
HotellingsT2(cbind(steel$Yield, steel$Strength) ~ factor(steel$Temp)) twosamp_fit2 ## ## Hotelling's two sample T2-test ## ## data: cbind(steel$Yield, steel$Strength) by factor(steel$Temp)
## T.2 = 10.76, df1 = 2, df2 = 9, p-value = 0.004106
## alternative hypothesis: true location difference is not equal to c(0,0)
# reject null. Hence, there is a difference in the means of the bivariate normal distributions