Comparision of Multivariate Means

Description:

• Measurements are compared by a set of units instead

Pair comparisions:

• Suppose a set of $p$ reponses comes from each of $n$ units for 2 different experiment conditions
  • Both experiments are provided identical set of units
• For the $j$ units $X_{1j1}$ is the first variable of the response of experiment 1 and $X_{2j1}$ is for experiment 2
• Define $D_{ji}=X_{1ji}-X_{2ji}$ for $i=1,...,p$ as the paired-difference random variables
• Define $D_j^{⊺}=[D_{j1},...,D_{jp}]$ for $j=1,..,n$ as random variable of difference in reponses of 2 experiments
  • $E(D_j)=\delta =\left[\begin{array}{c}{\delta }_1\\ .\\ .\\ {\delta }_p\end{array}\right]$ and $Cov(D_j)={\Sigma }_d$
• If $D_1,...,D_n$ are independent $N_p(\delta ,{\Sigma }_d)$, inferences about the vector of mean differences $\delta$ can be based upon a T-squared Statistic, $T^2=n(\bar{D}-\delta {)}^{⊺}{S}_d^{-1}(\bar{D}-\delta )$ where:
  • $\bar{D}=\frac{1}{n}{\Sigma }_{j=1}^n{D}_j$ and $S_d=\frac{1}{n-1}\sum _{j=1}^n(D_j-\bar{D})(D_j-\bar{D}{)}^{⊺}$
• Let the differences $D_1,...,D_n$ be a random sample from a population $N_p(\delta ,{\Sigma }_d)$. Then $T^2\sim \frac{(n-1)p}{n-p}{F}_{p,n-p}$ whatever true $\delta$ and ${\Sigma }_d$
  • if $n$ and $n-p$ are both large, $T^2$ is approximately distributed as a ${\chi }_p^2$ random variable, regardless of the form of the underlying population of differences.
• Test for mean difference is 0, $\delta =0$
  • Given the observed differences $d_j^{⊺}=[d_{j1},...,d_{jp}]$ corresponding to the random variables $D_{jp}$, an $\alpha$ level test of $H_0:\delta =0$ vs $H_1$
  • For an $N_p(\delta ,{\Sigma }_d)$ population, reject $H_0$ if the observerd $T^2=n{\bar{d}}^{⊺}{S}_d^{-1}\bar{d}>\frac{(n-1)p}{n-p}{F}_{p,n-p}(\alpha )$
• A $100(1-\alpha )\%$ confidence region for $\delta$ consists of all $\delta$ such that $(\bar{d}-\delta {)}^{⊺}{S}_d^{-1}(\bar{d}-\delta )\le \frac{(n-1)p}{n(n-p)}{F}_{p,n-p}(\alpha )$
• Indivisual mean differences ${\delta }_i$‘s confidence intervals are given by ${\delta }_i:{\bar{d}}_i\mp \sqrt{\frac{(n-1)p}{n-p}{F}_{p,n-p}(\alpha )}\sqrt{\frac{s_{d_i}^2}{n}}$
  • For $n-p$ large, $\frac{(n-1)p}{n-p}{F}_{p,n-p}(\alpha )\doteq {\chi }_p^2(\alpha )$ and normality need not to be assumed
• Drawing elipsoid chart of difference vector:
  • Find the errors of all units, $D_1,...,D_n$
  • Use Ellipsoid Format Chart

Comparing mean vectors from 2 populations:

• Pooled Statistics
• Consider a random sample of size $n_1$ 1 from population 1 and a sample of size $n_2$ from population 2. We want to make inferences about ${\mu }_1-{\mu }_2$
• We need the following assumptions.
  1. The sample $X_{11},...,X_{1n_1}$ is a random sample of size $n_1$ from a $p$-variate population with mean vector ${\mu }_1$ and covariance matrix $\Sigma$
  2. The sample $X_{21},...,X_{2n_2}$ is a random sample of size $n_2$ from a $p$-variate population with mean vector ${\mu }_2$ and covariance matrix $\Sigma$
  3. $X_{11},...,X_{1n_1}$ are independent of $X_{21},...,X_{2n_2}$
  4. Further, both populations are multivariate normal and ${\Sigma }_1={\Sigma }_2$
• The matrix $S_{pooled}=\frac{n_1-1}{n_1+n_2-2}{S}_1+\frac{n_2-1}{n_1+n_2-2}{S}_2$
• As $S_{pooled}$ is an estimate for $\Sigma ,(\frac{1}{n_1}+\frac{1}{n_2})S_{pooled}$ is an estimator of $Cov({\bar{X}}_1-{\bar{X}}_2)$
• The likelihood ratio test of $H_0:{\mu }_1-{\mu }_2={\delta }_0$ is based on the square of the statistical distance $T^2$
  • reject $H_0$ if $T^2=({\bar{x}}_1-{\bar{x}}_2-{\delta }_0{)}^{⊺}[(\frac{1}{n_1}+\frac{1}{n_2})S_{pooled}{]}^{-1}({\bar{x}}_1-{\bar{x}}_2-{\delta }_0)>c^2$ where the critical distance $c^2$ is determined from the distribution of the two-sample $T^2$-statistics T-squared Statistic
• Proposition: If $X_{11},...,X_{1n_1}$ is a random sample of size $n_1$ from $N_p({\mu }_1,\Sigma )$ and $X_{21},...,X_{2n_2}$ is a random sample of size $n_2$ from $N_p({\mu }_2,\Sigma )$ then $T^2=[{\bar{X}}_1-{\bar{X}}_2-({\mu }_1-{\mu }_2){]}^{⊺}[(\frac{1}{n_1}+\frac{1}{n_2})S_{pooled}{]}^{-1}[{\bar{X}}_1-{\bar{X}}_2-({\mu }_1-{\mu }_2)]$ is distrbuted as $\frac{(n_1+n_2-2)p}{n_1+n_2-p-1}{F}_{p,n_1+n_2-p-1}$
  • F.INV.RT(0.01)
  • Consequencetly, $P\{{\left[{\mathbf{X}}_1-{\overline{\mathbf{X}}}_2-\left({\mathbf{\mu }}_1-{\mathbf{\mu }}_2\right)\right]}^T{\left[\left(\frac{1}{n_1}+\frac{1}{n_1}\right)S_{\text{pooled }}\right]}^{-1}$ $\times \left[{\overline{\mathbf{X}}}_1-{\overline{\mathbf{X}}}_2-\left({\mathbf{\mu }}_1-{\mathbf{\mu }}_2\right)\right]\le c^2\}=1-\alpha$
    • where $c^2=\frac{\left(n_1+n_2-2\right)p}{n_1+n_2-p-1}{F}_{p,n_1+n_2-p-1}(\alpha )$

Simultaneous confidence intervals:

• Let $c^2=\left[\left(n_1+n_2-2\right)p/\left(n_1+n_2-p-1\right)\right]F_{p,n_1+n_2-p-1}(\alpha )$.
• With probability $1-\alpha$, $a^T\left({\overline{\mathbf{X}}}_1-{\overline{\mathbf{X}}}_2\right)\mp c\sqrt{{\mathbf{a}}^T\left(\frac{1}{n_1}+\frac{1}{n_2}\right)S_{\text{pooled }}a}$ will cover $a^T\left({\mathbf{\mu }}_1-{\mathbf{\mu }}_2\right)$ for all $a$. In particular, ${\mu }_{1i}-{\mu }_{2i}$ will be covered by $\left({\bar{X}}_{1i}-{\bar{X}}_{2i}\right)\mp c\sqrt{\left(\frac{1}{n_1}+\frac{1}{n_2}\right)s_{ii,\text{ pooled }}}i=1,\dots ,p$

The two-sample situation when ${\Sigma }_1\ne {\Sigma }_2$

• Let the sample sizes be such that $n_1-p$ and $n_2-p$ are large.
• Then, an approximate $100(1-\alpha )\%$ confidence ellipsoid for ${\mathbf{\mu }}_1-{\mathbf{\mu }}_2$ is given by all ${\mathbf{\mu }}_1-{\mathbf{\mu }}_2$ satisfying:
  • ${\left[{\bar{x}}_1-{\bar{x}}_2-\left({\mu }_1-{\mu }_2\right)\right]}^T{\left[\frac{1}{n_1}{S}_1+\frac{1}{n_2}{S}_2\right]}^{-1}\times \left[{\bar{x}}_1-{\bar{x}}_2-\left({\mu }_1-{\mu }_2\right)\right]\le {\chi }_p^2(\alpha )$
  • where ${\chi }_p^2(\alpha )$ is the upper $(100\alpha )$-th percentile of a chi-square distribution with $p$ d.f.
• Also, $100(1-\alpha )\%$ simultaneous confidence intervals for all linear combinations $a^T\left({\mu }_1-{\mu }_2\right)$ are provided by $a^T\left({\mu }_1-{\mu }_2\right)$ belongs to $a^T\left({\bar{x}}_1-{\bar{x}}_2\right)\mp \sqrt{{\chi }_{\rho }^2(\alpha )}\sqrt{a^T\left(\frac{1}{n_1}{S}_1+\frac{1}{n_2}{S}_2\right)a}$

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