More Info The academic variables are standardized tests scores in
reading (read), writing (write), and science (science), as well as a categorical
variable (prog) giving the type of program the student is in (general,
academic, or vocational). A researcher is interested in determining what factors influence
the health African Violet plants. Null is that all treatments have the same average effect. 2D correlation spectroscopy, due to high spectral resolution and good spectral interpretation capabilities, is suitable for the analysis of complex biological data. If your testing tool allows, you can also lower your statistical significance levels. .
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. 5) = . \sigma_{1p} \\
\sigma_{21} \sigma_{22} . (Significant simplification occurs in th special case where \(\sigma_1 = \sigma_2 = \sigma^2\))Thus, we classify into population 1 if\[
(y – \mu_2)^2 (y moved here \mu_1)^2
\]or\[
|y- \mu_2| |y – \mu_1|
\]and\[
-2 \log (\Lambda) = -2y \frac{(\mu_1 – \mu_2)}{\sigma^2} + \frac{(\mu_1^2 – \mu_2^2)}{\sigma^2} = \beta y + \alpha
\]Thus, we classify into population 1 if this is less than 0. ,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 \(\sum_{i=1}^k (n_i -1)\)(a modification of the likelihood ratio test). \[
H_0: \mathbf{1′}_p \mu_1 = \mathbf{1′}_p \mu_2 = \dots = \mathbf{1′}_p \mu_h
\]Equivalently,\[
H_0: \mathbf{LBM} = \mathbf{0}
\]where for website link cell means parameterization\[
\mathbf{L} =
\left[
\begin{array}
{ccc}
1 0 -1 \\
0 1 -1
\end{array}
\right]
\]and\[
\mathbf{M} =
\left[
\begin{array}
{cccc}
1 1 1 1
\end{array}
\right]’
\]multiplication yields\[
\left[
\begin{array}
{c}
(\mu_{11} + \mu_{12} + \mu_{13} + \mu_{14}) – (\mu_{31} + \mu_{32} + \mu_{33} + \mu_{34}) \\
(\mu_{21} + \mu_{22} + \mu_{23} + \mu_{24}) – (\mu_{31} + \mu_{32} + \mu_{33} + \mu_{34})
\end{array}
\right]
=
\left[
\begin{array}
{c}
0 \\
0
\end{array}
\right]
\]Different choices of \(\mathbf{L}\) and \(\mathbf{M}\) can yield the same resultGiven that we can’t reject the null hypothesis that all \(h\) profiles are the same, we can ask whether all of the elements of the common profile equal? (i.
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An estimate fo the misclassficaiton probability would be the fraction of \(n_j\) observations which were misclassifiedrepeat the process for other \(i \neq j\) populationsThis method is more reliable than the others, but also computationally intensiveCross-ValidationSummaryConsider the group-specific densities \(f_j (\mathbf{x})\) for multivariate vector \(\mathbf{x}\). These are the orthogonal projections of the data onto the “new principal component axesVariances of \(y_1\) are greater than those for any other possible projectionCovariance matrix decomposition and projection onto orthogonal axes = PCA\(p \times 1\) vectors \(\mathbf{x}_1, \dots , \mathbf{x}_n\) which are iid with \(var(\mathbf{x}_i) = \mathbf{\Sigma}\)The first PC is the linear combination \(y_1 = \mathbf{a}_1′ \mathbf{x} = a_{11}x_1 + \dots + a_{1p}x_p\) with \(\mathbf{a}_1′ \mathbf{a}_1 = 1\) such that \(var(y_1)\) is the maximum of all linear combinations of \(\mathbf{x}\) which have unit lengthThe second PC is the linear combination \(y_1 = \mathbf{a}_2′ \mathbf{x} = a_{21}x_1 + \dots + a_{2p}x_p\) with \(\mathbf{a}_2′ \mathbf{a}_2 = 1\) such that \(var(y_1)\) is the maximum of all linear combinations of \(\mathbf{x}\) which have unit length and uncorrelated with \(y_1\) (i. MVT is a highly sophisticated testing methodology. 4586839 ][ 0.
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A researcher has collected data on three psychological variables,
four academic variables (standardized test scores), and the type of educational
program the student is in for 600 high school students. MVT enables you to conduct an in-depth analysis on visitor behavior and their preference patterns. 97014443][ 1. A rule for which there is no better alternative is called admissibleBayes Discriminant Rules are admissibleIf we utilized prior probabilities, then we can form the posterior probability of a correct allocation, \(\sum_{i=1}^h \pi_i p_{ii}\)Bayes Discriminant Rules have the largest possible posterior probability of correct allocation with respect to the priorThese properties show that Bayes Discriminant rule is our best approach. The classification accuracy rates of unknown samples for K-OPLS, NPLS-DA, and unfolded PLS-DA models were 95%, 92.
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= \mu_6
\]Equivalently,\[
\mu_1 – \mu_2 = 0 \\
\mu_2 – \mu_3 = 0 \\
. .