In the univariate case, the data can often be arranged in a table as shown in the table below:

The columns correspond to the responses to g different treatments or from g different populations. And, the rows correspond to the subjects in each of these treatments or populations.

Notations:

• \(Y_{ij}\) = Observation from subject j in group i
• \(n_{i}\) = Number of subjects in group i
• \(N = n_{1} + n_{2} + \dots + n_{g}\) = Total sample size.

Assumptions for the Analysis of Variance are the same as for a two-sample t-test except that there are more than two groups:

1. The data from group i has common mean = \(\mu_{i}\); i.e., \(E\left(Y_{ij}\right) = \mu_{i}\) . This means that there are no sub-populations with different means.
2. Homoskedasticity: The data from all groups have common variance \(\sigma^2\); i.e., \(var(Y_{ij}) = \sigma^{2}\). That is, the variability in the data does not depend on group membership.
3. Independence: The subjects are independently sampled.
4. Normality: The data are normally distributed.

The hypothesis of interest is that all of the means are equal. Mathematically we write this as:

\(H_0\colon \mu_1 = \mu_2 = \dots = \mu_g\)

The alternative is expressed as:

\(H_a\colon \mu_i \ne \mu_j \) for at least one \(i \ne j\).

i.e., there is a difference between at least one pair of group population means. The following notation should be considered:

Here we are looking at the average squared difference between each observation and the grand mean. Note that if the observations tend to be far away from the Grand Mean then this will take a large value. Conversely, if all of the observations tend to be close to the Grand mean, this will take a small value. Thus, the total sum of squares measures the variation of the data about the Grand mean.

An Analysis of Variance (ANOVA) is a partitioning of the total sum of squares. In the second line of the expression below, we are adding and subtracting the sample mean for the ith group. In the third line, we can divide this out into two terms, the first term involves the differences between the observations and the group means, \(\bar{y}_i\), while the second term involves the differences between the group means and the grand mean.

\(\begin{array}{lll} SS_{total} & = & \sum_{i=1}^{g}\sum_{j=1}^{n_i}\left(Y_{ij}-\bar{y}_{..}\right)^2 \\ & = & \sum_{i=1}^{g}\sum_{j=1}^{n_i}\left((Y_{ij}-\bar{y}_{i.})+(\bar{y}_{i.}-\bar{y}_{..})\right)^2 \\ & = &\underset{SS_{error}}{\underbrace{\sum_{i=1}^{g}\sum_{j=1}^{n_i}(Y_{ij}-\bar{y}_{i.})^2}}+\underset{SS_{treat}}{\underbrace{\sum_{i=1}^{g}n_i(\bar{y}_{i.}-\bar{y}_{..})^2}} \end{array}\)

The first term is called the error sum of squares and measures the variation in the data about their group means.

Note that if the observations tend to be close to their group means, then this value will tend to be small. On the other hand, if the observations tend to be far away from their group means, then the value will be larger. The second term is called the treatment sum of squares and involves the differences between the group means and the Grand mean. Here, if group means are close to the Grand mean, then this value will be small. While, if the group means tend to be far away from the Grand mean, this will take a large value. This second term is called the Treatment Sum of Squares and measures the variation of the group means about the Grand mean.

The Analysis of Variance results is summarized in an analysis of variance table below:

 Hover over the light bulb to get more information on that item.

The ANOVA table contains columns for Source, Degrees of Freedom, Sum of Squares, Mean Square and F. Sources include Treatment and Error which together add up to the Total.

The degrees of freedom for treatment in the first row of the table are calculated by taking the number of groups or treatments minus 1. The total degree of freedom is the total sample size minus 1.  The Error degrees of freedom are obtained by subtracting the treatment degrees of freedom from the total degrees of freedom to obtain N-g.

The formulae for the Sum of Squares are given in the SS column. The Mean Square terms are obtained by taking the Sums of Squares terms and dividing them by the corresponding degrees of freedom.

The final column contains the F statistic which is obtained by taking the MS for treatment and dividing it by the MS for Error.

Under the null hypothesis that the treatment effect is equal across group means, that is \(H_{0} \colon \mu_{1} = \mu_{2} = \dots = \mu_{g} \), this F statistic is F-distributed with g - 1 and N - g degrees of freedom:

\(F \sim F_{g-1, N-g}\)

The numerator degrees of freedom g - 1 comes from the degrees of freedom for treatments in the ANOVA table. This is referred to as the numerator degrees of freedom since the formula for the F-statistic involves the Mean Square for Treatment in the numerator. The denominator degrees of freedom N - g is equal to the degrees of freedom for error in the ANOVA table. This is referred to as the denominator degrees of freedom because the formula for the F-statistic involves the Mean Square Error in the denominator.

We reject \(H_{0}\) at level \(\alpha\) if the F statistic is greater than the critical value of the F-table, with g - 1 and N - g degrees of freedom, and evaluated at level \(\alpha\).

\(F > F_{g-1, N-g, \alpha}\)

Now we will consider the multivariate analog, the Multivariate Analysis of Variance, often abbreviated as MANOVA.

Suppose that we have data on p variables which we can arrange in a table such as the one below:

In this multivariate case the scalar quantities, \(Y_{ij}\), of the corresponding table in ANOVA, are replaced by vectors having p observations.

Notation

Assumptions

The assumptions here are essentially the same as the assumptions in Hotelling's \(T^{2}\) test, only here they apply to groups:

1. The data from group i has common mean vector \(\boldsymbol{\mu}_i = \left(\begin{array}{c}\mu_{i1}\\\mu_{i2}\\\vdots\\\mu_{ip}\end{array}\right)\)
2. The data from all groups have a common variance-covariance matrix \(\Sigma\).
3. Independence: The subjects are independently sampled.
4. Normality: The data are multivariate normally distributed.

Here we are interested in testing the null hypothesis that the group mean vectors are all equal to one another. Mathematically this is expressed as:

\(H_0\colon \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2 = \dots = \boldsymbol{\mu}_g\)

The alternative hypothesis is:

\(H_a \colon \mu_{ik} \ne \mu_{jk}\) for at least one \(i \ne j\) and at least one variable \(k\)

This says that the null hypothesis is false if at least one pair of treatments is different on at least one variable.

Notation

The scalar quantities used in the univariate setting are replaced by vectors in the multivariate setting:

Sample Mean Vector

\(\bar{\mathbf{y}}_{i.} = \frac{1}{n_i}\sum_{j=1}^{n_i}\mathbf{Y}_{ij} = \left(\begin{array}{c}\bar{y}_{i.1}\\ \bar{y}_{i.2} \\ \vdots \\ \bar{y}_{i.p}\end{array}\right)\) = sample mean vector for group i . This sample mean vector is comprised of the group means for each of the p variables. Thus, \(\bar{y}_{i.k} = \frac{1}{n_i}\sum_{j=1}^{n_i}Y_{ijk}\) = sample mean vector for variable k in group i .

Grand Mean Vector

\(\bar{\mathbf{y}}_{..} = \frac{1}{N}\sum_{i=1}^{g}\sum_{j=1}^{n_i}\mathbf{Y}_{ij} = \left(\begin{array}{c}\bar{y}_{..1}\\ \bar{y}_{..2} \\ \vdots \\ \bar{y}_{..p}\end{array}\right)\) = grand mean vector. This grand mean vector is comprised of the grand means for each of the p variables. Thus, \(\bar{y}_{..k} = \frac{1}{N}\sum_{i=1}^{g}\sum_{j=1}^{n_i}Y_{ijk}\) = grand mean for variable k.

Total Sum of Squares and Cross Products

In the univariate Analysis of Variance, we defined the Total Sums of Squares, a scalar quantity. The multivariate analog is the Total Sum of Squares and Cross Products matrix, a p x p matrix of numbers. The total sum of squares is a cross products matrix defined by the expression below:

\(\mathbf{T = \sum\limits_{i=1}^{g}\sum_\limits{j=1}^{n_i}(Y_{ij}-\bar{y}_{..})(Y_{ij}-\bar{y}_{..})'}\)

Here we are looking at the differences between the vectors of observations \(Y_{ij}\) and the Grand mean vector. These differences form a vector which is then multiplied by its transpose.

Here, the \(\left (k, l \right )^{th}\) element of T is

\(\sum\limits_{i=1}^{g}\sum\limits_{j=1}^{n_i} (Y_{ijk}-\bar{y}_{..k})(Y_{ijl}-\bar{y}_{..l})\)

For k = l, this is the total sum of squares for variable k and measures the total variation in the \(k^{th}\) variable. For \(k ≠ l\), this measures the dependence between variables k and l across all of the observations.

We may partition the total sum of squares and cross products as follows:

\(\begin{array}{lll}\mathbf{T} & = & \mathbf{\sum_{i=1}^{g}\sum_{j=1}^{n_i}(Y_{ij}-\bar{y}_{..})(Y_{ij}-\bar{y}_{..})'} \\ & = & \mathbf{\sum_{i=1}^{g}\sum_{j=1}^{n_i}\{(Y_{ij}-\bar{y}_i)+(\bar{y}_i-\bar{y}_{..})\}\{(Y_{ij}-\bar{y}_i)+(\bar{y}_i-\bar{y}_{..})\}'} \\ & = & \mathbf{\underset{E}{\underbrace{\sum_{i=1}^{g}\sum_{j=1}^{n_i}(Y_{ij}-\bar{y}_{i.})(Y_{ij}-\bar{y}_{i.})'}}+\underset{H}{\underbrace{\sum_{i=1}^{g}n_i(\bar{y}_{i.}-\bar{y}_{..})(\bar{y}_{i.}-\bar{y}_{..})'}}}\end{array}\)

where E is the Error Sum of Squares and Cross Products, and H is the Hypothesis Sum of Squares and Cross Products.

The \(\left (k, l \right )^{th}\) element of the error sum of squares and cross products matrix E is:

\(\sum_\limits{i=1}^{g}\sum\limits_{j=1}^{n_i}(Y_{ijk}-\bar{y}_{i.k})(Y_{ijl}-\bar{y}_{i.l})\)

For k = l, this is the error sum of squares for variable k, and measures the within treatment variation for the \(k^{th}\) variable. For \(k ≠ l\), this measures the dependence between variables k and l after taking into account the treatment.

The \(\left (k, l \right )^{th}\) element of the hypothesis sum of squares and cross products matrix H is

\(\sum\limits_{i=1}^{g}n_i(\bar{y}_{i.k}-\bar{y}_{..k})(\bar{y}_{i.l}-\bar{y}_{..l})\)

For k = l, this is the treatment sum of squares for variable k and measures the between-treatment variation for the \(k^{th}\) variable. For \(k ≠ l\), this measures the dependence of variables k and l across treatments.

The partitioning of the total sum of squares and cross-products matrix may be summarized in the multivariate analysis of the variance table:

We wish to reject

\(H_0\colon \boldsymbol{\mu_1 = \mu_2 = \dots =\mu_g}\)

if the hypothesis sum of squares and cross products matrix H is large relative to the error sum of squares and cross products matrix E.

SAS uses four different test statistics based on the MANOVA table:

1. Wilks Lambda

   \(\Lambda^* = \dfrac{|\mathbf{E}|}{|\mathbf{H+E}|}\)

   Here, the determinant of the error sums of squares and cross-products matrix E is divided by the determinant of the total sum of squares and cross-products matrix T = H + E. If H is large relative to E, then |H + E| will be large relative to |E|. Thus, we will reject the null hypothesis if Wilks lambda is small (close to zero).

2. Hotelling-Lawley Trace

   \(T^2_0 = trace(\mathbf{HE}^{-1})\)

   Here, we are multiplying H by the inverse of E; then we take the trace of the resulting matrix. If H is large relative to E, then the Hotelling-Lawley trace will take a large value. Thus, we will reject the null hypothesis if this test statistic is large.

3. Pillai Trace

   \(V = trace(\mathbf{H(H+E)^{-1}})\)

   Here, we are multiplying H by the inverse of the total sum of squares and cross products matrix T = H + E. If H is large relative to E, then the Pillai trace will take a large value. Thus, we will reject the null hypothesis if this test statistic is large.

4. Roy's Maximum Root: Largest eigenvalue of HE^-1

   Here, we multiply H by the inverse of E and then compute the largest eigenvalue of the resulting matrix. If H is large relative to E, then Roy's root will take a large value. Thus, we will reject the null hypothesis if this test statistic is large.

Recall: The trace of a p x p matrix

\(\mathbf{A} = \left(\begin{array}{cccc}a_{11} & a_{12} & \dots & a_{1p}\\ a_{21} & a_{22} & \dots & a_{2p} \\ \vdots & \vdots & & \vdots \\ a_{p1} & a_{p2} & \dots & a_{pp}\end{array}\right)\)

is equal to

\(trace(\mathbf{A}) = \sum_{i=1}^{p}a_{ii}\)

Statistical tables are not available for the above test statistics. However, each of the above test statistics has an F approximation: The following details the F approximations for Wilks lambda. Details for all four F approximations can be found on the SAS website.

1. Wilks Lambda

\begin{align} \text{Starting with }&& \Lambda^* &= \dfrac{|\mathbf{E}|}{|\mathbf{H+E}|}\\  \text{Let, }&& a &= N-g - \dfrac{p-g+2}{2},\\ &&\text{} b &= \left\{\begin{array}{ll} \sqrt{\frac{p^2(g-1)^2-4}{p^2+(g-1)^2-5}}; &\text{if } p^2 + (g-1)^2-5 > 0\\ 1; & \text{if } p^2 + (g-1)^2-5 \le 0 \end{array}\right. \\ \text{and}&& c &= \dfrac{p(g-1)-2}{2} \\ \text{Then}&& F &= \left(\dfrac{1-\Lambda^{1/b}}{\Lambda^{1/b}}\right)\left(\dfrac{ab-c}{p(g-1)}\right) \overset{\cdot}{\sim} F_{p(g-1), ab-c} \\ \text{Under}&& H_{o} \end{align}

Differences among treatments can be explored through pre-planned orthogonal contrasts. Contrasts involve linear combinations of group mean vectors instead of linear combinations of the variables.

The contrast

\(\mathbf{\Psi} = \sum_{i=1}^{g}c_i \mu_i\)

is estimated by replacing the population mean vectors with the corresponding sample mean vectors:

\(\mathbf{\hat{\Psi}} = \sum_{i=1}^{g}c_i\mathbf{\bar{Y}}_i.\)

Because the estimated contrast is a function of random data, the estimated contrast is also a random vector. So the estimated contrast has a population mean vector and population variance-covariance matrix. The population mean of the estimated contrast is \(\mathbf{\Psi}\). The variance-covariance matrix of \(\hat{\mathbf{\Psi}}\)¸ is:

\(\left(\sum\limits_{i=1}^{g}\frac{c^2_i}{n_i}\right)\Sigma\)

which is estimated by substituting the pooled variance-covariance matrix for the population variance-covariance matrix

\(\left(\sum\limits_{i=1}^{g}\frac{c^2_i}{n_i}\right)\mathbf{S}_p = \left(\sum\limits_{i=1}^{g}\frac{c^2_i}{n_i}\right) \dfrac{\mathbf{E}}{N-g}\)

The following shows two examples to construct orthogonal contrasts. In each example, we consider balanced data; that is, there are equal numbers of observations in each group.

The suggestions dealt with in the previous page are not backed up by appropriate hypothesis tests. Consider hypothesis tests of the form:

\(H_0\colon \Psi = 0\) against \(H_a\colon \Psi \ne 0\)

For the univariate case, we may compute the sums of squares for the contrast:

\(SS_{\Psi} = \frac{\hat{\Psi}^2}{\sum_{i=1}^{g}\frac{c^2_i}{n_i}}\)

This sum of squares has only 1 d.f. so that the mean square for the contrast is

\(MS_{\Psi}= SS_{\Psi}\)

Then compute the F-ratio:

\(F = \frac{MS_{\Psi}}{MS_{error}}\)

Reject \(H_{0} \colon \Psi = 0\) at level \(\alpha\) if

\(F > F_{1, N-g, \alpha}\)

For the multivariate case, the sums of squares for the contrast is replaced by the hypothesis sum of squares and cross-products matrix for the contrast:

\(\mathbf{H}_{\mathbf{\Psi}} = \dfrac{\mathbf{\hat{\Psi}\hat{\Psi}'}}{\sum_{i=1}^{g}\frac{c^2_i}{n_i}}\)

\(\Lambda^* = \dfrac{|\mathbf{E}|}{\mathbf{|H_{\Psi}+E|}}\)

Compute the F-statistic

\(F = \left(\dfrac{1-\Lambda^*_{\mathbf{\Psi}}}{\Lambda^*_{\mathbf{\Psi}}}\right)\left(\dfrac{N-g-p+1}{p}\right)\)

Reject H_o : \(\mathbf{\Psi = 0} \) at level \(α\) if

\(F > F_{p, N-g-p+1, \alpha}\)

Within randomized block designs, we have two factors:

1. Blocks, and
2. Treatments

A randomized complete block design with a treatments and b blocks is constructed in two steps:

1. The experimental units (the units to which our treatments are going to be applied) are partitioned into b blocks, each comprised of a units.
2. Treatments are randomly assigned to the experimental units in such a way that each treatment appears once in each block.

Randomized block designs are often applied in agricultural settings. The example below will make this clearer.

In general, the blocks should be partitioned so that:

• Units within blocks are as uniform as possible.
• Differences between blocks are as large as possible.

These conditions will generally give you the most powerful results.

A model is formed for a two-way multivariate analysis of variance.

Two-way MANOVA Additive Model

\(\underset{\mathbf{Y}_{ij}}{\underbrace{\left(\begin{array}{c}Y_{ij1}\\Y_{ij2}\\ \vdots \\ Y_{ijp}\end{array}\right)}} = \underset{\mathbf{\nu}}{\underbrace{\left(\begin{array}{c}\nu_1 \\ \nu_2 \\ \vdots \\ \nu_p \end{array}\right)}}+\underset{\mathbf{\alpha}_{i}}{\underbrace{\left(\begin{array}{c} \alpha_{i1} \\ \alpha_{i2} \\ \vdots \\ \alpha_{ip}\end{array}\right)}}+\underset{\mathbf{\beta}_{j}}{\underbrace{\left(\begin{array}{c}\beta_{j1} \\ \beta_{j2} \\ \vdots \\ \beta_{jp}\end{array}\right)}} + \underset{\mathbf{\epsilon}_{ij}}{\underbrace{\left(\begin{array}{c}\epsilon_{ij1} \\ \epsilon_{ij2} \\ \vdots \\ \epsilon_{ijp}\end{array}\right)}}\)

In this model:

• \(\mathbf{Y}_{ij}\) is the p × 1 vector of observations for treatment i in block j;

This vector of observations is written as a function of the following

• \(\nu_{k}\) is the overall mean for variable k; these are collected into the overall mean vector \(\boldsymbol{\nu}\)
• \(\alpha_{ik}\) is the effect of treatment i on variable k; these are collected into the treatment effect vector \(\boldsymbol{\alpha}_{i}\)
• \(\beta_{jk}\) is the effect of block j on variable k; these are collected in the block effect vector \(\boldsymbol{\beta}_{j}\)
• \(\varepsilon_{ijk}\) is the experimental error for treatment i, block j, and variable k; these are collected into the error vector \(\boldsymbol{\varepsilon}_{ij}\)

These are the standard assumptions.

We could define the treatment mean vector for treatment i such that:

\(\mu_i = \nu +\alpha_i\)

Here we could consider testing the null hypothesis that all of the treatment mean vectors are identical,

\(H_0\colon \boldsymbol{\mu_1 = \mu_2 = \dots = \mu_g}\)

or equivalently, the null hypothesis that there is no treatment effect:

\(H_0\colon \boldsymbol{\alpha_1 = \alpha_2 = \dots = \alpha_a = 0}\)

This is the same null hypothesis that we tested in the One-way MANOVA.

We would test this against the alternative hypothesis that there is a difference between at least one pair of treatments on at least one variable, or:

\(H_a\colon \mu_{ik} \ne \mu_{jk}\) for at least one \(i \ne j\) and at least one variable \(k\)

We will use standard dot notation to define mean vectors for treatments, mean vectors for blocks, and a grand mean vector.

We define a mean vector for treatment i:

\(\mathbf{\bar{y}}_{i.} = \frac{1}{b}\sum_{j=1}^{b}\mathbf{Y}_{ij} = \left(\begin{array}{c}\bar{y}_{i.1}\\ \bar{y}_{i.2} \\ \vdots \\ \bar{y}_{i.p}\end{array}\right)\) = Sample mean vector for treatment i.

In this case, it is comprised of the mean vectors for ith treatment for each of the p variables and it is obtained by summing over the blocks and then dividing by the number of blocks. The dot appears in the second position indicating that we are to sum over the second subscript, the position assigned to the blocks.

For example, \(\bar{y}_{i.k} = \frac{1}{b}\sum_{j=1}^{b}Y_{ijk}\) = Sample mean for variable k and treatment i.

We  define a mean vector for block j:

\(\mathbf{\bar{y}}_{.j} = \frac{1}{a}\sum_{i=1}^{a}\mathbf{Y}_{ij} = \left(\begin{array}{c}\bar{y}_{.j1}\\ \bar{y}_{.j2} \\ \vdots \\ \bar{y}_{.jp}\end{array}\right)\) = Sample mean vector for block j.

Here we will sum over the treatments in each of the blocks so the dot appears in the first position. Therefore, this is essentially the block means for each of our variables.

For example, \(\bar{y}_{.jk} = \frac{1}{a}\sum_{i=1}^{a}Y_{ijk}\) = Sample mean for variable k and block j.

Finally, we define the Grand mean vector by summing all of the observation vectors over the treatments and the blocks. So you will see the double dots appearing in this case:

\(\mathbf{\bar{y}}_{..} = \frac{1}{ab}\sum_{i=1}^{a}\sum_{j=1}^{b}\mathbf{Y}_{ij} = \left(\begin{array}{c}\bar{y}_{..1}\\ \bar{y}_{..2} \\ \vdots \\ \bar{y}_{..p}\end{array}\right)\) = Grand mean vector.

This involves dividing by a × b, which is the sample size in this case.

For example, \(\bar{y}_{..k}=\frac{1}{ab}\sum_{i=1}^{a}\sum_{j=1}^{b}Y_{ijk}\) = Grand mean for variable k.

As before, we will define the Total Sum of Squares and Cross Products Matrix. This is the same definition that we used in the One-way MANOVA. It involves comparing the observation vectors for the individual subjects to the grand mean vector.

\(\mathbf{T = \sum_{i=1}^{a}\sum_{j=1}^{b}(Y_{ij}-\bar{y}_{..})(Y_{ij}-\bar{y}_{..})'}\)

Here, the \( \left(k, l \right)^{th}\) element of T is

\(\sum_{i=1}^{a}\sum_{j=1}^{b}(Y_{ijk}-\bar{y}_{..k})(Y_{ijl}-\bar{y}_{..l}).\)

• For \( k = l \), this is the total sum of squares for variable k and measures the total variation in variable k.
• For \( k ≠ l \), this measures the association or dependency between variables k and l across all observations.

In this case, the total sum of squares and cross products matrix may be partitioned into three matrices, three different sums of squares cross-product matrices:

\begin{align} \mathbf{T} &= \underset{\mathbf{H}}{\underbrace{b\sum_{i=1}^{a}\mathbf{(\bar{y}_{i.}-\bar{y}_{..})(\bar{y}_{i.}-\bar{y}_{..})'}}}\\&+\underset{\mathbf{B}}{\underbrace{a\sum_{j=1}^{b}\mathbf{(\bar{y}_{.j}-\bar{y}_{..})(\bar{y}_{.j}-\bar{y}_{..})'}}}\\ &+\underset{\mathbf{E}}{\underbrace{\sum_{i=1}^{a}\sum_{j=1}^{b}\mathbf{(Y_{ij}-\bar{y}_{i.}-\bar{y}_{.j}+\bar{y}_{..})(Y_{ij}-\bar{y}_{i.}-\bar{y}_{.j}+\bar{y}_{..})'}}} \end{align}

As shown above:

• H is the Treatment Sum of Squares and Cross Products matrix;
• B is the Block Sum of Squares and Cross Products matrix;
• E is the Error Sum of Squares and Cross Products matrix.

The \( \left(k, l \right)^{th}\) element of the Treatment Sum of Squares and Cross Products matrix H is

\(b\sum_{i=1}^{a}(\bar{y}_{i.k}-\bar{y}_{..k})(\bar{y}_{i.l}-\bar{y}_{..l})\)

• If  \(k = l\), is the treatment sum of squares for variable k, and measures variation between treatments.
• If \( k ≠ l \), this measures how variables k and l vary together across treatments.

The \( \left(k, l \right)^{th}\) element of the Block Sum of Squares and Cross Products matrix B is

\(a\sum_{j=1}^{a}(\bar{y}_{.jk}-\bar{y}_{..k})(\bar{y}_{.jl}-\bar{y}_{..l})\)

• For \( k = l \), is the block sum of squares for variable k, and measures variation between or among blocks.
• For \( k ≠ l \), this measures how variables k and l vary together across blocks (not usually of much interest).

The \( \left(k, l \right)^{th}\) element of the Error Sum of Squares and Cross Products matrix E is

\(\sum_{i=1}^{a}\sum_{j=1}^{b}(Y_{ijk}-\bar{y}_{i.k}-\bar{y}_{.jk}+\bar{y}_{..k})(Y_{ijl}-\bar{y}_{i.l}-\bar{y}_{.jl}+\bar{y}_{..l})\)

• For \( k = l \), is the error sum of squares for variable k, and measures variability within treatment and block combinations of variable k.
• For \( k ≠ l \), this measures the association or dependence between variables k and l after you take into account treatment and block.

The partitioning of the total sum of squares and cross products matrix may be summarized in the multivariate analysis of variance table as shown below:

SSP stands for the sum of squares and cross products discussed above.

To test the null hypothesis that the treatment mean vectors are equal, compute a Wilks Lambda using the following expression:

\(\Lambda^* = \dfrac{|\mathbf{E}|}{|\mathbf{H+E}|}\)

This is the determinant of the error sum of squares and cross-products matrix divided by the determinant of the sum of the treatment sum of squares and cross-products plus the error sum of squares and cross-products matrix.

Under the null hypothesis, this has an F-approximation. The approximation is quite involved and will not be reviewed here. Instead, let's take a look at our example where we will implement these concepts.

In this lesson we learned about:

• The 1-way MANOVA for testing the null hypothesis of equality of group mean vectors;
• Methods for diagnosing the assumptions of the 1-way MANOVA;
• Bonferroni corrected ANOVAs to assess the significance of individual variables;
• Construction and interpretation of orthogonal contrasts;
• Wilks lambda for testing the significance of contrasts among group mean vectors; and
• Simultaneous and Bonferroni confidence intervals for the elements of contrast.

In general, a thorough analysis of data would be comprised of the following steps:

1. Step 1:

   Perform appropriate diagnostic tests for the assumptions of the MANOVA. Carry out appropriate normalizing and variance-stabilizing transformations of the variables.

2. Step 2:

   Perform a one-way MANOVA to test for equality of group mean vectors. If this test is not significant, conclude that there is no statistically significant evidence against the null hypothesis that the group mean vectors are equal to one another and stop. If the test is significant, conclude that at least one pair of group mean vectors differ on at least one element and go on to Step 3.

3. Step 3:

   Perform Bonferroni-corrected ANOVAs on the individual variables to determine which variables are significantly different among groups.

4. Step 4:

   Construct up to g-1 orthogonal contrasts based on specific scientific questions regarding the relationships among the groups.

5. Step 5:

   Use Wilks lambda to test the significance of each contrast defined in Step 4.

6. Step 6:

   For the significant contrasts only, construct simultaneous or Bonferroni confidence intervals for the elements of those contrasts. Draw appropriate conclusions from these confidence intervals, making sure that you note the directions of all effects (which treatments or group of treatments have the greater means for each variable).

In this lesson we also learned about:

• How to perform multiple factor MANOVAs;
• What conclusions may be drawn from the results of a multiple-factor MANOVA;
• The Bonferroni corrected ANOVAs for the individual variables.

Just as in the one-way MANOVA, we carried out orthogonal contrasts among the four varieties of rice. However, in this case, it is not clear from the data description just what contrasts should be considered. If a phylogenetic tree were available for these varieties, then appropriate contrasts may be constructed.