MANOVA Follow up using Contrasts

Univariate case

As we describe in Planned Comparisons, for a sample with m groups, contrasts are linear combinations of the group means \sum_j c_j \mu_j based on contrast coefficients (i.e. weights) cj such that \sum_j c_j = 0. Hypothesis testing using contrasts consists of using a null hypothesis in the form of  \sum_j c_j \mu_j = 0 based on the appropriate values of the contrast coefficients as weights.

As usual we use \sum_j c_j \bar{x}_j as an estimate for \sum_j c_j \mu_j, and note that \sum_j c_j \bar{x}_j is a random variable with zero mean and variance MSW \sum_j \frac{c_j^2}{n_j}. Thus, when the null hypothesis is true the test statistic

image9144

has distribution T(dfW ). Since t ~ T(df) is equivalent to t2 ~ F(1, df), it follows that this test is equivalent to

image9145

Contrasts \sum_j c_j \bar{x}_j and \sum_j d_j \bar{x}_j are orthogonal provided \sum_j \frac{c_j d_j}{n_j} = 0, which in the balanced case, where n1 = n2 = ⋯ = nm, is the same as stating that the vectors (c1, …, cm) and (d1, …, dm) are orthogonal.

Note that the null hypotheses of orthogonal contrasts are independent of one another; i.e. the result of one has no impact on the result of the other.

If ψ1, …, ψm-1 are m –1 contrasts that are pairwise orthogonal, then any other contrast can be expressed as a linear combination of these contrasts. Thus you only ever need to look at m – 1 orthogonal contrasts. Since \sum_{j=1}^m c_j = 0 for any contrast, each of the ψj is orthogonal to the unit vector (1, …, 1) and so m –1 contrasts (and not m) are sufficient. Also by Corollary 1 of Orthogonal Vectors and Matrices, we can always find a set of m –1 contrasts.

Note that if ψ1, …, ψm-1 are pairwise orthogonal contrasts, then SS between groups, SSB = SS_{\psi_1} + ⋯ + SS_{\psi_{m-1}}. Also each df_{\psi_j} = 1, and so m – 1 = df_{\psi_1} + ⋯ + df_{\psi_{m-1}}. Thus any m – 1 pairwise orthogonal contrasts partition SSB.

Multivariate case

Multivariate contrasts are linear combinations of the group mean vectors \sum_j c_j \mu_j based on contrast coefficients (i.e. weights) cj such that \sum_j c_j = 0. Hypothesis testing using contrasts consists of using a null hypothesis in the form of  \sum_j c_j \mu_j = 0 based on the appropriate values of the contrast coefficients as weights.

As usual we use \sum_j c_j \bar{X}_j as an estimate for \sum_j c_j \mu_j, and note that \sum_j c_j \bar{X}_j is a random vector with zero mean and covariance matrix 

image9146

Contrasts \sum_j c_j \bar{X}_j and \sum_j d_j \bar{X}_j are orthogonal provided \sum_j \frac{c_j d_j}{n_j} = 0, which in the balanced case, where n1 = n2 = ⋯ = nm, is the same as stating that the vectors (c1, …, cm) and (d1, …, dm) are orthogonal.

Note that the null hypotheses of orthogonal contrasts are independent of one another; i.e. the result of one has no impact on the result of the other.

If ψ1, …, ψm-1 are m –1 contrasts that are pairwise orthogonal, then any other contrast can be expressed as a linear combination of these contrasts. Thus you only ever need to look at m – 1 orthogonal contrasts. Since \sum_{j=1}^m c_j = 0 for any contrast, each of the ψj is orthogonal to the unit vector (1, …, 1) and so m –1 contrasts (and not m) are sufficient. Also by Corollary 1 of Orthogonal Vectors and Matrices, we can always find a set of m –1 contrasts.

Let ψ\sum_j c_j \bar{X}_j be a contrast. Then we can define the hypothesis SSCP matrix for  as follows:

image9147

Note that if ψ1, …, ψm-1 are pairwise orthogonal contrasts, then the hypothesis matrix H can be partitioned H = H_{\psi_1} + ⋯ + H_{\psi_{m-1}}. Also each df_{\psi_j} = 1, and so m – 1 = df_{\psi_1} + ⋯ + df_{\psi_{m-1}}.

Since in our representation of the data we exchange the roles of rows and columns and represent contrasts as column vectors, we actually calculate

image9148

The test we use to determine whether to retain or reject the null hypothesis is similar to the omnibus MANOVA tests, and uses the test statistic

Wilk’s LambdaΛψ \frac{|E|}{|H_\psi+E|}

For the Wilks Lambda test, we use df1 = k and b = 1 and so the test becomes

image9149where df1 = k and df2 = n – m – k + 1.

Example 1: We have seen that there are significant differences between the four groups in Example 1 of Manova basic Concepts, but we still don’t know where these differences are. Based on the observations we made when looking at the charts in Figure 3 of Manova Basic Concepts, we would like to answer the following questions:

  1. Is there a significant difference between the clay and salty groups?
  2. Is there a significant difference between the loam and sandy groups?
  3. Do the loam/sandy groups have a higher mean vector than the clay/salty groups?

To answer the first question we select the Contrasts option in the Real Statistics MANOVA data analysis tool (see Figure 1 of Real Statistics Manova Support). The result is as shown in Figure 1.

Manova contrast blank Excel

Figure 1 – Contrasts option of MANOVA data analysis tool

Don’t worry about the #DIV/0! And zero entries. Remember we still haven’t fill in the contrast coefficients in the shaded area.

For question (a), we use the following coefficients: 1 for clay, -1 for salty and 0 for loam and sandy. The results are shown in the top half of Figure 2.

Manova contrasts Excel

Figure 2 – Contrasts for questions (a) and (b)

This time we get non-error values for all the cells. Key formulas used to create the contrast table in Figure 2 are given in Figure 3.

manova-contrasts-formulas-excel

Figure 3 – Representative formulas from Figure 2

The p-value (cell S10) = 0.896 > .05 = α, and so there is no significant difference between the mean vectors of the clay and salty groups.

For question (b) we can again select the Contrasts option in the Real Statistics MANOVA data analysis tool and enter the contrast with coefficients 1 for loam, -1 for sandy and 0 for clay and salty. Alternatively you can simply copy the output for the contrast for question (a) right below it (as in the bottom part of Figure 2) and simply change the contrast coefficients. As we can see from Figure 2, the p-value for question (b) is .060268, which is again larger than α = .05, and so we conclude there is no significant difference between the mean vectors of the loam and sandy groups.

For question (c) we use contrasts 1/2 for loam and sandy and -1/2 for clay and salty. This time, as we see in Figure 4, we get a p-value of .004132 < .05 = α, which shows there is a significant difference between the mean vectors.

Manova contrast data analysis

Figure 4 – Contrasts for question (c)

Observation: It turns out that the three contrasts evaluated above are mutually orthogonal. You don’t necessarily have to use only orthogonal contrasts. In fact, it is best to use the contrasts that correspond to the tests that you believe are necessary to perform based on the actual experiment under consideration.

Example 2: We have seen that the loam/sandy groups have a significantly higher mean vector than the clay/salty groups. Which of the factors (yield, water and herbicide) account for this difference?

When we ran the Contrasts option of MANOVA for Example 1, we saw that there were two other tests conducted for question (c), as shown in Figure 4, namely the simultaneous confidence intervals and the Bonferroni confidence intervals. These two tests were also produced for questions (a) and (b), but we didn’t show the results since there weren’t significant differences in the mean vectors and so these tests were not useful.

The two tests are extensions of the same follow-up tests conducted for Hotelling’s T2 tests as described in Hotelling’s T-square Tests. We now described the MANOVA version of these tests.

For each p, 1 ≤ p ≤ k

image9150

The 1 – α simultaneous confidence interval is

image9151

where E = [epq]

image9152

image9153

Here Fcrit = FINV(α, df1, df2), df1 = k and df2 = n – m – k + 1.

The results (see range A40:L43 of Figure 4) show that zero lies in the confidence interval for all three dependent variables, and so no one of the independent variables accounts for the difference in mean vectors observed in Example 1(c), but it is the combination of the independent variables.

Generally we use the Bonferroni confidence intervals instead of the simultaneous confidence intervals since the intervals tend to be narrower. The 1 – α Bonferroni confidence interval is

 ψp ± tcrit ∙ sep

where the contrast mean for the pth dependent variable ψp and standard error sep are as defined above and tcrit = TINV(α/k, df) where df = n – m.

The results (see range A45:L48 of Figure 4) show that although the Bonferroni confidence intervals are indeed narrower than the simultaneous confidence intervals, zero still lies in the confidence interval for all three dependent variables, and so no one of the independent variables accounts for the difference in mean vectors observed in Example 1(c), but it is the combination of the independent variables.

Effect size

We use the following value for the Mahalanobis distance squared as a measure of effect size for the multivariate contrast ψ:

image9154

For example, the effect size for Example 1(a) is 0.16071 (cell S11 in Figure 3).

3 Responses to MANOVA Follow up using Contrasts

  1. Roberto Mioni says:

    Dear Dr. Charles Zaiontz, thanks for answering my question.
    However the contrasts that I suggest do not produce the same simultaneous confidence intervals and, probably, are incorrect. In fact the sum of positive contrasts must be equal to +1, while the sum of negative contrasts must be equal to -1, just as you say.
    I was not aware of this rule!
    Congratulations again for your program.

    Roberto Mioni.

  2. Roberto Mioni says:

    Dear Dr. Charles Zaiontz, I am a physician practicing in Udine (Italy). I am studying the statistics by utilizing your very interesting program “Real Statistics Using Excel”. In “MANOVA follow up using Contrasts” and in “Simultaneous confidence intervals”, you suggest to use the contrast [-0.5 0.5 -0.5 0.5] to calculate the “p – standard error”, sep.
    If I use the contrast [1 -1 -1 1], while I get the same Wilk’s Lambda and the same F – value (0.606269 and 5.628408, respectively), the “Simultaneous confidence intervals” values, [-4.126 – +22.264; -12.421 – +9.634; -0.803 – +3.590], double: [-8.2526 – +44.5276; -24.8428 – +19.2678; -1.6056 – 7.181]. Why should I use [-0.5 0.5 -0.5 0.5] contrasts and not [-1 1 -1 1]?
    Best regards, Roberto Mioni.

    • Charles says:

      Roberto,
      I typically choose contrasts such that the sum of the negative contrasts equals -1 and the sum of the positive contrasts equals +1. The contrasts that you suggest will probably produce the same outcome.
      Charles

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