WHAT IS MULTIVARIATE ANALYSIS OF VARIANCE?


Multivariate analysis of variance is the multivariate extension of the univariate techniques for assessing the differences between group means. The univariate procedures include the t test for two-group situations and ANOVA for situations with three or more groups defined by two or more independent variables . Before proceeding with our discussion of the unique aspects of MANOVA, let us review the basic principles of the univariate techniques.

UNIVARIATE PROCEDURES FOR ASSESSING GROUP DIFFERENCES

These procedures are classified as univariate not because of the number of independent variables, but instead because of the number of dependent variables. In multiple regression, the terms univariate and multivariate refer to the number of independent variables, but for ANOVA and MANOVA, the terminology applies to the use of single or multiple dependent variables. The following discussion addresses the two most common types of univariate procedures, the t test, which compares a dependent variable across two groups, and ANOVA, which is used whenever the number of groups is three or more.

The T Test

The t test assesses the statistical significance of the difference between two independent sample means. For example, an experimenter may expose two groups of respondents to different advertisements reflecting different advertising messages ” one informational and one emotional ” and subsequently ask each group about the appeal of the message on a 10-point scale, with 1 being poor and 10 being excellent . The two different advertising messages represent a treatment with two levels (informational versus emotional). A treatment, also known as a factor, is a nonmetric independent variable, experimentally manipulated or observed , that can be represented in various categories or levels. In our example, the treatment is the effect of emotional versus informational appeals.

To determine whether the two messages are viewed differently (meaning that the treatment has an effect), a t statistic is calculated. The t statistic is the ratio of the difference between the sample means ( ¼ 1 - ¼ 2 ) to their standard error. The standard error is an estimate of the difference between means to be expected because of sampling error, rather than real differences between means. This can be shown in the equation

where ¼ 1 = mean of group 1, ¼ 2 = mean of group 2, and SE ¼ 1 , ¼ 2 = standard error of the difference in group means.

By forming the ratio of the actual difference between the means to the difference expected due to sampling error, we quantify the amount of the actual impact of the treatment that is due to random sampling error. In other words, the t value, or t statistic, represents the group difference in terms of standard errors. If the t value is sufficiently large, then statistically we can say that the difference was not due to sampling variability but represents a true difference. This is done by comparing the t statistic to the critical value of the t statistic ( t crit ) If the absolute value of the t statistic is greater than the critical value, this leads to rejection of the null hypothesis of no difference in the appeals of the advertising messages between groups. This means that the actual difference due to the appeals is statistically larger than the difference expected from sampling error. We determine the critical value ( t crit ) for our t statistic and test the statistical significance of the observed differences by the following procedure:

  1. Compute the t statistic as the ratio of the difference between sample means and their standard error.

  2. Specify a Type I error level (denoted as ± , or significance level), which indicates the probability level the experimenter will accept in concluding that the group means are different when in fact they are not.

  3. Determine the critical value ( t crit ) by referring to the t distribution with N 1 + N 2 - 2 degrees of freedom and a specified a, where N 1 and N 2 are sample sizes.

  4. If the absolute value of the computed t statistic exceeds t crit , the experimenter can conclude that the two advertising messages have different levels of appeal (i.e., ¼ 1 ‰  ¼ 2 ), with a Type I error probability of ± . The researcher can then examine the actual mean values to determine which group is higher on the dependent value.

Analysis of Variance

In our example for the t test, an experimenter exposed two groups of respondents to different advertising messages and subsequently asked them to rate the appeal of the advertisements on a 10-point scale. Suppose we were interested in evaluating three advertising messages rather than two. Respondents would be randomly assigned to one of three groups, and we would have three sample means to compare. To analyze these data, we might be tempted to conduct separate t tests for the difference between each pair of means (i.e., group 1 versus group 2; group 1 versus group 3; and group 2 versus group 3).

However, multiple t tests inflate the overall Type I error rate. ANOVA avoids this Type I error inflation due to making multiple comparisons of treatment groups by determining in a single test whether the entire set of sample means suggests that the samples were drawn from the same general population. That is, ANOVA is used to determine the probability that differences in means across several groups are due solely to sampling error.

The logic of an ANOVA test is fairly straightforward. As the name "analysis of variance" implies, two independent estimates of the variance for the dependent variable are compared, one that reflects the general variability of respondents within the groups (MS W ) and another that represents the differences between groups attributable to the treatment effects (MS B ):

  1. Within-groups estimate of variance (MS W : mean square within groups): This is an estimate of the average random respondent variability on the dependent variable within a treatment group and is based on deviations of individual scores from their respective group means. MS W is comparable to the standard error between two means calculated in the t test as it represents variability within groups. The value MS W is sometimes referred to as the error variance.

  2. Between-groups estimate of variance (MS B : mean square between groups): The second estimate of variance is the variability of the treatment group means on the dependent variable. It is based on deviations of group means from the overall grand mean of all scores. Under the null hypothesis of no treatment effects (i.e., ¼ 1 = ¼ 2 = ¼ 3 = ... = ¼ k ), this variance estimate, unlike MS W , reflects any treatment effects that exist; that is, differences in treatment means increase the expected value of MS B .

Given that the null hypothesis of no group differences is true, MS W and MS B represent independent estimates of population variance. Therefore, the ratio of MS B to MS W is a measure of how much variance is attributable to the different treatments versus the variance expected from random sampling. The ratio of MS B to MS w , gives us a value for the F statistic. This is similar to the calculation of the t value and can be shown as

Because group differences tend to inflate MS B , large values of the F statistic lead to rejection of the null hypothesis of no difference in means across groups. If the analysis has several different treatments (independent variables), then estimates of MS B are calculated for each treatment and F statistics are calculated for each treatment. This allows for the separate assessment of each treatment.

To determine if the F statistic is sufficiently large to support rejection of the null hypothesis, follow a process similar to the t test. First, determine the critical value for the F statistic (F crit ) by referring to the F distribution with ( k - 1 ) and ( N - k ) degrees of freedom for a specified level of a (where N = N 1 + ... + N k and k = number of groups). If the value of the calculated F statistic exceeds F crit , conclude that the means across all groups are not all equal.

Examination of the group means then allows the experimenter to assess the relative standing of each group on the dependent measure. Although the F statistic test assesses the null hypothesis of equal means, it does not address the question of which means are different. For example, in a three-group situation, all three groups may differ significantly, or two may be equal but differ from the third. To assess these differences, the experimenter can employ either planned comparisons or post hoc tests. We examine some of these methods in a later section.

MULTIVARIATE ANALYSIS OF VARIANCE

As statistical inference procedures, both the univariate techniques ( t test and ANOVA) and MANOVA are used to assess the statistical significance of differences between groups. In the t test and ANOVA, the null hypothesis tested is the equality of dependent variable means across groups. In MANOVA, the null hypothesis tested is the equality of vectors of means on multiple dependent variables across groups. In the univariate case, a single dependent measure is tested for equality across the groups. In the multivariate case, a variate is tested for equality. In MANOVA, the experimenter actually has two variates, one for the dependent variables and another for the independent variables. The dependent variable variate is of more interest because the metric dependent measures can be combined in a linear combination, as we have already seen in multiple regression and discriminant analysis. The unique aspect of MANOVA is that the variate optimally combines the multiple dependent measures into a single value that maximizes the differences across groups.

The Two-Group Case: Hotelling's T 2

In our earlier univariate example, experimenters were interested in the appeal of two advertising messages. But what if they also wanted to know about the purchase intent generated by the two messages? If only univariate analyses were used, the experimenters would perform separate t tests on the ratings of both the appeal of the messages and the purchase intent generated by the messages. Yet the two measures are interrelated; thus, what is really desired is a test of the differences between the messages on both variables collectively. This is where Hotelling's T 2 , a specialized form of MANOVA that is a direct extension of the univariate t test, can be used.

Hotelling's T 2 provides a statistical test of the variate formed from the dependent variables that produces the greatest group difference. It also addresses the problem of "inflating" the Type I error rate that arises when making a series of t tests of group means on several dependent measures. It controls this inflation of the Type I error rate by providing a single overall test of group differences across all dependent variables at a specified ± level.

The computational formula for Hotelling's T 2 represents the results of mathematical derivations used to solve for a maximum t statistic (and, implicitly, the most discriminating linear combination of the dependent variables). This is equivalent to saying that if we can find a discriminant function for the two groups that produces a significant T 2 , the two groups are considered different across the mean vectors.

How does Hotelling's T 2 provide a test of the hypothesis of no group difference on the vectors of mean scores? Just as the t statistic follows a known distribution under the null hypothesis of no treatment effect on a single dependent variable, Hotelling's T 2 follows a known distribution under the null hypothesis of no treatment effect on any of a set of dependent measures. This distribution turns out to be an F distribution with p and N 1 + N 2 - 2 - 1 degrees of freedom after adjustment (where p = the number of dependent variables). To get the critical value for Hotelling's T 2 , we find the tabled value for F crit at a specified a level and compute as follows:

Differences between MANOVA and Discriminant Analysis

So far we have discussed the basic elements of both the univariate and multivariate tests for assessing differences between groups on one or more dependent variables. In doing so, we noted the calculation of the discriminant function, which in the case of MANOVA is the variate of dependent variables that maximizes the difference between groups. The question may arise: What is the difference between MANOVA and discriminant analysis? In some aspects, MANOVA and discriminant analysis are "mirror images." The dependent variables in MANOVA (a set of metric variables) are the independent variables in discriminant analysis, and the single nonmetric dependent variable of discriminant analysis becomes the independent variable in MANOVA. Moreover, both use the same methods in forming the variates and assessing the statistical significance between groups.

The differences, however, center on the objectives of the analyses and the role of the nonmetric variable(s). Discriminant analysis employs a single nonmetric variable as the dependent variable. The categories of the dependent variable are assumed as given, and the independent variables are used to form variates that maximally differ between the groups formed by the dependent variable categories. In MANOVA, the metric variables now act as the dependent variables and the objective becomes finding groups of respondents that exhibit differences on the set of dependent variables. The groups of respondents are not prespecified; instead, the experimenter uses one or more independent variables (nonmetric variables) to form groups. MANOVA, even while forming these groups, still retains the ability to assess the impact of each nonmetric variable separately.




Six Sigma and Beyond. Statistics and Probability
Six Sigma and Beyond: Statistics and Probability, Volume III
ISBN: 1574443127
EAN: 2147483647
Year: 2003
Pages: 252

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