Multivariate analysis (MVA) is based on the statistical principle of multivariate statistics, which involves observation and analysis of more than one statistical outcome variable at a time. In design and analysis, the technique is used to perform trade studies across multiple dimensions while taking into account the effects of all variables on the responses of interest.
Uses for multivariate analysis include
- Design for capability (also known as capability-based design)
- Inverse design, where any variable can be treated as an independent variable
- Analysis of Alternatives (AoA), the selection of concepts to fulfill a customer need
- Analysis of concepts with respect to changing scenarios
- Identification of critical design drivers and correlations across hierarchical levels.
Multivariate analysis can be complicated by the desire to include physics-based analysis to calculate the effects of variables for a hierarchical “system-of-systems.” Often, studies that wish to use multivariate analysis are stalled by the dimensionality of the problem. These concerns are often eased through the use of surrogate models, highly accurate approximations of the physics-based code. Since surrogate models take the form of an equation, they can be evaluated very quickly. This becomes an enabler for large-scale MVA studies: while a Monte Carlo simulation across the design space is difficult with physics-based codes, it becomes trivial when evaluating surrogate models, which often take the form of response surface equations.
Multivariate Data Analysis refers to any statistical technique used to analyze data that arises from more than one variable. This essentially models reality where each situation, product, or decision involves more than a single variable. The information age has resulted in masses of data in every field. Despite the quantum of data available, the ability to obtain a clear picture of what is going on and make intelligent decisions is a challenge. When available information is stored in database tables containing rows and columns, Multivariate Analysis can be used to process the information in a meaningful fashion.
Multivariate analysis consists of a collection of methods that can be used when several measurements are made on each individual or object in one or more samples. We will refer to the measurements as variables and to the individuals or objects as units (research units, sampling units, or experimental units) or observations. In practice, multivariate data sets are common, although they are not always analyzed as such. But the exclusive use of univariate procedures with such data is no longer excusable, given the availability of multivariate techniques and inexpensive computing power to carry them out.
Ordinarily the variables are measured simultaneously on each sampling unit. Typically, these variables are correlated. If this were not so, there would be little use for many of the techniques of multivariate analysis. We need to untangle the overlapping information provided by correlated variables and peer beneath the surface to see the underlying structure. Thus the goal of many multivariate approaches is simplification. We seek to express what is going on in terms of a reduced set of dimensions. Such multivariate techniques are exploratory; they essentially generate hypotheses rather than test them.
On the other hand, if our goal is a formal hypothesis test, we need a technique that will allow several variables to be tested and still preserve the significance level.
MVA Tools
In order to understand multivariate analysis, it is important to understand some of the terminology. A variate is a weighted combination of variables. The purpose of the analysis is to find the best combination of weights. Non-metric data refers to data that are either qualitative or categorical in nature. Metric data refers to data that are quantitative, and interval or ratio in nature.
Data Quality
Before launching into an analysis technique, it is important to have a clear understanding of the form and quality of the data. The form of the data refers to whether the data are nonmetric or metric. The quality of the data refers to how normally distributed the data are. The first few techniques discussed are sensitive to the linearity, normality, and equal variance assumptions of the data. Examinations of distribution, skewness, and kurtosis are helpful in examining distribution. Also, it is important to understand the magnitude of missing values in observations and to determine whether to ignore them or impute values to the missing observations. Another data quality measure is outliers, and it is important to determine whether the outliers should be removed. If they are kept, they may cause a distortion to the data; if they are eliminated, they may help with the assumptions of normality. The key is to attempt to understand what the outliers represent.
Multiple Regression Analysis
Multiple Regression is the most commonly utilized multivariate technique. It examines the relationship between a single metric dependent variable and two or more metric independent variables. The technique relies upon determining the linear relationship with the lowest sum of squared variances; therefore, assumptions of normality, linearity, and equal variance are carefully observed. The beta coefficients (weights) are the marginal impacts of each variable, and the size of the weight can be interpreted directly. Multiple regression is often used as a forecasting tool.
Logistic Regression Analysis
Sometimes referred to as “choice models,” this technique is a variation of multiple regression that allows for the prediction of an event. It is allowable to utilize nonmetric (typically binary) dependent variables, as the objective is to arrive at a probabilistic assessment of a binary choice. The independent variables can be either discrete or continuous. A contingency table is produced, which shows the classification of observations as to whether the observed and predicted events match. The sum of events that were predicted to occur which actually did occur and the events that were predicted not to occur which actually did not occur, divided by the total number of events, is a measure of the effectiveness of the model. This tool helps predict the choices consumers might make when presented with alternatives.
Discriminant Analysis
The purpose of discriminant analysis is to correctly classify observations or people into homogeneous groups. The independent variables must be metric and must have a high degree of normality. Discriminant analysis builds a linear discriminant function, which can then be used to classify the observations. The overall fit is assessed by looking at the degree to which the group means differ (Wilkes Lambda or D2) and how well the model classifies. To determine which variables have the most impact on the discriminant function, it is possible to look at partial F values. The higher the partial F, the more impact that variable has on the discriminant function. This tool helps categorize people, like buyers and non-buyers.
Multivariate Analysis of Variance (MANOVA)
This technique examines the relationship between several categorical independent variables and two or more metric dependent variables. Whereas analysis of variance (ANOVA) assesses the differences between groups (by using T tests for two means and F tests between three or more means), MANOVA examines the dependence relationship between a set of dependent measures across a set of groups. Typically this analysis is used in experimental design, and usually a hypothesized relationship between dependent measures is used. This technique is slightly different in that the independent variables are categorical and the dependent variable is metric. Sample size is an issue, with 15-20 observations needed per cell. However, too many observations per cell (over 30) and the technique loses its practical significance. Cell sizes should be roughly equal, with the largest cell having less than 1.5 times the observations of the smallest cell. That is because, in this technique, normality of the dependent variables is important. The model fit is determined by examining mean vector equivalents across groups. If there is a significant difference in the means, the null hypothesis can be rejected and treatment differences can be determined.
Factor Analysis
When there are many variables in a research design, it is often helpful to reduce the variables to a smaller set of factors. This is an independence technique, in which there is no dependent variable. Rather, the researcher is looking for the underlying structure of the data matrix. Ideally, the independent variables are normal and continuous, with at least three to five variables loading onto a factor. The sample size should be over 50 observations, with over five observations per variable. Multicollinearity is generally preferred between the variables, as the correlations are key to data reduction. Kaiser’s Measure of Statistical Adequacy (MSA) is a measure of the degree to which every variable can be predicted by all other variables. An overall MSA of .80 or higher is very good, with a measure of under .50 deemed poor.
There are two main factor analysis methods: common factor analysis, which extracts factors based on the variance shared by the factors, and principal component analysis, which extracts factors based on the total variance of the factors. Common factor analysis is used to look for the latent (underlying) factors, whereas principal component analysis is used to find the fewest number of variables that explain the most variance. The first factor extracted explains the most variance. Typically, factors are extracted as long as the eigenvalues are greater than 1.0 or the Scree test visually indicates how many factors to extract. The factor loadings are the correlations between the factor and the variables. Typically a factor loading of .4 or higher is required to attribute a specific variable to a factor. An orthogonal rotation assumes no correlation between the factors, whereas an oblique rotation is used when some relationship is believed to exist.
Cluster Analysis
The purpose of cluster analysis is to reduce a large data set to meaningful subgroups of individuals or objects. The division is accomplished on the basis of similarity of the objects across a set of specified characteristics. Outliers are a problem with this technique, often caused by too many irrelevant variables. The sample should be representative of the population, and it is desirable to have uncorrelated factors. There are three main clustering methods: hierarchical, which is a treelike process appropriate for smaller data sets; nonhierarchical, which requires specification of the number of clusters a priori; and a combination of both. There are four main rules for developing clusters: the clusters should be different, they should be reachable, they should be measurable, and the clusters should be profitable (big enough to matter). This is a great tool for market segmentation.
Multidimensional Scaling (MDS)
The purpose of MDS is to transform consumer judgments of similarity into distances represented in multidimensional space. This is a decompositional approach that uses perceptual mapping to present the dimensions. As an exploratory technique, it is useful in examining unrecognized dimensions about products and in uncovering comparative evaluations of products when the basis for comparison is unknown. Typically there must be at least four times as many objects being evaluated as dimensions. It is possible to evaluate the objects with nonmetric preference rankings or metric similarities (paired comparison) ratings. Kruskal’s Stress measure is a “badness of fit” measure; a stress percentage of 0 indicates a perfect fit, and over 20% is a poor fit. The dimensions can be interpreted either subjectively by letting the respondents identify the dimensions or objectively by the researcher.
Correspondence Analysis
This technique provides for dimensional reduction of object ratings on a set of attributes, resulting in a perceptual map of the ratings. However, unlike MDS, both independent variables and dependent variables are examined at the same time. This technique is more similar in nature to factor analysis. It is a compositional technique, and is useful when there are many attributes and many companies. It is most often used in assessing the effectiveness of advertising campaigns. It is also used when the attributes are too similar for factor analysis to be meaningful. The main structural approach is the development of a contingency (crosstab) table. This means that the form of the variables should be nonmetric. The model can be assessed by examining the Chi-square value for the model. Correspondence analysis is difficult to interpret, as the dimensions are a combination of independent and dependent variables.
Conjoint Analysis
Conjoint analysis is often referred to as “trade-off analysis,” since it allows for the evaluation of objects and the various levels of the attributes to be examined. It is both a compositional technique and a dependence technique, in that a level of preference for a combination of attributes and levels is developed. A part-worth, or utility, is calculated for each level of each attribute, and combinations of attributes at specific levels are summed to develop the overall preference for the attribute at each level. Models can be built that identify the ideal levels and combinations of attributes for products and services.
Canonical Correlation
The most flexible of the multivariate techniques, canonical correlation simultaneously correlates several independent variables and several dependent variables. This powerful technique utilizes metric independent variables, unlike MANOVA, such as sales, satisfaction levels, and usage levels. It can also utilize nonmetric categorical variables. This technique has the fewest restrictions of any of the multivariate techniques, so the results should be interpreted with caution due to the relaxed assumptions. Often, the dependent variables are related, and the independent variables are related, so finding a relationship is difficult without a technique like canonical correlation.
Structural Equation Modeling
Unlike the other multivariate techniques discussed, structural equation modeling (SEM) examines multiple relationships between sets of variables simultaneously. This represents a family of techniques, including LISREL, latent variable analysis, and confirmatory factor analysis. SEM can incorporate latent variables, which either are not or cannot be measured directly into the analysis. For example, intelligence levels can only be inferred, with direct measurement of variables like test scores, level of education, grade point average, and other related measures. These tools are often used to evaluate many scaled attributes or to build summated scales.
