Although the roots of multivariate analysis lie in univariate and bivariate statistics, the extension to the multivariate domain introduces additional concepts and issues of particular relevance. These concepts range from the need for a conceptual understanding of the basic building block of multivariate analysis—the variate—to specific issues dealing with the types of measurement scales used and the statistical issues of significance testing and confidence levels. Each concept plays a significant role in the successful application of any multivariate technique.
tHe VARiAte
As previously mentioned, the building block of multivariate analysis is the variate, a linear combination of variables with empirically determined weights. The variables are specified by the researcher whereas the weights are deter- mined by the multivariate technique to meet a specific objective. A variate of n weighted variables (X1 to Xn) can be stated mathematically as:
Variate value5w1X11w2X21w3X31c1wnXn
where Xn is the observed variable and wn is the weight determined by the multivariate technique.
The result is a single value representing a combination of the entire set of variables that best achieves the objective of the specific multivariate analysis. In multiple regression, the variate is determined in a manner that maximizes the correlation between the multiple independent variables and the single dependent variable. In discriminant analysis, the variate is formed so as to create scores for each observation that maximally differentiates between groups of observations. In exploratory factor analysis, variates are formed to best represent the underlying structure or patterns of the variables as represented by their intercorrelations.
In each instance, the variate captures the multivariate character of the analysis. Thus, in our discussion of each technique, the variate is the focal point of the analysis in many respects. We must understand not only its
collective impact in meeting the technique’s objective but also each separate variable’s contribution to the overall variate effect.
MeASUReMent ScALeS
Data analysis involves the identification and measurement of variation in a set of variables, either among themselves or between a dependent variable and one or more independent variables. The key word here is measurement because the researcher cannot identify variation unless it can be measured. Measurement is important in accurately repre- senting the concept of interest and is instrumental in the selection of the appropriate multivariate method of analysis.
Data can be classified into one of two categories—nonmetric (qualitative) and metric (quantitative)—based on the type of attributes or characteristics they represent.
The researcher must define the measurement type—nonmetric or metric—for each variable. To the computer, the values are only numbers. As we will see in the following section, defining data as either metric or nonmetric has substantial impact on what the data can represent and how it can be analyzed.
Nonmetric Measurement Scales Nonmetric data describe differences in type or kind by indicating the presence or absence of a characteristic or property. These properties are discrete in that by having a particular feature, all other features are excluded; for example, if a person is male, he cannot be female. An “amount” of gender is not possible, just the state of being male or female. Nonmetric measurements can be made with either a nominal or an ordinal scale.
noMinAL ScALeS A nominal scale assigns numbers as a way to label or identify subjects or objects. The numbers assigned to the objects have no quantitative meaning beyond indicating the presence or absence of the attribute or characteristic under investigation. Therefore, nominal scales, also known as categorical scales, can only provide the number of occurrences in each class or category of the variable being studied.
For example, in representing gender (male or female) the researcher might assign numbers to each category (e.g., 2 for females and 1 for males). With these values, however, we can only tabulate the number of males and females;
it is nonsensical to calculate an average value of gender.
Nominal data only represent categories or classes and do not imply amounts of an attribute or characteristic.
Commonly used examples of nominally scaled data include many demographic attributes (e.g., individual’s sex, reli- gion, occupation, or political party affiliation), many forms of behavior (e.g., voting behavior or purchase activity), or any other action that is discrete (happens or not).
oRDinAL ScALeS Ordinal scales are the next “higher” level of measurement precision. In the case of ordinal scales, variables can be ordered or ranked in relation to the amount of the attribute possessed. Every subject or object can be compared with another in terms of a “greater than” or “less than” relationship. The numbers utilized in ordinal scales, however, are really non-quantitative because they indicate only relative positions in an ordered series. Ordinal scales provide no measure of the actual amount or magnitude in absolute terms, only the order of the values. The researcher knows the order, but not the amount of difference between the values.
For example, different levels of an individual consumer’s satisfaction with several new products can be illustrated, first using an ordinal scale. The following scale shows a respondent’s view of three products.
Product A
Very Satisfied Not At All Satisfied
Product B Product C
When we measure this variable with an ordinal scale, we “rank order” the products based on satisfaction level.
We want a measure that reflects that the respondent is more satisfied with Product A than Product B and more
satisfied with Product B than Product C, based solely on their position on the scale. We could assign “rank order”
values 115most satisfied, 25next most satisfied, etc.2 of 1 for Product A (most satisfaction), 2 for Product B, and 3 for Product C.
When viewed as ordinal data, we know that Product A has the most satisfaction, followed by Product B and then Product C. However, we cannot make any statements on the amount of the differences between products (e.g., we cannot answer the question whether the difference between Products A and B is greater than the difference between Products B and C). We have to use an interval scale (see next section) to assess what is the magnitude of differences between products.
In many instances a researcher may find it attractive to use ordinal measures, but the implications for the types of analyses that can be performed are substantial. The analyst cannot perform any arithmetic operations (no sums, averages, multiplication or division, etc.), thus nonmetric data are quite limited in their use in estimating model coefficients. For this reason, many multivariate techniques are devised solely to deal with nonmetric data (e.g., correspondence analysis) or to use nonmetric data as an independent variable (e.g., discriminant analysis with a nonmetric dependent variable or multivariate analysis of variance with nonmetric independent variables).
Thus, the analyst must identify all nonmetric data to ensure that they are used appropriately in the multivariate techniques.
Metric Measurement Scales In contrast to nonmetric data, metric data are used when subjects differ in amount or degree on a particular attribute. Metrically measured variables reflect relative quantity or degree and are appro- priate for attributes involving amount or magnitude, such as the level of satisfaction or commitment to a job. The two different metric measurement scales are interval and ratio scales.
inteRVAL ScALeS Interval scales and ratio scales (both metric) provide the highest level of measurement precision, permitting nearly any mathematical operation to be performed. These two scales have constant units of measure- ment, so differences between any two adjacent points on any part of the scale are equal.
In the preceding example in measuring satisfaction, metric data could be obtained by measuring the distance from one end of the scale to each product’s position. Assume that Product A was 2.5 units from the left end, Product B was 6.0 units, and Product C was 12 units. Using these values as a measure of satisfaction, we could not only make the same statements as we made with the ordinal data (e.g., the rank order of the products), but we could also see that the difference between Products A and B was much smaller 16.022.553.52 than was the difference between Products B and C 112.026.056.02.
The only real difference between interval and ratio scales is that interval scales use an arbitrary zero point, whereas ratio scales include an absolute zero point. The most familiar interval scales are the Fahrenheit and Celsius temperature scales. Each uses a different arbitrary zero point, and neither indicates a zero amount or lack of temperature, because we can register temperatures below the zero point on each scale. Therefore, it is not possible to say that any value on an interval scale is a multiple of some other point on the scale.
For example, an 80°F day cannot correctly be said to be twice as hot as a 40°F day, because we know that 80°F, on a different scale, such as Celsius, is 26.7°C. Similarly, 40°F on a Celsius scale is 4.4°C. Although 80°F is indeed twice 40°F, one cannot state that the heat of 80°F is twice the heat of 40°F because, using different scales, the heat is not twice as great; that is, 4.4°C32226.7°C.
RAtio ScALeS Ratio scales represent the highest form of measurement precision because they possess the advantages of all lower scales plus an absolute zero point. All mathematical operations are permissible with ratio-scale measurements. The bathroom scale or other common weighing machines are examples of these scales, because they have an absolute zero point and can be spoken of in terms of multiples when relating one point on the scale to another; for example, 100 pounds is twice as heavy as 50 pounds.
The Impact of Choice of Measurement Scale Understanding the different types of measurement scales is import- ant for two reasons:
1 The researcher must identify the measurement scale of each variable used, so that nonmetric data are not incorrectly used as metric data, and vice versa (as in our earlier example of representing gender as 1 for male and 2 for female). If the researcher incorrectly defines this measure as metric, then it may be used inappropri- ately (e.g., finding the mean value of gender).
2 The measurement scale is also critical in determining which multivariate techniques are the most applicable to the data, with considerations made for both independent and dependent variables. In the discussion of the techniques and their classification in later sections of this chapter, the metric or nonmetric properties of inde- pendent and dependent variables are the determining factors in selecting the appropriate technique.
MeASUReMent eRRoR AnD MULtiVARiAte MeASUReMent
The use of multiple variables and the reliance on their combination (the variate) in multivariate techniques also focuses attention on a complementary issue—measurement error. Measurement error is the degree to which the observed values are not representative of the “true” values. Measurement error has many sources, ranging from data entry errors to the imprecision of the measurement (e.g., imposing 7-point rating scales for attitude measurement when the researcher knows the respondents can accurately respond only to a 3-point rating) to the inability of respondents to accurately provide information (e.g., responses as to household income may be reasonably accurate but rarely totally precise). Thus, all variables used in multivariate techniques must be assumed to have some degree of measurement error. The measurement error adds “noise” to the observed or measured variables. Thus, the observed value obtained represents both the “true” level and the “noise.” When used to compute correlations or means, the
“true” effect is partially masked by the measurement error, causing the correlations to weaken and the means to be less precise. The specific impact of measurement error and its accommodation in dependence relationships is covered in more detail in Chapter 9.
Validity and Reliability The researcher’s goal of reducing measurement error can follow several paths. In assessing the degree of measurement error present in any measure, the researcher must address two important characteristics of a measure:
VALiDitY Validity is the degree to which a measure accurately represents what it is supposed to. For example, if we want to measure discretionary income, we should not ask about total household income. Ensuring validity starts with a thorough understanding of what is to be measured and then making the measurement as “correct” and accurate as possible. However, accuracy does not ensure validity. In our income example, the researcher could precisely define total household income, but it would still be “wrong” (i.e., an invalid measure) in measuring discretionary income because the “correct” question was not being asked.
ReLiABiLitY If validity is assured, the researcher must still consider the reliability of the measurements. Reliability is the degree to which the observed variable measures the “true” value and is “error free”; thus, it is the opposite of measurement error. If the same measure is asked repeatedly, for example, more reliable measures will show greater consistency than less reliable measures. The researcher should always assess the variables being used and, if valid alternative measures are available, choose the variable with the higher reliability.
Employing Multivariate Measurement In addition to reducing measurement error by improving individual variables, the researcher may also choose to develop multivariate measurements, also known as summated scales, for which several variables are joined in a composite measure to represent a concept (e.g., multiple-item personality
scales or summed ratings of product satisfaction). The objective is to avoid the use of only a single variable to repre- sent a concept and instead to use several variables as indicators, all representing differing facets of the concept to obtain a more well-rounded perspective. The use of multiple indicators enables the researcher to more precisely specify the desired responses. It does not place total reliance on a single response, but instead on the “average” or typical response to a set of related responses.
For example, in measuring satisfaction, one could ask a single question, “How satisfied are you?” and base the analysis on the single response. Or a summated scale could be developed that combined several responses of satis- faction (e.g., finding the average score among three measures—overall satisfaction, the likelihood to recommend, and the probability of purchasing again). The different measures may be in different response formats or in differing areas of interest assumed to comprise overall satisfaction.
The guiding premise is that multiple responses reflect the “true” response more accurately than does a single response. The researcher should assess reliability and incorporate scales into the analysis. For a more detailed intro- duction to multiple measurement models and scale construction, see further discussion in Chapter 3 (Exploratory Factor Analysis) and Chapter 9 (Structural Equations Modeling Overview) or additional resources [48]. In addition, compilations of scales that can provide the researcher a “ready-to-go” scale with demonstrated reliability have been published in recent years [2, 9].
The Impact of Measurement Error The impact of measurement error and poor reliability cannot be directly seen because they are embedded in the observed variables. The researcher must therefore always work to increase reliability and validity, which in turn will result in a more accurate portrayal of the variables of interest. Poor results are not always due to measurement error, but the presence of measurement error is guaranteed to distort the observed relationships and make multivariate techniques less powerful. Reducing measurement error, although it takes effort, time, and additional resources, may improve weak or marginal results and strengthen proven results as well.