University of Connecticut
In 1978, Joseph Renzulli published an article in Phi Delta Kappan in which he proposed a new way of looking at giftedness—the Three Ring Conception. In gross summary, this model highlighted that giftedness can be dissected into three components (creativity, task commitment, and above-average ability)—all of which must come together at relatively high but not necessarily equal levels in denoting giftedness. These three components were embedded in a hounds tooth background, whose interlocking graphic represented intricately connected personality and environmental factors—including intuition, character, socioeconomic status, and zeitgeist, for example—that influence giftedness and gifted behaviors (Renzulli, 1986). From this hounds tooth background the research at hand evolved.
Renzulli is making use of the zeitgeist of the new millennium to elucidate the personality and environmental factors represented within the hounds tooth pattern. Social issues descriptive of this point in history have primed the public and research communities for positive psychology (Csikszentmihalyi & Seligman, 2000) and renewed interest in values and the effect of environment. We are beginning research (“Operation Houndstooth”) into how six non-intellective factors pertaining to “positive human concerns” (a sense of vision, a sense of destiny, physical/mental energy, romance with a topic/discipline, courage, and optimism) contribute to the development of giftedness.
Supporting our identification of the factors and organization within the factors is challenging, let alone quantifying how these factors vary among individuals and contribute to giftedness. In order to check our conceptual, organizational structure, I need to know how the factors (referred to as “constructs” from this point forward) compare. Because a substantial literature review served to reaffirm difficulty in defining optimism, I specifically wanted and needed to find out how laypeople perceive optimism – what descriptors best characterize individuals’ understanding and emotions about optimism? Can or should a definition of optimism be parceled out into smaller units of meaning? What emotions, characteristics, or descriptors best ascribe meaning to optimism?
To this end, I designed a semantic differential, for which data are typically analyzed via factor analysis. The research question is can common factor analysis of a semantic differential yield a meaningful, useful structure for describing the construct of optimism? (How do factors and factor scores representing the construct of optimism operationally define the meaning of optimism?) Philosophical debates about factor analysis not withstanding, I have come to develop confidence that this research approach does yield interpretable, valuable results for quantitative analysis of optimism.
The semantic differential originated from the work of Charles Osgood in the 1950s as a technique for scaling people on their responses to pairs of bipolar adjectives in relation to concepts (Gable, 1993). It is a technique for measuring meaning that grew out of research at Dartmouth College in the late 1930’s on synesthesia (Osgood, Suci, & Tannenbaum, 1957). Typically, a single word (or short phrase) is the construct of interest, and individuals help the researcher differentiate the meaning of that construct by responding to several pairs of bipolar adjectives which are scored on a continuum running from +X to -X or from X to X + Y (likert style). In theory, each bipolar pair (“scale”) can be represented by a straight-line (“semantic space”); several such pairs or scales form a multidimensional geometric space (Gable, 1993). Thus, when individuals respond to a set of pairs or scales as they rate a concept, those individuals are, in effect, differentiating the meaning of that concept in intensity (degree from the origin along each semantic space) and direction (positive or negative along each semantic space). The larger and more representative the sample, the more thoroughly that space is defined as whole; determination of the minimum number of orthogonal dimensions (or axes) which exhausts the dimensionality of the semantic space allows maximum efficiency in defining that semantic space (Osgood, et al., 1957).
Osgood, Suci, and Tannenbaum (1957) provide many examples of bipolar pairs used in Osgood’s original work and in later work with factor analysis of such scales in a thesaurus study. The authors suggest that the semantic differential be used to examine differences among: different groups in the meaning of one concept, different individuals in the meaning of one concept, and different concepts for the same group or individual.
Data from semantic differentials are analyzed with factor analysis (Gable, 1993; Osgood, Suci, & Tannenbaum, 1957). (Suggested additional analyses include item and reliability analyses and use of the D statistic—a measure of generalized distance that allows a form of profile similarity.) The factors and distances assist the researcher in clarifying how sets of pairs or scales define the multidimensional semantic space, yielding increased measurement of meaning and permitting comparison of meaning among concepts.
Factor analysis simplifies complex sets of data based on correlations between variables (scales, in this case). There are two major types of factor analysis: exploratory and confirmatory. The goal of exploratory factor analysis is description and summary of data through the grouping of correlated variables (Tabachnik & Fidell, 2001); the discussions herein pertain solely to exploratory factor analysis. The factors generated by factor analysis are condensed statements of the relationships among variables (Kline, 1994). Tabachnik and Fidell (2001) define factors as variables that are correlated with one another but that are largely independent of other subsets of variables. Royce (1963, in Kline, 1994) describes factors more narrowly as dimensions operationally defined by their factor loadings, which are, in turn, correlations of each variable with each dimension. Employing this definition, a researcher could utilize individuals’ factor scores on the dimensions to make a prediction, answer a question, or summarize a definition (depending on the research agenda and instrumentation) rather than relying on the scores on the entire set of variables from which the factors emerged. Generally speaking, each factor should have between three and six marker variables (relatively “pure” measures—high loading on one and only one factor); this range of suggested marker variables is reflected by Tabachnik and Fidell (2001) and Green, Salkind, and Akey (2000). One may find, with shorter measurement instruments, even one or two marker variables, although this is not the preference (Ann O’Connell, personal communication, December, 2000).
Data on a semantic differential on the construct of optimism were obtained from attendees of a special session introducing the research described above at the National Association of Gifted Children annual convention in Atlanta, Georgia, November, 2000, in accordance with recommendations in Osgood, Suci, and Tannenbaum (1957). Attendees (estimated attendance=150) represented professionals in the field of education—teachers, administrators, counselors, and learning consultants—as well as academicians and researchers in education-related fields and a very limited number of individuals attending strictly as interested parents of a gifted child. As a result of insufficient time, double the anticipated attendance, and collection of instruments at the conclusion of the session as attendees were leaving, the number of completed returns was approximately half the number distributed.
The original dataset consisted of 86 individuals’ responses to 20 bipolar adjective pairs on a semantic differential about optimism. Each “response” was one mark along a 7-step, Likert-type scale for each bipolar pair (Osgood, Suci, & Tannenbaum, 1957, as cited in Gable, 1993). The instrument presented is the result of several months of development, critique, revision, and mini-pilot tests with colleagues at National Research Centers on the Gifted and Talented at Yale University, the University of Virginia, and the University of Connecticut. An SPSS 10.0 spreadsheet was designed and all data were entered. After substantial consideration and debate with colleagues, the responses for one pair (“children cannot develop this without seeing it modeled”) were reverse-coded, according to interpretation of response orientation guidelines (positive, active, desirable vs. negative, inactive, undesirable) in Osgood et al. (1957).
A representative sample requires approximately 6-10 times the number of people as scales (pairs) used (Gable, 1993). (Grimm and Yarnold (1995)call this the subjects-to-variables (STV) ratio; they suggest a minimum ratio of 5 and a minimum N of 100 regardless of the ratio.) Therefore, in order to adequately analyze the results for a sample of 86, I reasoned that the number of pairs should be reduced to 8-14. The 20 pairs went back to colleagues for reevaluation of their relevance to optimism. A correlation matrix of all 20 pairs was run using SPSS 10.0 in order to help ascertain which pairs were most highly correlated and appeared to be most significant as generalized indices of attitudes toward optimism. The final dataset was comprised of 13 bipolar adjective pairs that are presented in Table 1. Respondents rated (“scored”) each pair by placing a mark (“X” or a checkmark) in boxes along a continuum between the two bounded ends for each pair; I scored these from 1 to 7 (the “1” boundary denoting active/positive/desirable, the “7” boundary denoting inactive/negative/undesirable), making ‘4’ the midpoint. Because I wanted all pairs to appear as neutral as possible so that respondents were maximally focused on how each pair relates or describes optimism and maximally comfortable responding at or near either bounded end, the “scores” (1-7) were only attached during data entry for analysis purposes.
Final Dataset -Bipolar Pairs Scored 1 through 7
*Pairs from Osgood, Suci, & Tannenbaum’s (1957) Thesaurus Study.
Principal components analysis was run on the semantic differential in order to investigate correlations among subsets of responses to bipolar pairs. This analysis reduced the semantic differential to a smaller number of components representing subsets of bipolar pairs measuring similar aspects of optimism. Principal components analysis provided an initial number of possible factors (based on components with eigenvalues greater than 1) derived from patterns of correlation and intercorrelation of variables (pairs). Varimax rotation—a form of orthogonal rotation that forces items to correlate or load with one and only one factor—is typically used with principal components analysis (Tabachnik & Fidell, 2001, p. 619) and so was used for analyses of these data.
Correlations among all original 20 bipolar pairs were generated; significant correlations were color-coded according to strength, and the resulting patterns of correlations afforded good initial insight into comparative response patterns and relationships among the variables (pairs). Univariate statistics were run using SPSS 10.0 to examine the means and standard deviations of the bipolar pairs as well as to check for possible outliers or entry errors. No outliers were found; two errors in data entry were corrected. Table 2 presents means and standard deviations for the 13 bipolar pairs in the final dataset.
Means and Standard Deviations
Examination of standard deviations and corresponding means suggested that these scores would not reflect a normal distribution. To investigate normality, skewness and kurtosis statistics were run; TEACH was the only variable whose skewness and kurtosis were within the normal distribution range (statistic/standard error<=| 2 |), QUAN was normally distributed with respect to skewness, and DEF was normally distributed with respect to kurtosis. Normality data are not typically presented within factor analysis results because the value of the solution may be worthwhile whether variables are normally distributed or not. However, I include skewness and kurtosis reports herein because of what may be a situation—philosophically, if not practically—unique to affective instrument data such as these presented from the semantic differential. Elaboration of this point will be appropriately addressed in the discussion section below. Several analyses were run; the five factor solution presented below was most interpretable.
Principal factors extraction with varimax rotation was performed using SPSS 10.0 principal axis factoring analysis on 13 bipolar adjective pairs from a semantic differential on the construct “optimism.” The sample was drawn from professionals and interested parties in education-related fields attending a research presentation (N=86). Prior to principal axis factoring, principal components extraction with varimax (orthogonal) rotation was conducted to extract the number of factors and the factorability of the correlation matrices, providing initial insight into how the respondents’ ratings may be contributing to generalized indices of attitudes toward optimism (Gable, 1993).
Principal Components Analysis
In principal components analysis (PCA), all variance is distributed to components, including error and unique variance for each observed variable (bipolar pair). PCA duplicates the correlation matrix, with the components explain 100% of the variance (Tabachnik & Fidell, 2001). The goal of this analysis is twofold: it maximizes variance extracted by orthogonal components and serves to suggest the number of factors for the optimal solution if subsequent common factor analysis is conducted. Analysis of the correlation matrix reveals several correlations greater than .3, suggesting factorability of these data (Tabachnik & Fidell, 2000). The determinant (| R |=0.0647) approaches singularity (0) but is acceptably distant, relatively speaking. The Bartlett’s Test of Sphericity is extremely sensitive to the hypothesis that correlations within the correlation matrix (R) are zero; even with a relatively small sample, PCA shows that we are able to reject the null hypothesis of no correlation (X2(136)=218.609, p=0.000). The Kaiser-Meyer-Olkin Measure of Sampling adequacy (KMO=.518) is less than the suggested .6 value (Tabachnik & Fidell, 2001), indicating that the correlation matrix may be difficult to factor. Despite the low KMO value, .518 is within an acceptable range to proceed (Ann O’Connell, personal communication, December, 2000). Communalities are presented in Table 3; these explain the amount of variance (within each pair) accounted for by the factors.
Communalities in PCA
PCA revealed five components (plausible factors) with eigenvalues greater than 1. Table 4 provides both extraction sums of squared loadings (SSL) and rotation SSL for five components solution.
Extraction and Rotation SSL for 5 Components with Eigenvalues >1
*These five components explain approximately 66% of the variance among responses to bipolar pairs.
The screen plot, used to gain insight into the number of possible components (ultimately, in this case, factors) lacks a clear “elbow” but confirms five components. Based on the results of the PCA, factor analysis (FA) was conducted using principal axis factoring with 5 factors and varimax rotation.
Principal Axis Factoring
Whereas in PCA total variance is analyzed, in FA only common, or shared, variance is analyzed. Principal axis factoring (PAF), set for five factors, was run using varimax rotation. The correlation matrix, determinant, KMO, and results of the Bartlett’s Test for Sphericity were the same as for PCA; thus, please refer to the PCA section for these values. Communalities were somewhat lower in PAF than they were in PCA. Table 5 presents the communalities for PAF.
Communalities in PAF
Because PAF analyzes only shared variance, one would expect that the total variance explained by each of the five factors to be less than the variance explained by the five components in PCA. Results reflect this. Table 6 highlights the results of the five factor solution (including interpretive labels) with PAF. The fifth factor has an eigenvalue less than 1; factor 5 is kept in the solution for reasons addressed in the discussion section below.
PAF Results for a Five Factor Solution on the Optimism Semantic Differential
*These five factors explain approximately 47% of the shared variance among responses to bipolar pairs describing optimism.
Residuals are the differences between actual and reproduced correlations; they provide insight into the goodness of fit for the solution. Nineteen percent (15) nonredundant residuals had differences greater than .05. This reflects a moderate goodness of fit.
The five factor solution initially suggested via PCA and interpretable with PAF explains approximately 66% of the total and 47% of the shared variance (respectively) among responses to bipolar pairs on the semantic differential for the construct optimism. This solution reflects a factor structure that adequately reflects empirical examination of the interrelationships among bipolar pairs for optimism and identifies subsets sharing sufficient variation to justify their existence as factors measuring the construct of optimism.
This section is divided into four subsections for organizational and clarification purposes: Interpretation; Problems and Limitations; Issues; Suggestions for Future Research.
Five factors, labeled Relevance, Credibility, Faith, Certainty, and Durability for ease of interpretation, tell us about how people perceive optimism, naturally reflecting their attitudes toward the construct. While this factor structure is not optimal (I call it “moderate,” and discuss the rationale below), the factors do help clarify the meaning of optimism. These factors tell me what major subsets of information—characteristics, descriptors, emotions, perceptions—define the meaning of optimism. Relevance, accounting for 12.4% of the total shared variance explained, is best represented by respondents’ “scores” representing where they fall on continua pertaining to whether optimism is essential for satisfaction in life, whether it is teachable, and whether it is necessary for success. Individuals’ factor scores on these variables inform me about their own relative position with respect to each variable. Using the information from just this factor, I could compare individuals on total scores for this factor. Similarly, I could compare a group’s score on this factor to their score on another construct as measured by a semantic differential with the same factor to see how people perceive the relevance across constructs. Of course, the same is true for the rest of the factors—each factor can provide information on specific, individual bases, as well as contribute information to the broader interpretation and use of responses from the entire instrument. That is, the factor structure provides “micro” information as well as “macro” information, which is only meaningful in relation to other factors within one solution.
In a practical sense, the goal of factor analysis is reduction of variables reliably able to yield meaningful results from a relatively large set of variables. Unfortunately here, the number of variables was relatively small to start with—I certainly would not feel comfortable representing “Relevance” with just one variable (although, if I had to, I would use SUCCESS because the reliability alpha would be reduced to .1719 if it were removed). Given this situation, the primary goal of factor analysis, then, is to provide a better understanding of what variables truly measure or differentiate individuals on the construct of optimism. The results of these analyses serve that purpose well.
The solution presented above is moderate. The five factors explain less than half of the shared variance among responses to bipolar pairs of adjectives designed to help elucidate the meaning of optimism. Relatively low communalities (average=.421) reflect the proportion of variance in each item that can be explained by the factors. While the communalities are not unacceptable, neither are they very high. Perhaps the most salient reason for the seemingly moderate results pertains to sampling—lack of sufficient responses to better analyze patterns and correlations. The Kaiser-Meyer-Olkin Measure of Sampling Adequacy is quite low (.518), possibly speaking to this issue of sampling relative to the number of variables, if not sampling alone.
The percent of non-redundant residuals above the absolute value or .05 (19% in PAF) suggests that the five factor solution is only a moderate fit. One would expect that the differences between the actual and reproduced correlations would be smaller if the solution was optimal. (It makes sense that this percentage is much higher with PCA wherein total variance—including error, therefore—is analyzed.)
The one confounding factor here is the nature of the instrument and the rather amorphous construct it is trying to measure. Considering only PCA for a moment, the five factors (Relevance, Credibility, Faith, Certainty, and Durability) were capable of accounting for almost 70% of the total variance among responses pertaining to the meaning of optimism. When one considers this in light of the fact that psychologists and philosophers remain ensconced in debate about what optimism is and how to best define it, the results seem better than good.
Problems and Limitations
Limitations for this project include the sample size for the dataset. As stated above, samples should be a minimum of 100, according to some researchers. A sample size of 86 may be limiting the ability of factor analysis to best explain what is happening with the instrument and people’s responses. Because the subjects-to-variables ratio should be 6-10, a sample of 86 individuals limits the number of variables to a maximum of 14—with more appropriate variables, one might anticipate more intercorrelation and perhaps a stronger, more interpretable factor structure. Finally, a five factor solution with 13 variables creates a situation in which each factor is best characterized by high loadings from only 2 or 3 variables. Ideally, the number of variables loading highly on one factor should be higher, perhaps 4 – 6. Presumably a better factor structure would result with the latter situation.
There are several interesting issues to discuss. First, there is the issue of normality of variables. Tabachnik and Fidell (2001) say that variables lacking normal distribution may degrade the solution to the extent of their non-normality, despite the fact that the solution may remain worthwhile (p=0.588). I noted earlier that I expected non-normal variables; I believed that the means and standard deviations supported my expectation, and subsequent analyses of skewness and kurtosis confirmed it. What intrigues me is that, while normal variables may enhance typical factor analyses, I contend that non-normal variables for a semantic differential such as this are not only expected but desired. One should expect that people will have a fairly standard response pattern to a concept well-known in popular culture, despite technical difficulties in truly defining it. I would be more concerned about whether I could ever find reliable descriptors or factors elucidating the meaning of optimism if people responded to the bipolar pairs in such a way that the data followed normal distribution.
I chose to follow up PCA with PAF because of a notion related to item response theory. An individual’s actual score on any instrument is comprised of the true score plus some measure of error. This is not unlike variance as explained by PCA. Total variance is comprised of true variance (unique plus shared) as well as variance due to error. PCA does not ferret out the error variance from the true variance. Thus, the variance explained by five factors in PCA is substantially more the variance explained by those same five factors in PAF. Which is more accurate? Which better answers the research question? PAF essentially attempts to remove error from the variance equation by focusing on that variance which is shared among variables—this is why, for example, the cumulative variance explained fell from ~66% to ~46% when I employed PAF with the five factors suggested by PCA. But, I remain unconvinced that shared variance is really the only variance of value with regard to my purposes with the semantic differential—perhaps I am losing more than I gain in terms of interpretation and utility when I go beyond PCA to PAF.
I have debated about whether this analysis should have been limited to principal components analysis rather than taking the additional step into common factor analysis. I am aware that the choice should be rooted in the research question, but even then it seems difficult to know the best answer. For example, my guiding research question was: how do factors and factor scores representing the construct of optimism operationally define the meaning of optimism? The five factors derived from my analysis distill attitudes and responses about optimism into five primary descriptors—each of which can differentiate people or, ultimately, this construct from others. However, I question whether a better question—the more appropriate question—may simply have been, what orthogonal (non-overlapping) components best describe the meaning of optimism? Future research with similar instruments may be better suited to use of PCA to simply break down the construct of optimism (and the other constructs) into the most influencing, substantial components. I imagine that comparison across constructs might be easier and more interpretable and defensible were I to revise my analysis to reflect this PCA-only approach.
I selected varimax rotations for both analyses for three reasons. Philosophically, as is the case with many semantic differentials, I was only interested in orthogonal factors—that is, I wanted to know what factors existed when I asked SPSS to maximize variance by making the high loadings higher and the low loadings lower, ultimately providing me with the simplest, most interpretable factors. Varimax rotations were also selected because of their relatively easy interpretability and utility. The third reason is likely a result of the second—varimax rotations are the most commonly used rotations. However, perhaps an oblique rotation, with a more extensive dataset or better designed semantic differential, would yield a stronger solution. This remains to be seen, as all preliminary analyses (several of which included trial runs of oblique rotations) provided solutions of lesser quality than the solution provided herein.
Suggestions for Future Research
Harkening back to the issue of PCA versus PAF for these data, I would like to step back and reconsider the choice of PAF rather than PCA alone. Furthermore, I plan to reconsider the bipolar pairs best suited to our research agenda. Assuming that those issues get resolved, I anticipate developing similar semantic differentials for the other 5 non-intellective factors comprising our research in Operation Houndstooth. Once datasets are available for those factors, I anticipate using the D statistic (distances) and perhaps applying discriminate function analysis (DFA) to the data (Osgood, Suci, & Tannenbaum, 1957). With inclusion/application of PCA alone, the D statistic, and DFA, the multidimensional semantic space for comparison of groups, individuals, or constructs may be even better represented.