Measuring the Unobservable Scaling Measuring the Unobservable

Scaling Scaling involves the construction of an instrument that associates qualitative constructs with quantitative metric units. Scaling evolved out of efforts to measure "unmeasurable" constructs like authoritarianism and self esteem. Note: We are talking about constructed scales involving multiple items, not a response scale for a particular question.

Scaling How do we define or “capture” or measure a nebulous concept? By “taking stabs” from several directions, we can get a more complete picture of a concept we know exists but cannot see.

Scaling In scaling, we have several items that are intended to “capture” a piece of the underlying concept. The items are then combined in some form to create the scale. Quite technically, we will talk about scales and indexes interchangeably. Scales are composed of items caused by an underlying construct, whereas indexes are composed of items that indicate the level of a construct and might be useful together to predict outcomes.

Scaling Graphical depiction of a scale: Latent Variable Observed Item 1 Observed Item 2 Observed Item 3 Observed Item 4 e1 e2 e3 e4

Scaling + Form an Index Graphical depiction of an index: Observed Item 1 Observed Item 2 Observed Item 3 Observed Item 4

Scaling In most scaling, the objects are text statements, usually statements of attitude or belief.

Scaling A scale can have any number of dimensions in it. Most scales that we develop have only a few dimensions. What's a dimension? If you think you can measure a person's self-esteem well with a single ruler that goes from low to high, then you probably have a unidimensional construct.

Scaling

Scaling Many familiar concepts (height, weight, temperature) are actually unidimensional. But, if the concept you are studying is in fact multidimensional in nature, a unidimensional scale or number line won't describe it well. E.g., academic achievement: how do you score someone who is a high math achiever and terrible verbally, or vice versa? A unidimensional scale can't capture that type of achievement.

Scaling Factor analysis can tell you whether you have a unidimensional or multidimensional scale—helping you discover the number of dimensions or scales that exist among a group of variables. Factor analysis is typically an exploratory process, but it can be confirmatory. Exploratory factor analysis helps you reduce data by grouping variables into sets that tap the same phenomena.

Scaling Steps in factor analysis (what the computer does): Assumes one factor and checks the correlation of each item with the proposed factor and compares the proposed inter-item correlations with the actual inter-item correlations. Compared with Do they Match? Proposed Model Actual Data Item 1 A Item 1 Factor Sum of 1,2 B Correlation Item 2 A = 1’s correlation with factor B = 2’s correlation with factor By definition, Item 1 & 2’s correlation is A * B Item 2

Scaling Steps in factor analysis (what the computer does): If the single concept is not a good model, the computer rejects one factor and forms a residual correlation matrix (real 1,2 – proposed A*B) Identifies a second concept that may explain some of the remaining correlation and checks the proposed inter-item correlation against the real correlations. And so on until the correlations match.

Scaling In actuality, factor analysis will give K factors for K variables. The last residual correlation matrix will result in zeros. So, how many factors should you use? You could use statistical criteria: extract factors until matrix is not statistically significant from zero. Historically, number of factors has been determined by substantial needs, intuition, and theory.

Scaling Guideline for subjective analysis: A group of factors should be able to explain a high proportion of total covariance among a set of items. Eigenvalue test Scree Test

Scaling Eigenvalues An eigenvalue represents the number of units of information that a factor explains in a k set of variables with k units of information. E.g., when k = 10, an eigenvalue of 3 represents 30% of information is explained by the factor. An eigenvalue of 1 corresponds with a variable’s worth of information. Therefore, factors with an eigenvalue of 1 or less do not help to reduce data. Get rid of factors with eigenvalues less than 1

Scaling Scree Test Most researchers are looking for stronger, fewer factors (they want to reduce data). Therefore, they tend to use the scree plot. Plot the eigenvalues relative to each other Strong factors form a steep slope, weaker factors form a plateau Retain those factors that lie above the “elbow” of the plot—like with gangrene, cut off the elbow! 2 1 Scree plot for 5 variables

Scaling In addition, factors should be composed of similar, logically linked items. This is an especially helpful rule when the number of factors is not that obvious.

Scaling Factor Rotation Factor rotation involves using an algorithm to maximize the correlation of items to a factor—making each item appear most relevant to a single factor. The point is to identify variables that most similarly form indicators of the same factor—each factor’s variables being most clearly highlighted.

Scaling Factor Rotation The best-scenario (never happens) is when all items load (correlate with) as 1 on a single factor and 0 on all the rest. This is called simple structure. Factor rotation mathematically takes the items as close as possible to simple structure.

Scaling Factor Rotation Orthogonal versus oblique rotation Orthogonal rotation makes factors completely independent of each other. This is preferred for finding the most unique factors. Use if factors ought not be related. Any item’s variation explained by one factor can be added to that of another factor to get the total variation explained by the two. If you find lots of cross-loading, you should consider “Oblique.” Oblique rotation makes factors that are allowed to be correlated with each other to some degree. Use if the factors ought to be related. There is redundancy in the variation of any item explained by one factor versus another, such that they have overlapping explanatory power. You might want to try both and look for simple structure. Strong loadings on two factors may indicate a single factor, high correlation of two factors may indicate a single factor.

Scaling Factor Rotation Items with a high loading on (high correlation with) a factor form the factor’s variable for research purposes. Common elements of the items is likely what the factor represents.

Scaling Type of analysis in extracting factors: Principal components analysis produces specified proportion of total variance among items explained. Common factor analysis produces specified proportion of shared variance among items explained. Bottom line: report which you used.

Scaling Exploratory versus Confirmatory Analysis Exploratory is that which we have been discussing. If using exploratory, with new samples you “rediscover” a structure in each sample—you have persuasive evidence of the structure. Confirmatory typically refers to models generated by Structural Equation Modeling where items are specified to form a factor in advance. The question becomes, “How well do the data fit a specified model using statistical inference?” You have to be careful not to overproduce many meaningless factors.

Scaling Validity and Reliability Like other measures, scales and indexes must be valid and reliable to be useful. Validity: Face, Content, Criterion, Construct A particular kind of reliability that is particularly useful for scales and indexes is inter-item reliability (internal consistency or high inter-item correlation) To the degree that the items are correlated, the common correlation is attributable to the true score of the latent variable.

Scaling Inter-item Reliability—Alpha Variation in each item is caused by the latent variable and error (unique for each) Common variation is caused by the latent variable. Using the variance/covariance matrix, you can see total variance in the sum of components. The diagonal (variance) represents unique variation for each item. The off-diagonal represents co-variation of items. This also equals 1 – (Unique/Total)

Scaling Inter-item Reliability—Alpha The off-diagonal represents co-variation of items. This also equals 1 – (Unique/Total) To correct for the ways variance/covariance matrices change with number of items, the formula above is adjusted by k/k-1, where k = number of items. This constrains alpha to range from 0 to 1. k Unique variance  = k – 1 Total variance

Scaling Inter-item Reliability—Alpha Some characteristics of alpha Holding correlation constant, alpha goes up with more scale items To improve a scale, look for effect on alpha if an item were dropped. Reliability is not good unless it is .65 or above. “Best” reliability would be around .9. Good scales require a balance between reliability and length.

Scaling Creating a scale Determine what you want to measure Clarity Specificity Generate an item pool Scales may be generated from 40 to 100 items Good reason to reuse scales Writing Positive and negative items Stay on topic Avoid lengthy items Keep wording simple Avoid multiple negatives No double-barreled items

Scaling Creating a scale Determine format for measurement. Response options Broad versus narrow Review of item pool by experts Include scale validation items Administer to a development sample Evaluate items Differences between items You need item variance Look for means in the middle of the scale Item-scale correlations