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1、ScalingMeasuring the UnobservableScalinglScaling involves the construction of an instrument that associates qualitative constructs with quantitative metric units. lScaling evolved out of efforts to measure unmeasurable constructs like authoritarianism and self esteem. Note: We are talking about cons
2、tructed scales involving multiple items, not a response scale for a particular question.ScalinglHow do we define or “capture” or measure a nebulous concept?lBy “taking stabs” from several directions, we can get a more complete picture of a concept we know exists but cannot see.ScalinglIn scaling, we
3、 have several items that are intended to “capture” a piece of the underlying concept. lThe 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 indexe
4、s are composed of items that indicate the level of a construct and might be useful together to predict outcomes.ScalingLatentVariableObserved Item 1Observed Item 2Observed Item 3Observed Item 4e1e2e3e4Graphical depiction of a scale:Scaling+Form an IndexObserved Item 1Observed Item 2Observed Item 3Ob
5、served Item 4Graphical depiction of an index:ScalinglIn most scaling, the objects are text statements, usually statements of attitude or belief. ScalinglA scale can have any number of dimensions in it. Most scales that we develop have only a few dimensions. lWhats a dimension? lIf you think you can
6、measure a persons self-esteem well with a single ruler that goes from low to high, then you probably have a unidimensional construct. ScalingScalinglMany familiar concepts (height, weight, temperature) are actually unidimensional. lBut, if the concept you are studying is in fact multidimensional in
7、nature, a unidimensional scale or number line wont describe it well. E.g., academic achievement: how do you score someone who is a high math achiever and terrible verbally, or vice versa? lA unidimensional scale cant capture that type of achievement.ScalinglFactor analysis can tell you whether you h
8、ave a unidimensional or multidimensional scalehelping you discover the number of dimensions or scales that exist among a group of variables.lFactor analysis is typically an exploratory process, but it can be confirmatory.lExploratory factor analysis helps you reduce data by grouping variables into s
9、ets that tap the same phenomena.ScalinglSteps in factor analysis (what the computer does):1.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. Item 1Item 2FactorSum of 1,2Propo
10、sed ModelActual DataA = 1s correlation with factorB = 2s correlation with factorBy definition, Item 1 & 2s correlation is A * BABCompared with Do they Match? Item 1Item 2CorrelationScalinglSteps in factor analysis (what the computer does):2.If the single concept is not a good model, the computer rej
11、ects one factor and forms a residual correlation matrix (real 1,2 proposed A*B)3.Identifies a second concept that may explain some of the remaining correlation and checks the proposed inter-item correlation against the real correlations.4.And so on until the correlations match.ScalinglIn actuality,
12、factor analysis will give K factors for K variables. The last residual correlation matrix will result in zeros.lSo, 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 det
13、ermined by substantial needs, intuition, and theory.ScalinglGuideline for subjective analysis: A group of factors should be able to explain a high proportion of total covariance among a set of items.Eigenvalue testScree TestScalinglEigenvaluesAn eigenvalue represents the number of units of informati
14、on 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 variables worth of information. Therefore, factors with an eigenvalue of 1 or less do not
15、 help to reduce data.Get rid of factors with eigenvalues less than 1ScalinglScree TestMost 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 otherStrong factors form a steep slope, weaker f
16、actors form a plateauRetain those factors that lie above the “elbow” of the plotlike with gangrene, cut off the elbow!Scree plot for 5 variables21ScalinglIn addition, factors should be composed of similar, logically linked items. This is an especially helpful rule when the number of factors is not t
17、hat obvious.ScalinglFactor RotationFactor rotation involves using an algorithm to maximize the correlation of items to a factormaking each item appear most relevant to a single factor.The point is to identify variables that most similarly form indicators of the same factoreach factors variables bein
18、g most clearly highlighted.ScalinglFactor RotationThe 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.ScalinglF
19、actor RotationOrthogonal versus oblique rotationOrthogonal 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 items variation explained by one factor can be added to that of another factor to get
20、 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
21、 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.ScalinglFactor RotationItems with a high
22、 loading on (high correlation with) a factor form the factors variable for research purposes.Common elements of the items is likely what the factor represents.ScalinglType of analysis in extracting factors:Principal components analysis produces specified proportion of total variance among items expl
23、ained.Common factor analysis produces specified proportion of shared variance among items explained.Bottom line: report which you used.ScalinglExploratory versus Confirmatory AnalysisExploratory is that which we have been discussing. If using exploratory, with new samples you “rediscover” a structur
24、e in each sampleyou 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 h
25、ave to be careful not to overproduce many meaningless factors.ScalinglValidity and ReliabilityLike other measures, scales and indexes must be valid and reliable to be useful.Validity: Face, Content, Criterion, ConstructA particular kind of reliability that is particularly useful for scales and index
26、es 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.ScalinglInter-item ReliabilityAlpha Variation in each item is caused by the latent variable a
27、nd 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
28、1 (Unique/Total)ScalinglInter-item ReliabilityAlpha 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
29、 alpha to range from 0 to 1.k Unique variance = k 1 Total varianceScalinglInter-item ReliabilityAlpha Some characteristics of alphaHolding correlation constant, alpha goes up with more scale itemsTo improve a scale, look for effect on alpha if an item were dropped.Reliability is not good unless it i
30、s .65 or above. “Best” reliability would be around .9.Good scales require a balance between reliability and length. ScalinglCreating a scale1.Determine what you want to measurelClaritylSpecificity2.Generate an item poollScales may be generated from 40 to 100 itemslGood reason to reuse scaleslWriting
31、lPositive and negative itemslStay on topiclAvoid lengthy itemslKeep wording simplelAvoid multiple negativeslNo double-barreled itemsScalinglCreating a scale3.Determine format for measurement.lResponse optionslBroad versus narrow4.Review of item pool by experts5.Include scale validation items6.Administer to a development sample7.Evaluate itemslDifferences between itemslYou need item variancelLook for means in the middle of the scalelItem-scale correlations