Much of item response theory (IRT) is based on the assumption of unidimensionalty; namely, the associations among the item responses are explained completely by a single underlying latent variable, representing the target construct being measured. While this is often justified in many areas of educational measurement, more recent interest in measuring patientreported outcomes (Gibbons et al., 2008, 2012) involves items that are drawn from multiple uniquely correlated subdomains violating the usual conditional independence assumption inherent in unidimensional IRT models.
Much of item response theory (IRT) is based on the assumption of unidimensionalty; namely, the associations among the item responses are explained completely by a single underlying latent variable, representing the target construct being measured. While this is often justified in many areas of educational measurement, more recent interest in measuring patientreported outcomes (Gibbons et al., 2008, 2012) involves items that are drawn from multiple uniquely correlated subdomains violating the usual conditional independence assumption inherent in unidimensional IRT models. As alternatives, both unrestricted item factor analytic models (Bock & Aitkin, 1981) and restricted or confirmatory item factor analytic models (Gibbons & Hedeker, 1992) have been used to accommodate the multidimensionality of constructs for which the unidimensionality assumption is untenable. A concrete example is the measurement of depressive severity, where items are drawn from mood, cognition, and somatic impairment subdomains. While this is a somewhat extreme example, there are many more borderline cases where the choice between a unidimensional model and a multidimensional model is less clear, orthequestion of how many dimensions is “enough” is of interest. In this chapter, we explore the issue of determining the dimensionality of a particular measurement process. We begin by discussing multidimensional item factor analysis models, and then consider the consequences of incorrectly fitting a unidimensional model to multidimensional data. We also discuss nonparametric approaches such as DIMTEST (Stout, 1987). We then examine different approaches to testing dimensionality of a given measurement instrument, including approximate or heuristic approaches such as eigenvalue analysis, as well as more statistically rigorous limitedinformation and fullinformation alternatives. Finally, we illustrate the use of these various techniques for dimensionality analysis using a relevant example.
Multiple factor analysis as formulated by Thurstone (1947) assumes that the test scores are continuous measurements standardized to mean zero and standard deviation one in the sample. (Numberright scores on tests with 30 or more items are considered close enough to continuous for practical work.) The Pearson productmoment correlations between all pairs of tests are then sufficient statistics for factor analysis when the population distribution of the scores is multivariate normal. Because the variables are assumed standardized, the mean of the distribution is the null vector and the covariance matrix is a correlation matrix. If the dimensionality of the factor space is D, the assumed statistical model for the pth observed score y is
Just as the observed variables, the latent variables are assumed to follow a standard multivariate normal distribution but are uncorrelated; that is, their covariance matrix is a D × D identity matrix. The residual term (also called unique factor), ϵ_{p} , that accounts for all remaining variation in y_{p} is assumed to be normal with mean 0 and variance
, whereOn the above assumptions, efficient statistical estimation of the factor loadings from the sample correlation matrix is possible and available in published computer programs. In fact, only the communalities need to be estimated: once the communalities are known, the factor loadings can be calculated directly from the socalled “reduced” correlation matrix via matrix decompositions, in which the diagonal elements of the sample correlation matrix are replaced by the corresponding communalities (Harman, 1967).
In item factor analysis, the observed item responses are assigned to one of twoormore predefined categories. For example, test items marked right or wrong are assigned to dichotomous categories; responses to essay questions may be assigned to ordered polytomous categories (grades) A, B, C, D in order of merit; responses in the form of best choice among multiple alternatives may be assigned to nominal polytomous categories.
To adapt the factor analysis model for test scores to the analysis of categorical item responses, we assume that the yvariables are also unobservable. We follow Thurstone in referring to these underlying variables as response processes. In the dichotomous case, a process gives rise to an observable correct response when y_{p} exceeds some threshold γ_{i} specific to item i. On the assumption that y_{p} is standard normal, γ_{i} divides the area under the normal curve into two sections corresponding to the probability that a respondent with given value of θ will respond in the first or second category. Designating the categories 1and 2, we may express these conditional probabilities given θ as
The unconditional response probabilities, on the other hand, are the areas under the standard normal curve above and below −γ_{i} in the population from which the sample of respondents is drawn. The area above this threshold is the classical item difficulty, p_{i} , and the standard normal deviate at p_{i} is a large sample estimator of −γ_{i} (Lord & Novick, 1968, Chapter 16).
These relationships generalize easily to ordered polytomous categories. Suppose item j has m_{j} ordered categories; we then replace the single threshold of the dichotomous case with m_{i −1} thresholds, say, γ_{i1}, γ_{i2}, …, γ_{i, mi −1 } . The category response probabilities conditional on θ are the m_{j} areas under the normal curve corresponding to the intervals from minus to plus infinity bounded by the successive thresholds:
Because the response processes are unobserved, their productmoment correlation matrix cannot be calculated directly. Classical methods of multiple factor analysis do not apply directly to item response data. Full maximum likelihood estimation of the item correlation matrix requires calculating the normal orthant probabilities involving integrals over as many dimensions as there are items. While it is theoretically possible, actual computation remains difficult even with modern estimation approaches (Song & Lee,2003).
However, an approximation to the item correlations can be inferred from the category jointoccurrence frequencies tallied over the responses in the sample. Assuming in the twodimensional case that the marginal normal distribution of the processes is standard bivariate normal, the correlation value that best accounts for the observed joint frequencies can be estimated, for example, using pairwise maximum likelihood. If both items are scored dichotomously, the result is the wellknown tetrachoric correlation coefficient, an approximation for which was given by Divgi (1979). If one or both items are scored polytomously, the result is the less common polychoric correlation (Jöreskog, 2002). The correlations for all distinct pairs of items can then be assembled into a correlation matrix and unities inserted in the diagonal to obtain an approximation to the item correlation matrix. Because the calculation of tetrachoric and polychoric correlations breaks down if there is a vacant cell in the jointoccurrence table, a small positive value such as 0.5 (i.e., a continuity correction) may be added to each cell of the joint frequency table.
For the purpose of determining dimensionality, the correlation matrix described above can be subjected to the method of principal components or principal factor analysis with iterated communalities (Harman, 1967, p. 87). Classical principal factor analysis of item responses can be useful in its own right, or as a preliminary to more exact and more computationally intensive IRT procedures such as maximum marginal likelihood item factor analysis. In the latter role, the classical method provides a quick way of giving an upper bound on a plausible number of factors in terms of the total amount of association accounted for. It also gives good starting values for the iterative procedures discussed in the following section.
IRTbased item factor analysis makes use of all information in the original categorical responses and does not depend on pairwise indices of association such as tetrachoric or polychoric correlation coefficients. For that reason, it is referred to as fullinformation item factor analysis. It works directly with item response models giving the probability of the observed categorical responses as a function of latent variables descriptive of the respondents and parameters descriptive of the individual items. It differs from the classical formulation in its scaling, however, because it does not assume that the response process has unit standard deviation and zero mean; rather, it assumes that the residual term has unit standard deviation and zero mean. The latter assumption implies that the response processes have zero mean and standard deviation equal to
Inasmuch as the scale of the model affects the relative size of the factor loadings and thresholds, we rewrite the model for dichotomous responses in a form in which the factor loadings are replaced by factor slopes, a_{id} , and the threshold is absorbed in the intercept,c_{i}:
To convert factor slopes into loadings, we divide by the above standard deviation and similarly convert the intercepts to thresholds:
Conversely, to convert to factor analysis units, we change the standard deviation of the residual from one to
For polytomous responses, the model generalizes as
In the context of item factor analysis, this is the multidimensional generalization of the graded model introduced by Samejima (1969). Similarly, the rating scale model of Andrich (1978), in which all items have the same number of categories and the thresholds are assumed to have the same spacing but may differ in overall location, can be generalized by setting the above linear form to z_{i} + e_{i} + c_{h} , where e_{i} is the location intercept.
There is a long history, going back to Fechner (1860), of methods for estimating the slope and intercept parameters of models similar to the above—that is, models in which the response process is normally distributed and the deviate is a linear form. These socalled normal transform models differ importantly from the IRT models, however, in assuming that the θ variables are manifest measurements of either observed or experimentally manipulated variables. In Fechner's classic study of the sensory discrimination thresholds for lifted weights, the subjects were required to lift successively each of a series of two small, identical appearing weights differing by fixed amounts and say which feels heavier. Fechner fitted graphically the inverse normal transforms of the proportion of subjects who answered correctly and used the slope of the fitted line to estimate the standard deviation as a measure of sensory discrimination. Much later, R. A. Fisher (Bliss, 1935) provided a maximum likelihood method of fitting similar functions used in the field of toxicology to determine the socalled 50% lethal dose of pesticides. This method eventually became known as probit analysis (Bock & Jones, 1968; for behavioral applications, see Finney, 1952).
To apply Fisher's method of analysis to item factor analysis, one must find a way around the difficulty that the variable values (i.e., the θs) in the linear predictor are unobservable. The key to solving this problem lies in assuming that the values have a specifiable distribution in the population from which the respondents are drawn (Bock & Lieberman, 1970). This allows us to integrate over that distribution numerically to estimate the expected numbers of respondents located at given points in the latent space who respond in each of the categories. These expected values can then be subjected to a multidimensional version of probit analysis. The socalled EM method of solving this type of estimation problem (Aitkin, Volume Two, Chapter 12; Bock & Aitkin, 1981) is an iterative procedure starting from given initial values. It involves calculating expectations (the Estep) that depend on both the parameters and the observations, followed by likelihood maximization (the Mstep) that depends on the expectations. These iterations can be shown to converge on the maximum likelihood estimates under very general conditions (Dempster et al., 1977). In IRT and similar applications, this approach is called maximum marginal likelihood estimation because it works with the marginal probabilities of response rather than the conditional probabilities (Glas, Volume Two, Chapter 11). Details in the context of item factor analysis are given in Bock and Aitkin (1981) and Bock and Gibbons (2010, Appendix).
There are two major limitations of the unrestricted or exploratory factor analysis model described above. First, interpretation of the final solution depends on selecting the appropriate rotation of the final solution (e.g., varimax, quartimin, etc.; for a review, see Browne, 2001). Second, modern simulationbased estimation approaches notwithstanding (e.g., Cai, 2010a,b), the fullinformation IRT approach remains demanding in terms of the number of dimensions that can be evaluated because the computational complexity associated with the integrals in the likelihood equations is exponentially increasing in the number of factors. In confirmatory factor analysis, the first limitation (indeterminacy due to rotation) is resolved by assigning arbitrary fixed values to certain loadings of each factor during maximum likelihood estimation. In general, fixing of loadings will imply nonzero correlations of the latent variables, but this does not invalidate the analysis. The correlations may also be estimated if desired by selecting an oblique rotation criterion. An important example of confirmatory item factor analysis—which resolves the second problem of limitation of the number of dimensions that can be numerically evaluated—is the bifactor patterns for general and group factors, which applies to tests and scales with item content drawn from several welldefined subareas of the domain in question. Two prominent examples are tests of educational achievement consisting of reading, mathematics and science areas, and selfreports of health status covering both physical and emotional impairment. The main objective in the use of such instruments is to estimate a single score measuring, in these examples, general educational achievement or overall health status.
To analyze these kinds of structures for dichotomously scored item responses, Gibbons and Hedeker (1992) developed fullinformation item bifactor analysis for binary item responses, and Gibbons extended it to the polytomous case (Gibbons et al., 2007). Cai et al. (2011) further generalized the model to handle multiple groups. To illustrate, consider a set of n test items for which a Dfactor solution exists with one general factor and D − 1 group or methodrelated factors. The bifactor solution constrains each item j to a nonzero loading α_{i1} on the primary dimension and a second loading (α_{id}, d = 2,…, D) on not more than one of the D −1 group factors. For four items, the bifactor pattern matrix might be
This structure, which Holzinger and Swineford (1937) termed the “bifactor” pattern, also appears in the interbattery factor analysis of Tucker (1958) and is one of the confirmatory factor analysis models considered by Jöreskog (1969). In the latter case, the model is restricted to test scores assumed to be continuously distributed. However, the bifactor pattern might also arise at the item level (Muthén, 1989). Gibbons and Hedeker (1992) showed that paragraph comprehension tests, where the primary dimension represents the targeted process skill and additional factors describe content area knowledge within paragraphs, were described well by the bifactor model. In this context, they showed that items were conditionally independent between paragraphs, but conditionally dependent within paragraphs. More recently, the bifactor model has been applied to problems in patientreported outcomes in physical and mental health measurement (Gibbons et al., 2008, 2012, 2014). As shown by Gibbons and Hedeker (1992), the bifactor model always reduces the dimensionality of the likelihood equation to two, regardless of the number of secondary factors.
In the bifactor case, the graded response model (Gibbons et al., 2007) is
A natural question is whether there is any adverse consequence of applying unidimensional IRT models to multidimensional data. To answer this question, Stout and coworkers (e.g., Stout, 1987; Zhang & Stout, 1999) took a distinctly nonparametric approach to characterize the specific conditions under which multidimensional data may be reasonably well represented by a unidimensional latent variable. They emphasized a core concept that subsequently became the basis of a family of theoretical and practice devices for studying dimensionality, namely, local independence as expressed using conditional covariances. To begin, given n items in a test, the strong form of local independence states that the conditional response pattern probability factors into a product of conditional item response probabilities, that is,
The mathematical condition above suggests a statistical procedure for testing essential unidimensionality (Stout, 1987). In brief, the test is split into two subsets called assessment tests (AT1 and AT2), and a longer subset called the partitioning test (PT). The items for AT1 are chosen to be saturated with the same dominant latent trait, but are as dimensionally different as possible from the items in the PT. Then AT2 is selected such that the items have similar difficulty as AT1. Each test taker's total score on the PT is used to group the test takers into several homogeneous subgroups. The PT total score becomes the conditioning score (effectively as a surrogate of θ) to calculate required conditional covariances for statistical hypothesis testing using the AT1 and AT2 item responses. The procedure as formalized by Nandakumar and Stout (1993) is referred to as DIMTEST.
Gibbons et al. (2007) studied the consequences of fitting unidimensional models to multidimensional data empirically. The question they asked was slightly different. To the extent that the primary dimension of interest can be preserved in a unidimensional model and in the primary factor of a bifactor model or possibly in an exploratory item factor analysis model, does the specific model used make a difference in the results?
They conducted a simulation study to investigate the effects of applying Samejima's (1969) graded response model in unidimensional and bifactor form to multidimensional data. Conditions studied were (a) test length, 50 items or 100 items; (b) number of dimensions, 5 or 10; (c) primary loadings, 0.50 or 0.75; and (d) domain loadings, 0.25 or 0.50. Outcome results include standard deviation of expected a posteriori (EAP) estimates of θ, posterior standard deviations (PSDs, or standard errors) of Bayes EAP scores, loglikelihood (model fit), differences between EAP and actual θ, and percentage change between unidimensional and bifactor models of these variables. The generated data were based on a fourpoint categorical scale, and the examinee distribution was assumed to be normal, N(0, 1), based on 1000 replications. In the following, we summarize the key findings of this study.
Figure 3.1 reports the standard deviations of the θ estimates for the unidimensional and bifactor models across the 12 simulated conditions. Inspection of the figure indicates that the EAP estimates based on the unidimensional model were more varied across all conditions. The magnitude of the difference decreased when the primary and secondary loadings decreased, leading to a more unidimensional solution. As shown, as the number of items increased from 50 to 100, the EAP estimates from both models became more varied, but not as severe for the bifactor model.
Figure 3.1 Mean standard deviations of θ of the unidimensional and bifactor models based on 1000 replications per condition (number of items [NI] = 50 or 100, number of dimensions [ND] = 5 or 10, primary loadings [PL] = 0.50 or 0.75, domain loadings [DL] = 0.25 or 0.50).
Figure 3.2 reports the mean PSD of the Bayes EAP estimates. As shown, the differences in the PSD between the models can be dissected in terms of the dimensionality of the underlying data. Specifically, in the conditions in which the primary loadings are 0.75 and the domain loadings are 0.50, the PSD of the unidimensional model substantially underestimates the PSDs from the bifactor model. As shown, the largest PSD for the unidimensional model occurs with 100 items and 5 dimensions. The PSD estimated by the bifactormodel remains fairly consistent across the conditions in which the underlying structure can be regarded as strongly multidimensional (i.e., primary loadings = 0.75, domain loadings = 0.50). For the conditions in which the primary loadings are 0.75 and the domain loadings are 0.25, the PSD for the unidimensional approaches that for the bifactor model but, nevertheless, continues to underestimate the bifactor result, which is the correct value in this case. The largest discrepancies between the PSD of these models occur when the number of dimensions is 5 and the number of items is 50 and 100. The smallest difference between the mean PSDs for the unidimensional and bifactor models occurs when the number of dimensions is 10 with 50 items. For the bifactor model, the PSD decreases slightly when the number of items increases from 50 to 100. However, the number of dimensions does not seem to significantly influence the PSD of the bifactor model.
Figure 3.2 Mean PSDs of Bayes expected a posterior scores of the unidimensional and bifactor models based on 1000 replications per condition (number of items [NI] = 50 or 100, number of dimensions [ND] = 5 or 10, primary loadings [PL] = 0.50 or 0.75, domain loadings [DL] = 0.25 or 0.50).
The results of this study illustrate the consequences attached to applying a unidimensional IRT model to data with varying degrees of multidimensionality compared to the bifactor model. The first set of results addressed the variability in estimated θ values, or examinees'standing on the latent trait. Compared to the unidimensional model, the bifactor model yielded θ estimates that were more homogeneous across simulated data structures. As a consequence, studies that are designed to evaluate educational or clinical interventions will have increased statistical power to detect meaningful effects when scores are based on a bifactor model and the underlying data are the result of a multidimensional response process.
PSD estimates were found to be underestimated across all conditions for the unidimensional model. For the bifactor model, PSD values were consistently below 0.20 across conditions, except when the total test length was 50 and the primary loadings were 0.50 and the domain loadings were 0.25. One setting in which the underestimation of PSDs could affect test scores is in computer adaptive testing, in which each item is intentionally selected to provide the most information for estimating examinee ability in the sense of greatest reduction of PSD. Using PSD estimates based on the unidimensional model may therefore lead to suboptimal estimates of examinee ability. Used as measurement error variance, the inverse squared unidimensional PSDs are not valid for weighting observations in statistical analyses using the scores as data.
The nonparametric indices based on conditional covariances such as DIMTEST do not explicitly specify a distribution of the θs. Hence, they require the use of external conditioning subscores such as the partitioning total score. When an item factor analysis model is fitted using standard estimation methods such as maximum marginal likelihood, population distributions of θ are routinely assumed. Therefore, upon finding the maximum likelihood solution, the model yields expected probabilities for each single item, as well as joint probabilities for item pairs, triplets, quadruplets, etc. When contrasted against the observed probabilities, the residuals may be used to derive goodnessoffit statistics. Most of the time, univariate and bivariate association information is used.
In the context of IRT, statistics based on (mostly) univariate and bivariate subtables are referred to as limitedinformation goodnessoffit statistics, in contrast to fullinformation statistics (e.g., the Pearson's chisquare statistic) that are based on residuals of the full item by itembyitem crossclassifications. Despite the apparent loss of information due to collapsing the full contingency table into a series of first and secondorder association tables, limitedinformation test statistics have been suggested as a potential solution to the Achilles' heel of fullinformation statistics, namely, the sparseness of the underlying multiway contingency table upon which the IRT model is defined (Bartholomew & Tzamourani, 1999). The number of cells in the table is exponentially increasing in the number of items, and for tests of realistic length, the table will become extremely sparse for any conceivable sample size. The sparseness invalidates the usual asymptotic chisquare approximations to the distribution of Pearson's statistic or the likelihood ratio statistic, making model fit testing decisions based on fullinformation statistics untrustworthy in practical situations. On the other hand, test statistics based on univariate and bivariate subtables maintain Type I error rate control and have adequate power (see, e.g., Cai et al., 2006). In particular, MaydeuOlivares and Joe's (2005) M _{2} family of test statistics has witnessed increasing popularity.
In the context of multidimensional IRT, Cai and Hansen (2012) extended the dimension reduction technique, already used in parameter estimation of bifactor models, to limitedinformation goodnessoffit testing. For example, for a bifactor model, the probabilities and derivatives for computing limitedinformation test statistics require at most twodimensional numerical integration, regardless of the number of factors in the model, making it feasible to test much larger models with many latent variables. In addition, Cai and Hansen (2012) developed a new quadratic form test statistic, which they call
, that is based on the general limitedinformation testing principles proposed by Joe and MaydeuOlivares (2010). The statistic is best understood as a further reduction (or concentration) of the univariate and bivariate subtables. When the item responses are polytomous, this new statistic can be substantially better calibrated and more powerful than M_{2} . In addition, the chisquare distributed test statistics can be used to calculate fit measures such as the root mean square error of approximation (RMSEA; Browne & Cudeck, 1993) that are free from the influence of sample size.The details of limitedinformation goodnessoffit testing are more substantial than those can be covered in this chapter. In brief, the development begins with the realization that the IRT model can be written as a function of the (marginal) response pattern probability π⋃(γ) for pattern ⋃, where γ is a notational shorthand for the collection of free and estimable parameters in the model. Suppose there are C possible response patterns. Let us define the C × 1 vector of modeled probabilities as π ( γ ) and the corresponding C × 1 vector of observed proportions as p . Let the C × 1 population cell probabilities be π . The null hypothesis being evaluated in the goodnessoffit testing situation is H _{0}: π ( γ ) = π , for some γ , versus the alternative H_{A} : π ( γ )≠ π , for any γ .
Suppose the total sample size is N. Treating p as the fixed observed data, maximizing (e.g., using the EM algorithm) the multinomial likelihood with cell probabilities given by π ( γ ) leads to the maximum marginal likelihood estimator
. Let the fitted cell probabilities be . The cell residuals are . Standard discrete multivariate analysis results (Rao, 1973) suggest that the cell residuals are asymptotically Cvariate normal under the null hypothesis:Subtable probabilities such as the univariate and bivariate probabilities are linear functions of the cell probabilities (Cai et al., 2006). The relationship can be conveniently expressed using reduction operator matrices (Joe & MaydeuOlivares, 2010). Let T be a particular fixed q × C matrix with full row rank that achieves the reduction of π into lowerorder probabilities. The new vector of residuals retains asymptotic normality
As an illustration, we analyze data obtained with the Quality of Life Interview for the Chronically Mentally Ill (Lehman, 1988) from 586 chronically mentally ill patients. The instrument consists of one global lifesatisfaction item followed by 34 items in seven subdomains, namely, Family, Finance, Health, Leisure, Living, Safety, and Social, with four, four, six, six, five, five, and four items, respectively. Respondents were instructed to rate each item in turn on a sevenpoint scale consisting of ordered response categories: terrible, unhappy, mostly dissatisfied, mixed, about equally satisfied and dissatisfied, mostly satisfied, pleased, delighted. Both the multiple content areas of the subdomains and their labeling as such encourage responses to the set of items as a whole rather than considered responses to the individual items. This effect creates dependencies between responses within the sets that violate the assumption of conditional independence of response required for conventional onedimensional IRT analysis.
Given that the items are clustered within seven content domains, for the purpose of dimensionality assessment, we considered models containing one through eight factors, to determine if an additional factor explained any significant additional variation in item responses over the seven specified subdomains. Chisquare statistics for the addition of each successively added factor are shown in Table 3.1. Very roughly, a chisquare value is significant if it is at least twice as large as its degrees of freedom. By this rule, even the addition of an eighth orthogonal factor shows no danger of overfactoring, although its contribution to improved goodness of fit is the smallest of any factor. Notice that the decreases are not monotonic; unlike traditional factor analysis of productmoment correlations, the marginal probabilities of the response patterns (which determined the marginal likelihood) reflect changes in all parameters jointly, including the category parameters and not just the factor loadings. Because our inspection of the signs of the loadings of the first seven factors showed relationships to the item groups and the eighth factor did not (see Table 3.2), the sevenfactor model is likely the most parsimonious choice.
Decrease of −2 log L of Solutions with 1–8 Factors 


Solution 
−2 log L 
Decrease 
DF 
1 
66837.1  
2 
66045.0 
792.1 
34 
3 
65089.5 
955.5 
33 
4 
64118.4 
971.5 
32 
5 
63509.1 
609.3 
31 
6 
63063.7 
445.4 
30 
7 
62677.5 
386.2 
29 
8 
62370.5 
307.4 
28 
Factors 


Item Group 
Item 
1 
2 
3 
4 
5 
6 
7 
0 
1 
0.769 
0.021 
0.082 
−0.054 
0.054 
−0.002 
0.097 
Family  
1 
2 
0.614 
−0.269 
0.044 
−0.461 
0.272 
−0.081 
0.044 
3 
0.687 
−0.181 
−0.007 
−0.380 
0.159 
0.007 
−0.058 

4 
0.703 
−0.245 
−0.045 
−0.522 
0.257 
0.029 
0.008 

5 
0.729 
−0.214 
−0.004 
−0.505 
0.279 
0.046 
0.019 

Finance  
2 
6 
0.606 
0.468 
−0.391 
0.116 
0.284 
0.003 
−0.101 
7 
0.515 
0.405 
−0.300 
0.097 
0.220 
0.021 
−0.032 

8 
0.647 
0.511 
−0.342 
0.101 
0.276 
0.048 
−0.092 

9 
0.632 
0.510 
−0.305 
0.072 
0.242 
0.023 
−0.069 

Health  
3 
10 
0.568 
−0.123 
0.132 
0.201 
0.049 
−0.236 
0.095 
11 
0.644 
0.007 
0.195 
0.128 
0.038 
−0.443 
−0.139 

12 
0.627 
0.087 
0.289 
0.074 
−0.026 
−0.390 
−0.167 

13 
0.668 
−0.052 
0.156 
0.138 
−0.005 
−0.383 
−0.232 

14 
0.678 
−0.004 
0.154 
0.116 
0.061 
−0.288 
0.054 

15 
0.701 
0.044 
0.249 
0.045 
0.071 
−0.154 
0.054 

Leisure  
4 
16 
0.741 
0.215 
0.150 
0.030 
−0.138 
0.156 
0.155 
17 
0.657 
0.149 
0.142 
−0.017 
−0.128 
−0.054 
0.285 

18 
0.721 
0.223 
0.101 
−0.005 
−0.173 
0.019 
0.331 

19 
0.749 
0.313 
0.144 
−0.059 
−0.199 
0.095 
0.301 

20 
0.670 
0.192 
0.078 
−0.101 
−0.162 
−0.030 
0.295 

21 
0.522 
−0.002 
−0.056 
−0.002 
−0.099 
−0.049 
0.042 

Living  
5 
22 
0.664 
−0.241 
−0.401 
0.038 
−0.191 
0.048 
−0.008 
23 
0.549 
−0.332 
−0.325 
0.118 
−0.140 
−0.013 
−0.028 

24 
0.611 
−0.253 
−0.529 
0.006 
−0.190 
−0.112 
0.042 

25 
0.626 
−0.347 
−0.446 
0.079 
−0.285 
−0.127 
0.030 

26 
0.568 
−0.213 
−0.439 
0.066 
−0.177 
−0.018 
0.034 

Safety  
6 
27 
0.679 
−0.241 
0.221 
0.299 
0.232 
0.341 
−0.004 
28 
0.688 
−0.387 
0.051 
0.317 
0.141 
0.250 
−0.040 

29 
0.594 
−0.065 
0.231 
0.145 
0.109 
0.123 
−0.044 

30 
0.670 
−0.253 
0.181 
0.276 
0.196 
0.223 
−0.003 

31 
0.702 
−0.264 
0.140 
0.336 
0.064 
0.197 
0.006 

Social  
7 
32 
0.688 
0.189 
0.169 
−0.180 
−0.399 
0.197 
−0.375 
33 
0.696 
0.254 
0.099 
−0.192 
−0.317 
0.212 
−0.218 

34 
0.620 
0.203 
0.149 
−0.118 
−0.218 
0.161 
−0.232 

35 
0.494 
−0.163 
0.122 
−0.056 
−0.179 
0.046 
−0.202 
As expected, all first principal factor loadings are positive, clearly identifying the factor with the overall QualityofLife variable (see Table 3.2). In fact, the largest loading is that of item number one, which asks for the respondent's rating of overall quality of life. Only one item, the last, has a loading less than 0.5. As for the six bipolar factors, the significant feature is that the sign patterns of loadings of appreciable size conform to the item groups. Factor 2 strongly contrasts the Finance group with Family Living and Safety; to as lesser extent, Leisure is working in the same direction as Finance. In other words, persons who tend to report better financial positions and quality of leisure are distinguished by this factor from those who report better family relationships and safety. Factor 3 then combines Living and Finance and contrasts them primarily with a combination of Health and Safety. Factor 4 contrasts a combination of Family and Social with Finance, Health, and Safety. Factor 5 combines Social, Living, and Safety versus Family, Finance, and Safety. Factor 6 primarily contrasts Health with Safety. Finally, Factor 7 contrasts Social verses Leisure. The fact that the sevenfactor solution has the expected all positive first factor and patterns for the remaining bipolar factors that contrast item groups rather than items within groups clearly supports a bifactor model for these data. Estimated thresholds for this analysis are reported by Bock and Gibbons (2010).
The bifactor model produced a value of −2 log L = 64233.3, which is similar to that obtained for a fourfactor model. While the sevenfactor unrestricted model provides significant improvement in fit, inspection of the estimated factor loading in Table 3.3 shows that the bifactor model provides the most parsimonious and easily interpretable results. Because the group factor loadings are not constrained to orthogonality with those of the general factor, they are all positive and their magnitudes indicate the strength of the effect of items belonging to common domains. The effects of Family and Finance, for example, are stronger than those of Health and Leisure. It is interesting to note that the empirical reliability for the primary dimension for the bifactor model is 0.9, but is overestimated as 0.95 for a unidimensional model applied to these same data (standard errors of 0.322 and 0.232, respectively). As expected, reliability is overestimated and uncertainty in estimated scale scores is underestimated when the conditional dependencies are ignored. Avoiding this type of bias is a major motivation for item bifactor analysis.
Factors 


Item Group 
Item 
1 
2 
3 
4 
5 
6 
7 
0 
1 
0.789  
Family  
1 
2 
0.535 
0.620  
3 
0.576 
0.509  
4 
0.575 
0.586  
5 
0.631 
0.547  
Finance  
6 
0.476 
0.634  
2 
7 
0.437 
0.553  
8 
0.544 
0.617  
9 
0.535 
0.622  
Health  
3 
10 
0.560 
0.256  
11 
0.528 
0.504  
12 
0.486 
0.505  
13 
0.529 
0.473  
14 
0.650 
0.286  
15 
0.714 
0.141  
Leisure  
4 
16 
0.694 
0.285  
17 
0.565 
0.413  
18 
0.628 
0.451  
19 
0.635 
0.506  
20 
0.571 
0.473  
21 
0.479 
0.208  
Living  
5 
22 
0.536 
0.549  
23 
0.484 
0.530  
24 
0.497 
0.668  
25 
0.508 
0.688  
26 
0.508 
0.672  
Safety  
6 
27 
0.557 
0.517  
28 
0.593 
0.474  
29 
0.533 
0.501  
30 
0.558 
0.538  
31 
0.591 
0.383  
Social  
7 
32 
0.545 
0.438 

33 
0.586 
0.351 

34 
0.520 
0.466 

35 
0.446 
0.296 
Limitedinformation goodnessoffit testing lends additional support for the appropriateness of the bifactor solution. Take the unidimensional model for instance; the CaiHansen modified
statistic is equal to 2674.85 on 385° of freedom, p < 0.0001. The statistic uses both univariate and bivariate residual tables. Since there are 35 items, there are 35 + 35 × (35 − 1)/2 = 630 residuals available for testing model fit. The unidimensional graded model contains 35 × 7 = 245 free item parameters, resulting in 630 − 245 = 385° of freedom. The null hypothesis of exact unidimensionality is rejected and the unidimensional model is untenable for this dataset. We may compute the RMSEA index, which is a widely used measure of fit in factor analysis and structural equation modeling, and it is equal to 0.08 and a 90% confidence interval of RMSEA is (0.076,0.084). Adopting established conventions in factor analysis (Browne & Cudeck, 1993), an RMSEA that exceeds 0.05 cannot be taken as an indication of good fit. On the other hand, the statistic is equal to 546.88 on 351° of freedom. While it remains significant at the 0.05 level, the RMSEA index for the bifactor model is equal to 0.03 and the 90% confidence interval is (0, 0.035), indicating substantially improved fit.We have shown that for many applications of IRT, multidimensionality rather than unidimensionality should represent the null hypothesis. There are a variety of limitedinformation and fullinformation methods for determining the goodness of fit and the underlying dimensionality of a particular test. As it turns out, the bifactor model produces excellent results for a variety of different IRT applications because it (a) uses expert judgement to define the underlying factor structure and (b) evaluates the likelihood in an always computationally tractable way because it reduces it to a twodimensional integral, which is relatively easy to evaluate in practice. Traditional methods based on eigenvalues have a tendency to identify factors that are not indicative of the underlying trait of interest. By contrast, goodnessoffit statistics which compare various nested and nonnested statistical models can be used efficiently to evaluate the dimensionality of a particular test. In general, unrestricted item factor analysis should not be used to evaluate multidimensionality. It is poorly specified and is subject to considerable rotational variance leading to a plethora of different conclusions regarding the latest variables. This is not the case for the bifactor model which provides essentially the same answer regardless of small changes in the model specification. Finally, one should be extremely cautious regarding the fitting of a unidimensional model to what are inherently multidimensional data. The net result is an underestimate of the point at which adaptive testing should terminate (i.e., underestimates of the posterior variance of the latent variable estimate), and increases in the empirical variance of the resulting test score. Neither of these conditions is good. As a consequence, the best possible approaches to determining dimensionality should always be used.