Partial versus total invariance

Varying degrees of measurement and structural invariance are possible across groups with respect to any or all of the invariance hypotheses, ranging from the complete absence of invariance to total invariance. Partial invariance exists when some but not all of the parameters being compared are equivalent across groups [25]. Either full or partial measurement invariance is necessary in order to permit interpretable comparisons of factor means across groups.

Configural invariance

An initial omnibus test of measurement invariance often entails a comparison of the covariance matrix of item variances and covariances between groups. However, numerous statistical analysts [24, 26] have recommended against this overall test of equality because excellent multigroup fit in one part of the measurement model may mask departures from invariance in other parts of the model and produce Type II errors concerning overall group differences.

For this reason, focused tests of invariance typically begin by assessing the issue of equal factorial form or configural invariance-that is, whether the same factors and patterns of loadings are appropriate for both groups [23, 27]. Configural invariance is assessed by determining whether the same congeneric measurement model provides a reasonable goodness-of-fit to each group's data [28]. Thus, whereas the tests of other forms of invariance are based on estimated p values associated with inferential null-hypothesis testing, the test of configural invariance is merely descriptive.

Metric invariance

Given configural invariance, more rigorous tests are conducted concerning first the hypothesis of equal factor loadings across groups, or metric invariance [23, 27, 29]. Also known as weak invariance [27], the issue here concerns the degree to which a one-unit change in the underlying factor is associated with a comparable change in measurement units for the same given item in each group. Items that have different factor loadings across groups represent instances of "non-uniform" differential item functioning [30, 31]. Numerous theorists [23, 32, 33] have argued that between-group equivalence in the magnitude of factor loadings is necessary in order to conclude that the underlying constructs have the same meaning across groups.

Scalar invariance

Given some degree of metric invariance, a third form of measurement equivalence concerns scalar invariance, or the degree to which the items have the same predicted values across groups when the underlying factor mean is zero [23, 27, 29]. Differences in item intercepts when holding the latent variable mean constant at zero reflect instances of "uniform" differential item functioning [30, 31, 34, 35], and indicate that the particular items yield different mean responses for individuals from different groups who have the same value on the underlying factor. Scalar invariance is tested only for items that show metric invariance [26]. Strong invariance is said to exist when equivalent form (configural invariance), equivalent loadings (metric invariance), and equivalent item intercepts (scalar invariance) are all found across groups [27].

Equivalence of factor variances and covariances

An additional test of structural invariance concerns the degree to which the underlying factors have the same amount of variance and covary to the same extent across groups. Although this form of invariance is unnecessary for interpretable between-group comparisons of factor means [24], the equivalence of factor variances indicates that the particular groups being compared report a comparable range of values with respect to the underlying measurement constructs; and the equivalence of factor covariances indicates that the underlying constructs interrelate to a comparable degree in each group.

Invariance of item unique error variances

A second test of structural invariance concerns the degree to which the underlying factors produce the same amount of unexplained variance in the items across groups. Although this form of invariance is not a technical requirement for valid between-group comparisons of factor means [32, 34], the invariance of unique errors indicates that the levels of measurement error in item responses are equivalent across groups. Strict invariance is said to exist when configural invariance, metric invariance, scalar invariance, and invariance in unique errors are all found across groups [27].

Equivalence of factor means

A final test of structural invariance concerns whether the multiple groups have equivalent means on each underlying factor in the measurement model. The primary advantages of using CFA to compare latent variable means across groups, as opposed to comparing group means on composite indices of unit-weighted summary scores via t tests or ANOVAs, are that CFA allows researchers to: (a) operationalize constructs in ways that are appropriately invariant or noninvariant across groups; (b) correct mean levels of constructs for attenuation due to item unreliability; and (c) adjust between-group mean differences for differential item reliability across groups.

VETERAN Sample

The VETERAN sample consisted of baseline data of 314 males between the ages of 28-85 years from a large prospective observational study of Veterans dialyzing at Department of Veterans Affairs (VA) facilities or in the private sector during 2001-2003 [36]. Veterans who had received care at a VA facility within the prior 3 years and were receiving hemodialysis for end-stage renal disease were eligible for enrollment. Patients were excluded if they: 1) had a live kidney donor identified; 2) required skilled nursing facility care; 3) had a life expectancy less than 1 year, determined by a nephrologist; 4) were cognitively impaired; 5) had a severe speech or hearing impairment; 6) were not fluent in English; or 7) had no access to a telephone for follow-up contact.

Participants were recruited from eight VA Medical Centers with outpatient dialysis facilities from 2001 to 2003 and followed for at least six months. Health-related quality of life questionnaires were completed via a phone interview. Institutional review board (IRB) approval was obtained from all VA sites. Coordinators at each site explained the study and obtained written informed consent from patients who were interested in participating.

Non-Veteran Sample

The non-Veteran data are from the first phase of the Dialysis Outcomes and Practice Patterns Study (DOPPS) [37, 38]. The DOPPS is an international, prospective, observational study of the care and outcomes of patients receiving hemodialysis in seven countries including France, Germany, Italy, Japan, Spain, the United Kingdom, and the U.S. A detailed description of DOPPS Phase 1 has been published [37, 38]. Health-related quality of life data was collected by a written questionnaire. In the U.S., 6,609 patients from 142 dialysis facilities completed baseline data between 1996 and 2001. For the present analyses, only males living in the U.S. between the ages of 28 and 85 who had completed quality of life data were included resulting in a sample size of 3,300.

Instruments

Demographic information such as patient age, gender, marital status, race, work status, and educational level were collected using an investigator developed questionnaire.

Kidney Disease Quality of Life

Health-related quality of life was measured with the Kidney Disease Quality of Life Instrument -Short Form (KDQOL-SF). The KDQOL was developed as a self-report, health-related quality of life measurement tool designed specifically for patients with CKD [22]. The 134-item KDQOL was later condensed into the 80-item Kidney Disease Quality of Life Instrument-Short Form (KDQOL-SF) [39]. The questionnaire consists of the generic SF-36 [40] as well as 11 multi-item scales focused on quality of life issues specific to patients with kidney disease (Figure 1). Subscales of the KDCS are (1) symptoms/problems (6 items), (2) effects of kidney disease (4 items), (3) burden of kidney disease (3 items), (4) work status (2 items), (5) cognitive function (3 items), (6) quality of social interaction (3 items), (7) sexual function (2 items), (8) sleep (4 items), (9) social support (2 items), (10) dialysis staff encouragement (2 items), and (11) patient satisfaction. For example, related to the effects of kidney disease, participants are asked how true or false (using a 5 point Likert scale ranging from "definitely true" to "definitely false" the following statements are for them: (1) "My kidney disease interferes too much with my life;" and (2) "Too much of my time is spent dealing with my kidney disease" [22, 39]. All kidney disease subscales are scored on a 0 to 100 scale, with higher numbers representing better HRQOL. The 11 kidney disease-specific subscales can be averaged to form the Kidney Disease Component Summary (KDCS) [21, 41–44]. The KDQOL-SF has been widely used in several studies of patients with kidney disease, including the ongoing, international DOPPS [21, 45–50], and has demonstrated good test-retest reliability on most dimensions [2, 22]. Published reliability statistics for all subscales range from 0.68 to 0.94 with the subscale of social interaction (0.68) being the only subscale with an internal consistency reliability of less than the recommended 0.70 [22].

Data Analysis

Missing values occurred between 1% and 10% for all items except for sexual function which was missing greater than 50% of data. The Veteran data set contained less missing data than the DOPPS data set (between 0 to 5% for the Veterans versus 6% to 10% missing data for the DOPPS data set). This difference may have been attributed to the Veteran data being collected over the telephone whereas DOPPS data were collected via written questionnaire. Because of the large amounts of missing data from both the VETERANS and DOPPs samples for the sexual function subscale, sexual function was not included in the calculation of the KDCS. For all other items, missing data were replaced for the KDQOL-SF variables using the SAS 9.2 (Cary, NC) multiple imputation procedure [51]. The multiple imputation procedure consisted of using a regression model fitted for each variable with missing data with 3 imputed data sets [52]. A one-factor confirmatory factor analysis of the KDCS demonstrated weak factor loadings of the subscales of work status, patient satisfaction and dialysis staff encouragement, suggesting that these three subscales measure something other than HRQOL. These findings are consistent with CFA findings from a previous study [53]. Therefore, the 7-subcale KDCS comprising the subscales measuring symptoms, effects of kidney disease on daily life, quality of social interaction, burden of kidney disease, cognitive function, support, and sleep was used for analyses in this study. Descriptive statistics (mean, range, standard deviation) were calculated using SAS (Cary, NC).

Assessing model fit

We used four different statistical criteria to judge the goodness-of-fit of the hypothesized two-factor CFA model. As measures of absolute fit, we examined the root mean square error of approximation (RMSEA) and the standardized root mean residual (SRMR). RMSEA reflects the size of the residuals that result when using the model to predict the data, adjusting for model complexity, with smaller values indicating better fit. According to Browne and Cudeck [61], RMSEA < .05 represents "close fit," RMSEA between .05 and .08 represents "reasonably close fit," and RMSEA > .10 represents "an unacceptable model." SRMR reflects the average standardized absolute value of the difference between the observed covariance matrix elements and the covariance matrix elements implied by the given model, with smaller values indicating better fit. Hu and Bentler [62] suggested that SRMR < .08 represents acceptable model fit. As measures of relative fit, we used the non-normed fit index (NNFI) and the comparative fit index (CFI). NNFI and CFI indicate how much better the given model fits the data relative to a "null" model that assumes sampling error alone explains the covariation observed among items (i.e., no common variance exists among measured variables). Bentler and Bonett [63] recommended that measurement models have NNFI and CFI > .90. More recently, Hu and Bentler [64] suggested that relative fit indices above 0.95 indicate acceptable model. However, Marsh et al., [65] have strongly cautioned researchers against accepting Hu and Bentler's (1999) [64] more stringent criterion for goodness-of-fit indices, and have provided a strong conceptual and statistical rationale for retaining Bentler and Bonett's [63] long-standing criterion for judging the acceptability of goodness-of-fit indices. Therefore, following Marsh et al.'s [65] recommendation, we have adopted Bentler and Bonett's [63] criterion of relative fit indices > .90 as reflective of acceptable model fit.

Assessing invariance

We followed Vandenberg and Lance's [23] recommended sequence for conducting tests of measurement invariance. Given an acceptable fit for the hypothesized two-factor CFA model in each group (i.e., configural invariance), we tested five different hypotheses about measurement invariance between the VETERAN and DOPPS samples. These structural hypotheses concerned between-group differences (versus equivalence) in: (a) the magnitude of the factor loadings (metric invariance); (b) the intercepts of the measured subscales (scalar invariance); (c) the variances and covariance of the KDCS and SF-36 factors; (d) the unique error variances of the measured subscales; and (e) the latent means of the KDCS and SF-36Q factors.

We used the difference in chi-square values and degrees of freedom, i.e., the likelihood ratio test [66], to test hypotheses about differences in goodness-of-fit between nested CFA models. Because the goodness-of-fit chi-square is inflated by large sample size [66], we also examined differences in CFI across nested models, with difference in the CFI (ΔCFI) ≤ .01 considered evidence of measurement invariance [67]. In addition, we computed the effect size for each probability-based test of invariance expressed in terms of w², or the ratio of chi-square divided by N [68], which is analogous to R-squared (i.e., the proportion of explained variance) in multiple regression. Cohen [68] suggested that w² ≤ 0.01 is small, w² = 0.09 is medium, and w² ≥ 0.25 is large.

In testing invariance hypotheses, there is disagreement in the literature about whether researchers should test invariance hypotheses globally across all relevant parameters simultaneously (e.g., a single test of whether all factor loadings show between-group invariance) versus test invariance hypotheses separately across relevant sets of parameters (e.g., separate tests of the equivalence of factor loadings for each factor). Although omnibus tests of parameter equivalence reduce Type I errors by decreasing the number of statistical tests when the null hypothesis is true, Bontempo and Hofer [24] have suggested that perfectly invariant factors can obscure noninvariant factors and make multivariate global tests of invariance misleading. For this reason, we chose to examine the between-group equivalence of factor loadings, item intercepts, and unique error variances separately for each factor in our two-factor CFA model.

To further reduce the likelihood of capitalizing on chance, we corrected the Type I error rate for probability-based tests of invariance (see Cribbie) [69], by imposing a sequentially-rejective Bonferroni adjustment to the generalized p value for each statistical test [70]. Specifically, we used a Sidak step-down adjustment procedure [71, 72] to ensure an experimentwise Type I error rate of p < .05, correcting for the number of statistical comparisons made.

In drawing inferences from tests of measurement or structural invariance, we examined four different statistical criteria: (a) the unadjusted p-value associated with the likelihood-ratio test; (b) the sequentially-rejective Bonferroni adjusted p-value associated with the likelihood-ratio test; (c) the difference in CFI values (ΔCFI); and (d) effect size (w²). Research comparing the likelihood-ratio test and ΔCFI as criteria for judging measurement invariance [73] suggests that these two criteria produce highly inconsistent conclusions. Because the likelihood-ratio test is biased against finding invariance when sample sizes are large [67, 73, 74], we expected that likelihood-ratio tests using unadjusted p-values would more often support the rejection of invariance hypotheses relative to the other statistical criteria, given the large sample size for our multigroup analyses (N = 3,614). Because the large number of anticipated invariance tests (i.e., 40-50) will produce a more stringent adjusted p-value, we expected that using Bonferroni-adjusted p values would reduce the bias toward rejecting invariance hypotheses via the likelihood-ratio test.

Configural invariance

In the completely standardized CFA solution, the KDCS and SF-36 factors correlated 0.924 in the VETERAN sample and 0.879 in the DOPPS sample. Although these factor intercorrelations reflect a high degree of overlap between the two HRQOL instruments in both the VETERAN (0.924² = 85% shared variance) and DOPPS (0.879² = 77% shared variance) samples, they also indicate that roughly one-seventh of the variance in each instrument for the VETERAN sample, and one-quarter of the variance in each instrument for the DOPPS sample, has nothing to do with the other instrument. Furthermore, a one-factor model, representing overall HRQOL, fit the combined KDCS and SF-36 data significantly worse than did the two-factor model for both the VETERAN, Δχ²(1, N = 314) = 20.287, p < .0001, and DOPPS samples, Δχ²(1, N = 314) = 524.571, p < .0001; and a one-factor model did not yield an acceptable model fit with respect to RMSEA for either the VETERAN, χ²(90, N = 314) = 352.459, RMSEA = .102, SRMR = .0589, NNFI = .946, CFI = .954, or DOPPS sample, χ²(90, N = 3,300) = 2989.164, RMSEA = .107, SRMR = .0564, NNFI = .942, CFI = .951.

Although the two-factor model fit the data well, we also tested a three-factor model that consisted of a single second-order factor for the KDCS and two second-order factors, representing the physical and mental component summary scores of the SF-36 [40]. The two second-order factors were evaluated by allowing the four physical subscales (PF, RP, BP, & GH) to load on the second-order physical component summary and the four mental health subscales (MH, RE-SF, & VT) to load on the second-order mental health component summary and then estimating these loadings. This three-factor model fit the data of both the DOPPS and VETERAN sample slightly better than the two-factor model. However, the SF-36 physical component summary factor correlated very highly with the SF-36 mental component summary in the CFA solution for both the DOPPS sample (r=.957) and the VETERAN sample (r=.997).

Metric invariance

In the next step, we examined the magnitude of factor loadings or metric invariance. As seen in Table 3 (Model 3), the likelihood-ratio test revealed invariant factor loadings for the SF-36 subscales according to both unadjusted (p < .29) and Bonferroni-adjusted (p = ns) criteria. In addition, the effect size of group differences in loadings on the SF-36 factor was modest (w² = .05), and the change in CFI (ΔCFI = .0002) also suggested invariant SF-36 factor loadings. In contrast, the likelihood-ratio test revealed significant group differences in loadings for the KDCS factor (Model 2) according to both unadjusted (p < .00085) and Bonferroni-adjusted (p < .025) criteria. However, this effect approached only medium size (w² = .08), and the change in CFI (ΔCFI = .0003) suggested that the VETERAN and DOPPS samples had equivalent loadings on the KDCS factor.

Tests of the invariance of factor loadings for each of the non-referent KDCS subscales revealed statistically significant differences in loadings for two subscales (Sleep and Social Support) using the unadjusted criterion (p < .0069), but for only the Sleep subscale using the adjusted criterion (p < .0036; see Table 3, Model 8). All six tests of invariance in KDCS subscale factor loadings produced modest effect sizes (w²s ≤ .06), and all ΔCFIs were within the recommended 0.01 threshold for inferring invariance (ΔCFIs ≤ .0005).

Adopting the most conservative criterion for assessing invariance (i.e., unadjusted p-value), we thus sought to establish a partially metric invariant measurement model that constrained the factor loadings for all seven non-referent SFQ subscales and four of the six non-referent KDCS subscales (all except the Sleep and Social Support subscales) to be invariant across the VETERAN and DOPPS samples. This partially metric invariant model fit the data well and provided an equivalent goodness-of-fit compared to the initial unconstrained baseline model, Δ(11, N = 3,614) = 14.342, unadjusted p < .22, Bonferroni-adjusted p = ns, ΔCFI = .0003, w² = .06 (see Model 10, Table 3). These results support the conclusion that the VETERAN and DOPPS samples used the SF-36 subscales in largely equivalent ways to define the subjective quality of their lives (full metric equivalence). Thus, quality of life, as measured by the KDCS and SF-36, has mostly the same meaning for the VETERAN and DOPPS samples (weak invariance).

Scalar invariance

As discussed, scalar invariance is the magnitude of item intercepts. According to the likelihood-ratio test (unadjusted p < .000001, Bonferroni-adjusted p < .00005), an omnibus test of scalar invariance suggested that the five metric-invariant KDCS subscales (including the referent subscale) had different intercepts for the VETERAN and DOPPS samples (see Table 3, Model 11). Although the size of this overall effect was between medium and large (w² = .15), the difference in CFI again did not reach the 0.01 threshold (ΔCFI = .0019). As seen in Table 3, individual follow-up tests of scalar invariance revealed that only the KDCS Burden subscale (Model 13) had equivalent intercepts for the two groups, as reflected in a nonsignificant likelihood-ratio test (unadjusted p < .14, Bonferroni-adjusted p = ns), ΔCFI <0.01, and a small effect size (w² = .03).

Likewise, an omnibus test of scalar invariance suggested that the eight metric-invariant SF-36 subscales (including the referent subscale) had different intercepts for the VETERAN and DOPPS samples, according to the likelihood-ratio test (unadjusted p < .000001, Bonferroni-adjusted p < .00005). Although the size of this overall effect was relatively large (w² = .21), the difference in CFI again did not reach the 0.01 threshold (ΔCFI = .004). As seen in Table 3 (Models 18 thru 25), individual follow-up tests of scalar invariance revealed that four of the SF-36 subscales (PF, BP, GH, & VT) had equivalent intercepts for the two groups, as reflected in nonsignificant Bonferroni-adjusted likelihood-ratio tests, ΔCFIs < 0.01, and small effect sizes (w²s ≤ .05).

Strong invariance

Thus, supporting partial scalar invariance (magnitude of item intercepts), half of the eight SF-36 subscales, but only one of the seven KDCS subscales, showed evidence of scalar invariance for the VETERAN and DOPPS groups. These results suggest that strong invariance (i.e., configural, metric, and scalar combined) exists in partial (50%) form for the SF-36, but only minimal form (one subscale) for the KDCS. Inspection of CFA solutions revealed that subscales with nonequivalent intercepts had higher values for the VETERAN sample relative to the DOPPS sample.

Invariance of factor variances and covariance

As seen in Table 3 (Model 26), all four statistical criteria suggested that the variances and covariance of the KDCS and SF-36 factors were equivalent for the VETERAN and DOPPS samples. These results indicate the two groups used an equivalent range of the latent construct continuum in responding to the subscales for each instrument [23]. Because both the factor variances and the covariance are invariant, the correlation between the factors is also invariant [26].

The finding of equal factor variances across groups is important for at least two reasons. First, the invariance of factor variances is a precondition for using the comparison of item unique variances as a test of equal subscale reliabilities across groups [23, 75]. Second, because factor variances are equal across groups, regression coefficients obtained when predicting the factors from other constructs are not biased by differential range restriction across groups [24].

Invariance of item unique error variances

As seen in Table 3 (Model 27), the omnibus hypothesis of invariance in unique error variances for the KDCS subscales (w² = .12) was rejected according to the likelihood-ratio test (adjusted p < .000001, Bonferroni-adjusted p < .00005), but not according to the difference in CFI values (ΔCFI = .0007). Individual follow-up tests (w²s ≤ .05) revealed that five of the KDCS subscales-Social Interaction, Cognitive Functioning, Symptoms, Effects, and Social Support-had invariant unique error variances according to the Bonferroni-adjusted likelihood-ratio test (all p s = ns) and the difference in CFI (all ΔCFIs < .001). These results suggest that five (71%) of the seven KDCS subscales were equally reliable for the VETERAN and DOPPS samples. Both of the KDCS subscales (Burden and Sleep) that showed evidence of differential reliability had greater unique error variance for the VETERAN sample than for the DOPPS sample in the partially invariant CFA model.

Also seen in Table 3 (Model 28), the omnibus hypothesis of invariance in unique error variances for the SF-36 subscales (w² = .09) was rejected according to the unadjusted likelihood-ratio test (p < .0025), but not according to the Bonferroni-adjusted likelihood-ratio test (p = ns) or the difference in CFI (all ΔCFIs < .0001). Individual follow-up tests (w²s ≤ .05) revealed that five of the SF-36 subscales-PF, RP, BP, GH, and MH-had invariant unique error variances according to the likelihood-ratio test (all unadjusted p s > .11, all Bonferroni-adjusted p s = ns) and the difference in CFI (all ΔCFIs < .001). In addition, the RE subscale (w² = .09) and SF subscale (w² = .04) of the SF-36 had invariant unique error variances according to the Bonferroni-adjusted likelihood ratio test (p s = ns), but not according the unadjusted likelihood-ratio test (p s < .035) or the difference in CFI (ΔCFIs < .0012). And the VT subscale of the SF-36 had a different unique error variance for the two groups according to the likelihood-ratio test (unadjusted p < .0017, Bonferroni-adjusted p < .045), but had a small ΔCFI (.0002) and a small effect size (w² = .05). These results suggest that at least seven (88%) of the eight SF-36 subscales were equally reliable for the VETERAN and DOPPS samples. The one SF-36 subscale that showed evidence of differential reliability (VT) had greater unique error variance for the VETERAN sample than for the DOPPS sample.

Invariance in factor means

As a final step in our invariance analyses, we tested for group differences in latent means for the KDCS and SF-36 factors, using a multigroup CFA model that specified partially invariant factor loadings (all loadings invariant except for the Sleep and Social 2 subscales of the KDCS), fully invariant factor variances and covariances, and partially invariant unique error variances (all unique error variances invariant except for the Burden and Sleep subscales of the KDCS and the VT subscale of the SF-36). Because of problems of under-identification, item intercepts and factor means cannot both be estimated in the same model [76] nor can the factor means of both groups be estimated simultaneously [28]. Accordingly, to test invariance in factor means, we followed standard practice in the structural equation modeling literature by: (a) constraining all item intercepts to be equal across groups; (b) fixing at zero the factor means for the DOPPS group; (c) estimating the factor means for the VETERAN group; and (d) using Wald tests to assess whether the factor means for the VETERAN group were significantly different from zero (i.e., from the factor means for the DOPPS group).

This final partially invariant multigroup CFA model fit the data of the two groups reasonably well. Inspection of the LISREL solution revealed that the latent means were significantly higher for the VETERAN sample, compared to the DOPPS sample, for both the KDCS (Z = 5.253, p < .000001, Cohen's d = 0.18) and SF-36 (Z = 7.240, p < .000001, Cohen's d = 0.23) factors. Thus, given that higher scores reflect higher quality of life, the VETERAN sample reported higher overall levels of subjective life quality on both the general and specific measures of QOL, when controlling for differences in the ways in which the two groups used the subscales to define QOL and for between-group differences in the reliabilities of the subscales.