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Establishing Measurement Invariance: English and Spanish Paediatric Asthma Quality of Life Questionnaire

Sousa, Karen H.; West, Stephen G.; Moser, Stephanie E.; Harris, Judy A.; Cook, Susanne W.

doi: 10.1097/NNR.0b013e3182544750

Background: Registered nurses and nurse researchers often use questionnaires to measure patient outcomes. When questionnaires or other multiple-item instruments have been developed using a relatively homogeneous sample, the suitability of even a psychometrically well-developed instrument for the new population comes into question. Bias or lack of equivalence can be introduced into instruments through differences in perceptions of the meaning of the measured items, constructs, or both in the two groups.

Objective: To explain measurement invariance and illustrate how it can be tested using the English and Spanish versions of the Paediatric Asthma Quality of Life Questionnaire (PAQLQ).

Methods: A sample of 607 children from the Phoenix Children’s Hospital Breathmobile was selected for this analysis. The children were of ages 6–18 years; 61.2% completed the PAQLQ in Spanish. Testing measurement invariance using multiple-group confirmatory factor analysis, a series of hierarchical nested models, is demonstrated. In assessing the adequacy of the fit of each model at each stage, both χ2 tests and goodness-of-fit indexes were used.

Results: The test of measurement invariance for the one-factor model showed that the English and Spanish versions of the scale met the criteria for measurement invariance. The level of strict invariance (equal factor loadings, intercepts, and residual variances between groups) was achieved.

Discussion: Confirmatory factor analysis is used to evaluate the structural integrity of a measurement instrument; multiple confirmatory factor analyses are used to assess measurement invariance across different groups and to stamp the data as valid or invalid. The PAQLQ, a widely used instrument having evidence to support reliability and validity was used separately in English- and Spanish-speaking groups. Traditional methods for evaluating measurement instruments have been less than thorough, and this article demonstrates a well-developed approach, allowing for confident comparisons between populations.

Karen H. Sousa, PhD, RN, FAAN, is Professor and Associate Dean for Research and Extramural Affairs, College of Nursing, University of Colorado, Aurora.

Stephen G. West, PhD, is Professor, Department of Psychology, Arizona State University, Tempe.

Stephanie E. Moser, PhD, is Data Analyst, Department of Veterans Affairs Ann Arbor Healthcare System, Michigan.

Judy A. Harris, MS, RN, CPNP, is Director, Breathmobile, Phoenix Children’s Hospital, Arizona.

Susanne W. Cook, PhD, RN, is a retired Pediatric Asthma Consultant.

Accepted for publication March 5, 2012.

This study was supported partially by funds from the National Institute of Nursing Research (NIH 1R15NR010632-010).

The authors have no conflicts of interest to disclose.

Corresponding author: Karen H. Sousa, RN, PhD, FAAN, College ofNursing, University of Colorado Denver, 13120 E 19th Ave., Rm. 4320, Mailstop C288-04, Aurora, CO 80045 (e-mail:

Registered nurses and nurse researchers often use questionnaires to measure patient outcomes. When questionnaires or other multiple-item instruments have been developed using a relatively homogeneous sample, the suitability of even a psychometrically well-developed instrument for the new population comes into question (Corless, Nicholas, & Nokes, 2001; Staniszewska, Ahmed, & Jenkinson, 1999). This is problematic for any research involving comparisons among different groups. The questionnaire then has limited utility for providing interpretable findings both within the new population group (e.g., different ethnic, gender, or language groups) and for comparisons between the original and the new populations. The performance of the questionnaire in each subgroup must be considered. Introducing measurement invariance to validity discussions in nursing research can eliminate this limitation.

Two psychometric methods have been developed for establishing measurement invariance between different population groups: item response theory (Embretson & Reise, 2000) and confirmatory factor analysis (CFA; Meredith, 1993; Millsap, 2011; Widaman & Reise, 1997); the focus here is on methods of establishing measurement invariance using CFA. Although both approaches are useful, CFA is adapted more easily to instruments measuring multidimensional constructs; it can be applied also to questionnaires having binary, ordered category, or continuous response formats (Bollen, 1989; Muthén, 1984). Available structural equation modeling software (e.g., AMOS, EQS, LISREL, and Mplus) permits estimation of the multiple-group CFA models needed to study measurement invariance.

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Juniper et al. (1996) developed the Paediatric Asthma Quality of Life Questionnaire (PAQLQ) as a conceptual model and measurement scale to measure quality of life in children living with asthma. They proposed that quality of life for a child living with asthma has three dimensions: symptoms, emotional function, and activity limitations. Decisions related to treatment, health resource allocations, and interventions are often based on PAQLQ-generated data. The PAQLQ has been translated into over 20 languages including Spanish. However, little is known about the measurement quality of the PAQLQ and the items as measured across groups because careful studies of the questionnaire’s measurement invariance have not been undertaken. Juniper and colleagues at the Mapi Research Institute, Lyon, France (see used an extensive forward and backward translation procedure to develop the Spanish for Mexico version. They asserted that their version was equivalent to the original and that it could be used in studies of Spanish-speaking children. However, they did not suggest that it was appropriate to use the Spanish version for between-group comparisons with children answering the English version.

Many questionnaires, including the PAQLQ, could be conceptually problematic when used to compare population groups (Johnson & Wolinsky, 1994; Markides et al., 1996; Mutran, Reed, & Sudha, 2001; Stewart & Napoles-Springer, 2000). Testing instruments for measurement invariance becomes a critical issue if outcomes are to be compared between groups, in this example, English- and Spanish-speaking children living with asthma. Clearly, confidence in the validity of the measurement instrument is critical when investing confidence in the fidelity of the between-group measure yielded by that instrument (Tann, Sousa, & Kwok, 2005). The PAQLQ influences the theory and practice of healthcare; however, the theory and practice are built upon an instrument whose validity has been tested insufficiently, resulting in potential measurement invariance.

Instruments must be designed to yield quantitative and replicable findings, specifically between cultural and language groups (Flaherty et al., 1988). Stewart and Napoles-Springer (2000) stated that a valid comparison of self-reported measures like the PAQLQ requires that the constructs have a similar meaning across groups. This semantic equivalence (identical meaning of each item after translation) is difficult to achieve and evaluate (Flaherty et al., 1988). The International Test Commission has emphasized that measurement equivalence cannot be assumed across language and cultural groups (Hambleton, Merenda, & Spielberger, 2005; Tanzer & Sim, 1999). “Test developers/publishers should apply appropriate statistical techniques to (a) establish the equivalence of the different versions of the test or instrument, and (b) identify problematic components or aspects of the test or instrument which may be inadequate to one or more of the intended populations” (International Test Commission, 2010, Guideline D7). Invariance analysis is used to address this issue.

Bias or lack of equivalence can be introduced into instruments through group differences in perceptions of the meaning of the measured items, differences in the constructs, or both. These differences in perception can result from group differences in the cognitive processes of answering items and use of response scales or from item differences resulting from inadequate translations (Hahn & Cella, 2003; Rogler, 1989). Observed lower levels of self-rated health in Latinos compared with other ethnic groups (Marin & Marin, 1991; Osmond, Vranizan, Schillinger, Stewart, & Bindman, 1996; Shetterly, Baxter, Mason, & Hamman, 1996) may be a function of a response pattern specific to Latinos (e.g., comparing one’s health to that of others rather than viewing health as a personal state). Measures derived from instruments that do not take into account differences in processing between differing groups may not reflect real differences between those groups; this data-related uncertainty is antithetical to the translation of healthcare research into healthcare practice (Johnson et al., 1996).

In traditional two-group comparisons, there is a desire to draw conclusions as to how an intervention may affect each of the separate groups. Group comparisons of the mean scale scores of the measured items are justified to the extent that psychometric work indicates these comparisons approximate comparisons of means on the theoretical true score for the construct. Unambiguous interpretation of observed mean differences is dependent on the between-group equivalence of the underlying measurement model, and a comparison between different population groups—even with instruments that have been shown to have adequate reliability and validity within each group—requires making several traditionally untested assumptions described below (Flaherty et al., 1988; Hui & Triandis, 1985; Vandenberg & Lance, 2000).

If a measure is used to assess one thing in one group and something different in another group, comparing group means is like comparing apples to oranges (Chen, 2008). Differences in scores between groups cannot be used to infer group differences in the theoretical attribute unless the measured items accord with a particular set of invariance restrictions (Borsboom, 2006); if evidence supporting a measure’s invariance is lacking, conclusions are at best ambiguous and at worst erroneous (Steenkamp & Baumgartner, 1998). Measurement invariance analysis provides a tool for developing psychometrically appropriate instruments for between-population-group comparisons.

With continuous variables, the most frequently used technique for testing measurement invariance is multiple-group CFA. The early statistical developments of this technique (e.g., Alwin & Jackson, 1981; Jöreskog, 1971) and the applications that followed were limited to the comparison of covariance structures; this work permitted researchers to establish that the theoretical constructs and the units in which they were measured were equivalent in the two populations, important for correlational studies. In later research (Meredith, 1993; Widaman & Reise, 1997), the technique was developed further, permitting the comparison of mean structures between the groups by establishing that the origin (0-point) of the theoretical constructs were equivalent in the two populations.

Meredith (1993), Widaman and Reise (1997), and Millsap (2011) developed a set of hierarchical tests to identify the extent to which measurement invariance can be established in the two populations. The most basic level of measurement invariance is configural invariance (Horn, McArdle, & Mason, 1983). The central requirement is that the same item must be an indicator of the same latent factor in each group; however, the magnitude of the factor loadings can differ across groups. When this level of invariance is achieved, roughly similar, but not identical, latent variables are present in the groups (Widaman & Reise, 1997), precluding clear comparisons between groups.

The second level of invariance is factor loading invariance (weak invariance). Factor loadings represent the strength of the linear relation between each factor and each of the associated items (Bollen, 1989; Jöreskog & Sörbom, 1999). When the loading of each item on the underlying factor is equal in two (or more) groups, the unit of measurement of the underlying factor is identical. This level of invariance does not require that the scales of the factors have a common origin. When factor loading invariance is met, relations between the factors can be compared across groups; one unit of change in one group is equal to one unit of change in another. However, the factor means of the scale still cannot be compared across groups, because the origin of the scale may differ.

The third level of invariance is intercept invariance (strong invariance). Intercepts represent the origin of the scale. In testing this form of invariance, each of the intercepts of the measured variables, in addition to factor loadings of the latent variables, is constrained to be equal across groups. This level of invariance is required for comparing latent means across groups (Millsap, 2011; Widaman & Reise, 1997). When this level of invariance is achieved, the scores from different groups have the same unit of measurement (factor loading) as well as the same origin (intercept). Without this level of invariance, it cannot be determined whether any difference between groups on factor means is a true group difference or a measurement artifact.

The fourth form of invariance is residual invariance (strict invariance). In testing this form of invariance, the residual (uniqueness or measurement error) associated with each measured variable, in addition to the factor loadings of the latent variables and the intercepts of measured variables, is constrained to be equal across groups. When this level of invariance holds, all group differences on the items are due solely to group differences on the common factors.

In summary, configural, factor loading, intercept, and residual invariance are the most commonly tested forms of measurement invariance. Modifications of the testing procedure for basic measurement invariance have been proposed to address instruments having more complex factor structures (e.g., second-order factor models; Chen, West, & Sousa, 2006) or binary or ordinal response formats (e.g., Muthén & Christofferson, 1981).

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The goal of this study was to explain measurement invariance and illustrate how it can be tested using the English and Spanish language versions of the PAQLQ. These questionnaires have been used widely with pediatric asthma patients and have shown evidence of reliability and validity separately in both English and Spanish populations (Juniper et al., 1996; see Initially, the measurement structure of each of the scales was studied, and then analyses were conducted to establish the extent to which the same constructs are being assessed in the English and Spanish language forms of the instrument. Although different language groups were used, these procedures can be used to assess equivalence of constructs across age, gender, and ethnic groups.

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Since January 2000, the pulmonary division of Phoenix Children’s Hospital has operated a mobile asthma clinic, the Phoenix Children’s Hospital Breathmobile, for medically underserved, inner-city school age children. The program provides asthma case detection, education, and interventional treatment programs. The population consists of children at 5–18 years old with asthma symptoms, identified through a case detection process. A pediatric nurse practitioner, a respiratory therapist, an eligibility worker, and a bilingual pulmonary nurse staff the Breathmobile. All children referred are given the PAQLQ on their first visit. Depending on the primary language of the child, the PAQLQ is administered during the face-to-face interviews in either English or Spanish, following the protocol originally developed by Juniper (see

For this illustration, medical records for all the children seen by the Breathmobile were abstracted. A sample of 607 primarily Hispanic children (of Mexican origin), with a completed PAQLQ, were used. The children’s ages ranged from 6 to 18 years (M = 10.85 years, SD = 2.14 years), and 51.4% of the sample were boys. Children from other ethnic backgrounds were excluded in the sample to permit valid group comparisons. Kirkman-Liff and Mondragón (1991) investigated language of interview in a large sample of Hispanics in the Southwest. They found that there were significant differences between Hispanics interviewed in English with those interviewed in Spanish on several health status items. Therefore, to demonstrate measurement invariance, we were confident that our group distinction was appropriate. A total of 61.2% of the Hispanic children completed the questionnaire in Spanish, and 38.8% completed the questionnaire in English.

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The English language version of the PAQLQ and its Spanish language version were made available to each of the children. Children were interviewed face-to-face in their preferred language based on the pediatric nurse practitioner’s assessment. Norris et al. (1996) supported Marin and Marin’s (1991) assumption that the language factor alone provides a valid and reliable indicator of acculturation. Both the English and Spanish versions of the scale have 23 items used to assess three domains: symptoms (10 items), emotional function (8 items), and activity limitation (5 items). Three of the five activity items potentially reflect different respondent-generated activities that are not consistent across children. For these items, each child is asked to nominate three important activities from a list of 35 potential activities (e.g., running and shopping) that are performed commonly, are important to the child, and are limiting because of the threat of exacerbating their asthma: “How much have you been bothered by your asthma in [the activity] during the past week?” For ease of presenting this example, these items were excluded initially in the analysis because of their unique structure and the necessity to make the strong assumption that the nature of the activity is comparable both within and across children (i.e., Respondent 1’s response to Item 1 “running” is equivalent to Respondent 2’s response to Item 1 “shopping”). Questions 5, 6, 7, 9, 11, 13, 15, 16, 17 18, 19, 20, 21, and 23 had a 7-point response format anchored by 1 = all of the time and 7 = none of the time. For Questions 8, 10, 12, 14, and 22, the anchors were 1 = extremely bothered and 7 = not bothered (Table 1). The validity tests to which the PAQLQ has been subjected so far have shown it to have adequate internal consistency and predictive validity (Clarke et al., 1999; Juniper et al., 1996; Raat et al., 2005; Tauler et al., 2001).



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Procedure and Statistical Analysis

Tests for the structure of the overall PAQLQ, for its measurement invariance between English- and Spanish-speaking children, and for the latent mean differences were based on the analysis of mean and covariance structures using multiple-group CFA. All CFAs were estimated using Mplus 5.2 (Múthén & Múthén, 2010).

Testing measurement invariance entails testing a series of hierarchically nested models; each pair of models in the sequence is nested because a set of parameters is constrained to be equal across groups in the more restricted, but not in the less restricted, model. For example, in the configural invariance model, no constraints are placed on the values of the hypothesized factor loadings across groups, whereas the factor loading invariance model constrains the corresponding factor loadings to be equal in each group.

In assessing the adequacy of the fit of each model, both χ2 tests and goodness-of-fit indexes were used. The χ2 test assesses the magnitude of the discrepancy between the sample and fitted covariance matrices and the sample and fitted mean vectors. A significant test result indicates that the fit is poor and that the model may not be appropriate for the data. However, moderate discrepancies from normality in the data also lead to rejection of the model using the χ2 test (West, Finch, & Curran, 1995). Furthermore, when the sample size is large, a small discrepancy that may be of no practical or theoretical interest can lead to similar rejection. Consequently, the χ2 test of fit was supplemented with three fit indexes that showed good performance in a simulation study by Hu and Bentler (1998). The root mean square error of approximation (RMSEA; Steiger, 1990) is a measure of the estimated discrepancy between the population and model implied population covariance matrices per degree of freedom. Browne and Cudeck (1993) suggested that values of the RMSEA of .05 or less indicate a close fit and .08 or less indicate adequate fit. The standardized root mean square residual (SRMR; Hu & Bentler, 1998) is a measure of the average of the standardized fitted residuals. It ranges from 0 to 1.00, and a value of less than .08 is conventionally taken as an indication of adequate fit. The comparative fit index (CFI; Bentler, 1990) ranges from 0 (poor fit) to 1.00 (perfect fit) and is derived from a comparison of a restricted model in which restrictions are imposed on the data with a baseline model in which all pairs of observed variables are assumed to be mutually uncorrelated. Hu and Bentler (1999) suggested the use of .95 as a criterion for adequate fit.

To compare the fit for the two nested models representing different levels in the hierarchy of measurement invariance, the likelihood ratio test (the χ2 difference test; Bentler & Bonett, 1980) was used. If significant, the likelihood ratio test (the difference between the χ2 statistics associated with the more and less restricted models) suggests that the constraints on the more restricted model may be too strict. Large sample sizes can lead to the detection of significant differences between two models of a tiny magnitude that is of no practical importance. On the basis of a simulation study, Cheung and Rensvold (2002) concluded that a difference greater than .01 in the CFI would indicate a meaningful change in model fit.

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Latent Mean Difference Test

Once measurement invariance is established, multigroup CFA may be used to test whether the latent factor means differ across the groups. To test the latent construct mean differences, a combined mean and covariance structure model must be used (Bentler, 1989; Bollen, 1989; Sörbom, 1978). To estimate the difference between the factor means, one group is chosen as a reference or baseline group and the latent means are set to zero. The latent means of the other group, which represent the difference between the factor means in the two groups, are estimated. The significance test (Wald or z test) for the latent means of the second group provides a test for significance of the difference between the means of the two groups on the latent construct (Aiken, Stein, & Bentler, 1994).

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Qualitative Itemmetric Analyses

Because previous work on the structure of the PAQLQ had not been reported, separate itemmetric analysis were run (Angleitner, John, & Löhr, 1986) of the English and Spanish language versions of the PAQLQ. The checks on the Spanish version were conducted by bilingual research assistants from Mexico who represented the local dialects of Spanish in the Southwest United States, where the study took place.

For the English language version, each of the items was hypothesized to measure a different facet of asthma quality of life: symptoms, emotional function, or activity limitations (see Juniper et al., 1996, for list of items and domain assessed). Each pair of items was examined for similarity of wording, identifying items that were potentially more similar than would be expected based on their assessment of a single construct. Two pairs of items (Items 16 and 20 and Items 15 and 19; Table 1) were identified as having potentially high correlated residuals due to their almost identical wording and content.



Two bilingual Spanish speakers of Mexican origin examined each pair of items in the Spanish version for similarity of wording and identified potentially problematic items (see for wording of items). Again, two pairs (Items 16 and 20 and Items 15 and 19) were identified as possibly having correlated residuals due to their almost identical wording and content.

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Quantitative Analyses

Items that were common across respondents were analyzed, and the three participant-nominated items (Items 1–3) were deleted. Several models were tested to assess and establish the overall factor structure of the scale separately in English and Spanish. Also estimated was a two-factor model in which Items 1–3 were allowed to load on a separate factor to determine whether the three self-nominated items (Items 1–3) were related to the 20 items (Items 4–23) that were consistent across respondents. The correlations between the two factors were .292 in the English-speaking group and .147 in the Spanish-speaking group. These results support the decision to discard the three self-nominated items because they were measuring a different factor than the remaining items.

Following Juniper et al. (1996), the scale was hypothesized to consist of three related domains that would be represented by a structure with three correlated factors (Figure 1). The hypothesized factor model was specified as follows: (a) each item would have a nonzero loading on the factor (symptoms, emotional function, and activity limitations) that it was designed to measure and a zero loading on each of the other factors and (b) error terms associated with each item would be uncorrelated. Identification of the model is required for estimation in CFA. A model is identified if there is a unique numerical solution for each of the parameters (Ullman, 2001). In the context of multiple-group CFA, the marker variable strategy is used to identify the scale of the measurement models. One of the factor loadings (marker variable) is set to a value of 1 for each factor. Because there was no theoretical basis for choosing a marker, the first indicator of each construct was used as the marker.



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Both the English and Spanish PAQLQ questionnaires were hypothesized to assess three domains that are represented as three correlated factors. The model depicted in Figure 1, in which each of the items was expected to load on one factor corresponding to the domain, was used as specified in Juniper et al. (1996, p. 46). Each pair of the three factors was allowed to correlate. To preserve the possibility of having different factor structures in the English and Spanish groups, the configural model was tested (Horn & McArdle, 1992), in which factor loadings, correlations between factors, and unique variances could differ between the English- and Spanish-speaking groups. The factor loadings for English- and Spanish-speaking children are found in Table 2. The overall fit of the three-factor model was not acceptable based on the χ2 and fit indexes [χ2(334) = 848.89, p < .001; CFI = .92; RMSEA = .07; SRMR = .04]. Furthermore, the solution was improper: There were correlations greater than 1.0 between the latent factors activity limitation and emotional function and activity limitation and symptoms in both the English- and Spanish-speaking groups. These results suggested that a single general quality-of-life factor may be sufficient to account for the data. No additional reliable information appears to be represented by the three specific domains.





To probe this observation, the fit of the overall quality-of-life single-factor model was tested (Figure 2). Included were Items 4–23 for both the English and Spanish versions. The overall fit of the model was not adequate in terms of the chi-square and CFI indexes [χ2(340) = 912.05, p < .001; CFI = .91; RMSEA = .07; SRMR = .05]. The factor loadings for the single-factor model of the English and Spanish versions are presented in Table 3.

Observations from the itemmetric analyses were used to develop highly restricted hypotheses that might lead to improvements in the factor structure, proceeding one step at a time in model modification (MacCallum, 1986). The single-factor model was tested first, allowing the residuals of Items 16 and 20 to correlate in both English and Spanish. The fit of the model improved [χ2(338) = 727.12, p < .001; CFI = .94; RMSEA = .06; SRMR = .04] and was significantly better than the fit of the original single-factor model [Δχ2(2) = 184.93, p < .001]. The correlated residuals were significant for both English and Spanish (r = .33, p < .001; r = .30, p < .001, respectively). Next, residuals of Items 15 and 19 were allowed to correlate for both the English and Spanish groups. The fit of the model again improved [χ2(336) = 593.19, p < .001; CFI = .96; RMSEA = .05; SRMR = .04] and was significantly better than the fit of the previous model [Δχ2(2) = 133.93, p < .001]. The correlated residuals were significant for both English and Spanish (r = .17, p < .001; r = .28, p < .001, respectively). Item 15 represented the emotional functioning domain, whereas Item 19 represented the activity limitation domain. Because these items loaded on different factors, they represent a likely source of some of the problems in estimating the earlier models noted above.

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Measurement Invariance of the One-Factor Model

The measurement invariance of the final one-factor model from the CFA was tested with the correlated residuals of Items 16 and 20 and Items 15 and 19 for both the English and Spanish scales. The factor loading of Item 4 was set to 1.0, and the intercept for Item 4 was set equal across the two language groups to identify the model. Step 1 began with the baseline configural model, which allows all the other factor loadings and intercepts as well as the measurement error variances and correlated measurement errors to be estimated freely. The pattern of fixed and free factor loadings was constrained to be the same across groups, but different estimates were allowed for the corresponding parameters in the different groups. The fit of this model was acceptable [χ2(336) = 593.19, p < .001; CFI = .96; RMSEA = .05; SRMR = .04].

In Step 2, the factor loadings were constrained to be equal across the English and Spanish scales. This level of invariance was nested within Model 1. The fit of this model was also acceptable [χ2(355) = 604.53, p < .001; CFI = .96; RMSEA = .05; SRMR = .04] and did not differ from that of the baseline configural model [Δχ2(19) = 11.34, ns]. This result indicates that a one-unit change on the latent quality-of-life factor is not associated with differential change on any corresponding items on the English and Spanish questionnaires. The factor loadings were invariant across the two groups.

In Step 3, the intercepts were constrained to be equal across groups. This condition is required to detect potential differences in the intercepts of the measured variable between the groups. The fit of this model was also acceptable [χ2(374) = 621.47, p < .001; CFI = .96; RMSEA = .05; SRMR = .04] and did not differ from that of the previous model [Δχ2(19) = 16.94, ns]. The result indicates that the zero-point on the corresponding latent factor for the English and Spanish items did not differ appreciably and that comparisons of means may be conducted.

In Step 4, the correlated residuals were constrained to be equal across groups. The fit of this model was still acceptable [χ2(376) = 627.14, p < .001; CFI = .96; RMSEA = .05; SRMR = .04] and did not differ significantly from that of the model in Step 3 [Δχ2(2) = 5.67, ns].

In Step 5, the residual variances were constrained to be equal across groups. The fit of this model was acceptable [χ2(396) = 638.83, p < .001; CFI = .96; RMSEA = .05; SRMR = .05] and did not differ significantly from that of the previous model [Δχ2(20) = 11.69, ns]. These results indicate that strict measurement invariance was obtained for the one-factor model with two correlated residuals across the English and Spanish language versions of the PAQLQ (Table 4).



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Equality of Factor Means

After establishing invariance between the English and Spanish language groups, the equality of latent factor means of the two groups was tested. In the preceding model, the latent factor mean for the English group was set to 0 and the latent factor mean for the Spanish group was free to vary. The estimated factor mean for the Spanish group is the mean difference between the latent factors for English and Spanish. The estimated value was .185, p = .054. Thus, although the Spanish-speaking group had a higher factor mean than the English-speaking group, this difference was only marginally statistically significant.

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Confirmatory factor analysis is used to evaluate the structural integrity of a measurement instrument; multigroup CFA is used to assess measurement invariance across different age, cultural, ethnic, or language groups. The PAQLQ, a widely used instrument with evidence supporting its reliability and validity separately in English and Spanish language groups, was used. The hypothesized measurement structure of the PAQLQ had not been systematically tested previously. There were problems with the hypothesized three-factor structure; a one-factor structure with the same two correlated residuals in both language groups to account for wording similarity provided an adequate fit in each language. The test of measurement invariance for the one-factor model showed that the English and Spanish versions of the scale met the criteria for measurement invariance. The level of strict invariance (equal factor loadings, intercepts, and residual variances between groups) was achieved, which indicates that any differences between the two language groups are accounted for by group differences on the common factor. This finding is encouraging, indicating that mean differences in the quality of life construct assessed by the English and Spanish language versions of the PAQLQ may be compared directly. The slightly higher mean level in the Spanish-speaking group relative to the English-speaking group can be interpreted as indicating that the less acculturated Spanish-speaking group perceived a higher quality of life. However, this difference was only marginally significant.

Although strict measurement invariance was established with the PAQLQ with respect to test language, an outcome of strict measurement invariance will often not be the case. When a test of a level of measurement invariance fails, it is only appropriate to interpret the meaning of the scale at the previous level in the hierarchy at which measurement invariance was established.

For example, if the test of the equality of the intercepts failed (Step 3), then the measurement scales have only indicated that scales could be interpreted at the level of weak invariance (Step 2). Given such an outcome, the difference between slopes representing relationships between constructs within each group could be compared, for example, using moderated multiple regression (Cohen, Cohen, West, & Aiken, 2003). Alternatively, Yoon and Millsap (2007) have developed a specification search procedure to identify the noninvariant items when only partial measurement invariance is achieved. Simulation studies have shown that this procedure works well if the number of invariant items is small and group differences and the sample size in each group are large.

In this study, only the most prominent potential source of invariance was investigated: test language. Space limitations precluded the investigation of invariance across age groups (e.g., young children vs. teenagers) or gender. It is particularly important to investigate potential group differences if there is theory or empirical work to suggest that the constructs may differ in the subgroups and the goal of the research is to compare between group differences. Other limitations are that no formal measurement was used to assess acculturation in the children and the data used for this illustration were collected for routine screening of the children during a visit to the Breathmobile, not for research.

In the absence of measurement invariance procedures, comparisons between different age, cultural, ethnic, gender, language, or other groups are difficult to make. Vandenberg and Lance (2000) note that comparisons between groups still require several assumptions regarding equivalence. These assumptions are examined rarely and, if violated, can render interpretations of between-population-group comparisons on the nonequivalent measures highly suspect (Bollen, 1989; Vandenberg & Lance, 2000). Measurement invariance analysis provides tests of each of these assumptions. Traditional methods for evaluating measurement instruments have been less thorough, and a well-developed approach was presented here to address these concerns. Testing for and assuring measurement invariance is essential for a profession such as nursing, wherein group comparisons are frequently of interest.

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confirmatory factor analysis; measurement bias; patient outcomes

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