## Longitudinal invariance analysis of the satisfaction with life scale

### Chia-Huei Wu

a### , Lung Hung Chen

b### , Ying-Mei Tsai

c,*a

Institute of Business and Management, National Chiao Tung University, 4F, No. 147, Sec. 1, Xinguang Road, Wenshan District, Taipei City 116, Taiwan, ROC b

Graduate Institute of Physical Education, National Taiwan Sport University, Taoyuan County, P.O. Box 59182, Taipei 10012, Taiwan, ROC

c_{Ofﬁce of Physical Education, Central Taiwan University of Science and Technology, No. 11, Buzih Lane, Beitun District, Taichung City 40601, Taiwan, ROC}

### a r t i c l e

### i n f o

Article history: Received 8 August 2008

Received in revised form 26 October 2008 Accepted 3 November 2008

Available online 24 December 2008 Keywords:

Satisfaction with life scale Measurement invariance Longitudinal analysis

### a b s t r a c t

This research examined longitudinal measurement invariance in the satisfaction with life scale using two studies. The ﬁrst followed a sample of 236 university students (93 male and 143 female) who completed the SWLS twice over a two-month interval. The second used a sample of 242 adolescent athletes (133 male and 109 female) who completed the SWLS three times over a period of six months. Conﬁrmatory factor analysis was used to examine longitudinal measurement invariance. For the university student sample, results showed that the SWLS is partial strict invariant (equality of factor patterns, loadings and intercepts across time for all items and equality of item uniqueness for Items 1, 2, 4 and 5 across time). For the adolescent athlete student sample, the SWLS is partial strong invariant (equality of factor patterns, loadings across time for all items and equality of intercepts for Items 2, 3, and 4 across time). For both samples, stability coefﬁcients across time were moderately high and latent factor means were not signiﬁcantly different across time. Generally, these results suggest that the SWLS has satisfactory psycho-metric properties for longitudinal measurement invariance.

Ó 2008 Published by Elsevier Ltd.

1. Introduction

Subjective well-being (SWB) has been an important area of re-search in psychology, and can be instrumental in helping to im-prove the lives of individuals. Researchers have put much effort into deﬁning and measuring SWB and have developed indicators to determine why some people have higher SWB than others and how to promote and maintain individuals’ SWB (see Diener, 1984; Diener, Suh, Lucas, & Smith, 1999for a review). To date, two main aspects of SWB have been distinguished: affective and cognitive (Lucas, Diener, & Suh, 1996). The affective aspect of SWB refers to the emotional component whereby levels of positive and negative affect are used to indicate the level of SWB. People who experienced more positive affect than negative affect were re-garded as having higher SWB. The cognitive aspect of SWB refers to a conscious cognitive judgment of life in which individuals com-pare their life circumstances with a self-imposed standard; it is operationalized as life satisfaction (Diener, Emmons, Larsen, & Grifﬁn, 1985). That is, individuals will report high life satisfaction if their perceived life circumstances are in line with their own standard.

In order to measure an individual’s SWB for the cognitive as-pect,Diener et al. (1985)developed the satisfaction with life scale (SWLS). Because different people may have very different ideas about what constitutes a good life, the SWLS was developed to

as-sess satisfaction with one’s life as a whole (Diener et al., 1985). The SWLS has been used extensively since 1985 and has good psycho-metric properties (Pavot & Diener, 1993). Its internal consistency reliability coefﬁcients range from 0.79 to 0.89 (Pavot & Diener, 1993). Test–retest reliability coefﬁcients of the SWLS were 0.83 for a two-week interval and 0.84 for a one-month interval (Pavot & Diener, 1993). The SWLS also demonstrated adequate construct validity, convergent validity and discriminant validity (Arrindell, Heesink, & Feij, 1999; Lucas et al., 1996; Pavot & Diener, 1993; Sachs, 2003). In addition to these basic psychometric properties, the property of measurement invariance across different groups has also been examined for the SWLS in recent years.

Measurement invariance is an important property, since inter-pretation of mean differences may be problematic unless the underlying constructs are the same across groups. Previous studies have discussed measurement invariance of the SWLS for gender, age and culture.Shevlin, Brunsden, and Miles (1998)found that the SWLS has the property of strict measurement invariance (i.e., equality of factor loadings, factor variances, item uniqueness, item intercepts and factor means) across gender in a sample of British university students.Wu and Yao (2006)reported the same ﬁnding for a sample of Taiwanese university students, ﬁnding that male and female groups had the same factor loadings, variance and item uniqueness. However, in a sample of Spanish junior high school students, measurement invariance of the SWLS (Spanish version) was not obtained (Atienza, Balaguer, & Garcia-Merita, 2003). With regard to measurement invariance across ages, Pons, Atienza, Balaguer, and Garcia-Merita (2000) analyzed the measurement

0191-8869/$ - see front matter Ó 2008 Published by Elsevier Ltd. doi:10.1016/j.paid.2008.11.002

*Corresponding author.

E-mail address:fjudragon@yahoo.com.tw(Y.M. Tsai).

Contents lists available atScienceDirect

## Personality and Individual Differences

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / p a i dinvariance of the SWLS (Spanish version) across samples of adoles-cents (aged 11–15 yrs) and the elderly (aged 60–91 yrs). The re-sults indicated that factor loadings and variances were not invariant between the two samples, suggesting that the SWLS is sensitive to age. Recently, Daniel and Petter (2008)tested mea-surement invariance across sex and age at the same time and re-ported similar results. They found that the SWLS is invariant across gender in factor loadings and item intercepts, but not across age. Finally, with regard to cultural comparison, Tucker, Ozer, Lyubomirsky, and Boehm (2006)examined measurement invari-ance in the SWLS across student and community groups from the United States and Russia. They found that the two groups had the same factor loadings when student and community samples were combined. When student and community samples were ana-lyzed separately, the student groups from the US and Russia had the same factor loadings and item slopes, but the community sam-ples did not show invariance.

Although the property of measurement invariance in the SWLS has been investigated, existing studies focus only on measurement invariance across different groups. Longitudinal measurement invariance (i.e., measurement invariance across time) has not been considered for the SWLS. Similar to measurement invariance across different groups, longitudinal measurement invariance analysis examines the equality of factor structure for a measurement, but its focus is on equality across time. Longitudinal measurement invariance is desirable for a measurement because it ensures that the same construct can be assessed across time and provides a so-lid basis for mean comparison. In the existing literature, the SWLS was used in longitudinal studies to assess individuals’ SWB across time in order to examine the possible longitudinal mechanisms be-hind SWB (e.g.,Lai, Bond, & Hui, 2007) or to monitor intervention effects among patients (e.g.,Chan, Ungvari, Shek, & Leung, 2003). However, these studies did not examine whether the SWLS has longitudinal invariance.

Hence, the main purpose of this research is to examine longitu-dinal measurement invariance in the SWLS to determine whether the SWLS has satisfactory properties for longitudinal comparison. There are two studies. The ﬁrst is a university student sample that completed the SWLS two times over a two-month interval. The second is an adolescent sample that completed the SWLS three times over a period of six months (three months between two con-tiguous tests). The test interval of two-months or three-months is selected because participants of both samples are students, choos-ing a two-months or three-months interval for them would not be too short to test the stability and, on the other hand, it can also en-sure that they would not have had a dramatic life change during the study period. The interval of two- or three-months is ideal for the research purpose and the current samples.

For the sake of mean comparison across time, it is desirable for the SWLS to have (at least partial) a strong invariance property (i.e., equality of factor patterns, factor loadings and item intercepts). Thus, this research aims to determine whether the SWLS can sat-isfy the requirements of (partial) strong invariance. In addition, the stability coefﬁcient across time is computed and latent factor means are compared across time.

2. Study 1 2.1. Method

2.1.1. Participants and procedure

Two hundred and thirty-six (93 male and 143 female) under-graduates from Central Taiwan University of Science and Technol-ogy and Nan Kai Institute of TechnolTechnol-ogy voluntarily participated in this study. The research project was announced to students by

lec-tures (they are also researchers in the research team) in class. Stu-dents were told that they can participate in a study according to their willingness and they can obtain extra course credits for their participation. Because they are recruited in classes, it is easy for us to obtain longitudinal data from students. Their ages ranged from 18 to 23 years old (M = 19.62, SD = 1.29). A set of self-report ques-tionnaires was administered to the participants in a classroom set-ting. Participants’ conﬁdentiality and anonymity were assured. After completing the questionnaires, participants returned them to the administrator directly. Participants completed the question-naires twice during a two-month interval.

2.1.2. Instrument

Satisfaction with life scale – Taiwanese version. The satisfaction with life scale (SWLS), developed byDiener et al. (1985)is a widely used measure of subjective well-being.Diener et al. (1985)deﬁne life satisfaction as a conscious cognitive judgment of life in which individuals compare their life circumstances to a self-imposed standard. Their scale contained ﬁve items and employed a seven-point Likert scale, with higher values corresponding to greater sat-isfaction. The ﬁve global evaluation items are: (a) in most ways, my life is close to my ideal; (b) the conditions of my life are excellent; (c) I am satisﬁed with my life; (d) so far, I have gotten the impor-tant things I want in life; and (e) if I could live my life over, I would change almost nothing. The SWLS has shown good reliability and validity (seePavot & Diener, 1993).Wu and Yao (2006)conﬁrmed the single-factor structure of the SWLS-Taiwan version and re-vealed that the SWLS-Taiwan version was measurement invariant across gender.

2.1.3. Data analysis

In longitudinal measurement invariance analysis, a baseline model needs to be established prior to any invariance constraints to see if patterns of factor structures at different times are the same (conﬁgural invariance). Thus, the ﬁrst step is to build a model to test conﬁgural invariance across time. In this model, the ﬁve items of the SWLS assessed at T1 are inﬂuenced by a latent factor, and the other ﬁve items of the SWLS assessed at T2 are inﬂuenced by a sec-ond latent factor. The two factors may be correlated. In addition, item uniqueness is correlated across time to account for the spe-ciﬁc effect associated with each item. The latent factor scale is ﬁxed by setting the ﬁrst factor loading of each factor at 1. If the baseline model (conﬁgural invariance) is supported, further restrictive con-straints can then be imposed on the model. First, factor loadings are constrained to be equal across time to test invariance of factor loadings. A chi-square difference test is conducted to see if the baseline model is signiﬁcantly different from the loading-con-strained model. A non-signiﬁcant chi-square difference test means that factor loadings are invariant across time, satisfying weak invariance. Further, based on the weak invariance model, inter-cepts are constrained to be equal across time. Chi-square differ-ence tests between the weak invariance model and intercept-constrained model are also conducted. A non-signiﬁcant chi-square difference test means that intercepts are invariant across time, satisfying strong invariance. Finally, based on the strong invariance, item uniqueness is constrained to be equal across time. A chi-square difference test between the strong invariance model and uniqueness-constrained model is then conducted. A non-sig-niﬁcant chi-square difference test means that, in addition to factor loadings and intercepts, item uniqueness is invariant across time, satisfying strict invariance. All the models are estimated by a max-imum likelihood estimator with robust correction using Mplus (Muthén & Muthén, 2007). Because the chi-square test is sensitive to sample size, information about other ﬁt indices (CFI, TLI, RMSEA and SRMR) is used to evaluate each model. The general cutoffs for accepting a model for CFI and TLI were equal to or greater than

0.95, and equal to or less than 0.05 for the RMSEA, and less than 0.08 for the SRMR (Hu & Bentler, 1999). However,Hu and Bentler (1999) also mentioned that model ﬁt evaluation based on the above criteria should not be over-generalized. Therefore, in the current study, rules proposed by them were used for reference. Model comparison for invariance analysis relies on a chi-square difference test. A Satorra-Bentler-scaled chi-square statistic is used in the current study; its difference test is conducted according to Satorra and Bentler (2001).

2.2. Results

2.2.1. Descriptive analysis of statistics

Descriptive statistics for each item over time are presented in Table 1, including the mean, standard deviation, skewness, and kurtosis.Table 2presents correlations among items over time. 2.2.2. Longitudinal invariance analysis

Longitudinal invariance analysis is conducted by several steps. Table 3presents the results of model ﬁts and comparisons. First, the baseline model for conﬁgural invariance is acceptable because of its satisﬁed values on ﬁt indices (CFI = 0.972; TLI = 0.956; RMSEA = 0.070; SRMR = 0.036), although the SB-

### v

2_{test is }

signiﬁ-cant (SB-

### v

2_{(29) = 67.28, p < .01).}

Factor loadings are constrained to be equal across time to test for weak invariance. The weak invariance model is acceptable be-cause of its satisﬁed values on ﬁt indices as well (CFI = 0.970;

TLI = 0.959; RMSEA = 0.068; SRMR = 0.051), although the SB-

### v

2test is signiﬁcant (SB-

### v

2_{(33) = 69.24, p < .01). The SB-}

_{v}

2_{difference}

test between conﬁgural invariance and weak invariance models is not signiﬁcant (4SB-

### v

2_{(4) = 6.69, p > .05), revealing that weak}

invariance is supported.

Further, equality of intercepts across time is imposed on the model to test for strong invariance. The strong invariance model also has satisfactory values on ﬁt indices (CFI = 0.965; TLI = 0.959; RMSEA = 0.068; SRMR = 0.060), although the SB-

### v

2test is signiﬁcant (SB-

### v

2_{(38) = 79.95, p < .01). The SB-}

_{v}

2_{difference}

test between the weak invariance and strong invariance models is not signiﬁcant (DSB-

### v

2_{(5) = 10.75, p > .05), showing that strong}

invariance is supported.

Finally, equality of item uniqueness across time is further im-posed to test strict invariance. The strict invariance model also has satisﬁed values on ﬁt indices (CFI = 0.953; TLI = 0.951; RMSEA = 0.075; SRMR = 0.070), although the SB-

### v

2_{test is }

signiﬁ-cant (SB-

### v

2_{(43) = 99.43, p < .01). The SB-}

_{v}

2_{difference test between}

strong invariance and strict invariance models is signiﬁcant (DSB-

### v

2(5) = 99.43, p < .01), showing that strict invariance is not supported. We then tested the partial strict invariance of the SWLS using a backward method by removing the constraints that contributed more chi-square values to the model until the partial strong invari-ance model did not differ signiﬁcantly from the strong invariinvari-ance model. By this procedure, a model for partial strict invariance on Items 1, 2, 4 and 5 across two waves is retained. The model ﬁt of partial strict invariance model is also satisfactory (CFI = 0.963; TLI = 0.960; RMSEA = 0.067; SRMR = 0.062), although the SB-

### v

2test is signiﬁcant (SB-

### v

2(42) = 86.38, p < .01). 2.2.3. Stability coefﬁcient across time

The stability coefﬁcient (correlation between two wave factors) across time is computed using a partial strict invariance model. In order to compute the factor correlation, factor variances are set to 1, and the ﬁrst factor loading for each factor is freely estimated. The resulting estimated factor correlation is 0.57.

2.2.4. Latent factor mean comparison

Because the strong invariance model is supported, latent factor means across time can be compared. In the partial strict invariance

Table 1

Descriptive statistics of items across time for Study 1 (n = 236).

Mean SD Skewness Kurtosis

T1-Item1 4.32 1.19 0.11 0.00 T1-Item2 4.32 1.22 0.04 0.18 T1-Item3 4.52 1.30 0.12 0.56 T1-Item4 4.47 1.28 0.11 0.22 T1-Item5 3.70 1.68 0.19 0.74 T2-Item1 4.34 1.28 0.17 0.07 T2-Item2 4.47 1.19 0.20 0.03 T2-Item3 4.58 1.23 0.19 0.24 T2-Item4 4.53 1.26 0.23 0.09 T2-Item5 4.06 1.65 0.15 0.76 Table 2

Correlation matrix among items across time for Study 1 (n = 236).

T1-Item1 T1-Item2 T1-Item3 T1-Item4 T1-Item5 T2-Item1 T2-Item2 T2-Item3 T2-Item4

T1-Item1 T1-Item2 0.778 T1-Item3 0.639 0.763 T1-Item4 0.572 0.604 0.628 T1-Item5 0.453 0.466 0.480 0.577 T2-Item1 0.478 0.494 0.444 0.374 0.288 T2-Item2 0.407 0.447 0.391 0.326 0.218 0.817 T2-Item3 0.433 0.479 0.466 0.422 0.286 0.792 0.826 T2-Item4 0.452 0.439 0.389 0.415 0.258 0.744 0.716 0.736 T2-Item5 0.326 0.304 0.276 0.328 0.300 0.491 0.476 0.480 0.525 Table 3

Model ﬁt of various invariance models for Study 1.

Model df SB-v2

4SB-v2

CFI TLI RMSEA SRMR

Conﬁgural invariance 29 62.78 – 0.972 0.956 0.070 0.036

Weak invariance 33 69.24 6.69 0.970 0.959 0.068 0.051

Strong invariance 38 79.95 10.75 0.965 0.959 0.068 0.060

Strict invariance 43 99.43 17.19** _{0.953} _{0.951} _{0.075} _{0.070}

Partial strict invariancea

42 86.38 7.66 0.963 0.960 0.067 0.062

** _{p < .01.}

a

model, we set the factor mean at Time 1 to zero and freely estimate the factor mean at Time 2. The estimated factor mean at Time 2 is 0.090, which is not signiﬁcantly different from zero. Hence, the re-sult shows an equality of latent factor means across time. 3. Study 2

3.1. Method

3.1.1. Participants and procedure

Two hundred and forty-two (133 male and 109 female) student athletes from six senior high schools participated in this study vol-untarily. The research project was announced to students by teach-ers (they are also researchteach-ers in the research team) at practice time. Students were told that they can participate in a study according to their willingness and they can obtain a small gift for their participation. Because they are recruited in school, it is easy for us to obtain longitudinal data from them. Their ages ranged from 15 to 18 years old (M = 16.08, SD = 0.75). The same procedure was applied. Participants completed questionnaires three times over a period of three months.

3.1.2. Instrument

Satisfaction with life scale – Taiwanese version. The same scale from Study 1 is used, but with a six-point Likert scale.

3.1.3. Data analysis

Analysis procedures are similar to Study 1, but in the current sample, a three-wave model is built. In the baseline model, the ﬁve items of the SWLS assessed at T1 are inﬂuenced by a latent factor, the ﬁve items assessed at T2 are inﬂuenced by the second latent fac-tor, and the ﬁnal ﬁve items of the SWLS assessed at T3 are inﬂu-enced by the third latent factor. The three factors are allowed to be correlated. In addition, item uniqueness of the same items is also set to be correlated across time to account for the speciﬁc effect associated with each item. The latent factor scale is ﬁxed by setting the ﬁrst factor loading of each factor at 1. If the baseline model (con-ﬁgural invariance) is supported, further restrictive constraints can then be imposed on the model following the same steps as Study 1. 3.2. Results

3.2.1. Descriptive analysis of statistics

Descriptive statistics for each item across time are presented in Table 4, including the mean, standard deviation, skewness, and kurtosis.Table 5presents correlations between items across time.

Table 4

Descriptive statistics of items across time for Study 2 (n = 242).

Mean SD Skewness Kurtosis

T1-Item1 3.77 1.22 0.24 0.27 T1-Item2 3.86 1.29 0.20 0.35 T1-Item3 4.29 1.17 0.29 0.30 T1-Item4 3.05 1.49 0.31 0.63 T1-Item5 3.42 1.43 0.05 0.75 T2-Item1 3.60 1.17 0.09 0.09 T2-Item2 3.88 1.30 0.32 0.10 T2-Item3 4.15 1.20 0.33 0.20 T2-Item4 3.07 1.42 0.37 0.46 T2-Item5 3.57 1.30 0.01 0.31 T3-Item1 3.71 1.05 0.06 0.13 T3-Item2 3.85 1.23 0.26 0.17 T3-Item3 4.27 1.09 0.22 0.01 T3-Item4 3.28 1.34 0.22 0.36 T3-Item5 3.68 1.36 0.08 0.40 Table 5 Correlation matrix among items across time for Study 2 (n = 242). T1-Item1 T1-Item2 T1-Item3 T1-Item4 T1-Item5 T2-Item1 T2-Item2 T2-Item3 T2-Item4 T2-Item5 T3-Item1 T3-Item2 T3-Item3 T3-Item4 T1-Item1 T1-Item2 0.341 T1-Item3 0.476 0.564 T1-Item4 0.394 0.317 0.388 T1-Item5 0.409 0.245 0.354 0.416 T2-Item1 0.547 0.244 0.345 0.340 0.310 T2-Item2 0.337 0.456 0.383 0.301 0.248 0.564 T2-Item3 0.393 0.353 0.461 0.331 0.351 0.631 0.649 T2-Item4 0.392 0.331 0.372 0.455 0.330 0.586 0.501 0.592 T2-Item5 0.450 0.204 0.296 0.277 0.494 0.524 0.340 0.417 0.434 T3-Item1 0.280 0.252 0.266 0.303 0.139 0.359 0.233 0.285 0.276 0.153 T3-Item2 0.137 0.273 0.210 0.127 0.070 0.183 0.373 0.297 0.229 0.087 0.453 T3-Item3 0.118 0.142 0.249 0.226 0.119 0.241 0.204 0.370 0.289 0.093 0.505 0.574 T3-Item4 0.246 0.122 0.197 0.227 0.141 0.278 0.315 0.318 0.352 0.195 0.433 0.458 0.598 T3-Item5 0.277 0.210 0.259 0.248 0.347 0.322 0.279 0.281 0.308 0.463 0.460 0.356 0.431 0.511

3.2.2. Longitudinal invariance analysis

Longitudinal invariance analysis is conducted following several steps.Table 6presents the results of model ﬁts and comparisons. First, the baseline model for conﬁgural invariance is acceptable be-cause of its satisﬁed values on ﬁt indices (CFI = 0.966; TLI = 0.951; RMSEA = 0.048; SRMR = 0.051), although the SB-

### v

2_{test rejected}

the model (SB-

### v

2_{(72) = 111.82, p < .01).}

Then, factor loadings are constrained to be equal across time to test weak invariance. The weak invariance model is acceptable be-cause of its satisﬁed values on ﬁt indices as well (CFI = 0.964; TLI = 0.953; RMSEA = 0.047; SRMR = 0.059), although the SB-

### v

2test rejected the model (SB-

### v

2_{(80) = 122.87, p < .01). The SB-}

_{v}

2
difference test between conﬁgural invariance and weak invariance models is not signiﬁcant (4SB-

### v

2_{(8) = 11.09, p > .05), revealing}

that weak invariance is supported.

Further, equality of intercepts across time is imposed in the model to test strong invariance. The strong invariance model also has satisﬁed values on ﬁt indices (CFI = 0.952; TLI = 0.944; RMSEA = 0.051; SRMR = 0.062), although the SB-

### v

2test rejected the model (SB-

### v

2_{(90) = 146.78, p < .01). However, the SB-}

_{v}

2_{}

differ-ence test between weak invariance and strong invariance models is signiﬁcant (4SB-

### v

2_{(10) = 25.34, p < .01), showing that strong}

invariance is not supported.

We then test the partial strong invariance in the SWLS using the backward method by dropping constraints that contribute more chi-square value to the model until the partial strong invariance model did not have signiﬁcant differences from the weak ance model. By this procedure, a model for partial strong invari-ance on Items 2, 3 and 4 across three waves is retained. The model ﬁt of the partial strong invariance model is also satisfactory (CFI = 0.959; TLI = 0.950; RMSEA = 0.048; SRMR = 0.060), although the SB-

### v

2test is signiﬁcant (SB-

### v

2(86) = 134.55, p < .01). 3.2.3. Stability coefﬁcient across time

Stability coefﬁcient (correlation between three wave factors) across time is computed using partial strong invariance model. In order to compute the factor correlation, factor variances are set to 1, and the ﬁrst factor loading for each factor is freely estimated. The resulting factor correlation between Time 1 and Time 2 is 0.69, the factor correlation between Time 1 and Time 3 is 0.41 and the factor correlation between Time 2 and Time 3 is 0.47.

3.2.4. Latent factor mean comparison

Because the partial strong invariance model is still supported, latent factor means across time can be compared. In the partial strong invariance model, we set the factor mean at Time 1 as zero and freely estimate factor means at Times 2 and 3. The estimated factor mean at Time 2 is 0.005, and the factor mean at Time 3 is 0.033. Both do not differ signiﬁcantly from zero. Hence, the re-sult shows an equality of latent factor means across time. 4. Discussion

This study examines longitudinal measurement invariance in the SWLS across time. For the university student sample, results

show that the SWLS is partial strict invariant (equality of factor patterns, loadings and intercepts across time for all items and equality of item uniqueness for Items 1, 2, 4 and 5 across time), thus ensuring that the SWLS is an adequate measurement for lon-gitudinal mean comparisons over a two-month interval for univer-sity student. The results also reveal that Items 1, 2, 4 and 5 have the same reliability across time because the item uniqueness of these items is time-invariant. Moreover, the SWLS also has a moderate stability with a coefﬁcient of 0.57 over a two-month interval. The factor means across time are also invariant.

For the adolescent student athlete sample, results showed only that the SWLS is partial strong invariant (equality of factor pat-terns, loadings across time for all items and equality of intercepts for Items 2, 3, and 4 across time), which has already provided a ba-sis for longitudinal mean comparisons. Moreover, in the sample, the SWLS also has moderate stability, with coefﬁcients ranged from 0.41 to 0.69. The stability coefﬁcient between Time 1 and Time 2 is 0.69 (three-month interval), higher than that between Time 1 and Time 3 (0.41, six-month interval) and Time 2 and Time 3 (0.47, three-month interval). Finally, factor means across three waves are invariant.

Study 1 and Study 2 suggest using different items of the SWLS in longitudinal usage for an undergraduate and adolescent student sample. Under the basic requirement of a strong invariant property for longitudinal usage, Study 1 reveals that all items in the SWLS are strong invariant for undergraduate student sample, but Study 2 shows that only Item 2, 3, and 4 are strong invariant for adoles-cent students. This could be that the contents of Item 1 (In most ways, my life is close to my ideal) and Item 5 (If I could live my life over, I would change almost nothing) are not suitable for adoles-cent students because these two statement are far from their life experiences and result in an unstable interpretation and response to these two items.

However, more studies are needed to examine longitudinal measurement invariance in the SWLS because the sample and de-sign of the current study are limited. For example, the current study uses only university and adolescent student samples. The SWLS, although used mainly for student samples in most psycho-logical research, is also widely used in other samples, such as com-munity adults in different cultures (Tucker et al., 2006), the elderly (Pons et al., 2000), and schizophrenia patients (Chan et al., 2003). In addition to the limitations of sample characteristics, the time interval in the current study is two to three months. Because not all longitudinal studies are conducted within a certain time inter-val (like the time interinter-val used here), different time spans are desirable for testing the duration of invariance properties.

Moreover, we investigated partial invariance properties in the SWLS for the current two samples through a backward method. Although we did ﬁnd partial invariance properties in the SWLS, this partial invariance investigation should be treated with caution as it is more exploratory than conﬁrmatory. This is another reason more studies are needed to clarify longitudinal measurement invariance in the SWLS.

Given the limitations of sample, time interval, and exploratory-wise partial invariance structure, we are not able to generalize our

Table 6

Model ﬁt of various invariance models for Study 2.

Model df SB-v2

4SB-v2

CFI TLI RMSEA SRMR

Conﬁgural invariance 72 111.82 – 0.966 0.951 0.048 0.051

Weak invariance 80 122.87 11.09 0.964 0.953 0.047 0.059

Strong invariance 90 146.78 25.34** _{0.952} _{0.944} _{0.051} _{0.062}

Partial strong invariancea

86 134.55 12.14 0.959 0.950 0.048 0.060

** _{p < .01.}

a

ﬁndings to other samples and designs; instead, we would mention that longitudinal measurement invariance is an important psycho-metric property of the SWLS, especially when it is administered in a longitudinal study. Future studies should pay more attention to this property.

Acknowledgment

This research was partially supported by a Grant form the Cen-tral Taiwan University of Science and Technology (CTU97-P-24) to Ying-Mei Tsai.

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