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Supplemental Materials Section 1
Use of Target Rotation.
For present purposes we used a target rotation, taking advantage of the fact that a few items from
each factor are relatively pure measures of the factor (i.e., factor cross-loadings are near-zero). As
emphasized by Browne (2001; also see Asparouhov & Muthén, 2009; Dolan, Oort, Stoel &
Wicherts, 2009) this strategy reflects a compromise between the rationales for ESEM and ICM-CFA,
based on partial knowledge of the factor structure. However, the important difference is that in ICMCFA, factor loadings specified to be zero are forced to assume this value. In contrast, for target
rotations the factor loadings specified to be zero are made to be as close to zero as possible, but are
not constrained to be zero. Thus, whilst the targets influence the final rotated solution, the factor
loadings for target items can end up very different from zero if the zero loadings are not appropriate
(Asparouhov & Muthén, 2009). Based on simulated data, Asparouhov and Muthén (2009) suggested
that the target rotation was more appropriate for more complex factor models (but ones that still have
approximately simple structure) in which several or many of the items load on each of the different
factors. The target rotation is particularly appropriate when there is a clearly defined a priori factor
structure and at least a reasonable approximation to simple structure. The target rotation also has
strategic advantages in multi-group analyses in that it greatly increases the likelihood that the order
of the factors is the same for each group-something that is not guaranteed by other rotations.
Consequences of a lack of Invariance.
Most readers of Psychological Assessment are familiar with the consequences of a lack of
invariance. Hence we have briefly outlined some of these issues here rather than in the published
article.
Tests of the invariance of factor loadings (Weak measurement invariance, Model 2 in
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Appendix 2) are particularly important, both for relating passion factors to other constructs for
different groups with cross-sectional data, and for evaluating patterns of relations among variables in
the same group over time with longitudinal data. Indeed, all models except the configural invariance
model (Model 1) assume the invariance of factor loadings. Unless the factor loadings are reasonably
invariant over groups, any comparisons must be considered suspect, as the constructs themselves
differ (i.e., the apples and oranges problem). However, if there is a sufficient number of items, tests
of partial invariance might be warranted, such that invariance of factor loadings is supported for
almost all the items for each factor. Such tests of invariance might also be on the basis of selecting
items to be retained in early stages of instrument development.
If applied researchers want to compare latent mean differences across groups, then tests of
item intercept invariance (Strong measurement invariance, Model 5), in addition to factor loading
invariance, are critical. For example, assume for six items designed to measure a particular trait that
three clearly favor one group and three clearly favor the second group. These results provide no basis
for evaluating mean differences on the factor, in that even the direction of differences would depend
on the particular items used to measure the trait. Furthermore, because these 6 items are only a small
sample of items that could be used to evaluate this trait, the results provide only a weak basis for
knowing what would happen if a larger, more diverse sample of items were sampled. Support for the
invariance of item intercepts would mean that differences based on each of the items considered
separately are reasonably consistent in terms of magnitude as well as direction. These results would
provide a stronger basis of support for the generalizability of the interpretation of the observed mean
differences. Although issues of non-invariance of item intercepts and differential item functioning
are well known, historically these issues have been used more frequently in the evaluation of
standardized achievement tests than for measures of psycho-social variables. In summary, a lack of
invariance of item intercepts would mean that the observed group differences are not consistent
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across even the items used to represent a latent factor on a particular instrument, and would provide
no basis for the generalizability of the results across a wider and more diverse set of items that could
be used to represent the trait.
In order to compare the Passion Scale (manifest) scale scores (or even factor scores), the
invariance of items’ uniquenesses also represents an important prerequisite (Strict measurement
invariance, Model 7). Indeed, the presence of differences in reliability (as represented or absorbed in
the item uniquenesses) across the multiple groups could distort mean differences on the observed
scores. However, for comparisons based on latent constructs that are corrected for measurement
error, the valid comparison of latent means only requires support for strong measurement invariance
and not for the additional assumption of the invariance of measurement error. Hence, comparison of
group mean differences based on latent-variable models like those considered here makes fewer
assumptions than those based on manifest scores.
A lack of invariance in relations between factors does not compromise comparisons of latent
mean differences over time or groups. However, particularly for multifactorial constructs like the
Passion Scale factors, the pattern of relations among factors has important practical and theoretical
implications; interpretations are likely to be complicated by the heterogeneity of relations between
Passion Scale factors and other variables. In more complex models, the invariance of other parameter
estimates (e.g., correlated uniquenesses or path coefficients) may also be relevant as a test of the
generalizability of the results.
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Supplemental Materials Figure 1
Figure 1. HP=Harmonious Passion. OP= Obsessive Passion. Age2 = Quadratic effect of age. A. Confirmatory
Factor Analysis (CFA) Model. B. Exploratory Factor Analysis Model (ESEM). C. ESEM with multiple groups
(which can incorporate invariance constraints to test measurement invariance and latent mean differences; see
Appendix 2). D. Multiple Indicator Multiple Cause (MIMIC) model used to test mean differences in latent
ESEM factors for manifest independent variables). Solid black lines represent target loadings. Broken grey
lines represent non/target loadings; see methods section for information on Target rotation).
.
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Supplemental Materials Appendix 1
Wording of Items on the Passion Scale and Validity Correlates
Harmonious Passion (HP): Reliability= .83; N = 3,571.
HP1 This activity is in harmony with the other activities in my life.
HP2 The new things that I discover with this activity allow me to appreciate it even more.
HP3 This activity reflects the qualities I like about myself.
HP4 This activity allows me to live a variety of experiences.
HP5 My activity is well integrated in my life.
HP6 My activity is in harmony with other things that are part of me.
Obsessive Passion (OP): Reliability = .86; N = 3,571.
OP1 I have difficulties controlling my urge to do my activity.
OP2 I have almost an obsessive feeling for this activity.
OP3 This activity is the only thing that really turns me on.
OP4 If I could, I would only do my activity.
OP5 This activity is so exciting that I sometimes lose control over it.
OP6 I have the impression that my activity controls me.
Life Satisfaction. Reliability = .85; N = 1,649
In most ways my life is close to my ideal.
The conditions of my life are excellent.
I am satisfied with my life.
So far I have got the important things I want in life.
If I could live my life over, I would change almost nothing.
Rummination. Reliability = .88; N = 364
In general, when I am not doing my activity, I analyse and think continually about my activity.
In general, when I am not doing my activity, I have difficulty to stop thinking about my activity.
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In general, when I am not doing my activity, I become so deeply thoughtful about my activity that I have
difficulty thinking about something else.
Conflict. Reliability = .84; N = 220
My activity is in conflict with other activities in my life.
I can sacrifice other life domains for my activity.
My activity is frequently overlapping with other life domains.
Sometimes, I think that I am spending too much time on my activity and not enough on other activities in my
life.
Single-Item Validity Criteria.
Time (N = 2,793): I spend a lot of time doing this activity.
Like (N = 2,793):I like this activity.
Value (N = 2,797): This activity is important for me.
Passion (N = 2789): This activity is a passion for me.
Note. Data are from the Passion Scale responses based on many different studies. Although respondents in all
studies completed the Passion Scale, the various multi-item and single-item validity correlates were only used
in some of the studies.
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Supplemental Materials Appendix 2
Taxonomy of Multiple Group Tests of Invariance Testable with ESEM and Nesting Relations (in brackets)
ModeParameters Constrained to Be Invariant
1
Model 1
none (configural invariance)
2
Model 2
FL [1] (weak factorial/measurement invariance)
3
Model 3
FL Uniq [1, 2]
4
Model 4
FL, FVCV [1, 2]
5
Model 5
FL, Inter [1, 2] (Strong factorial/measurement invariance)
6
Model 6
FL, Uniq, FVCV [1, 2, 3, 4]
7
Model 7
FL, Uniq, Inter [1, 2, 3, 5] (Strict factorial/measurement invariance)
8
Model 8
FL, FVCV, Inter [1, 2, 4, 5]
9
Model 9
FL, Uniq, FVCV, Inter [1-8]
10
Model 10
FL, Inter, FMn [1, 2, 5] (Latent mean invariance)
11
Model 11
FL, Uniq, Inter, FMn [1, 2, 3, 5, 7, 10] (Manifest mean invariance)
12
Model 12
FL, FVCV, Inter, FMn [1, 2, 4, 5, 6, 8, 10]
13
Model 13
FL, Uniq, FVCV, Inter, FMn [1-12] (complete factorial invariance)
Note. FL = Factor Loadings; FVCV = Factor variance-covariances; Inter = item intercepts; Uniq = item uniquenesses;
FMn = Factor Means. Models with latent factor means freely estimated constrain intercepts to be invariant across groups,
whilst models where intercepts are free, imply that mean differences are a function of intercept differences. Bracketed
values represent nesting relations in which the estimated parameters of the less general model are a subset of the
parameters estimated in the more general model under which it is nested. All models are nested under Model 1 (with no
invariance constraints) whilst Model 13 (complete invariance) is nested under all other models. Table 2 was adapted
from Marsh, et al. (2009)
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Supplemental Materials Appendix 3
Mplus Syntax For Model MG-M2p (Table 6);
Weak Factorial Invariance Across Passion Activity Groups
TITLE: MG-M2p Weak Factorial Invariance Across Passion Activity Groups;
DATA: FILE IS
'C:\herb\Mplus\passion\combpassion\passion hm 23nov11.dat';
VARIABLE: NAMES ARE
SAMPLE ACTIVITY PH1 PH2 PH3 PH4 PH5 PH6 PO1 PO2 PO3 PO4 PO5 PO6
PTIME PLIKE PIMPO PPASS
LS1 LS2 LS3 LS4 LS5 RUM1 RUM2 RUM3 RUM4 CON1 CON2 CON3 CON4
ActGrp ID phtot potot LStot RUMtot CONTot source;
usevariables are PH1 PH2 PH3 PH4 PH5 PH6 PO1 PO2 PO3 PO4 PO5 PO6;
missing are all(-9);
grouping is ActGrp (1=lesr 2=sprt 3=soc 4=work 5=educ);
ANALYSIS:
estimator=mlr;
ROTATION =target(OBLIQUE);
MODEL: f1 BY PH1-PO5 po1~0 po2~0 (*t1);
f2 BY PH1-PO5 ph1~0 ph5~0 ph6~0(*t1);
PO3 with PO4; ph1 with ph5;
po6 on F1; po6 on F2;
Model sprt: [f1-f2@0];[PH1-PO5];
Model soc: [f1-f2@0];[PH1-PO5];
Model work: [f1-f2@0];[PH1-PO5];
Model educ: [f1-f2@0];[PH1-PO5];
OUTPUT:
TECH1; stdyx; mod(0);
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TWO FACTOR PASSION SCALE_SUPPL
Supplemental Materials Appendix 4
Mplus Syntax For Model MG-M5p (Table 6);
Weak Factorial Invariance and Intercept Invariance Across Passion Activity Groups
TITLE: MIMIC 1A no sex on fac or items no-cu no-part 25feb09;
DATA: FILE IS
'C:\herb\Mplus\passion\combpassion\passion hm 23nov11.dat';
VARIABLE: NAMES ARE
SAMPLE ACTIVITY PH1 PH2 PH3 PH4 PH5 PH6 PO1 PO2 PO3 PO4 PO5 PO6
PTIME PLIKE PIMPO PPASS
LS1 LS2 LS3 LS4 LS5 RUM1 RUM2 RUM3 RUM4 CON1 CON2 CON3 CON4
ActGrp ID phtot potot LStot RUMtot CONTot source;
usevariables are PH1 PH2 PH3 PH4 PH5 PH6 PO1 PO2 PO3 PO4 PO5 PO6;
missing are all(-9);
grouping is ActGrp (1=lesr 2=sprt 3=soc 4=work 5=educ);
ANALYSIS:
estimator=mlr;
ROTATION=target(OBLIQUE);
MODEL: f1 BY PH1-PO5 po1~0 po2~0 (*t1);
f2 BY PH1-PO5 ph1~0 ph5~0 ph6~0(*t1);
PO3 with PO4; ph1 with ph5;
po6 on F1; po6 on F2; [po6](12);
Model lesr:[po6](12); [po5,ph6,po6,ph1];
Model sprt:[f1-f2]; [po6];[ph2,ph4];
Model soc:[po6](12); [f1-f2]; [po2,po3,po5,ph2];
Model work:[po6](12);[f1-f2];[po4];
Model educ:[po6](12);[f1-f2];
OUTPUT: TECH1; stdyx; mod (all 3); svalues;
Leisure: po5,ph6,po6,ph1; sport: po6, ph2,ph4; soc: po2,po3,po5,ph2; work: po4;
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