People differ in their expectations for the future. Some people—the optimists—expect that good things will happen to them; others—the pessimists—expect that bad things will happen to them (Scheier & Carver, 1985). Dispositional optimism and pessimism affect all areas of life and represent stable personality traits (Scheier & Carver, 1985; Scheier et al., 1994).

Empirical evidence confirms that optimists and pessimists differ in their general approach to life and, as a consequence, experience different life outcomes (for an overview, see, e.g., Carver et al., 2010). The general pattern of findings suggests that because optimists have better self-regulation strategies than pessimists, higher levels of optimism are linked to favorable life outcomes (Scheier & Carver, 1985). Optimists employ coping strategies more flexibly than pessimists do (Nes & Segerstrom, 2006; Pavlova & Silbereisen, 2013). Moreover, because they expect more positive outcomes, they show greater persistence in the pursuit of goals (Nes et al., 2011). Consistent positive associations have been found, for instance, between dispositional optimism and life satisfaction (e.g., Hajek & König, 2019), health (e.g., Hajek & König, 2019), employment (e.g., Hajek & König, 2019), and income (e.g., Kemper et al., 2013).

However, dispositional optimism may not be universally positive: (Male) Optimists engage more often in risky behavior than do (male) pessimists (Felton et al., 2003). In general, higher dispositional optimism is associated with higher risk proneness (e.g., Barel, 2019). Furthermore, dispositional optimism has been shown to be consistently related to other personality characteristics. Optimism correlates positively with the Big Five personality traits, especially with Emotional Stability and Extraversion (e.g., Sharpe et al., 2011). It also correlates positively with self-esteem (e.g., Hajek & König, 2019), interpersonal trust (e.g., Mealy et al., 2015), and general self-efficacy (e.g., Hajek & König, 2019). Concerning beliefs about the general effectiveness of one’s own behavior, optimism has been consistently found to be positively associated with internal locus of control and negatively with external locus of control (e.g., Guarnera & Williams, 1987).[1] With regard to beliefs about the political effectiveness of one’s own behavior, previous research found optimism to be positively associated with internal and external political efficacy (e.g., Groskurth et al., 2021).[2]

In sum, dispositional optimism is associated with a broad range of life outcomes and with structural and dynamic aspects of personality. These correlations underline the potential relevance of dispositional optimism to many researchers who study human behavior and its consequences on the individual and societal level. Up to now, the literature has not yet arrived at a final consensus on the dimensionality of the construct. There is evidence that either a bipolar unidimensional (e.g., Kam & Meyer, 2012; Rauch et al., 2007; Segerstrom et al., 2011) or highly correlated two-dimensional structure underly the optimism–pessimism construct (e.g., Benyamini, 2005; Chang et al., 1997; Herzberg et al., 2006; Segerstrom et al., 2011).

One option to measure dispositional optimism either as a unidimensional or two-dimensional construct is the 10-item Life Orientation Test–Revised (LOT-R; Scheier et al., 1994). Because large-scale surveys usually operate under severe monetary and time constraints, Kemper et al. (2013) developed an even more economical ultra-short scale, the two-item German-language Skala Optimismus–Pessimismus–2 (Optimism–Pessimism Short Scale–2; SOP2). The German-language SOP2 (Kemper et al., 2013; Kemper, Beierlein, Kovaleva, & Rammstedt, 2014) has proved to be a psychometrically sound measure that is strongly correlated to LOT-R (r = .68) and allows for the measurement of dispositional optimism in about 20 s. Optimism and pessimism can either be interpreted as separate or bipolar dimensions with SOP2. Because of the strong correlation between optimism and pessimism (r = –.86) and the two facets’ highly similar correlations with external variables (see Kemper et al., 2013), we interpreted SOP2 as a unidimensional, bipolar measure for dispositional optimism in the further course of this paper.

SOP2 can be applied in a variety of research areas in psychology, sociology, economics, and related disciplines. Versions of SOP2 are available in English, Spanish, Italian, and Greek (Kemper et al., 2015). However, evidence for the psychometric quality of the English-language version has been missing to date.

Item generation and selection

For the development of the original, German-language SOP2 scale, Kemper et al. (2013; see also Kemper, Beierlein, Kovaleva, & Rammstedt, 2014) aimed to measure the construct with as few items as possible while still capturing the essential aspects of the definition of dispositional optimism proposed by Scheier and Carver (1985). These authors’ definition is widely used and accepted in research on the construct. On that basis, Kemper et al. (2013) generated two items measuring confidence and generalized positive expectations about the future as well as doubt and generalized negative expectations about the future, respectively. Thereby, the wording of the items should contain the terms optimistic and pessimistic. These items underwent cognitive pretesting to ensure item clarity and comprehensibility for a broad range of potential study participants. The results of the pretest showed that there were individual differences in the interpretation of the terms optimistic and pessimistic. To reduce measurement error in item response and to increase content validity of the items, Kemper et al. (2013) prepended a brief definition of the respective construct to each item (for more detailed information, see Kemper et al., 2013). The authors thoroughly validated the German-language SOP2 based on a large and diverse random sample representative of the adult population in Germany in terms of age, gender, and educational attainment.

To enable social scientists to use SOP2 in English-language surveys, the scale was adapted to the English language by Kemper, Beierlein, Kovaleva, and Rammstedt, 2014. In a first step, the two items of SOP2 and their rating scales were translated into English following the TRAPD (Translation, Review, Adjudication, Pretesting, and Documentation; Harkness, 2003) approach. Two professional translators (English native speakers) translated the item wordings and the response scale labels independently of each other into British English and American English, respectively. Second, an adjudication meeting was held in which psychological experts, the two translators, and an expert in questionnaire translation reviewed the translation proposals and developed the final translation.

The empirical validation of this English-language SOP2 was hitherto lacking. The aim of the present study was to make this evidence for the psychometric quality of the English-language SOP2 available to researchers who intend to use the adaptation for their research in English-language samples or as part of cross-cultural comparisons. We addressed several aspects of psychometric quality, that is, scale reliability, different types of validity evidence (regarding the internal structure of the scale and the relationship between scores on SOP2 and on 10 scales measuring other constructs), and measurement invariance of the English-language adaptation by analyzing the data of two large quota samples from the UK and Germany. In view of the good psychometric quality of the German-language source version, we expected to find similar reliability estimates and validity evidence for the English-language adaptation, and to be able to confirm the internal structure of this adaptation.

Samples

To investigate the psychometric properties of the English-language adaptation of SOP2 and their comparability with those of the German-language source instrument, we assessed both versions in a Web-based survey (using computer-assisted self-administered interviewing [CASI]) that was conducted in the UK and Germany by the online access panel provider respondi AG. Data collection took place in January 2018. For both nations, quota samples were drawn that represented the heterogeneity of the adult population in terms of age, gender, and educational attainment. Data from the last German Census (2011) were used as a reference (https://ergebnisse.zensus2011.de/?locale=en). To avoid bias introduced by a lack of reading/language proficiency, only native speakers of the respective languages were recruited. We explained our research goal (investigation of the quality of several questionnaires) to the participants. Respondents received a small financial reward for their participation. In both nations, a subsample was reassessed after around 3 to 4 weeks (median time intervals: 28 days in the UK and 20 days in Germany).

Only respondents who completed the full questionnaire—that is, who did not abort the survey prematurely—were included in our analyses. To handle missing values on individual items (of SOP2 or other variables included in the survey), we used full-information maximum likelihood (FIML) estimation in our analyses. The gross sample sizes were N_UK = 508 (retest: N_UK = 117) and N_DE = 513 (retest: N_DE = 125). In the next step, invalid cases were excluded based on three complementary criteria: (a) ipsatized variance—that is, the within-person variance across items (Kemper & Menold, 2014)—if the person fell within the lower 5% of the sample distribution of ipsatized variance; (b) the Mahalanobis distance of a person’s response vector from the average sample response vector (Meade & Craig, 2012) if the person fell within the upper 2.5% of the sample distribution of the Mahalanobis distance[1]; and (c) implausibly short response times, namely, if the person took, on average, less than 1 s to respond to an item. Our intention in choosing relatively conservative cut-off values was to exclude as many careless responders as possible while preserving valid cases. All exclusion criteria were applied simultaneously—that is, any respondent who violated one or more of the three criteria was excluded from the analyses, and only those who met all three criteria were included.[2] This approach resulted in the exclusion of 40 cases (7.9%) from the UK subsample and 39 cases (7.6%) from the German subsample, thereby yielding net sample sizes of N_UK = 468 (retest: N_UK = 111) and N_DE = 474 (retest: N_DE = 117). Table 2 depicts in detail the sample characteristics and their distribution.

Table 2

Sample Characteristics by Nation

Note. The equivalent German educational levels were as follows (from low to high): ohne Bildungsabschluss/Hauptschulabschluss [no educational qualification/basic school-leaving qualification]; Mittlere Reife [intermediate school-leaving qualification]; and Fachhochschulreife/Abitur [entrance qualification for a university of applied sciences/general higher education entrance qualification].

Material

The online surveys were conducted in German for the German sample and in English for the UK sample. Study questionnaires comprised the respective language version of SOP2, a set of questions on sociodemographic characteristics (i.e., gender, age, highest level of education, income[3], and employment status), and numerous measures to enable us to subsequently examine the relationship between scores on SOP2 and on scales measuring other constructs. As SOP2 was part of a comprehensive multi-theme survey, our choice of correlates was driven by theoretical considerations and data availability.

On theoretical grounds, we selected, first, constructs that reflect general psychological dispositions and resources: (a) the Big Five personality traits, (b) risk proneness, (c) general self-efficacy, (d) self-esteem, (e) internal and external locus of control, (f) interpersonal trust, and (g) internal and external political efficacy. Second, we selected constructs that reflect quality of life variables: (h) life satisfaction and (i) health. As outlined in the theoretical background, previous research has found that all these constructs consistently correlate with dispositional optimism. Accordingly, we expected dispositional optimism to correlate positively with the Big Five personality traits (the highest with Emotional Stability and Extraversion), risk proneness, general self-efficacy, self-esteem, internal locus of control, interpersonal trust, internal and external political efficacy, life satisfaction and health, as well as negatively with external locus of control. Third, we examined the susceptibility of SOP2 to two aspects of (j) socially desirable responding (exaggerating positive qualities and minimizing negative qualities) and, hence, a possible distortion of respondents’ answers.[4] All short-scale measures, which were also administered as part of the survey in the respective language version, exhibit satisfactory psychometric quality criteria and are described in more detail in Table 3.

Before computing the correlations, we recoded Item 2 of SOP2, the minimizing negative qualities dimension of socially desirable responding (for both language versions), the health variable (for both language versions), and the self-esteem scale (UK only) so that high values represented dispositional optimism, high socially desirable responding, high health values, and high self-esteem, respectively. Because the Big Five dimension Emotional Stability is negatively worded in relation to the construct Negative Emotionality in the BFI-2-XS, we also recoded the respective items so that high values represented the positive pole of this dimension—that is, Emotional Stability. In addition, we recoded the employment status variable and tested four contrasts: (a) self-employed versus employed, (b) unemployed (out of work and looking for work/out of work but not currently looking for work) versus (self-)employed, (c) retired/homemaker versus (self-)employed, and (d) pupil/student/apprentice/intern versus (self-)employed

Table 3 

Short-Scale Measures Used in the Survey Correlated with SOP2

Item analyses

We investigated the factorial structure of SOP2 separately in the UK and Germany by means of confirmatory factor analyses (CFA) using the R package “lavaan” (Rosseel, 2012). We tested three different models, which are depicted in Figure 1, and estimated all models with robust maximum likelihood (MLR) estimation. Our rationale was to examine whether the item scores captured sufficient common variance to warrant a unidimensional interpretation of the construct and the aggregation of both single-item scale scores. Because SOP2 comprises only two items, a unidimensional model with equal loadings (i.e., an essentially tau-equivalent model) is just-identified. Therefore, we tested models with the items measured at two occasions to be able to evaluate model fit and to test whether the models developed for the original German-language scale were replicable.

The first models we estimated were two hierarchical measurement models for repeated measures initially developed by Kemper, Beierlein, Kovaleva, and Rammstedt (2014) and Kemper et al. (2013) in their publications of the German-language source version of SOP2. The third model we estimated was the unidimensional model.

Model I

The first model consisted of two first-order factors and one second-order factor capturing generalized positive expectations. The first-order optimism factor consists of the optimism item at two measurement occasions; the first-order pessimism factor consists of the pessimism item at two measurement occasions. We identified the model by constraining the first factor loading of each factor to 1 and the factor loadings within each factor to equality. The fit indices refer to the commonly used MLR-scaled CFI and RMSEA, which are functions of the MLR-adjusted chi-square statistic and lead to biased population values (Brosseau-Liard et al., 2012; Brosseau-Liard & Savalei, 2014). According to the benchmarks of Hu and Bentler (1999), the model fit was good for Germany, whereas it was borderline acceptable for the UK, with a slightly too high RMSEA. Following Browne and Cudeck (1992), who reported more differentiated cut-offs for RMSEA, RMSEA was still acceptable: UK—χ²(3) = 15.291, p < .01, CFI = .935, RMSEA = .094, SRMR (standardized root mean square residual) = .069, BIC (Bayesian information criterion) = 4,131.665; DE—χ²(3) = 4.637, p = .200, CFI = .993, RMSEA = .034, SRMR = .054, BIC = 3,949.892.[5] The size of the items’ factor loadings, the measurement errors, and the mean structure are depicted in Figure 1A.

Model II

Because the fit of the previous model was only borderline acceptable in the UK, we next tested another hierarchical model. Following Kemper et al. (2013), besides the content factors, we additionally specified wave-specific method factors, in the sense that the optimism and pessimism items of the first measurement time load on one factor and both items of the second measurement time load on a second factor. In line with Kemper et al. (2013), we applied the following model constraints: (a) equalizing the factor loadings within each factor (i.e., content and method factors), (b) equalizing the error variances of the items of one factor (i.e., content factor), and (c) not allowing the content and method factors to correlate (i.e., setting these correlations to 0). We diverged slightly from Kemper et al.’s (2013) approach because we (d) identified our model by setting the factor loadings instead of the factor variances to 1 and (e) captured the correlation between the (first-order) optimism and pessimism factors in a second-order factor with equal loadings. However, our model is technically equivalent to the model of Kemper et al. (2013). The fit indices should not diverge between the different identification and modeling strategies and proved to be better than the previous model for both nations: UK—χ²(3) = 10.728, p < .05, CFI = .959, RMSEA = .074, SRMR = .053, BIC = 4,135.192; DE—χ²(3) = 1.419, p = .702, CFI = 1.000, RMSEA = .000, SRMR = .018, BIC = 3,944.921.[6] The size of the items’ factor loadings, the measurement errors, and the mean structure are depicted in Figure 1B.

Model III

The aforementioned model developed by Kemper et al. (2013) showed good fit. However, it requires two measurement occasions with the same respondents, which is not always feasible. Therefore, we additionally present an alternative unidimensional measurement model with one latent factor capturing generalized positive expectations. This model does not require a second measurement occasion. Because a latent measurement model with only two items and no further restrictions is not identified, we estimated an essentially tau-equivalent model setting the factor loadings to 1. Hence, the model is just-identified (df = 0), meaning that it calculates the closed-form solution for the parameters, and we could not evaluate a model fit. Only a multi-group model with equivalence restrictions provides an interpretable model fit (see the “Further quality criteria” section). The size of the items’ factor loadings, the measurement errors, and the mean structure are depicted in Figure 1C.

A

B

C

Figure 1. Measurement models of SOP2 with standardized coefficients and equalized factor loadings. The negatively worded item is reverse-scored. The coefficients of the German sample are presented after the double slash. N_UK = 468; N_DE = 474. A: Hierarchical two-factor measurement model. B: Hierarchical two-factor measurement model with wave-specific method factors. C: Unidimensional measurement model.

Item parameters

Table 4 shows the means, standard deviations, skewness, and kurtosis for the two items as well as for the aggregate score, separately for the UK and German samples. All descriptive statistics were comparable across the two languages. The inter-item correlations were as follows: UK—r₁₂ = .51, DE—r₁₂= .63 (for both correlations: p < .001).

Table 4

Descriptive Statistics by Nation for the SOP2 Items

Note. The rating scale ranged from 1 (low) to 7 (high). UK = United Kingdom (N = 468), DE = Germany (N = 474).

To empirically examine the English-language adaptation of SOP2 and to investigate its comparability with the German-language source version, we analyzed psychometric properties (objectivity, reliability, and validity) in both language versions as well as measurement invariance across both nations. We ran all statistical analyses with R.

Objectivity

A scale can be regarded as objective when it works (a) independently of the administrator (objectivity of application), (b) independently of the evaluator of the instrument (objectivity of evaluation), and (c) when unambiguous and user-independent rules are pro-vided (objectivity of interpretation). The standardized questionnaire format and written instructions, the fixed scoring rules and labeled response categories, and the reference rang-es ensured the objectivity of the application, evaluation, and interpretation of SOP2.

Reliability

As estimates for the reliability of SOP2, we computed McDonald’s omega (ω; McDonald, 1999; Raykov, 1997) using the R package “semTools” (Jorgensen et al., 2019). In addition, we computed the test–retest stability, r_tt, over a period of about 28 days (Mdn) in the UK (N_UK = 111) and of 20 days (Mdn) in Germany (N_DE = 117), respectively. Our reasoning was that this time span of 3 to 4 weeks was long enough to allow for meaningful test–retest stability estimates and short enough to preclude the occurrence of pronounced and systematic change in the true scores of dispositional optimism. Because the test–retest stability is sensitive not only to measurement error but also to state fluctuations in dispositional optimism (e.g., Allen & Yen, 2002), the resulting reliability coefficient is best understood as a lower-bound estimate of reliability.

The reliability estimates for SOP2 were ω = .68 and r_tt = .74 (CI_95%: .64 < r_tt < .81) for the UK and ω = .77 and r_tt = .77 (CI_95%: .68 < r_tt < .83) for Germany. As often occurs with (ultra-)short scales, test–retest reliability was minimally higher than internal consistency for the UK, whereas it was the same for the German sample. In detail, SOP2 proved to be slightly more reliable in the German sample than in the UK sample. Because internal consistency estimates vary across groups, test–retest correlations are recommended for a comparison of the reliability of scale scores. Especially given the small number of items, both reliability estimates (ω and r_tt) are satisfactory and sufficient for research purposes (Aiken & Groth-Marnat, 2006; Kemper et al., 2019).

Validity

Content-related validity

Content-related validity evidence was provided by Kemper et al. (2013) and Kem-per, Beierlein, Kovaleva, and Rammstedt (2014) during the original scale development process. In addition, we investigated two types of validity evidence—namely, evidence based on the internal structure of the scale and evidence based on the relationship between scores on the SOP2 scale and on scales measuring other variables.

Validity evidence based on the relationship between scores on SOP2 and on scales measuring other variables

We computed evidence based on the relationship between scores on SOP2 and on scales measuring other variables based on manifest indicators (scale scores). Therefore, the reported correlations are subject to attenuation and represent the lower bound of the true associations. The correlation coefficients between scores on SOP2 and on other scales are depicted in Table 5. Their interpretation is based on effect size guidelines proposed by Gignac and Szodorai (2016): relatively small effects (r ≥ .10), typical (medium) effects (r ≥ .20), and relatively large effects (r ≥ 30). According to these authors, a correlation of .20 corresponds to the 50th percentile of a meta-analytical distribution of correlations in individual differences research. Therefore, in Table 5, medium to large effects are highlighted. In order to investigate this type of evidence, we correlated scores on SOP2 with scores on the scales measuring the other constructs outlined in the Material section.

In both nations, dispositional optimism showed the strongest positive association (r > .50) with self-esteem, the Big Five dimension Emotional Stability, and life satisfaction. This is in line with previous findings that individuals high in self-esteem (e.g., Hajek & König, 2019; Kemper, Beierlein, Kovaleva, & Rammstedt, 2014), Emotional Stability (e.g., Kemper et al., 2013; Kemper, Beierlein, Kovaleva, & Rammstedt, 2014; Sharpe et al., 2011), and life satisfaction (e.g., Hajek & König, 2019; Kemper et al., 2013; Kemper, Beierlein, Kovaleva, & Rammstedt, 2014) have a higher propensity for also reporting more optimistic expectations.

Table 5

Correlations of SOP2 with Validation Measures and Sociodemographic Characteristics, by Nation

Note. UK = United Kingdom (N = 468, N_{Employment status} = 450, N_Income = 431); DE = Germany (N = 474, N_Self-esteem = 473, N_{Employment status} = 462, N_Income = 449); CI = confidence interval. Health: very bad (1) – very good (5). Gender: 1 = male, 2 = female. Coefficients with r ≥ |.20| are in bold type.

The second-highest correlations, which were again similar across both nations, were large positive associations (.30 < r < .50) between optimism and interpersonal trust (see, e.g., Beierlein, Kemper, et al., 2014a; Mealy et al., 2015; Schweer, 2006), health (see, e.g., Hajek & König, 2019; Kemper et al., 2013; Kemper, Beierlein, Kovaleva, & Rammstedt, 2014; Rasmussen et al., 2009), general self-efficacy (see, e.g., Hajek & König, 2019; Beierlein, Kovaleva, et al., 2014; Kemper, Beierlein, Kovaleva, & Rammstedt, 2014; Luszczynska et al., 2005), and Big Five Extraversion (in DE, it was only a medium effect; see, e.g., Kemper et al., 2013; Kemper, Beierlein, Kovaleva, & Rammstedt, 2014; Sharpe et al., 2011).

Regarding locus of control, previous research has suggested positive correlations between optimism and internal locus of control and negative correlations between optimism and external locus of control (e.g., Guarnera & Williams, 1987; Kemper et al., 2015; Kovaleva et al., 2014). We could replicate this pattern for both nations with small-to-large-sized effects.

With respect to the other constructs, we also found substantial small-to-medium-sized positive correlations between optimism and Big Five Agreeableness and Openness, and between optimism and risk proneness. This is consistent with previous findings indicating that individuals high in Agreeableness (e.g., Kemper et al., 2013; Kemper, Beierlein, Kovaleva, & Rammstedt, 2014; Sharpe et al., 2011), Openness (e.g., Kemper et al., 2013; Kemper, Beierlein, Kovaleva, & Rammstedt, 2014), and risk proneness (e.g., Barel, 2019; Beierlein, Kovaleva, Kemper, & Rammstedt, 2015; Kemper et al., 2015) also tend to have more optimistic expectations.

Furthermore, dispositional optimism also showed medium-sized positive correlations with internal political efficacy and the exaggerating positive qualities subdimension of socially desirable responding. The aforementioned subdimension depicts the self-deceptive enhancement component of communion-induced socially desirable responding (Nießen et al., 2019).

We calculated correlations between SOP2 and relevant sociodemographic characteristics, namely, employment status, income, educational level, age, and gender. In the present analyses, consistent across the two nations, we found medium-sized positive correlations with income. This finding is in line with evidence from Kemper et al. (2013) and Kemper, Beierlein, Kovaleva, and Rammstedt (2014) indicating that the tendency to report higher optimistic expectations increases with increasing income. Moreover, in both nations, there were small-to-medium negative associations between optimism and unemployment. Employed individuals had a higher propensity for dispositional optimism than unemployed ones (see, e.g., Hajek & König, 2019).

Overall, the pattern of correlations suggests that the evidence based on the relationship between scores on the English-language version of SOP2 and on scales/items measuring other variables was similar to that of the German-language source version. This evidence was also confirmed by the meta-correlation of r = .97, 95% CI [.94, .99], across the UK and Germany (i.e., the correlation of all correlations across nations).

Descriptive statistics

Table 6 provides the reference ranges in terms of the means, standard deviations, skewness, and kurtosis of the SOP2 scale scores for the total population as well as separately for gender and age groups in both nations.

Table 6

Reference Ranges of the SOP2 Scale Scores by Nation for the Total Population and Separately for Gender and Age Groups

Note. UK = United Kingdom (N = 468); DE = Germany (N = 474).

Further quality criteria

Owing to its good psychometric properties (reliability and validity) and its very short completion time (< 20 s), the instrument can be seen as highly economical.

We assessed the comparability of SOP2 across both nations via measurement invariance tests with multiple-group confirmatory factor analysis (MG-CFA; Vandenberg & Lance, 2000; Widaman & Reise, 1997). Metric invariance is given if the number of factors, the loading patterns, and the factor loadings are the same across groups, implying the comparability of correlations based on the latent factors across nations. Scalar invariance is given if, in addition, the item intercepts are the same across groups, implying the comparability of latent (co)variances and latent means across nations. Uniqueness invariance is given if, in addition, the residual variances are the same across groups, implying the comparability of manifest scale scores and correlations across nations without systematic bias (e.g., Bluemke, et al., 2016). In order to determine the level of measurement invariance, we used the cut-off values recommended by Chen (2007). According to these benchmarks, metric invariance must be rejected when ΔCFI [comparative fit index] ≤ −.010 in accordance with ΔRMSEA [root mean square error of approximation] ≥ .015, or ΔSRMR ≥ .030. Scalar and uniqueness invariance must be rejected when ΔCFI ≤ −.010 in accordance with ΔRMSEA ≥ .015, or ΔSRMR ≥ .010. A significant c² difference test is also an indicator of metric, scalar, and uniqueness non-invariance.

We assessed the comparability of SOP2 across the UK and Germany, across gender, and across age groups via measurement invariance tests. Because it is the most economical model, we based the measurement invariance tests on the essentially tau-equivalent unidimensional model with equal loadings (Model III) using MLR estimation. Therefore, the configural model and the metric model are equivalent. We identified the mean structure of the model by fixing the first intercept to 0 and the factor loadings to 1. As mentioned above, this model has zero degrees of freedom; therefore, no misfit can occur.

Table 7

Comparability of SOP2 Across Gender and Age Groups via Measurement Invariance Tests, by Nation

Note. MI = measurement invariance; UK = United Kingdom (N = 468, n_male = 222, n_female = 246, n₁₈₋₂₉ = 104, n₃₀₋₄₉ = 180, n₅₀₋₆₉ = 184); DE = Germany (N = 474, n_male = 237, n_female = 237, n₁₈₋₂₉ = 105, n₃₀₋₄₉ = 191, n₅₀₋₆₉ = 178). Δχ² is Satorra-Bentler corrected.

^a The intercept of the optimism item is freely estimated in age group 2 (30–49 years).

First, we assessed cross-national comparability. When comparing the scalar model with the metric model, SRMR as well as MLR-scaled CFI and RMSEA indicated that the scalar measurement invariance of SOP2 holds across the UK and Germany: Δχ²(1) = 0.162, p = .687, ΔCFI = .000, ΔBIC = –6.686, ΔRMSEA = .000, ΔSRMR = .004.[1] Uniqueness invariance did not hold when comparing the uniqueness model and the scalar model: Δχ²(3) = 15.911, p < .001, ΔCFI = –.088, ΔBIC = 8.455, ΔRMSEA = .102, ΔSRMR = .033.[2] The highest measurement invariance level we found—scalar measurement invariance—implies the comparability of latent (co)variances and latent means between both nations.

Second, we assessed cross-gender and cross-age comparability of SOP2 within both nations. Table 7 provides the results of these measurement invariance tests. In both nations, we found that uniqueness invariance holds for gender, implying the comparability of manifest scale scores (means and variances) and correlations across gender without systematic bias (e.g., Bluemke et al., 2016). For age, scalar invariance did not hold in both nations. Thus, the three age groups did not have the same point of origin. Freely estimating the intercept of the optimism item in age group two led to a good model fit, indicating partial scalar measurement variance of SOP2 across age. In the UK, the intercept of the optimism item was much lower in age group two (30–49 years) than in the other two age groups (18–29 and 50–69 years), whereas it was the opposite in Germany, namely a higher intercept for the intermediate age group two than for the youngest and oldest age groups.

Further literature

The scale was first published in the journal Measurement Instruments for the Social Sciences.

Acknowledgement

We would like to thank Melanie Partsch from GESIS – Leibniz Institute for the Social Sciences for helping us in preparing the data.

The study was funded by an internal grant provided by GESIS – Leibniz Institute for the Social Sciences.

Désirée Nießen, GESIS – Leibniz Institute for the Social Sciences, P.O. Box 12 21 55, 68072 Mannheim, Germany; e-mail: desiree.niessen@gesis.org

The dataset supporting the conclusions of this article is available in the GESIS SowiDataNet | datorium repository, https://doi.org/10.7802/2098.