Measurement model testing

The measurement model (see Chapter 4, Figure 4.3) was first tested before testing the structural model. The latent variable, mathematics anxiety, was measured by three indicators: students’ report of their anxiety (sanxiety), their confidence in learning mathematics (sconfidence), and their effectance motivation (smotivation). The second latent variable, attitude towards mathematics, was measured by four indicators namely, students’ report of their general attitude towards maths and success in mathematics (sattitude), students’ report of the usefulness of mathematics (susefulness), students’ report of their parents’ attitude towards maths and child’s maths study

(sparentattitude), and parents’ report of their own attitude towards maths and child’s maths study (pattitude). Lastly, the latent variable, parental involvement, was measured by a set of four indicators, students’ report of their parents’ behaviours in child’s maths learning (sparentbeh), parents’ report of their own behaviours in child’s maths learning (pparentbeh), their mathematics anxiety (panxiety), and expectations of their child’s mathematics learning and future education (pexpectation). The Rasch person estimates for each of the variables were used in this analysis.

As explained in detail in Section 4.8 in Chapter 4, a measurement model was tested for invariance using multi-group confirmatory factor analysis (MGCFA). The MGCFA was conducted in the sequence of configural invariance, metric invariance, scalar invariance, and invariance of latent means. Configural invariance required an identical factor structure across the two groups. Metric invariance needed the same factor loadings across the groups considered. Scalar invariance required invariant factor loadings and

intercepts across the two groups. Lastly, latent mean differences were tested by

constraining the latent mean for the male group to zero as the male group was kept as the reference group (Hancock, 1997). The model fit was assessed by examining the relative chi-square (i.e.,χ2_{/df) value, as the chi-square values were sensitive to sample}

size (Byrne, 2001; R. B. Kline, 1998). In addition, the comparative fit index (CFI) and the root mean square error approximation (RMSEA) were used to assess the overall fit of the model. A CFI value of 0.95 or greater (Hu & Bentler, 1999), and an RMSEA value of 0.08 or less was considered as a good fit of the model (Browne & Cudek, 1993). In the case of nested models, difference in CFI (i.e., ∆CFI) was considered to evaluate the fit of the models (G. W. Cheung & Rensvold, 2002). A non-significant change in CFI (∆CFI) with a value of less than 0.01 was used for testing the invariance (G. W. Cheung & Rensvold, 2002).

The hypothesised measurement model was first tested for configural invariance. That is, the model was tested to check whether the factor structure was identical for both boys and girls. The model provided a poor fit with fit values, χ2_{(98) = 196.98, p = 0.00,}

χ2_{/df = 2.0, CFI = 0.79, RMSEA = 0.098 (90% CI = 0.078, 0.118). The indicators}

students’ report of their parent attitude towards maths and child’s maths study and the parents’ report of their attitude towards maths and child’s maths study were removed due to poor factor loadings. The resultant measurement model was then evaluated and the model provided an acceptable fit, χ2 (60) = 76.36, p< 0.10,χ2/df = 1.27, CFI = 0.96, RMSEA = 0.051 (90% CI = 0.000, 0.083). All factor loadings showed identical pattern across boys and girls. Thus configural invariance has been established, showing that the model has the same factor structure for both boys and girls. The modified measurement model showing the relationship between the observed variables and the latent variables is shown in Figure 5.13.

Mathematicsanxiety sconfidence e1 sanxiety e2 smotivation e3 sparentbeh e4 pparentbeh e5 sattitude e10 susefulness e9 Achievement measure 1 panxiety e6 pexpectation e7 1 Attitudestowardsmathematics Parentalinvolvement 1 1 1 1 1 1 1 1 1

Figure 5.13. The modified measurement model showing the relationship between the

observed variables and the latent variables

The modified measurement model was next tested for metric invariance, where all the factor loadings were constrained to be equal for both boys and girls. The model fit was acceptable,χ2 _{(66) = 86.67, p < 0.05,χ}2_{/df = 1.31, CFI = 0.94, RMSEA = 0.055}

(90% CI = 0.009, 0.084). The differences in fit values between the configural and metric invariance model were significant, ∆χ2 _{(6) = 10.31, p< .001 ∆CFI = 0.01 (G. W. Cheung}

& Rensvold, 2002), showing that the full metric invariance was not established, indicating that all the items did not have the same relative contribution to the measure across the two groups. Consequently, partial metric invariance was tested by freely estimating the factor loadings one by one. The analysis yielded acceptable fit for the model, χ2_{(63) =}

78.91, p= 0.10, χ2_{/df = 1.25, CFI = 0.96, RMSEA = 0.049 (90% CI = 0.000, 0.080)}

mathematics anxiety and expectation of their child’s mathematics learning and future education were variant across groups of boys and girls. The differences in the fit values between this model and the configural invariant model were not significant, ∆χ2 _{(3) =}

2.55, ns, ∆CFI = 0.00 (G. W. Cheung & Rensvold, 2002), indicating that partial invariance was achieved.

The next step in the testing of the measurement model was to test for scalar invariance. Scalar invariance was tested with the factor loadings and intercepts equal across both groups. The model fit was acceptable, χ2 _{(72) = 102.75, p < .010, χ}2_{/df =}

1.43, CFI = 0.92, RMSEA = 0.064 (90% CI = 0.032, 0.091). The differences in the fit values between the scalar invariant model and the partial metric invariant model were significant, ∆χ2 _{(9) = 23.84,p<.001, ∆CFI = 0.04 (G. W. Cheung & Rensvold, 2002),}

showing that the full scalar invariance was not supported. Therefore, partial scalar invariance was evaluated by freely estimating the intercepts one after another. The fit values of the final model were good,χ2 _{(69) = 84.89, p = 0.10, χ}2_{/df = 1.23, CFI = 0.96,}

RMSEA = 0.047 (90% CI = 0.000, 0.078), where the intercepts of students’ report of their general attitude towards maths and success in mathematics, parents’ report of their anxiety, and parents’ report of their behaviours in their child’s maths learning were variant across the groups. Thus, partial scalar invariance was achieved based on the non-significant change in the fit indices, ∆χ2(6) = 5.98, ns, ∆CFI = 0.00 (G. W. Cheung & Rensvold, 2002).

Once partial scalar invariance was established, the model was tested to examine whether the latent construct means were different across the two groups. The invariance of latent means was achieved by fixing the latent means for the boys to be zero. The fit values obtained showed a good fit, χ2(66) = 81.05, ns, χ2/df = 1.23, CFI = 0.96, RMSEA = 0.047 (90% CI= 0.000, 0.078). The changes in fit values between this model

and the partial scalar invariant model were non-significant, ∆χ2 (3) = 3.84, ns, ∆CFI = 0.00 (G. W. Cheung & Rensvold, 2002), showing that all the latent means were invariant for the groups.