5. Results
The means and S.D. for all variables in the analysis are presented in Table 3. Note that price
has been scaled by a factor of 0.01 to make it more consistent with the scales of the other vari-
ables in the analysis. Large differences in scales among variables can produce operational prob-
lems in Prelis 2 Joreskog and Sorbom, 1996a such that the software adjusts the scaling of vari-
ables automatically.
From Table 3, it is evident that the mean level of agreement with the belief statements were
higher in the two samples having the lowest mean household income. The mean WTP responses
were also lower in these samples, despite the equivalent price vectors used in the Perth, Mel-
bourne and Sydney CV surveys. Across samples, highest agreement occurred for Protest 2 i.e. ‘the
government should use existing revenue to pay for stormwater pollution controls’ and Protest 3 i.e.
‘We would be able to afford better protection of receiving waters already if the government did
not waste so much money’.
Variance – covariance matrices were computed for each watershed sample, together with their
asymptotic variances – covariances. These matrices were calculated with bootstrapping so as to ob-
tain the best estimates given the relatively small sample sizes Joreskog and Sorbom, 1996a. The
inclusion of the asymptotic variances – covariances allowed the model parameters to be estimated,
without an assumption of multivariate normality. This method enabled the calculation of x
2
Sa- torra and Bentler, 1988 and S.E. Yuan and
Bentler, 1997 corrected for non-normality. The model was estimated in each group using
Lisrel 8.20 Joreskog and Sorbom, 1996b prior to tests of invariance across samples. If a model fails
to demonstrate an acceptable approximation to the data in each group, then invariance tests
would be unwarranted Byrne et al., 1989.
In all four samples, the error variances for Protest 2 and 3 were allowed to covary to test for
the presence of a residual attitude toward govern- ment. Moreover, the error variances of the single
indicator variables were fixed to appropriate val- ues based on the item variances and reliabilities.
WTP was assumed to have a reliability of 0.70 based on published data Mitchell and Carson,
1989; Jorgensen et al., 1998. Dichotomous choice WTP reliability studies have reported test – retest
coefficients ranging from 0.30 Stevens et al., 1994 to 0.75 Heberlein, 1986. Whitehead et al.
1995, using the equivalent forms method and assuming parallel measures, reported a reliability
coefficient of 0.92 for dichotomous choice WTP. If only congeneric measures are assumed for their
data a more reasonable assumption in their case, the reliability coefficient was approximately 0.85
since a measure cannot correlate with another variable more than the square root of its reliabil-
ity Carmines and Zeller, 1979.
The reliability of the income measure was as- sumed to be 0.80 on the basis of previous research
Withey, 1954; Stein et al., 1993. Importantly, the value at which the error variance is fixed does not
affect a model’s x
2
statistic, but does influence the variances and error variances of the relevant la-
tent variables. In general, assumptions of higher reliability in endogenous variables e.g. the perfect
reliability assumed in regression analysis produce larger disturbance terms within latent variable
equations. This is because higher indicator reli- ability suggests that more of the endogenous la-
tent variable’s variance is explicable by the exogenous latent variables.
The overall fit of the model in each sample was satisfactory Table 4. The Satorra – Bentler scaled
x
2
SBx
2
was associated with a probability level \
0.05 in all samples. The goodness-of-fit index GFI Tanaka Huba, 1984 which is based on
the SBx
2
statistic, had values in excess of the 0.90 criterion proposed by Browne and Cudeck 1993.
The comparative fit index CFI Bentler, 1990, like the GFI, can range between 0 and 1, with
higher values associated with better fit. The CFI indicates the extent to which the model fits better
than a baseline model in which all latent variables are orthogonal. The baseline model had 36 de-
grees of freedom and x
2
values, ranging from 286.10 Perth to 394.56 Melbourne.
Another different type of fit measure — the root mean square error of approximation RM-
SEA Steiger, 1990 — takes into account the
Table 4 Model goodness-of-fit within samples
Sample Fit statistic
Melbourne Sydney
Perth Brisbane
SBx
2
df
a
33.25 24 28.90 24
27.32 24 22.05 24
0.96 GFI
0.96 0.96
0.97 0.98
0.98 0.95
1.00 CFI
0.05 RMSEA
0.03 0.03
0.00 0.00, 0.08
90 CI RMSEA 0.00, 0.07
0.00, 0.07 0.00, 0.06
0.04 0.05
0.05 0.04
SRMR
a
All x
2
statistics are not significant at PB0.05.
error of approximation in the population as well as the model degrees of freedom. A 90 confidence
interval can be calculated for the RMSEA so as to determine the precision of the statistic. Browne
and Cudeck 1993 contended that point estimates
B 0.08 represent reasonable errors of approxima-
tion in the population. The RMSEA estimates in Table 4 indicate that even the upper bounds of
the confidence intervals are not \ 0.08 in each sample.
Finally, the standardized root mean-square residuals SRMR also indicated acceptable levels
of fit in each sample. The SRMS is the average standardized fitted residual and indicates the dis-
crepancy between the sample variance – covariance matrix and the fitted matrix. In each sample, the
SRMR was below the criterion of 0.08 proposed by Hu and Bentler 1995.
The factor loadings and standard errors for attitude are given in Table 5. The single indicator
loadings are not shown since they were fixed to unity and, therefore, not subject to significance
tests. When a variance covariance matrix is used in structural equation modeling, it is necessary to
fix the scale of the latent variables for the purpose of identification Anderson and Gerbing, 1988.
Setting one of the factor loadings l
ij
of a latent variable to unity does this. The scale of the latent
variable is then fixed to equal the scale of the reference indicator.
Protest 6 was chosen as the reference indictor for the attitude factor, so its value was also equal
to one in each sample. The parameter estimates, shown in the table, indicate little variation over
samples, and all were significantly different to zero at P B 0.001. In all samples, attitude toward
paying was best reflected in responses to the belief statements Protest 4 i.e. ‘It is my right to have
cleaner stormwater and not something I should have to pay extra for’ and Protest 1 i.e. It is
unfair to ask me to pay more money for stormwa- ter pollution controls’. The items having the
smallest relationship with attitude were Protest 2 i.e. ‘The government should use existing revenue
to pay for stormwater pollution controls’ and Protest 3 i.e. ‘We would be able to afford better
protection of receiving waters already, if the government did not waste so much money’.
Table 5 Unstandardized attitude loadings within samples
a
Sample Indicator
Melbourne Sydney
Brisbane Perth
1.08 Protest 1
0.98 0.98
1.25 0.11
0.13 0.15
0.18 0.48
0.68 Protest 2
0.63 0.72
0.09 0.12
0.12 0.10
0.54 0.63
Protest 3 0.47
0.49 0.12
0.12 0.11
0.12 1.27
1.14 Protest 4
1.06 1.18
0.16 0.11
0.13 0.10
1.01 Protest 5
0.84 0.90
0.98 0.16
0.10 0.11
0.13 Protest 6
b
1.00 1.00
1.00 1.00
a
All coefficients are significantly different to zero at PB 0.001.
b
Loading set to unity for purposes of identification.
Table 6 Unstandardized path coefficients within samples
Error Exogenous variables
R
2
Endogenous variables Price
Income Attitude
Perth −
0.09 –
0.44 0.07
Attitude 0.03
0.04 0.05
0.10 0.04
− 0.31
0.09 WTP
0.47 −
0.12 0.02
0.06 0.02
0.02 Melbourne
− 0.15
Attitude –
0.16 0.65
0.20 0.03
0.06 0.11
− 0.10
WTP 0.01
− 0.22
0.06 0.53
0.02 0.04
0.01 0.03
Sydney −
0.16 –
0.05 0.72
Attitude 0.20
0.06 0.03
0.13 0.01
− 0.22
− 0.13
0.08 WTP
0.47 0.02
0.04 0.02
0.03 Brisbane
− 0.19
Attitude –
0.02 0.54
0.21 0.03
0.04 0.12
0.01 −
0.29 −
0.03 0.11
WTP 0.38
0.02 0.01
0.05 0.02
PB0.05; PB0.01; PB0.001.
There was a significant covariance between the uniquenesses of Protest 2 and 3 in the Perth
o
2,3
= 0.19, P B 0.001 and Sydney o
2,3
= 0.25,
P B 0.01 samples. These items seem to measure both attitude toward paying and a smaller atti-
tude toward government factor. The error covari- ance was not significant in either the Brisbane or
Melbourne samples, suggesting that attitudes to- ward government efficiency and toward paying
for stormwater pollution were statistically indis- tinguishable in these groups.
Estimates for parameters in the structural com- ponent of the model are shown in Table 6. In
each sample, price and attitude were negatively associated with WTP, and the path from income
to attitude was also negative and significantly different to 0. In the Melbourne sample only,
attitude increased with price, suggesting that a small proportion of the protest variability in this
group was a function of the magnitude of the prices associated with the intervention.
Income had a positive direct effect on WTP in the Perth group only. However, this effect was
marginally significant at 0.05 t = 2.09. When the model was re-estimated in Perth, after setting this
path to 0, the increase in the x
2
was not signifi- cantly different to the model that included the
path. The x
2
difference test for the two models was equal to 2.05 with 1 degree of freedom and
was less than the critical value of 3.84 at the 5 level of significance. Since the path between In-
come and WTP g
4,2
in Fig. 1 was not robust in any of the samples, it was removed from the
model. The model explained between 7 Perth and
21 Brisbane of the variance in attitude toward paying for stormwater pollution abatement. Vari-
ability in WTP was better accounted for in com- parison with coefficients of determination ranging
from 0.38 Brisbane to 0.53 Melbourne. How- ever, these statistics are dependent upon the mag-
nitudes of the reliability estimates for the WTP
and income observed variables. An assumption of higher lower reliability would result in smaller
larger coefficients of determination. For exam- ple, when the reliability of WTP was assumed to
be perfect as is the case in regression analysis the coefficients of determination ranged from 0.26
Brisbane to 0.37 Melbourne.
The model was compared across samples in order to assess the extent to which attitude to-
ward paying and its relationship with WTP was invariant with respect to methodological differ-
ences in the stormwater CV surveys. The results of the various tests of each matrix of parameters
comprising the structural equation model are pre- sented in Table 7.
The comparisons across samples began with a test of equal form or configural invariance in
which only the reference indicators of the factors were held invariant over samples. The x
2
value for Model 1 served as a baseline for comparison with
Model 2 in which all the loadings of the factors were held invariant over groups. The x
2
difference between Model 1 and 2 was less than the critical
value of 11.07. This indicated that setting the attitude factor loadings to be equal across the
samples i.e. metric invariance did not produce a significant decrease in fit.
Next, the error variances of the attitude items were held invariant over samples Model 3 in
Table 7. The resulting x
2
was significantly larger than that associated with model 2. The modifica-
tion index in Lisrel 8.20, indicated that freeing the error variance of Protest 3 o
3
would produce a better model fit. Model 4 was the same as Model
3, except that o
3
was estimated in each group. The difference in the x
2
for Models 2 and 4 was not statistically significant.
With subsequent tests of invariance, only the relationship between price and WTP g
4,1
was significantly different between samples. This can
be seen in Table 8 showing the unstandardized coefficients for the structural parameters in each
group. In addition to the nonsignificant x
2
for this model Model 7, other measures of overall fit
were also satisfactory GFI = 0.96, CFI = 0.99, RMSEA = 0.01, SRMR = 0.06. Thus, despite
methodological variations in the CV surveys, the same attitude was present in all samples and had
an equivalent relationship with WTP in all four cases.
It is possible to scale the latent variables so that a weighted average of their covariance matrix is a
correlation matrix. This is different from stan- dardizing within groups where the covariance ma-
trix of the latent variables is rescaled to a correlation matrix for each group. When a
weighted average of the four sample covariance matrices is scaled to a correlation matrix, the
metric invariance of the unstandardized solution holds for the standardized solution, such that a
scale common to all groups is retained Joreskog and Sorbom, 1996b. By scaling the coefficients in
Model 7 to a standardized metric common across groups, all parameters could be compared be-
tween and within groups.
The parameter estimates of the standardized common metric solution for the final model
Model 7 are given in parentheses in Table 8. The largest predictor of WTP was attitude toward
paying. This variable had a larger effect on WTP in each sample than did price. However, the price
associated with the interventions was a relatively strong predictor of WTP in all samples, with the
exception of the Brisbane group. WTP in the Brisbane sample was much more a function of
whether
respondents felt
favorably disposed
toward paying for stormwater pollution abate- ment.
Table 7 Tests of invariance across city samples
SBx
2 D
df SBx
2
df Model description
1. Equal form
– 114.76 100
124.90 115 10.14 5
2. L invariant factor
loadings l
3,1
to l
3,5
3. U invariant error vari-
31.22 18 156.12 133
ances 5.59 15
4. o
3
free protest 3 130.49 130
5. B invariant path B
4,3
134.19 133 3.7 3
6. G invariant paths g
3,1
154.88 142 20.69 9
g
3,2
g
4,1
g
4,2
7. g
4,1
free price to WTP 141.21 139
7.02 3
Table 8 Unstandardized and common metric path coefficients between samples
a
Error Exogenous variables
R
2
Endogenous variables Price
Income Attitude
Perth −
0.15 –
0.46 0.15
Attitude 0.05
−0.40 0.10
0.66 −
0.12 WTP
– −
0.26 0.10
0.39 −0.56
0.64 −0.47
Melbourne −
0.15 –
0.05 0.61
Attitude 0.18
−0.40 0.87
0.01 −
0.26 −
0.10 0.06
WTP 0.54
−0.40 −0.56
0.38 Sydney
− 0.15
– 0.64
Attitude 0.20
0.05 −0.40
0.10 0.91
– −
0.26 0.08
WTP 0.50
− 0.12
−0.56 0.5.
−0.50 Brisbane
− 0.15
– 0.05
0.62 Attitude
0.15 −0.40
0.88 0.10
– −
0.26 −
0.03 0.11
WTP 0.35
−0.56 0.70
−0.12
a
Common metric coefficients are given in parentheses. PB0.05. PB0.01. PB0.001.
6. Discussion
Kategori