There are several tests involved in measurement model assessment. Table 3.7 presents the output of these tests. Before presenting that output, I describe the tests in some detail. Further discussion regarding measurement model assessment can be found in the Appendix.
Assessment of the measurement model aims to verify that the model as specified contains unidimensional and internally consistent blocks of manifest variables (i.e., survey items for a particular construct make up a block of manifest variables), and that those items reliably predict their assigned latent variable with both convergent validity and discriminant validity.
I have already established unidimensionality of each block of manifest variables in the factor analyses above. Internal consistency provides an indication that items in a block are homogenous. Internal consistency can be inferred from reliability statis- tics. In a PLS modelling context, there are two reliability statistics that are commonly used: Cronbach’s alpha (Cronbach, 1951, α) and Dillon-Goldstien’s rho (Werts, Linn & Joreskog, 1974, ρc). The critical values for α and ρcare equivalent. A value of 0.70 is
considered a minimum1, whereas a 0.80 is viewed as a more strict minimum threshold recommended for established scales (Hair, Anderson, Tatham & Black, 1998; Nunnally, 1978).
According to Chin (1998), Dillon-Goldstien’s rho (ρc) is preferable to Cronbach’s
alpha (α) when conducting PLS analyses. This is because α assumes tau equivalence of the indictors; items are given an equal weighting when calculating the α statistic. The ρcstatistic is computed using the loadings of the items on a construct.
The α statistic will always be less than or equal to ρc for any given dataset and
block of items. When item loadings do not vary within a block, α and ρc will be close.
When item loadings vary within a block, α will be less than ρc. So, although α is
more conservative, ρcrelies on the same assumption as PLS regarding latent variable
creation, and may be seen as more conceptually true to PLS. I will use only Dillon- Goldstien’s rho (ρc) throughout.
Indicator reliability is a test of the extent to which the variance of an indicator can be explained by a latent variable. It is analogous to a factor loading. The generally
accepted threshold is a loading greater than 0.707 (Hulland, 1999). Loadings less than 0.707 are allowable in newly developed scales, or when a block contains a large number of items, but in no case should it be less than 0.40 (Chin, 1998; Falk & Miller, 1992; Hulland, 1999). At values above 0.707, the shared variance between a construct and an indicator exceeds measurement error variance (Bohrnstedt, 1970). The critical value then is whether more than half of an indicator’s variance is explained by the latent factor, as opposed to error.
Convergent reliability is inferred when the Average Variance Extracted (AVE) is greater than 0.50 (Fornell & Larcker, 1981). The AVE is a measure of the variance that a latent variable has accounted for in the block as a whole. An AVE of 0.50 means that a latent variable has accounted for, on average, 50% of the variance in a block of items. Note that this 50% cut-off is the same critical value as that underlying the recommendation of 0.707 minimum indicator loading.
The final customary tests in measurement model assessment relate todiscriminant validity. For an item to exhibit discriminant validity, it should load more highly on its own latent variable than it does on other latent variables (Fornell & Larcker, 1981). Chin (1998) recommends also that a latent variable should attract the highest loadings from its own indicators, in a rank fashion. This can be easily assessed using a ‘cross- loadings table’, which contains all loadings, for all items, on all latent variables.
Fornell & Larcker (1981) also recommend the comparison of AVE to squared la- tent variable-latent variable correlations, called a Fornell-Larcker Table. This directly compares the average aggregated item-variance explained by a latent variable to the variance that latent variable shares with other latent variables. The AVE of a latent variable should exceed all squared correlations of that latent variable and other latent variables. In practice, this is quite a low standard, and is usually easily met. An alter- native presentation of the Fornell-Larcker Table is to compare the square root of the AVE with the latent variable to latent variable correlations. A rank order comparison can still be made, but the information regarding the latent variable correlations is more easily interpreted, as compared to the squared correlation.
The just-described metrics for measurement model assessment (of the model in Figure 3.4, using dataset T) are reported in Table 3.7. I discuss these in turn below.
Composite reliability (ρc) easily exceeds the 0.70 cut-off for all blocks of items, indi-
DL GBL PGL AS LS | ¯XLi| ρc 0.78 0.87 0.91 0.87 0.88 – DL01 0.73 -0.14 -0.07 -0.21 -0.21 0.16 DL02 0.71 -0.09 -0.13 -0.25 -0.18 0.16 DL03 0.76 -0.21 -0.07 -0.19 -0.21 0.17 GBL01 -0.13 0.88 0.46 0.23 0.19 0.25 GBL02 -0.23 0.88 0.37 0.27 0.28 0.29 PGL01 -0.10 0.36 0.86 0.22 0.11 0.20 PGL02 -0.13 0.34 0.78 0.25 0.17 0.22 PGL03 -0.10 0.40 0.85 0.21 0.16 0.22 PGL04 -0.02 0.41 0.80 0.22 0.10 0.19 PGL05 -0.14 0.42 0.79 0.29 0.17 0.26 AS01 -0.28 0.23 0.28 0.88 0.42 0.30 AS02 -0.24 0.27 0.23 0.88 0.32 0.27 LS01 -0.23 0.22 0.15 0.40 0.87 0.25 LS02 -0.19 0.22 0.17 0.28 0.80 0.22 LS03 -0.26 0.24 0.12 0.39 0.86 0.25 | ¯XLc| 0.17 0.27 0.18 0.27 0.21 – AVE 0.54 0.77 0.67 0.77 0.71 – – 0.74 – – – – – GBL -0.21 0.88 – – – – PGL -0.12 0.47 0.82 – – – AS -0.30 0.28 0.29 0.88 – – LS -0.27 0.27 0.17 0.43 0.84 –
Table 3.7: Measurement Model Assessment for the R-BIGL12+ using Dataset T -
DL=Disbelief in Luck; GBL=General Belief in Luck; PGL=Belief in Personal Good Luck; AS=Astrological Superstitions; LS=Lucky Superstitions. See Tables 3.4 and 3.5 for item content.
Provided at top are Composite Reliabilities (Dillon-Goldstein’s rho; ρc). In the middle
section are item loadings (in bold) and cross-loadings, for each item in the model (item labels are to the left). Vertically down the right side of the middle section, | ¯XLi| is the
average of the absolute values of the cross-loadings for a given item. Horizontally across the bottom of the middle section, | ¯XLc| is the average of the absolute values of the cross-
loadings for a given construct. In the lower section is the Fornell-Larcker table with AVE’s (horizontally in bold), the square root of the AVE (diagonally in bold) and latent variable to latent variable correlations.
the recommended level of 0.50, indicating that convergent validity is acceptable. In- dicator loadings all exceed the recommended value of 0.707, supporting the view that the indicators are reliable in their measurement of their assigned latent variables.
As regards discriminant validity, all cross-loadings are below the recommended threshold of 0.50, and in every case items load highest on their own latent variables. The items GBL01 and GBL02 load a little high on PGL (0.46 and 0.37). The items PGL03, PGL04 and PGL05 load a little high on GBL (0.40, 0.41, and 0.42). These cross-loadings are not surprising given the strong correlation between GBL and PGL of 0.47. The Fornell-Larcker table in the bottom section raises no concerns. The square root of the AVE (bolded on the diagonal) easily exceeds the latent variable correlations (both down the column and across the row from the bolded number on the diagonal).