WHY ARE THE HYPOTHESIZED ASSOCIATIONS NOT SIGNIFICANT?
A THREE-WAY INTERACTION?
Robert A. Ping, Wright State University, Dayton, OH
There is little guidance for estimating a latent variable (LV) "three-way" interaction (e.g., XZW). The paper explores these variables, and suggests their specification. It also provides a pedagogical example to suggest the utility of three-way interactions. Hypothesizing these LV’s is discussed, their reliability is derived, a remedy for their nonessential ill-conditioning (their high correlations with X, Z and W) in real-world data is suggested, and an approach to interpreting them is illustrated.
"Two-way" interactions in structural equation analysis (SEM) such as XZ, XW and ZW in
1) Y = β0 + β1X + β2Z + β3W + β4XZ + β5XW + β6ZW + β7XZW + ζY ,
where X, Z, W and Y are non-categorical variables, β1 through β7 are unstandardized structural coefficients, β0 is an intercept (typically ignored in SEM), and ζY is the structural disturbance (estimation error) term, have received considerable theoretical attention (see Aiken and West 1991). They also have been investigated with survey data in several substantive literatures (see Aiken and West 1991, p. 2; Bohrnstedt and Marwell 1978; Jaccard, Turissi and Wan 1990, p. 79; Lubinski and Humphreys 1990; and Podsakoff, Tudor, Grover and Huber 1984 for partial lists of citations).
However, non-categorical "three-way" interactions in survey data (e.g., XZW in Equation 1) have received little attention. They also have yet to appear in published SEM models, perhaps because there is little guidance for estimating them. This paper sheds additional light on these LV’s and their estimation. Specifically, it discusses their specification, estimation and interpretation. Along the way their utility is illustrated, and a remedy for a property of these LV's in real world data that apparently is not well known, their nonessential ill-conditioning, is proposed. Hypothesizing a three-way interaction is discussed, their reliability is derived, and an approach to interpreting these LV’s is illustrated.
To help motivate this topic, we will skip ahead to a pedagogical example. In studies of firms' Reactions to Dissatisfaction in Business-to-Business relationships, the relationship of the subject's Switching Costs (SC’s) (costs to replace the primary supplier) with the subject's Opportunism (OPP) (guileful self-interest seeking) was observed to be non-significant (NS) in Ping (1993), and positive in Ping (2007). Similarly, the OPP association with the subject's Investment (INV) (expenditures to maintain the relationship) was NS in Ping (1993), and positive in Ping (2007). This suggested the possibility that INV and SC were being moderated (Ping 1996d). Subsequently, it was judged plausible that INV moderated SC (argument omitted). In a reanalysis of one of the above studies' data sets, however, INVxSC was not significant.
Another possibility was that Alternatives (ALT) (attractive replacement relationships) moderated an interaction between INV and SC. Specifically, it was plausible that there was a three-way interaction among ALT, INV and SC: ALTxINVxSC. In the reanalysis data set ALTxINVxSC was significant.
Next, we will discuss two-way interactions, which will lead to a proposed specification of a three-way interaction involving LV’s, then the details of the above pedagogical example (that will illustrate their estimation and interpretation).
INTERACTIONS IN SURVEY MODELS
It will be important later to briefly discuss two-way interaction specification, in order to lay the groundwork for specifying XZW. There have been several proposals for specifying two-way LV interactions including (1) Kenny and Judd 1984; (2) Bollen 1995; (3) Jöreskog and Yang 1996; (4) Ping 1995; (5) Ping 1996a; (6) Ping 1996b; (7) Jaccard and Wan 1995; (8) Jöreskog 2000; (9) Wall and Amemiya 2001; (10) Mathieu, Tannenbaum and Salas 1992; (11) Algina and Moulder 2001; (12) Marsh, Wen and Hau 2004; (13) Klein and Moosbrugger 2000/Schermelleh-Engle, Kein and Moosbrugger 1998/Klein and Muthén 2002; and (14) Moulder and Algina 2002.
These proposed techniques are based on the Kenny and Judd (1984) product-of-indicators proposal (x1z1, x1z2, ... x1zm, x2z1, x2z2, ... x2zm, ... xnzm, where n and m are the number of indicators of X and Z respectively). However, in theoretical model tests using real world survey data, where models with several, usually over-determined, LV’s (i.e., LV's with four or more indicators), are the rule, specifying XZ with all the Kenny and Judd product indicators typically produces model-to-data fit problems. Specifically, in Techniques 1 and 5, the resulting specification of XZ in its single construct measurement model usually will not fit the data (i.e., this specification of XZ is inconsistent with the data), and full measurement and structural models containing this specification of XZ can exhibit unacceptable model-to-data fit.
Several proposals use subsets of the Kenny and Judd (1984) product indicators, or indicator aggregation, to avoid these inconsistency problems (Techniques 3, 5, 7, 9, 11, 12 and 14). Unfortunately, omitting Kenny and Judd product indicators raises questions about the face or content validity of the resulting interaction. Specifically, if all the indicators of X are not present in the itemization of XZ, is XZ still the "product of the LV X and the LV X"? (techniques 3, 7, 9, 11, 12 and 14). This specification has additional drawbacks: the reliability of XZ is unknown for a partially itemized XZ. As we shall see, the formula for the reliability of XZ is a function of X and Z with all their items (see Bohrnstedt and Marwell 1978). Further, a procedure for determining which product indicators to retain is unknown. And, deleting Kenny and Judd product indicators can produce interpretation problems because the X in XZ is no longer operationally the same as X in Equation 1, for example.
Some proposed techniques do not involve Maximum Likelihood estimation, or commercially available estimation software (Techniques 2, 6 and 13). And, several of these proposals have not been evaluated for possible bias and lack of efficiency (i.e., Techniques 8 and 10).
A SINGLE INDICATOR
The following will rely on the Ping (1995) proposal for specifying XZ because it has the fewest of the above drawbacks. This proposed specification uses a single indicator for XZ that is the product of sums of the indicators for X and Z. Specifically, for X with the indicators x1 and x2, and Z with indicators z1 and z2 the single indicator of XZ would be xz = (x1+x2)(z1+z2). Ping (1995) suggested that under the Kenny and Judd (1984) normality assumptions, a loading, λxz, and measurement error variance, θεxz, for this single indicator are
2) λxz = ΛXΛZ ,
2a) θεxz = ΛX2Var(X)θZ + ΛZ2Var(Z)θX + θXθZ ,
where ΛX = λx1 + λx2, Var indicates error disattenuated variance, θX = Var(εx1) + Var(εx2), εx1 is the measurement error of x1, εx2 is the measurement error of x2, Λz = λz1 + λz2, θz = Var(εz1) + Var(εz2), λxz = ΛXΛZ, θεxz = (ΛX)2Var(X)θZ + (ΛZ)2Var(Z)θX + θXθZ , and λ and θ are loadings and measurement error variances. The indicators xi and zj are mean-centered by subtracting the mean of xi, for example, from xi in each case, and the single indicator of XZ, xz, becomes
xczc = [Σ(xiu - M(xiu))] [Σ(zju - M(zju))] ,
where xiu and zju are uncentered indicators (denoted by the superscript “u”), M denotes a mean, and Σ is a sum taken before any multiplication. Centering xi and zj not only helps provide simplified Equations 2) and 2a), it reduces the high correlation or nonessential ill-conditioning (Marquardt, 1980; see Aiken and West, 1991) of X and Z with XZ that produces unstable (inefficient) structural coefficient estimates that can vary widely across studies.
Using simulated data sets and data conditions that were representative of those encountered in surveys, Ping's (1995) results suggested that the proposed single indicator for an interaction produced unbiased and consistent coefficient estimates.
This single-indicator specification can be estimated in two steps. First, the data for the single indicator of XZ is created by computing the sum of the indicators of X times the sum of the indicators of Z in each case. Next, the measurement parameters in Equations 2 and 2a (i.e., λx1, λx2, etc., Var(εx1), etc., Var(X), etc.) are estimated in a measurement model (MM) that excludes XZ. Then, the loadings and measurement error variances for XZ’s (λxz and θxz ) are computed using equations 2 and 2a, and using these parameter estimates. Finally, specifying the calculated loadings and error variances λxz and θxz for the product indicator as fixed values, the structural model is estimated.
If the structural model estimates of the measurement parameters for X and Z (i.e., λx1, λx2, etc., Var(εx1), etc., Var(X), etc.) do not approximate those from the MM (i.e., equality in the first two decimal places) the loadings and error variances of the product indicator are recomputed using the structural model estimates of the equation 2 and 2a measurement parameters. Experience suggests that with consistent LV’s zero to two of these iterations are sufficient to produce exact estimates (i.e., equal to “direct” estimates of XZ-- see Ping 1995).
THREE WAY INTERACTIONS
XZW also could be specified as the product of sums of indicators (e.g., xzw = (Σxi)(Σzj)(Σwk)). However, mean centering xi, zj and wk does not reduce the multicollinearity (nonessential ill-conditioning--see Marquardt 1980, and Aiken and West 1991) between XZW and X, Z and W that typically occurs in real-world data. Unfortunately, the bias from this multicollinearity can produce an apparently non-significant (NS) three-way interaction. For example, in the pedagogical example ALTxINVxSC specified with mean-centered X, Z and W was NS.
An alternative specification that avoids multicollinearity bias is to use the indicator
3) xc(zw)c = [Σxiu - M(Σxiu)][(Σzju)(Σwku) - M((Σzju)(Σwku))] ,
where xiu, zju and wku are uncentered indicators (denoted by the superscript “u”), M denotes a mean, and Σ is a sum taken before any multiplication, to spec the 3 way Xc(ZuxWu)c.
The loading and error variance of xc(zw)c is derived in Appendix A..
Returning to the pedagogical example, after developing a plausible argument for ALT moderating the Y-INVxSC association, the three-way interaction ALTc(INVuxSCu)c in
5) Y = aSAT + bALTc + cINVu + dSCu + gALTc(INVuxSCu)c + h(INVuxSCu)c + ζ ,
where a, b, etc. are structural coefficients, was specified in a measurement model for Equation 5 to gage each LV's psychometrics.
ALTc(INVuxSCu)c was specified by computing the Equation 3 single indicator, altc(inv∙sc)c = [Σaltiu - M(Σaltiu)][(Σinvju)(Σscku) - M((Σinvju)(Σscku))] in each case, then the loading and measurement error variance was computed using the Equations 4e and 4f (altiu, invju and sck are the uncentered indicators of ALT, INC and SC respectively, and M denotes a mean).
Specifically, estimates of the loading and measurement error variance of ALTc(INVuxSCu)c were computed using the loadings, variances, covariances and measurement error variances of ALTc, INVu and SCu from earlier measurement models (MM's) without the interactions, and SPSS estimates of the means.
Then, the measurement model for Equation 5 was estimated, using LISREL 8 and Maximum Likelihood estimation. The resulting loadings and measurement error variances of SAT, ALT, INV, SC and Y were sufficiently similar to those from previous MM's containing just SAT, ALT, INV, SC and Y, that a second Equation 5 measurement model estimation (to revise the computed interaction loadings and measurement error variances) was judged not necessary (see Ping 1996a). Because the Equation 5 measurement model fit the data, ALTc(INVuxSCu)c was judged to be externally consistent. Finally, ALTc(INVuxSCu)c was judged to be trivially internally consistent.
RELIABILITY AND VALIDITY
The reliability of Xc(ZW)c is unknown, and it is derived in Appendix B.
The reliability of ALTc(INVuxSCu)c was computed to be 0.89. Specifically, SPSS values for the means of INV and SC, the square root of the MM variances for the standard deviations of INV and SC, the MM value for the correlation between INV and SC, SPSS reliabilities of ALT, INV and SC, and the calculated error-disattenuated correlation of ALTc and (INVxSC)c (see Ping 1996c) were substituted into Equation 6. Then, ALTc(INVuxSCu)c was judged to be valid.
Next, the structural model was estimated using LISREL 8 and Maximum Likelihood estimation, and the abbreviated results are shown in Table B. Then, the Y associations with ALT, INV and SC were interpreted to account for their moderation. First, the plausible moderation of INVxSC by ALT was trivially "confirmed" by the significant ALTc(INVuxSCu)c coefficient in Table B.) Then, the (now moderated) Y-ALT association was interpreted. Specifically, Equation 5 was "factored" to produce the (full) structural coefficient ("simple slope"--see Aiken and West 1991) of ALTc:
Y = aSAT + (b + g(INVuxSCu)c)ALTc + cINVu + dSCu + h(INVuxSCu)c + ζ
and the coefficient of ALT was interpreted using (b' + g(INVuxSCu)c)--the results are shown in Table C.
Next, Equation 5 was re-factored to produce the moderated coefficient of INV,
Y = aSAT + bALTc + cINVu + dSCu + gALTc(INVuxSCu)c + h(INVuxSCu)c + ζ
= aSAT + bALTc + cINVu + dSCu + gALTc(INVuxSCu-m) c + h(INVuxSCu-m)c + ζ
= aSAT + bALTc + dSCu
+ cINVu + gALTc(SCu-m/INVu)cINVu + h(SCu-m/INVu)c INVu + ζ
= aSAT + bALTc + cINVu + dSCu
+ (c + gALTc(SCu-m/INVu)c + h(SCu-m/INVu)c)INVu + ζ ,
where INV is non zero and m = M((Σzju)(Σwku)), and INVu was interpreted (see Table D).
Finally, Equation 5 was again re-factored to produce a moderated coefficient of SC in,
Y = aSAT + bALTc + cINVu
+ (d + gALTc(INVu-m/SCu) + h(INVu-m/SCu)c)SCu + ζ
for interpretation, and the results (not reported) were similar to Table D.
Because these specifications of a three-way interaction have not been formally evaluated for possible bias and inconsistency, their threshold for significance proabably should be conservative (e.g., |t-value| > 2.10).
ALTcxINVcxSCc- was non significant while ALTc(INVuxSCu)c was significant, as previously reported. In addition, the ALTc(INVuxSCu)c structural coefficient was different from the INVc(ALTuxSCu)c and SCc(INVuxALTu)c structural coefficients (unstandardized beta=0.23, 0.01 and 0.29; t=2.64, 0.83 and 1.90 respectively). This suggests there are several specifications of a three-way interaction: an “all centered” specification (ALTcxINVcxSCc), and three “permutation” three-way interaction specifications, ALTc(INVuxSCu)c, INVc(ALTuxSCu)c and SCc(INVuxALTu)c. This, plus some authors’ preference for including ALTxINV, ALTxSC and INVxSC, appears to suggest that a “proper” disconfirmation test of a three-way interaction should include all the relevant two-way interactions, ALTxINV, ALTxSC and INVxSC, plus the “all centered” three-way interactions, and ALTcxINVcxSCc, plus the “permutation” three-ways interactions ALTc(INVuxSCu)c, INVc(ALTuxSCu)c and SCc(INVuxALTu)c.