resulted in a chi-square of 72.01 with 46 degrees of freedom. These results are not
directly comparable to the single sample solutions because twice as many parame-
ters are being estimated. While this is a statistically significant chi-square value, all
measures of fit are acceptable. The fit indices for the congeneric model are GFI =
.96, NNFI = .93, and RMSEA = .04.
Figure 2.
Comparison for father-child relationship quality and child well-being by
gender of child.
One conceptual limitation of the congeneric model is that it fails to test whether
the variables have the same meaning for father-daughter
dyads as they do for father-
son dyads. There are two ways of testing this. One method requires the same load-
ings of indicators on latent variables to be equal (constrained as invariant); the sec-
ond requires the loadings and also the corresponding error terms to be identical.
These are called
tau
equivalent and parallel form models, respectively, and results
for these models are shown in Table 2. Comparing fit indices,
the tau equivalent
model is not significantly worse than the congeneric model, and the parallel form
model is not significantly worse than the tau equivalent model. Just as important,
there is not a significant difference between these models. These results furnish
remarkable evidence that the meaning of the four variables—communication, con-
nection, relationship quality, and child well-being—can be described as equivalent
in both types of dyads. Given the measurement invariance for father-daughter and
C
ONNECTION AND
C
OMMUNICATION
201
Connection
Communication
Child
Well-Being
GFI = .95, NNFI = .95, RMSEA = .03
χ
2
(63) = 89.43;
p
= .05.
Unstandardized coefficients with
standardized coefficients
in parentheses, father-son followed by father-daughter.
*
p
< .05. **
p
< .01. ***
p
< .001.
R
2
Father-Child Relationship Quality .13.
R
2
Child Well-Being .06 (father-son) and .10 (father-
daughter).
.41***
(.37, .37)
.00
(.00, .00)
.09***
(.25, .32)
Father-Child
Relationship
Quality
father-son dyads, the loadings and error terms were constrained to be equal in the
subsequent analysis.
Since both types of dyads were shown to have equivalent measurement models,
a fourth step is to compare their structural models. This fourth step gets to the heart
of our research objective to evaluate the effects of the four variables on each other
and to compare these across father-daughter and father-son dyads. To do this, the
fourth model imposed the restriction that all structural parameters (gamma and beta
estimates) are identical for both types of dyads (see last row in Table 2).
This final
model resulted in a chi-square value of 89.43 with 63 degrees of freedom, which
was not significantly worse than the parallel form model that did not impose this
restriction. The difference in chi-square between these models was just .034, which,
with three degrees of freedom, has a significance level of .952. Of most importance,
goodness-of-fit indices for the final measure all demonstrated a good model fit
within an acceptable range with GFI = .95, NNFI = .95, and the RMSEA = .03.
The measurement model for the fourth step is available from the senior author.
In summary, all of the factor loadings were statistically significant and substantial.
The correlations and standard deviations from which all models were produced
appear in the appendix.
Our primary interest is in the structural solution, which is presented in Table 3
(also shown graphically in Figure 2). This is the result
that addresses our hypothe-
ses. In making comparisons across groups, father-daughter versus father-son, it is
appropriate to compare unstandardized estimates. In making comparisons of differ-
ent effects within groups, it is appropriate to compare standardized coefficients.
Because the final structural model constrained the unstandardized effects so that
they were equal, the corresponding unstandardized coefficients are identical for both
types of dyads. It is noted that when the unstandardized
coefficients are equal, the
standardized coefficients are necessarily unequal except when the corresponding
variances are equal.
Positive associations were hypothesized for the relationships between the latent
constructs of connection and communication on father-child relationship quality, as
well as for the relationship between quality and child well-being. The results indi-
cate that communication has virtually no effect (0.00, ns) on relationship quality for
both types of father-adolescent dyads. However, connection has a significant and
equal effect (0.41,
p
< .001) on relationship quality for both types of dyads. The
standardized coefficients for both father-son and father-daughter dyads is .37,
reflecting a moderate effect. Relationship quality has a significant effect (0.09,
p
<
.001) on child well-being for both types of dyads. The standardized coefficient for
father-son dyads is .25, and for father-daughter dyads the
standardized coefficient is
.32. Both of these show a moderate effect.
Thus, the results showed no association between father-child communication
and father-child relationship quality for both genders, when father-child connection
is controlled. Father-son and father-daughter dyads both showed a moderate, posi-
tive relationship between father-child connection and relationship quality. The
R
2
value of .13 signifies that the variable of connection accounts for approximately
13% of the variance in relationship quality for both father-son and father-daughter
dyads. Finally, the effect of relationship quality on child well-being was .25 for
B
ROTHERSON
et al.
202
