Additional Examples

Additional Examples

This section replicates step-by-step the analyses presented in the chapter "Social Relations Designs with Roles" of the book on dyadic data analysis written by Kenny et al. (2006, pp. 223-262).

Datasets of Kenny et al. (2006)

The first dataset is based on a study published by Cook (2000), where 208 four-person US families rated their fear of rejection in relation to their family members in a round-robin design. This build in dataset is called four.person in the fSRM package. It contains one measure of fear of rejection for each of the 12 relationships, and will be used here to replicate with fSRM the standard four-person model analysis described in the book.

The second dataset, labeled as three.person in fSRM, uses the same family data with the exceptions that relationships involving the oldest child were dropped and two indicators for each relationship are now included. The SRM-means, as described by Kenny et al. (2006, p. 245), are also estimated.

The standard four-person model: SRM variances and covariances

The four.person data in long format from the fSRM package is used. This build-in dataset can be loaded by typing data(four.person) after loading the package. One can also download the original dataset in wide format from and then convert it to the desired format by means of the pop-up window. For details on the exact procedure consult the main paper. Please note that the labels of the roles differ from the main paper. The mother, father, oldest and youngest child are labeled as "m", "f", "c" and "y", respectively. Anxiety, the outcome of interest is abbreviated as "anx".Below we mimic the analyses of the four.person data presented by Kenny et al. (2006).

# load the fSRM package
library(fSRM)

## Warning: package 'lavaan' was built under R version 3.0.3

# load the build in dataset
data(four.person)
# inspect the data by showing the first few lines of the dataset
head(four.person)

## family.id actor.id partner.id anx
## 1 1 c f 1.000
## 2 1 c m 1.000
## 3 1 c y 1.167
## 4 1 f c 1.500
## 5 1 f m 1.000
## 6 1 f y 1.833

# assign the result of the SRM analyses to the new object "fSRM.4.1"
fSRM.4.1 <-fSRM(anx ~actor.id*partner.id |family.id, data = four.person)
# show the output by asking for "fSRM.4.1"
fSRM.4.1

## fSRM version 0.6
## ======
##
## SRM with roles (Roles: c, f, m, y); DVs = anx
##
## Model summary:
## ------
## lavaan (0.5-17) converged normally after 42 iterations
##
## Number of observations 208
##
## Estimator ML
## Minimum Function Test Statistic 45.312
## Degrees of freedom 47
## P-value (Chi-square) 0.543
##
## Model Fit:
## ------
## Chi2 (df=47) = 45.312, p = 0.543
## CFI = 1
## TLI / NNFI = 1.004
## RMSEA = 0 [0;0.043]; Test of close fit: p(data | true value == .05) = 0.982
##
##
## Variance decomposition:
## ------
## component estimate se z p.value sig ci.lower ci.upper
## 1 FE ~~ FE 0.039 0.018 2.206 p = .014 * 0.010 0.067
## 2 A.c ~~ A.c 0.214 0.031 6.854 p < .001 *** 0.163 0.265
## 3 A.f ~~ A.f 0.216 0.034 6.348 p < .001 *** 0.160 0.272
## 4 A.m ~~ A.m 0.163 0.032 5.112 p < .001 *** 0.110 0.215
## 5 A.y ~~ A.y 0.231 0.037 6.261 p < .001 *** 0.170 0.291
## 6 P.c ~~ P.c 0.064 0.020 3.182 p < .001 *** 0.031 0.097
## 7 P.f ~~ P.f 0.055 0.021 2.641 p = .004 ** 0.021 0.090
## 8 P.m ~~ P.m 0.044 0.017 2.544 p = .005 ** 0.016 0.072
## 9 P.y ~~ P.y 0.078 0.021 3.792 p < .001 *** 0.044 0.113
## 10 R.c.f ~~ R.c.f 0.203 0.034 5.958 p < .001 *** 0.147 0.259
## 11 R.c.m ~~ R.c.m 0.107 0.026 4.142 p < .001 *** 0.064 0.149
## 12 R.c.y ~~ R.c.y 0.205 0.034 6.045 p < .001 *** 0.149 0.260
## 13 R.f.c ~~ R.f.c 0.169 0.033 5.086 p < .001 *** 0.114 0.224
## 14 R.f.m ~~ R.f.m 0.613 0.071 8.589 p < .001 *** 0.495 0.730
## 15 R.f.y ~~ R.f.y 0.204 0.036 5.719 p < .001 *** 0.145 0.263
## 16 R.m.c ~~ R.m.c 0.222 0.038 5.815 p < .001 *** 0.159 0.285
## 17 R.m.f ~~ R.m.f 0.489 0.062 7.887 p < .001 *** 0.387 0.591
## 18 R.m.y ~~ R.m.y 0.336 0.048 7.049 p < .001 *** 0.258 0.415
## 19 R.y.c ~~ R.y.c 0.354 0.050 7.111 p < .001 *** 0.272 0.436
## 20 R.y.f ~~ R.y.f 0.334 0.048 6.978 p < .001 *** 0.256 0.413
## 21 R.y.m ~~ R.y.m 0.203 0.036 5.582 p < .001 *** 0.143 0.263
##
## (p-values are for one-sided tests for variances; confidence level for CIs is 90 %)
##
##
##
## Relative variance decomposition:
## ------
## Family Actor Partner Relationship Error Explained
## c_f 8 42 11 40 0 100
## c_m 10 53 11 26 0 100
## c_y 7 40 15 38 0 100
## f_c 8 44 13 35 0 100
## f_m 4 24 5 67 0 100
## f_y 7 40 15 38 0 100
## m_c 8 33 13 46 0 100
## m_f 5 22 7 66 0 100
## m_y 6 26 13 55 0 100
## y_c 6 34 9 52 0 100
## y_f 6 35 8 51 0 100
## y_m 7 45 9 39 0 100
## mean 7 36 11 46 0 100
##
##
## Generalized reciprocity (actor-partner covariances):
## ------
## component estimate se z p.value sig ci.lower ci.upper r
## 1 A.c ~~ P.c 0.059 0.019 3.140 p = .002 ** 0.022 0.096 .507
## 2 A.f ~~ P.f 0.003 0.019 0.155 p = .877 -0.034 0.040 .027
## 3 A.m ~~ P.m 0.034 0.017 1.962 p = .050 † 0.000 0.068 .400
## 4 A.y ~~ P.y 0.073 0.021 3.531 p < .001 *** 0.032 0.113 .542
##
##
## Dyadic reciprocity (relationship covariances):
## ------
## component estimate se z p.value sig ci.lower ci.upper
## 1 R.c.f ~~ R.f.c 0.041 0.024 1.712 p = .087 † -0.006 0.088
## 2 R.c.m ~~ R.m.c 0.025 0.022 1.119 p = .263 -0.019 0.069
## 3 R.c.y ~~ R.y.c 0.014 0.029 0.493 p = .622 -0.043 0.072
## 4 R.f.m ~~ R.m.f 0.191 0.049 3.868 p < .001 *** 0.094 0.288
## 5 R.f.y ~~ R.y.f 0.050 0.030 1.702 p = .089 † -0.008 0.109
## 6 R.m.y ~~ R.y.m -0.001 0.030 -0.041 p = .968 -0.059 0.057
## r
## 1 .220
## 2 .163
## 3 .053
## 4 .349
## 5 .193
## 6 -.005

The heading "SRM with roles" shows the label names of the roles of the different family members together with the dependent variable. Next, a brief model summary is provided under the heading "Model Summary", followed by the fit indices under the header "Model Fit", revealing an excellent fit. Indeed, the p-value of the chi-square test equals 0.543, indicating that the default SRM fits the data well. The other fit indices confirm this conclusion. Under the header "variance decomposition" the estimates of the SRM variances are presented. In the first column the different labels of the SRM components are shown. The character(s) preceding the dot represent the labels of the different SRM components. The character(s) following the dot define the role of the family member (e.g., A.m. denotes the actor effect of the mother, P.f. the partner effect of the father). In case of relationship effects the character after the first dot represents the rater in the dyadic measurement and the character after the second dot the person who is being rated (e.g., the relationship effect of the mother towards the father is R.m.f). The second column with heading "estimate" shows the variance estimates of each SRM component. The remaining columns show the standard errors (i.e., the se-column), the z-value (i.e., the z-column), the 1-sided p-value (i.e., the p.value-column), a flag for the significance of the 1-sided test (i.e., the sig-column) and the 90% confidence interval (i.e., the ci.lower- and ci.upper-column).

Regardless of whether one performs a 1-sided test at the 5% significance level or looks at the 2-sided 90% confidence intervals, one finds significant variance for all SRM-components. Note that the estimated SRM-variances are identical to those presented by Kenny et al. (2006). In particular the section under the heading "variance decomposition" equals the table on page 241.

In order to gain a better understanding of the size of each SRM component the section "Relative variance decomposition" is printed in the output. The columns reflect the relative importance of each SRM component both per observed relationship, as well as averaged over the different relationships. Note that summing up the different components results in 100%. As there is only one indicator per relationship measure in this example, the residual variance is contained in the relationship-specific variance and therefore set to zero. One can easily see that on average the majority of the variance is explained by the relationship effects (46%), followed by the actor effects (36%). Note that this section is identical to the table on page 242 of the dyadic data analysis book (Kenny et al., 2006). In addition, it is possible to retrieve a graphical representation by means of the following line of code:

plot(fSRM.4.1)

In the last two sections, the generalized and dyadic reciprocities are displayed[1]. Here, the "estimate"-column contains the covariances and the "r"-column the correlations. In this study significant correlations between the actor and partner effects of the oldest (r = .507, p < .01) and youngest child (r = .542, p < .001) are found, but not for the father (r =.027, p = .877) nor the mother (r = .400, p = .050). At the dyadic level on the other hand, only a significant reciprocity effect is observed for the mother-father relationship (r =.349, p < .001). This indicates that the more fear of rejection the mother experiences in relation to her husband, the more fear of rejection he reports about the relationship with his wife. Such relation specific reciprocities could not be found between the other family members. Once again, these results are identical to the ones described in the book of Kenny et al. (2006, p. 243).

One can additionally explore the presence of intragenerational similarities (abbreviated as IGSIM) by simply inserting the IGSIM-argument to the fSRM()-function. As shown below, one defines the two generations by entering between quotes the labels of the roles of the same generation in a separate vector. In the four.person dataset, the first generation comprises the mother (m) and father (f) and the other generation the two children (c and y). In the following code this SRM analyses is run and the output is requested. One can perform an omnibus test to test the overall presence of intragenerational similarities by comparing the chi-square test of the default model to this model. Additionally each intragenerational similarity can be separately assessed at the 5% significance level with the standard Wald-test.

# specify that m and f form one generation and c and y the other
fSRM.4.1.g <-fSRM(anx ~actor.id*partner.id |family.id,
data = four.person,
IGSIM=list(c("m", "f"), c("c", "y")))
# omnibus test IGSIM
anova(fSRM.4.1$fit, fSRM.4.1.g$fit)

## Chi Square Difference Test
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## fSRM.4.1.g$fit 43 5167 5284 43.2
## fSRM.4.1$fit 47 5162 5265 45.3 2.13 4 0.71

# show the output
fSRM.4.1.g

## fSRM version 0.6
## ======
##
## SRM with roles (Roles: c, f, m, y); DVs = anx
##
## Model summary:
## ------
## lavaan (0.5-17) converged normally after 51 iterations
##
## Number of observations 208
##
## Estimator ML
## Minimum Function Test Statistic 43.181
## Degrees of freedom 43
## P-value (Chi-square) 0.464
##
## Model Fit:
## ------
## Chi2 (df=43) = 43.181, p = 0.464
## CFI = 1
## TLI / NNFI = 1
## RMSEA = 0.004 [0;0.047]; Test of close fit: p(data | true value == .05) = 0.966
##
##
## Variance decomposition:
## ------
## component estimate se z p.value sig ci.lower ci.upper
## 1 FE ~~ FE 0.032 0.019 1.722 p = .043 * 0.001 0.063
## 2 A.c ~~ A.c 0.210 0.033 6.407 p < .001 *** 0.156 0.264
## 3 A.f ~~ A.f 0.233 0.037 6.348 p < .001 *** 0.172 0.293
## 4 A.m ~~ A.m 0.177 0.034 5.208 p < .001 *** 0.121 0.233
## 5 A.y ~~ A.y 0.225 0.038 5.906 p < .001 *** 0.162 0.288
## 6 P.c ~~ P.c 0.052 0.022 2.365 p = .009 ** 0.016 0.088
## 7 P.f ~~ P.f 0.069 0.024 2.826 p = .002 ** 0.029 0.109
## 8 P.m ~~ P.m 0.058 0.021 2.766 p = .003 ** 0.024 0.093
## 9 P.y ~~ P.y 0.068 0.023 2.976 p = .001 ** 0.030 0.105
## 10 R.c.f ~~ R.c.f 0.207 0.034 6.021 p < .001 *** 0.150 0.264
## 11 R.c.m ~~ R.c.m 0.107 0.026 4.114 p < .001 *** 0.064 0.150
## 12 R.c.y ~~ R.c.y 0.208 0.035 5.954 p < .001 *** 0.151 0.266
## 13 R.f.c ~~ R.f.c 0.168 0.033 5.060 p < .001 *** 0.113 0.223
## 14 R.f.m ~~ R.f.m 0.611 0.072 8.492 p < .001 *** 0.493 0.729
## 15 R.f.y ~~ R.f.y 0.202 0.036 5.648 p < .001 *** 0.144 0.261
## 16 R.m.c ~~ R.m.c 0.224 0.038 5.857 p < .001 *** 0.161 0.287
## 17 R.m.f ~~ R.m.f 0.481 0.062 7.742 p < .001 *** 0.379 0.583
## 18 R.m.y ~~ R.m.y 0.331 0.047 6.991 p < .001 *** 0.253 0.409
## 19 R.y.c ~~ R.y.c 0.350 0.050 7.060 p < .001 *** 0.268 0.431
## 20 R.y.f ~~ R.y.f 0.335 0.048 6.990 p < .001 *** 0.256 0.413
## 21 R.y.m ~~ R.y.m 0.202 0.036 5.548 p < .001 *** 0.142 0.262
##
## (p-values are for one-sided tests for variances; confidence level for CIs is 90 %)
##
##
##
## Relative variance decomposition:
## ------
## Family Actor Partner Relationship Error Explained
## c_f 6 41 13 40 0 100
## c_m 8 52 14 26 0 100
## c_y 6 41 13 40 0 100
## f_c 7 48 11 35 0 100
## f_m 3 25 6 65 0 100
## f_y 6 43 13 38 0 100
## m_c 7 37 11 46 0 100
## m_f 4 23 9 63 0 100
## m_y 5 29 11 54 0 100
## y_c 5 34 8 53 0 100
## y_f 5 34 10 51 0 100
## y_m 6 43 11 39 0 100
## mean 6 37 11 46 0 100
##
##
## Generalized reciprocity (actor-partner covariances):
## ------
## component estimate se z p.value sig ci.lower ci.upper r
## 1 A.c ~~ P.c 0.069 0.021 3.343 p = .001 ** 0.029 0.109 .660
## 2 A.f ~~ P.f -0.001 0.020 -0.060 p = .952 -0.040 0.037 -.009
## 3 A.m ~~ P.m 0.028 0.018 1.587 p = .113 -0.007 0.063 .279
## 4 A.y ~~ P.y 0.080 0.022 3.604 p < .001 *** 0.037 0.124 .652
##
##
## Dyadic reciprocity (relationship covariances):
## ------
## component estimate se z p.value sig ci.lower ci.upper
## 1 R.c.f ~~ R.f.c 0.042 0.024 1.740 p = .082 † -0.005 0.089
## 2 R.c.m ~~ R.m.c 0.025 0.023 1.112 p = .266 -0.019 0.070
## 3 R.c.y ~~ R.y.c 0.013 0.030 0.436 p = .663 -0.046 0.073
## 4 R.f.m ~~ R.m.f 0.185 0.050 3.686 p < .001 *** 0.086 0.283
## 5 R.f.y ~~ R.y.f 0.051 0.030 1.717 p = .086 † -0.007 0.109
## 6 R.m.y ~~ R.y.m -0.002 0.029 -0.074 p = .941 -0.060 0.056
## r
## 1 .224
## 2 .163
## 3 .049
## 4 .340
## 5 .196
## 6 -.008
##
##
## Intragenerational similarity:
## ------
## component estimate se z p.value sig ci.lower ci.upper r
## 80 A.f ~~ A.m 0.026 0.026 0.979 p = .327 -0.026 0.077 .126
## 81 P.f ~~ P.m 0.020 0.017 1.178 p = .239 -0.013 0.054 .318
## 82 A.c ~~ A.y -0.006 0.027 -0.229 p = .819 -0.058 0.046 -.028
## 83 P.c ~~ P.y -0.019 0.018 -1.074 p = .283 -0.054 0.016 -.320

We can conclude that there is no evidence for intragenerational similarities at the 5% significance level (p > .05). Although intragenerational similarities are not present in this data, it is suggested that family researchers always examine these effects in their data (Kenny et al., 2006; p. 245).

Another feature of this package is that it also allows to estimate the SRM means. In order to obtain these estimates, the argument means=TRUE needs to be added to the fSRM()-function.

# Additionally, ask for the mean SRM components
fSRM.4.1.m <-fSRM(anx ~actor.id*partner.id |family.id,
data = four.person,
means =TRUE)
# show the output
fSRM.4.1.m

## fSRM version 0.6
## ======
##
## SRM with roles (Roles: c, f, m, y); DVs = anx
##
## Model summary:
## ------
## lavaan (0.5-17) converged normally after 632 iterations
##
## Number of observations 208
##
## Estimator ML
## Minimum Function Test Statistic 45.312
## Degrees of freedom 47
## P-value (Chi-square) 0.543
##
## Model Fit:
## ------
## Chi2 (df=47) = 45.312, p = 0.543
## CFI = 1
## TLI / NNFI = 1.004
## RMSEA = 0 [0;0.043]; Test of close fit: p(data | true value == .05) = 0.982
##
##
## Variance decomposition:
## ------
## component estimate se z p.value sig ci.lower ci.upper
## 1 FE ~~ FE 0.039 0.018 2.206 p = .014 * 0.010 0.067
## 2 A.c ~~ A.c 0.214 0.031 6.854 p < .001 *** 0.163 0.265
## 3 A.f ~~ A.f 0.216 0.034 6.348 p < .001 *** 0.160 0.272
## 4 A.m ~~ A.m 0.163 0.032 5.112 p < .001 *** 0.110 0.215
## 5 A.y ~~ A.y 0.231 0.037 6.261 p < .001 *** 0.170 0.291
## 6 P.c ~~ P.c 0.064 0.020 3.182 p < .001 *** 0.031 0.097
## 7 P.f ~~ P.f 0.055 0.021 2.641 p = .004 ** 0.021 0.090
## 8 P.m ~~ P.m 0.044 0.017 2.544 p = .005 ** 0.016 0.072
## 9 P.y ~~ P.y 0.078 0.021 3.792 p < .001 *** 0.044 0.113
## 10 R.c.f ~~ R.c.f 0.203 0.034 5.958 p < .001 *** 0.147 0.259
## 11 R.c.m ~~ R.c.m 0.107 0.026 4.142 p < .001 *** 0.064 0.149
## 12 R.c.y ~~ R.c.y 0.205 0.034 6.045 p < .001 *** 0.149 0.260
## 13 R.f.c ~~ R.f.c 0.169 0.033 5.086 p < .001 *** 0.114 0.224
## 14 R.f.m ~~ R.f.m 0.613 0.071 8.589 p < .001 *** 0.495 0.730
## 15 R.f.y ~~ R.f.y 0.204 0.036 5.719 p < .001 *** 0.145 0.263
## 16 R.m.c ~~ R.m.c 0.222 0.038 5.815 p < .001 *** 0.159 0.285
## 17 R.m.f ~~ R.m.f 0.489 0.062 7.887 p < .001 *** 0.387 0.591
## 18 R.m.y ~~ R.m.y 0.336 0.048 7.049 p < .001 *** 0.258 0.415
## 19 R.y.c ~~ R.y.c 0.354 0.050 7.111 p < .001 *** 0.272 0.436
## 20 R.y.f ~~ R.y.f 0.334 0.048 6.978 p < .001 *** 0.256 0.413
## 21 R.y.m ~~ R.y.m 0.203 0.036 5.582 p < .001 *** 0.143 0.263
##
## (p-values are for one-sided tests for variances; confidence level for CIs is 90 %)
##
##
##
## Relative variance decomposition:
## ------
## Family Actor Partner Relationship Error Explained
## c_f 8 42 11 40 0 100
## c_m 10 53 11 26 0 100
## c_y 7 40 15 38 0 100
## f_c 8 44 13 35 0 100
## f_m 4 24 5 67 0 100
## f_y 7 40 15 38 0 100
## m_c 8 33 13 46 0 100
## m_f 5 22 7 66 0 100
## m_y 6 26 13 55 0 100
## y_c 6 34 9 52 0 100
## y_f 6 35 8 51 0 100
## y_m 7 45 9 39 0 100
## mean 7 36 11 46 0 100
##
##
## Generalized reciprocity (actor-partner covariances):
## ------
## component estimate se z p.value sig ci.lower ci.upper r
## 1 A.c ~~ P.c 0.059 0.019 3.140 p = .002 ** 0.022 0.096 .507
## 2 A.f ~~ P.f 0.003 0.019 0.155 p = .877 -0.034 0.040 .027
## 3 A.m ~~ P.m 0.034 0.017 1.962 p = .050 † 0.000 0.068 .400
## 4 A.y ~~ P.y 0.073 0.021 3.531 p < .001 *** 0.032 0.113 .542
##
##
## Dyadic reciprocity (relationship covariances):
## ------
## component estimate se z p.value sig ci.lower ci.upper
## 1 R.c.f ~~ R.f.c 0.041 0.024 1.712 p = .087 † -0.006 0.088
## 2 R.c.m ~~ R.m.c 0.025 0.022 1.119 p = .263 -0.019 0.069
## 3 R.c.y ~~ R.y.c 0.014 0.029 0.493 p = .622 -0.043 0.072
## 4 R.f.m ~~ R.m.f 0.191 0.049 3.868 p < .001 *** 0.094 0.288
## 5 R.f.y ~~ R.y.f 0.050 0.030 1.702 p = .089 † -0.008 0.109
## 6 R.m.y ~~ R.y.m -0.001 0.030 -0.041 p = .968 -0.059 0.057
## r
## 1 .220
## 2 .163
## 3 .053
## 4 .349
## 5 .193
## 6 -.005
##
##
## Mean structure
## ------
## factor estimate se z p.value sig ci.lower ci.upper
## 1 FE 1.838 0.027 67.407 p < .001 *** 1.793 1.883
## 2 A.c -0.134 0.033 -3.998 p < .001 *** -0.189 -0.079
## 3 A.f 0.103 0.034 3.000 p = .003 ** 0.046 0.159
## 4 A.m -0.087 0.033 -2.638 p = .008 ** -0.141 -0.033
## 5 A.y 0.117 0.035 3.328 p < .001 *** 0.059 0.175
## 6 P.c 0.022 0.025 0.850 p = .395 -0.020 0.064
## 7 P.f 0.038 0.025 1.511 p = .131 -0.003 0.079
## 8 P.m -0.169 0.024 -6.996 p < .001 *** -0.208 -0.129
## 9 P.y 0.109 0.026 4.256 p < .001 *** 0.067 0.152
## 10 R.c.f -0.003 0.021 -0.143 p = .886 -0.038 0.032
## 11 R.c.m -0.059 0.020 -2.866 p = .004 ** -0.092 -0.025
## 12 R.c.y 0.062 0.023 2.631 p = .009 ** 0.023 0.100
## 13 R.f.c -0.066 0.022 -3.019 p = .003 ** -0.102 -0.030
## 14 R.f.m 0.116 0.026 4.432 p < .001 *** 0.073 0.159
## 15 R.f.y -0.050 0.022 -2.262 p = .024 * -0.086 -0.014
## 16 R.m.c -0.028 0.023 -1.189 p = .235 -0.066 0.011
## 17 R.m.f 0.040 0.026 1.536 p = .125 -0.003 0.082
## 18 R.m.y -0.012 0.024 -0.509 p = .611 -0.051 0.027
## 19 R.y.c 0.094 0.026 3.581 p < .001 *** 0.051 0.137
## 20 R.y.f -0.037 0.024 -1.533 p = .125 -0.076 0.003
## 21 R.y.m -0.057 0.023 -2.506 p = .012 * -0.095 -0.020

The SRM means are presented under the header "Mean structure". It shows the different SRM means (i.e., the factor-column), together with the estimated means (i.e., the estimate-column), the standard error (i.e., the se-column), the z-value (i.e., the z-column), the 2-sided p-value (i.e., p.value-column) and the 95% confidence interval (i.e., the ci.lower- and ci.upper-column). Those means correspond to the ones presented in Table 9.14 in the book of Kenny et al. (2006, p. 259). Additionally, one can request a graphical decomposition of the average dyadic measurements into these SRM-means by typing the following line. Please note that between brackets the fitted SRM object is specified.

plot(fSRM.4.1.m, means = T)

The Three Person Model: SRM Means and Multiple Indicators.

In this section we mimic the analyses of the three person example of Kenny et al. (2006, p. 245). When only three family members are considered, it is not possible to estimate all the SRM components. By default fSRM constrains the family variance to zero in a three-person family. However, the user can specify alternative models by adding the drop-argument to the fSRM()-function. One can choose to alternatively drop all the actor effects (and corresponding reciprocities), all the partner effects (and corresponding reciprocities) or the generalized reciprocities by adding drop= followed by "actor", "partner" or "GR", respectively.

Furthermore, when it is desired to partition the relationship variance from the error variance, multiple indicators for each dyadic measurement are indispensable. For the attachment data analyzed here, Cook (2000) created a first indicator by averaging the first half of the items of the scale and the second indicator by averaging the second half of the items. These two measures of relationship anxiety are labeled "anx1" and "anx2" in the three.person dataset. The following fSRM()-call models the default three person model with two indicators for the three.person dataset, and additionally requests the means.

# load the dataset
data(three.person)
# show the first few lines of the dataset
head(three.person)

## family.id actor.id partner.id anx1 anx2
## 1 1 f m 1.000 1.000
## 2 1 f y 1.667 2.000
## 3 1 m f 1.000 1.000
## 4 1 m y 1.667 1.333
## 5 1 y f 3.000 4.000
## 6 1 y m 4.667 3.000

# fit the model
fSRM.3.2 <-fSRM(anx1/anx2 ~actor.id*partner.id |family.id,
data = three.person,
means=TRUE)

## Three-member families: Dropping family variance per default.

# request the content of the fitted object
fSRM.3.2

## fSRM version 0.6
## ======
##
## SRM with roles (Roles: f, m, y); DVs = anx1, anx2
##
## Model summary:
## ------
## lavaan (0.5-17) converged normally after 246 iterations
##
## Number of observations 208
##
## Estimator ML
## Minimum Function Test Statistic 59.792
## Degrees of freedom 24
## P-value (Chi-square) 0.000
##
## Model Fit:
## ------
## Chi2 (df=24) = 59.792, p = 0
## CFI = 0.973
## TLI / NNFI = 0.927
## RMSEA = 0.085 [0.058;0.112]; Test of close fit: p(data | true value == .05) = 0.019
##
##
## Variance decomposition:
## ------
## component estimate se z p.value sig ci.lower ci.upper
## 1 A.f ~~ A.f 0.190 0.048 3.945 p < .001 *** 0.111 0.269
## 2 A.m ~~ A.m 0.206 0.048 4.280 p < .001 *** 0.127 0.286
## 3 A.y ~~ A.y 0.298 0.047 6.414 p < .001 *** 0.222 0.375
## 4 P.f ~~ P.f 0.072 0.046 1.552 p = .060 † -0.004 0.147
## 5 P.m ~~ P.m 0.072 0.043 1.696 p = .045 * 0.002 0.142
## 6 P.y ~~ P.y 0.142 0.039 3.614 p < .001 *** 0.077 0.206
## 7 R.f.m ~~ R.f.m 0.517 0.086 5.995 p < .001 *** 0.375 0.659
## 8 R.f.y ~~ R.f.y 0.134 0.062 2.169 p = .015 * 0.032 0.236
## 9 R.m.f ~~ R.m.f 0.394 0.083 4.744 p < .001 *** 0.257 0.530
## 10 R.m.y ~~ R.m.y 0.181 0.061 2.983 p = .001 ** 0.081 0.280
## 11 R.y.f ~~ R.y.f 0.213 0.067 3.179 p < .001 *** 0.103 0.323
## 12 R.y.m ~~ R.y.m 0.055 0.057 0.956 p = .170 -0.039 0.149
##
## (p-values are for one-sided tests for variances; confidence level for CIs is 90 %)
##
##
##
## Relative variance decomposition:
## ------
## Family Actor Partner Relationship Error Explained
## f_m 0 21 8 57 14 86
## f_y 0 30 23 21 26 74
## m_f 0 24 8 45 23 77
## m_y 0 31 21 27 21 79
## y_f 0 39 9 28 24 76
## y_m 0 47 11 9 33 67
## mean 0 32 13 31 24 76
##
##
## Generalized reciprocity (actor-partner covariances):
## ------
## component estimate se z p.value sig ci.lower ci.upper r
## 1 A.f ~~ P.f 0.028 0.036 0.783 p = .434 -0.042 0.099 .242
## 2 A.m ~~ P.m 0.094 0.035 2.708 p = .007 ** 0.026 0.161 .766
## 3 A.y ~~ P.y 0.094 0.031 3.062 p = .002 ** 0.034 0.155 .458
##
##
## Dyadic reciprocity (relationship covariances):
## ------
## component estimate se z p.value sig ci.lower ci.upper
## 1 R.f.m ~~ R.m.f 0.157 0.064 2.456 p = .014 * 0.032 0.282
## 2 R.f.y ~~ R.y.f 0.029 0.048 0.598 p = .550 -0.066 0.124
## 3 R.m.y ~~ R.y.m -0.071 0.044 -1.622 p = .105 -0.157 0.015
## r
## 1 .348
## 2 .172
## 3 -.715
##
##
## Mean structure
## ------
## factor estimate se z p.value sig ci.lower ci.upper
## 1 FE 1.894 0.032 59.803 p < .001 *** 1.842 1.946
## 2 A.f 0.093 0.034 2.683 p = .007 ** 0.036 0.149
## 3 A.m -0.067 0.036 -1.852 p = .064 † -0.127 -0.008
## 4 A.y -0.025 0.042 -0.595 p = .552 -0.095 0.044
## 5 P.f 0.061 0.028 2.155 p = .031 * 0.014 0.108
## 6 P.m -0.092 0.031 -3.004 p = .003 ** -0.142 -0.042
## 7 P.y 0.031 0.036 0.847 p = .397 -0.029 0.090
## 8 R.f.m 0.023 0.015 1.568 p = .117 -0.001 0.048
## 9 R.f.y -0.023 0.015 -1.568 p = .117 -0.048 0.001
## 10 R.m.f -0.023 0.015 -1.568 p = .117 -0.048 0.001
## 11 R.m.y 0.023 0.015 1.568 p = .117 -0.001 0.048
## 12 R.y.f 0.023 0.015 1.568 p = .117 -0.001 0.048
## 13 R.y.m -0.023 0.015 -1.568 p = .117 -0.048 0.001

First, note that the family variance is not shown as it is automatically fixed to zero. All SRM variances are significant at the one-sided 5% significance level, with exception of the partner effect of the father and the relationship effect of the youngest child with the mother. The results are identical to those presented in the Dyadic Data Analysis book (Kenny et al., 2006; p. 249). A graphical representation of the variance decomposition can be obtained by simply typing plot(fSRM.3.2). Because we have two indicators here, the relationship variance and residual error variance can be separated but the family variance is assumed to be zero.

In the last paragraph of the output of the fSRM.3.2 model, the SRM means are presented under the header "Mean structure". Note that although the variance of the family effect is set to zero, the mean of the family effect is calculated. A common mean is assumed for the two different indicators.

Testing whether all the actor, partner or relationship means are equal across roles can easily be done by the function equalMeans(). For the previously described example:

equalMeans(fSRM.3.2)

## Wald df p.value sig
## H0: Equal actor means 8.450 2 p = .015 *
## H0: Equal partner means 11.168 2 p = .004 **
## H0: Equal relationship means 2.458 1 p = .117

We find evidence against the equality in both actor and partner means between the three different roles at the 5% significance level.

While the equalMeans() function examines if each of the SRM components differs significantly over all roles, one can additionally ask for pairwise comparisons of the mean actor (and partner) effects between two roles. This is done by adding pairwise = TRUE in the fSRM()-function. For the fSRM.3.2.-example this results in the following output:

# fit the model and ask for the pairwise comparisons
fSRM.3.2.p <-fSRM(anx1/anx2 ~actor.id*partner.id |family.id,
data = three.person,
means=TRUE, pairwise=TRUE)

## Three-member families: Dropping family variance per default.

# request the output
fSRM.3.2.p

## fSRM version 0.6
## ======
##
## SRM with roles (Roles: f, m, y); DVs = anx1, anx2
##
## Model summary:
## ------
## lavaan (0.5-17) converged normally after 246 iterations
##
## Number of observations 208
##
## Estimator ML
## Minimum Function Test Statistic 59.792
## Degrees of freedom 24
## P-value (Chi-square) 0.000
##
## Model Fit:
## ------
## Chi2 (df=24) = 59.792, p = 0
## CFI = 0.973
## TLI / NNFI = 0.927
## RMSEA = 0.085 [0.058;0.112]; Test of close fit: p(data | true value == .05) = 0.019
##
##
## Variance decomposition:
## ------
## component estimate se z p.value sig ci.lower ci.upper
## 1 A.f ~~ A.f 0.190 0.048 3.945 p < .001 *** 0.111 0.269
## 2 A.m ~~ A.m 0.206 0.048 4.280 p < .001 *** 0.127 0.286
## 3 A.y ~~ A.y 0.298 0.047 6.414 p < .001 *** 0.222 0.375
## 4 P.f ~~ P.f 0.072 0.046 1.552 p = .060 † -0.004 0.147
## 5 P.m ~~ P.m 0.072 0.043 1.696 p = .045 * 0.002 0.142
## 6 P.y ~~ P.y 0.142 0.039 3.614 p < .001 *** 0.077 0.206
## 7 R.f.m ~~ R.f.m 0.517 0.086 5.995 p < .001 *** 0.375 0.659
## 8 R.f.y ~~ R.f.y 0.134 0.062 2.169 p = .015 * 0.032 0.236
## 9 R.m.f ~~ R.m.f 0.394 0.083 4.744 p < .001 *** 0.257 0.530
## 10 R.m.y ~~ R.m.y 0.181 0.061 2.983 p = .001 ** 0.081 0.280
## 11 R.y.f ~~ R.y.f 0.213 0.067 3.179 p < .001 *** 0.103 0.323
## 12 R.y.m ~~ R.y.m 0.055 0.057 0.956 p = .170 -0.039 0.149
##
## (p-values are for one-sided tests for variances; confidence level for CIs is 90 %)
##
##
##
## Relative variance decomposition:
## ------
## Family Actor Partner Relationship Error Explained
## f_m 0 21 8 57 14 86
## f_y 0 30 23 21 26 74
## m_f 0 24 8 45 23 77
## m_y 0 31 21 27 21 79
## y_f 0 39 9 28 24 76
## y_m 0 47 11 9 33 67
## mean 0 32 13 31 24 76
##
##
## Generalized reciprocity (actor-partner covariances):
## ------
## component estimate se z p.value sig ci.lower ci.upper r
## 1 A.f ~~ P.f 0.028 0.036 0.783 p = .434 -0.042 0.099 .242
## 2 A.m ~~ P.m 0.094 0.035 2.708 p = .007 ** 0.026 0.161 .766
## 3 A.y ~~ P.y 0.094 0.031 3.062 p = .002 ** 0.034 0.155 .458
##
##
## Dyadic reciprocity (relationship covariances):
## ------
## component estimate se z p.value sig ci.lower ci.upper
## 1 R.f.m ~~ R.m.f 0.157 0.064 2.456 p = .014 * 0.032 0.282
## 2 R.f.y ~~ R.y.f 0.029 0.048 0.598 p = .550 -0.066 0.124
## 3 R.m.y ~~ R.y.m -0.071 0.044 -1.622 p = .105 -0.157 0.015
## r
## 1 .348
## 2 .172
## 3 -.715
##
##
## Mean structure: Indices starting with 'C.' are pairwise comparisons between roles
## ------
## factor estimate se z p.value sig ci.lower ci.upper
## 1 FE 1.894 0.032 59.803 p < .001 *** 1.842 1.946
## 2 A.f 0.093 0.034 2.683 p = .007 ** 0.036 0.149
## 3 A.m -0.067 0.036 -1.852 p = .064 † -0.127 -0.008
## 4 A.y -0.025 0.042 -0.595 p = .552 -0.095 0.044
## 5 P.f 0.061 0.028 2.155 p = .031 * 0.014 0.108
## 6 P.m -0.092 0.031 -3.004 p = .003 ** -0.142 -0.042
## 7 P.y 0.031 0.036 0.847 p = .397 -0.029 0.090
## 8 R.f.m 0.023 0.015 1.568 p = .117 -0.001 0.048
## 9 R.f.y -0.023 0.015 -1.568 p = .117 -0.048 0.001
## 10 R.m.f -0.023 0.015 -1.568 p = .117 -0.048 0.001
## 11 R.m.y 0.023 0.015 1.568 p = .117 -0.001 0.048
## 12 R.y.f 0.023 0.015 1.568 p = .117 -0.001 0.048
## 13 R.y.m -0.023 0.015 -1.568 p = .117 -0.048 0.001
## 14 C.means.A.f.m 0.160 0.057 2.807 p = .005 ** 0.066 0.254
## 15 C.means.P.f.m 0.153 0.046 3.289 p = .001 ** 0.076 0.229
## 16 C.means.A.f.y 0.118 0.068 1.730 p = .084 † 0.006 0.229
## 17 C.means.P.f.y 0.030 0.057 0.529 p = .597 -0.064 0.125
## 18 C.means.A.m.y -0.042 0.071 -0.597 p = .551 -0.159 0.074
## 19 C.means.P.m.y -0.122 0.061 -2.016 p = .044 * -0.222 -0.023

Note that the pairwise comparisons are displayed in the section "Mean Structure". For example, C.means.A.f.m represents the pairwise comparison between the actor effects of the father and the mother. For this dataset, the mean actor and partner effects of both parents differ significantly, together with the mean partner effect of the mother and the youngest child (all p < .05).

References

Cook, W. L. (2000). Understanding Attachment Security in Family Context. Journal of Personality and Social Psychology, 78(2), 285-294. doi:10.1037//0022-3514.78.2.285

Kenny, D. A., Kashy, D. A., & Cook, W. L. (2006). Dyadic Data Analysis. New York: The Guilford Press.

[1]Please note that it is only useful to interpret these reciprocities when both corresponding variance component differ significantly from zero.