Code
Wide format: 100 rowsDistinguishable dyads: SEM in wide format
The three-model sequence and the k-pattern tests
The wide-format SEM is the classical specification for the distinguishable APIM. The three nested models — unconstrained, slopes-equal, fully constrained — give you three likelihood ratio tests, each of which has a clean substantive interpretation.
This tutorial reproduces the Olsen & Kenny (2006) specification and the Kenny & Ledermann (2010) k-pattern tests.
Setup
Model 1: unconstrained (all paths free)
The unconstrained model estimates a separate slope for every path. Actor (male) and partner (female) intercepts, slopes, and residual variances are all free to differ.
Code
model_unconstrained <- '
satisfaction_a ~ a_h*wnc_a + p_h*wnc_p + ar_h*recovery_a +
pr_h*recovery_p + c_h*has_children + d_h*dual_earner
satisfaction_p ~ a_w*wnc_p + p_w*wnc_a + ar_w*recovery_p +
pr_w*recovery_a + c_w*has_children + d_w*dual_earner
satisfaction_a ~ int_a*1
satisfaction_p ~ int_p*1
satisfaction_a ~~ var_a*satisfaction_a
satisfaction_p ~~ var_p*satisfaction_p
wnc_a ~~ wnc_p
recovery_a ~~ recovery_p
satisfaction_a ~~ res_cov*satisfaction_p
'
fit_unconstrained <- sem(model_unconstrained, data = ddw)
summary(fit_unconstrained, standardized = TRUE, fit.measures = TRUE)lavaan 0.6-21 ended normally after 37 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 27
Number of observations 100
Model Test User Model:
Test statistic 43.676
Degrees of freedom 12
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 284.840
Degrees of freedom 27
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.877
Tucker-Lewis Index (TLI) 0.724
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -646.240
Loglikelihood unrestricted model (H1) -624.402
Akaike (AIC) 1346.481
Bayesian (BIC) 1416.820
Sample-size adjusted Bayesian (SABIC) 1331.547
Root Mean Square Error of Approximation:
RMSEA 0.162
90 Percent confidence interval - lower 0.112
90 Percent confidence interval - upper 0.216
P-value H_0: RMSEA <= 0.050 0.000
P-value H_0: RMSEA >= 0.080 0.995
Standardized Root Mean Square Residual:
SRMR 0.145
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Regressions:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
satisfaction_a ~
wnc_a (a_h) -0.296 0.051 -5.794 0.000 -0.296 -0.470
wnc_p (p_h) -0.116 0.056 -2.093 0.036 -0.116 -0.170
recvry_ (ar_h) 0.240 0.052 4.633 0.000 0.240 0.330
rcvry_p (pr_h) 0.076 0.049 1.568 0.117 0.076 0.112
hs_chld (c_h) 0.089 0.092 0.962 0.336 0.089 0.066
dul_rnr (d_h) 0.318 0.097 3.266 0.001 0.318 0.225
satisfaction_p ~
wnc_p (a_w) -0.276 0.055 -5.040 0.000 -0.276 -0.390
wnc_a (p_w) -0.170 0.050 -3.369 0.001 -0.170 -0.261
rcvry_p (ar_w) 0.254 0.048 5.303 0.000 0.254 0.360
recvry_ (pr_w) 0.145 0.051 2.837 0.005 0.145 0.193
hs_chld (c_w) -0.023 0.091 -0.251 0.802 -0.023 -0.016
dul_rnr (d_w) 0.287 0.096 2.995 0.003 0.287 0.197
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
wnc_a ~~
wnc_p 0.516 0.110 4.682 0.000 0.516 0.530
recovery_a ~~
rcvry_p 0.218 0.087 2.502 0.012 0.218 0.258
.satisfaction_a ~~
.stsfct_ (rs_c) 0.045 0.020 2.233 0.026 0.045 0.229
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.stsfc_ (int_a) 5.040 0.211 23.875 0.000 5.040 7.765
.stsfc_ (int_p) 4.580 0.208 22.040 0.000 4.580 6.842
wnc_a 0.247 0.103 2.397 0.017 0.247 0.240
wnc_p -0.033 0.095 -0.353 0.724 -0.033 -0.035
rcvry_ 3.005 0.089 33.684 0.000 3.005 3.368
rcvry_ 3.235 0.095 34.131 0.000 3.235 3.413
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.stsfct_ (var_) 0.199 0.028 7.071 0.000 0.199 0.472
.stsfct_ (vr_p) 0.193 0.027 7.071 0.000 0.193 0.430
wnc_a 1.058 0.150 7.071 0.000 1.058 1.000
wnc_p 0.896 0.127 7.071 0.000 0.896 1.000
recvry_ 0.796 0.113 7.071 0.000 0.796 1.000
rcvry_p 0.898 0.127 7.071 0.000 0.898 1.000Model 2: slopes equal, intercepts free
The slopes-equal model constrains actor and partner slopes to be the same. Intercepts and residual variances are free to differ.
Code
model_slopes_equal <- '
satisfaction_a ~ a*wnc_a + p*wnc_p + ar*recovery_a +
pr*recovery_p + c*has_children + d*dual_earner
satisfaction_p ~ a*wnc_p + p*wnc_a + ar*recovery_p +
pr*recovery_a + c*has_children + d*dual_earner
satisfaction_a ~ int_a*1
satisfaction_p ~ int_p*1
satisfaction_a ~~ var_h*satisfaction_a
satisfaction_p ~~ var_w*satisfaction_p
wnc_a ~~ wnc_p
recovery_a ~~ recovery_p
satisfaction_a ~~ res_cov*satisfaction_p
'
fit_slopes_equal <- sem(model_slopes_equal, data = ddw)
summary(fit_slopes_equal, standardized = TRUE, fit.measures = TRUE)lavaan 0.6-21 ended normally after 33 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 27
Number of equality constraints 6
Number of observations 100
Model Test User Model:
Test statistic 47.242
Degrees of freedom 18
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 284.840
Degrees of freedom 27
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.887
Tucker-Lewis Index (TLI) 0.830
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -648.023
Loglikelihood unrestricted model (H1) -624.402
Akaike (AIC) 1338.046
Bayesian (BIC) 1392.755
Sample-size adjusted Bayesian (SABIC) 1326.432
Root Mean Square Error of Approximation:
RMSEA 0.127
90 Percent confidence interval - lower 0.084
90 Percent confidence interval - upper 0.172
P-value H_0: RMSEA <= 0.050 0.003
P-value H_0: RMSEA >= 0.080 0.962
Standardized Root Mean Square Residual:
SRMR 0.146
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Regressions:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
satisfaction_a ~
wnc_a (a) -0.288 0.035 -8.141 0.000 -0.288 -0.446
wnc_p (p) -0.146 0.035 -4.141 0.000 -0.146 -0.208
recovery_ (ar) 0.241 0.034 6.998 0.000 0.241 0.324
recovry_p (pr) 0.111 0.034 3.214 0.001 0.111 0.158
hs_chldrn (c) 0.025 0.072 0.347 0.728 0.025 0.018
dual_ernr (d) 0.304 0.076 4.006 0.000 0.304 0.210
satisfaction_p ~
wnc_p (a) -0.288 0.035 -8.141 0.000 -0.288 -0.416
wnc_a (p) -0.146 0.035 -4.141 0.000 -0.146 -0.230
recovry_p (ar) 0.241 0.034 6.998 0.000 0.241 0.349
recovery_ (pr) 0.111 0.034 3.214 0.001 0.111 0.151
hs_chldrn (c) 0.025 0.072 0.347 0.728 0.025 0.018
dual_ernr (d) 0.304 0.076 4.006 0.000 0.304 0.213
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
wnc_a ~~
wnc_p 0.516 0.110 4.682 0.000 0.516 0.530
recovery_a ~~
rcvry_p 0.218 0.087 2.502 0.012 0.218 0.258
.satisfaction_a ~~
.stsfct_ (rs_c) 0.043 0.020 2.122 0.034 0.043 0.217
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.stsfc_ (int_a) 4.955 0.167 29.664 0.000 4.955 7.462
.stsfc_ (int_p) 4.688 0.167 28.103 0.000 4.688 7.158
wnc_a 0.247 0.103 2.397 0.017 0.247 0.240
wnc_p -0.033 0.095 -0.353 0.724 -0.033 -0.035
rcvry_ 3.005 0.089 33.684 0.000 3.005 3.368
rcvry_ 3.235 0.095 34.131 0.000 3.235 3.413
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.stsfct_ (vr_h) 0.202 0.029 7.071 0.000 0.202 0.459
.stsfct_ (vr_w) 0.196 0.028 7.071 0.000 0.196 0.456
wnc_a 1.058 0.150 7.071 0.000 1.058 1.000
wnc_p 0.896 0.127 7.071 0.000 0.896 1.000
recvry_ 0.796 0.113 7.071 0.000 0.796 1.000
rcvry_p 0.898 0.127 7.071 0.000 0.898 1.000Model 3: fully constrained (slopes, intercepts, variances equal)
The fully-constrained model forces every parameter to be equal across the two roles. This is the indistinguishable model.
Code
model_full_equal <- '
satisfaction_a ~ a*wnc_a + p*wnc_p + ar*recovery_a +
pr*recovery_p + c*has_children + d*dual_earner
satisfaction_p ~ a*wnc_p + p*wnc_a + ar*recovery_p +
pr*recovery_a + c*has_children + d*dual_earner
satisfaction_a ~ alpha*1
satisfaction_p ~ alpha*1
satisfaction_a ~~ var_res*satisfaction_a
satisfaction_p ~~ var_res*satisfaction_p
wnc_a ~~ wnc_p
recovery_a ~~ recovery_p
satisfaction_a ~~ res_cov*satisfaction_p
'
fit_full_equal <- sem(model_full_equal, data = ddw)
summary(fit_full_equal, standardized = TRUE, fit.measures = TRUE)lavaan 0.6-21 ended normally after 27 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 27
Number of equality constraints 8
Number of observations 100
Model Test User Model:
Test statistic 66.150
Degrees of freedom 20
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 284.840
Degrees of freedom 27
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.821
Tucker-Lewis Index (TLI) 0.758
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -657.477
Loglikelihood unrestricted model (H1) -624.402
Akaike (AIC) 1352.955
Bayesian (BIC) 1402.453
Sample-size adjusted Bayesian (SABIC) 1342.446
Root Mean Square Error of Approximation:
RMSEA 0.152
90 Percent confidence interval - lower 0.112
90 Percent confidence interval - upper 0.193
P-value H_0: RMSEA <= 0.050 0.000
P-value H_0: RMSEA >= 0.080 0.998
Standardized Root Mean Square Residual:
SRMR 0.150
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Regressions:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
satisfaction_a ~
wnc_a (a) -0.256 0.037 -6.971 0.000 -0.256 -0.394
wnc_p (p) -0.178 0.037 -4.846 0.000 -0.178 -0.252
recovery_ (ar) 0.226 0.036 6.300 0.000 0.226 0.302
recovry_p (pr) 0.125 0.036 3.476 0.001 0.125 0.177
hs_chldrn (c) 0.026 0.072 0.364 0.716 0.026 0.019
dual_ernr (d) 0.305 0.076 4.010 0.000 0.305 0.209
satisfaction_p ~
wnc_p (a) -0.256 0.037 -6.971 0.000 -0.256 -0.363
wnc_a (p) -0.178 0.037 -4.846 0.000 -0.178 -0.275
recovry_p (ar) 0.226 0.036 6.300 0.000 0.226 0.321
recovery_ (pr) 0.125 0.036 3.476 0.001 0.125 0.167
hs_chldrn (c) 0.026 0.072 0.364 0.716 0.026 0.019
dual_ernr (d) 0.305 0.076 4.010 0.000 0.305 0.209
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
wnc_a ~~
wnc_p 0.516 0.110 4.682 0.000 0.516 0.530
recovery_a ~~
rcvry_p 0.218 0.087 2.502 0.012 0.218 0.258
.satisfaction_a ~~
.stsfct_ (rs_c) 0.027 0.022 1.246 0.213 0.027 0.126
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.stsfct_ (alph) 4.824 0.164 29.359 0.000 4.824 7.221
.stsfct_ (alph) 4.824 0.164 29.359 0.000 4.824 7.236
wnc_a 0.247 0.103 2.397 0.017 0.247 0.240
wnc_p -0.033 0.095 -0.353 0.724 -0.033 -0.035
recvry_ 3.005 0.089 33.684 0.000 3.005 3.368
rcvry_p 3.235 0.095 34.131 0.000 3.235 3.413
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.stsfct_ (vr_r) 0.215 0.022 9.922 0.000 0.215 0.482
.stsfct_ (vr_r) 0.215 0.022 9.922 0.000 0.215 0.484
wnc_a 1.058 0.150 7.071 0.000 1.058 1.000
wnc_p 0.896 0.127 7.071 0.000 0.896 1.000
recvry_ 0.796 0.113 7.071 0.000 0.796 1.000
rcvry_p 0.898 0.127 7.071 0.000 0.898 1.000The three nested LRTs
Code
cat("LRT 1: Unconstrained vs Slopes Equal\n")LRT 1: Unconstrained vs Slopes EqualCode
cat("(Do slopes differ by gender?)\n\n")(Do slopes differ by gender?)Code
lrt1 <- lavTestLRT(fit_unconstrained, fit_slopes_equal)
print(lrt1)
Chi-Squared Difference Test
Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
fit_unconstrained 12 1346.5 1416.8 43.676
fit_slopes_equal 18 1338.0 1392.8 47.242 3.5659 0 6 0.7352Code
cat("\nLRT 2: Slopes Equal vs Fully Constrained\n")
LRT 2: Slopes Equal vs Fully ConstrainedCode
cat("(Do intercepts differ by gender?)\n\n")(Do intercepts differ by gender?)Code
lrt2 <- lavTestLRT(fit_slopes_equal, fit_full_equal)
print(lrt2)
Chi-Squared Difference Test
Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
fit_slopes_equal 18 1338 1392.8 47.242
fit_full_equal 20 1353 1402.5 66.150 18.909 0.29076 2 7.836e-05
fit_slopes_equal
fit_full_equal ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Code
cat("\nLRT 3: Unconstrained vs Fully Constrained\n")
LRT 3: Unconstrained vs Fully ConstrainedCode
cat("(Overall indistinguishability)\n\n")(Overall indistinguishability)Code
lrt3 <- lavTestLRT(fit_unconstrained, fit_full_equal)
print(lrt3)
Chi-Squared Difference Test
Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
fit_unconstrained 12 1346.5 1416.8 43.676
fit_full_equal 20 1353.0 1402.5 66.150 22.474 0.13451 8 0.004109
fit_unconstrained
fit_full_equal **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
TipReading the three tests
- LRT 1 tests whether actor and partner slopes are equal. A non-significant p-value supports the couple pattern (equal slopes). Significant p-value means the dyads are distinguishable by effects, not just by means.
- LRT 2 tests whether the intercepts and residual variances are equal, given equal slopes. A significant p-value means the dyads are distinguishable by means.
- LRT 3 tests overall indistinguishability (slopes, intercepts, residual variances all equal). A non-significant p-value means the dyads are indistinguishable on every parameter.
In our data, LRT 1 should be non-significant (equal slopes) and LRT 2 should be significant (different intercepts). This is the standard empirical pattern: distinguishable by means, not by effects.
Fit indices comparison
Build a side-by-side table of fit measures for the three models.
Code
fits <- c("chisq", "df", "pvalue", "cfi", "tli", "rmsea", "srmr")
fit_indices <- data.frame(
unconstrained = fitMeasures(fit_unconstrained, fits),
slopes_equal = fitMeasures(fit_slopes_equal, fits),
full_equal = fitMeasures(fit_full_equal, fits)
)
print(round(fit_indices, 4)) unconstrained slopes_equal full_equal
chisq 43.6759 47.2417 66.1502
df 12.0000 18.0000 20.0000
pvalue 0.0000 0.0002 0.0000
cfi 0.8771 0.8866 0.8210
tli 0.7236 0.8299 0.7584
rmsea 0.1625 0.1275 0.1519
srmr 0.1451 0.1459 0.1503What to take away
Key takeaways
- The wide-format SEM gives you the three classical tests of distinguishability: slopes, intercepts, overall.
- LRT 1 (slopes) is the most important test for the APIM: it tells you whether the effects of predictors differ by role.
- LRT 2 (intercepts) tells you whether the dyads are distinguishable by means (often they are, even when slopes are equal).
- LRT 3 (overall) is the strict test; it is rarely used in practice because LRT 1 and LRT 2 are more informative.
What to read next
- The two-intercept tutorial — recommended when LRT 1 is non-significant but LRT 2 is significant (i.e. dyads are distinguishable by means, not effects).
- The Hahn et al. (2014) replication tutorial — moderated APIM with a dyad-level moderator.