Bivariate Social Relations Model

Path Diagram

alt bv.srm.xxmModel

xxM Model Matrices

Level-1 A(B): Within matrices

Residual Covariance Matrix

The residual-covaraince matrix \( (2 \times 2) \) at level-1, \( (\Theta_{1,1}) \) is .

\[ \Theta^{1,1}= \begin{bmatrix} \theta_{1,1}^{1,1} & \\ \theta_{2,1}^{1,1} & \theta_{2,2}^{1,1} \end{bmatrix} \]

Corresponding intercept vector is:

\[ \nu^1 = \begin{bmatrix} \nu^1_1 & \\ \nu^1_2 \end{bmatrix} \]

Level-2 B(A): Within matrices

A(B) and B(A) include reciprocal exchangable ratings of members of the same dyad. Hence, the within-level parameters are constrained to be equal.

Residual Covariance Matrix

Resudual covariance matrix at level-2 \( (\Theta_{2,2}) \) is contrained to be equal to level-1 residual covariance.

\[ \Theta^{2,2}= \begin{bmatrix} \theta_{1,1}^{2,2} & \\ \theta_{2,1}^{2,2} & \theta_{2,2}^{2,2} \end{bmatrix} = \begin{bmatrix} \theta_{1,1}^{1,1} & \\ \theta_{2,1}^{1,1} & \theta_{2,2}^{1,1} \end{bmatrix} \]

Similarly, intercepts are also constrained across the two levels.

\[ \nu^2 = \begin{bmatrix} \nu^2_1 & \\ \nu^2_2 \end{bmatrix} = \begin{bmatrix} \nu^1_1 & \\ \nu^1_2 \end{bmatrix} \]

Level-3 (Target): Within Matrices

At level-3, we have two latent variables: Target effects for \( y_1 \) and \( y_2 \). There is a single parameter matrix, latent factor covariance matrix at the target level.

Latent Factor Covariance Matrix

The latent covariance matrix is a \( (2 \times 2) \) matrix:

\[ \Psi^{3,3}= \begin{bmatrix} \psi_{1,1}^{3,3} & \\ \psi_{2,1}^{3,3} & \psi_{2,2}^{3,3} \end{bmatrix} \]

Level-4 (Rater): Within Matrices

Similarly, at level-4, there are two latent variables: Rater effects for \( y_1 \) and \( y_2 \) and a corresponding latent factor covariance matrix..

Latent Factor Covariance Matrix

The latent covariance matrix is a \( (2 \times 2) \) matrix with two variances and single covariance:

\[ \Psi^{4,4}= \begin{bmatrix} \psi_{1,1}^{4,4} & \\ \psi_{2,1}^{4,4} & \psi_{2,2}^{4,4} \end{bmatrix} \]

Across level matrices:

Across-level factor-loading matrices

There are four factor-loading matrices:

Target to A(B)
Target to B(A)
Rater to A(B)
Rater to B(A)

All four of these are \( 2 \times 2 \) identity matrices.

\[ \Lambda^{1,3}= \begin{bmatrix} 1.0 & 0.0 \\ 0.0 & 1.0 \end{bmatrix} \]

\[ \Lambda^{2,3}= \begin{bmatrix} 1.0 & 0.0 \\ 0.0 & 1.0 \end{bmatrix} \]

\[ \Lambda^{1,4}= \begin{bmatrix} 1.0 & 0.0 \\ 0.0 & 1.0 \end{bmatrix} \]

\[ \Lambda^{2,4}= \begin{bmatrix} 1.0 & 0.0 \\ 0.0 & 1.0 \end{bmatrix} \]

Across-level dyadic-covariances

Residuals of levels A(B) and B(A) covary across levels:

\[ \Theta^{1,2}= \begin{bmatrix} \theta_{1,1}^{1,2} & \theta_{1,2}^{1,2}\\ \theta_{2,1}^{1,2} & \theta_{2,2}^{1,2} \end{bmatrix} \]

However, across-level, across-variable residual-covariances are constrained to equality, \( \theta_{1,2}^{1,2} = \theta_{2,1}^{1,2} \).

Across-level Target-rater covariances

Target and Rater level latent variables levels are allowed to covary:

\[ \Psi^{3,4}= \begin{bmatrix} \psi_{1,1}^{3,4} & \psi_{1,2}^{3,4}\\ \psi_{2,1}^{3,4} & \psi_{2,2}^{3,4} \end{bmatrix} \]

Code Listing

xxM

Load xxM

xxM library needs to be loaded first.

library(xxm)

Load data ready for SRM analysis

SRM model has four real levels:

a(b)
a(b)
target
rater

and two virtual levels:

dyad
person

xxM requires separate datasets for six levels:

a(b)
a(b)
target
rater
dyad
person

Structure of each of these datasets is described below. Note:

Observations within levels A(B) and B(A) with a common ID represent reciprocal ratings within a dyad.
Observations for levels Target and Rater with a common ID represent the same person playing two different roles.

attach("nlsem.srm.xxm.RData")

A(B) includes rater B's ratings of target A and includes following columns:

a_b unique IDs for the response level A(B). Must be of type integer.
target target IDs. Must be of type integer.
rater rater IDs. Must be of type integer.
like dependent varaible like. Must be of type numeric.
warm dependent variable warm. Must be of type numeric.

str(A_B)

## 'data.frame':    209 obs. of  5 variables:
##  $ a_b   : int  2030202 2040202 2040203 2060201 2060202 2070202 2070203 2070204 2080205 2110205 ...
##  $ target: int  202 202 203 201 202 202 203 204 205 205 ...
##  $ rater : int  203 204 204 206 206 207 207 207 208 211 ...
##  $ like  : num  7 9 4 9 2 9 9 9 7 7 ...
##  $ warm  : num  9 7 3 7 2 9 8 9 8 7 ...

B(A) includes rater A's ratings of target B and includes following columns:

b_a unique IDs for the response level B(A). Must be of type integer.
target target IDs. Must be of type integer.
rater rater IDs. Must be of type integer.
like dependent varaible like. Must be of type numeric.
warm dependent variable warm. Must be of type numeric.

str(B_A)

## 'data.frame':    256 obs. of  5 variables:
##  $ b_a   : int  2030202 2040202 2040203 2060202 2060204 2070202 2080205 2100203 2100206 2110205 ...
##  $ target: int  203 204 204 206 206 207 208 210 210 211 ...
##  $ rater : int  202 202 203 202 204 202 205 203 206 205 ...
##  $ like  : num  7 8 8 1 1 8 8 4 1 7 ...
##  $ warm  : num  8 9 4 3 1 8 4 2 1 3 ...

target dataset includes a single column:

target unique target IDs. Must be of type numeric.

str(target)

## 'data.frame':    164 obs. of  1 variable:
##  $ target: int  201 202 203 204 205 206 207 208 209 210 ...

rater dataset includes a single column:

rater unique target IDs. Must be of type numeric.

str(rater)

## 'data.frame':    114 obs. of  1 variable:
##  $ rater: int  202 203 204 205 206 207 208 211 213 214 ...

dyad dataset includes a single column:

dyad unique target IDs. Must be of type numeric.

str(dyad)

## 'data.frame':    383 obs. of  1 variable:
##  $ dyad: int  2030202 2040202 2040203 2060201 2060202 2060204 2070202 2070203 2070204 2080205 ...

person dataset includes a single column:

person unique target IDs. Must be of type numeric.

str(person)

## 'data.frame':    172 obs. of  1 variable:
##  $ person: int  201 202 203 204 205 206 207 208 209 210 ...

Construct R-matrices

For each parameter matrix, construct three related matrices:

pattern matrix: A matrix indicating free or fixed parameters.
value matrix: with start or fixed values for corresponding parameters.
label matrix: with user friendly label for each parameter. label matrix is optional and used for imposing equality constraints. In this case, equality constraints are imposed on residual-covariance and intercepts for A(B) and B(A). Parameters with the same label are constrained to be equal.

# Within matrices for A(B) and B(A)
theta_wn_pat <- matrix(1, 2, 2)
theta_wn_val <- diag(3, 2)
theta_wn_lab <- matrix(c("like_like", "like_warm", "like_warm", "warm_warm"), 
    2, 2)

nu_pat <- matrix(1, 2, 1)
nu_val <- matrix(6, 2, 1)
nu_lab <- matrix(c("like_mean", "warm_mean"), 2, 1)

# Note: The use of common labels for nu (nu_lab) and theta (theta_wn_lab)
# for A(B) and B(A) imposes equality constraints on corresponding parameter
# estimates.

# Within latent covariance matrix for target
target_psi_pat <- matrix(1, 2, 2)
target_psi_val <- diag(0.1, 2)
# Within latent covariance matrix for rater
rater_psi_pat <- matrix(1, 2, 2)
rater_psi_val <- diag(0.1, 2)

# Across level dyadic residual-covaraince (A_B with B_A)
theta_bw_pat <- matrix(1, 2, 2)
theta_bw_val <- diag(0, 2)
theta_bw_lab <- matrix(c("like1_like2", "like1_warm2", "like1_warm2", "warm1_warm2"))
# Note: The use of common labels like1_warm2 imposes equality constraints on
# across-level [A(B) and B(A)] covariance between `like` and `warm`.

# Across level Target-Rater covariance
rater_target_psi_pat <- matrix(1, 2, 2)
rater_target_psi_val <- matrix(0, 2, 2)

# Across-level factor loadings are all (2x2) identity matrices
lambda_pat <- matrix(0, 2, 2)
lambda_val <- diag(1, 2)

Construct model

xxmModel() is used to declare level names. The function returns a model object that is passed as a parameter to subsequent statements. Variable name for the return value can be anything.

zz <- xxmModel(c("a_b", "b_a", "target", "rater"))

## A new model was created.

For SRM, only real levels have observed and/or latent variables. Only real levels need to be declared using the xxmModel command. Virtual levels are declared in a separate command.

In this case, we have four real levels:

a_b
b_a
rater
target

Note that the two levels with observed dependent variables come before the levels with latent independnet variables.

Define virtual levels and add virtual data

For each vitual level we need three pieces of information.

Name of the virtual level
List of real levels that are instances of the virtual level
Dataset for the virtual level with a column of IDs.

There are two commands that together accomplish these goals:

Virtual levels and role models xxmVirtuals()

xxmVirtuals() informs our model that there are virtual levels and binds each virtual level with the corresponding set of real levels. The function has two parameters.

model is the first parameter is the model for which we want to define virtual levels.
alias is a named list with as many elements as the number of virtual levels. The name of each element is the name of the virtual level. For each virtual level, a list of corresponding real levels is provided. In this case, there are two virtual levels:
1. person with corresponding real levels: rater and target
2. dyad with corresponding real levels: b_a and a_b

Virtual data: xxmVirtualIds()

xxmVirtualIds() adds data for a virtual level with a single column of IDs to our model. In this case, person and dyad datasets are added to our model.

alias = list(person = c("target", "rater"), dyad = c("a_b", "b_a"))
zz <- xxmVirtuals(zz, alias)

## Creating virtual `person` with 2 aliases
## Creating virtual `dyad` with 2 aliases
## Aliases created

zz <- xxmVirtualids(zz, person)

## Virtual level `person` was created

zz <- xxmVirtualids(zz, dyad)

## Virtual level `dyad` was created

Add submodels

For each declared level xxmSubmodel() is invoked to add corresponding submodel to the model object. The function adds three types of information to the model object:

parents declares a list of all parents of the current level.
- Level with the independent variable is the parent level.
- Level with the dependent variable is the child level.
variables declares names of observed dependent (ys), observed independent (xs) and latent variables (etas) for the level.
data R data object for the current level.

In this case, a submodel is declared for each of the four real levels. Note, virtual levels do not need a submodel! Their sole purpose is to bind real levels so that we can include across-level covariances.

# Level 1: A(B)
zz <- xxmSubmodel(model = zz, level = "a_b", parents = c("target", "rater"), 
    ys = c("like", "warm"), xs = , etas = , data = A_B, )

## Creating role model `a_b` for virtual `dyad`
## Submodel for level `a_b` was created.


# Level 1: B(A)
zz <- xxmSubmodel(model = zz, level = "b_a", parents = c("target", "rater"), 
    ys = c("like", "warm"), xs = , etas = , data = B_A, )

## Creating role model `b_a` for virtual `dyad`
## Submodel for level `b_a` was created.


# Level 3: Target
zz <- xxmSubmodel(model = zz, level = "target", parents = , ys = , xs = , etas = c("like-target", 
    "warm-target"), data = target, )

## Creating role model `target` for virtual `person`
## Submodel for level `target` was created.


# Level 3: Rater
zz <- xxmSubmodel(model = zz, level = "rater", parents = , ys = , xs = , etas = c("like-rater", 
    "warm-rater"), data = rater, )

## Creating role model `rater` for virtual `person`
## Submodel for level `rater` was created.


## @ knitr xxmWithinMatrix Level 1: A_B :: Intercepts
zz <- xxmWithinMatrix(model = zz, level = "a_b", type = "nu", pattern = nu_pat, 
    value = nu_val, label = nu_lab, name = )

## 
## 'nu' matrix does not exist and will be added.
##  Added `nu` matrix.

# Level 1: A_B :: Residual-covariance
zz <- xxmWithinMatrix(model = zz, level = "a_b", "theta", pattern = theta_wn_pat, 
    value = theta_wn_val, label = theta_wn_lab, )

## 
## 'theta' matrix does not exist and will be added.
##  Added `theta` matrix.

# Level 2: B_A :: Intercepts
zz <- xxmWithinMatrix(model = zz, level = "b_a", type = "nu", pattern = nu_pat, 
    value = nu_val, label = nu_lab, )

## 
## 'nu' matrix does not exist and will be added.
##  Added `nu` matrix.

# Level 2: B_A :: Residual-covariance
zz <- xxmWithinMatrix(model = zz, level = "b_a", type = "theta", pattern = theta_wn_pat, 
    value = theta_wn_val, label = theta_wn_lab, )

## 
## 'theta' matrix does not exist and will be added.
##  Added `theta` matrix.

# Level 3: Target
zz <- xxmWithinMatrix(model = zz, level = "target", type = "psi", pattern = target_psi_pat, 
    value = target_psi_val, , )

## 
## 'psi' matrix does not exist and will be added.
##  Added `psi` matrix.

# Level 4: Rater
zz <- xxmWithinMatrix(model = zz, level = "rater", type = "psi", pattern = rater_psi_pat, 
    value = rater_psi_val, , )

## 
## 'psi' matrix does not exist and will be added.
##  Added `psi` matrix.

## @ knitr xxmBetweenMatrix A(B)-B(A) residual covariance
zz <- xxmBetweenMatrix(model = zz, parent = "b_a", child = "a_b", type = "theta", 
    pattern = theta_bw_pat, value = theta_bw_val, label = theta_bw_lab, )

## 
## 'theta' matrix does not exist and will be added.
##  Added `theta` matrix.

### Target-Rater covariance
zz <- xxmBetweenMatrix(model = zz, parent = "rater", child = "target", type = "psi", 
    pattern = rater_target_psi_pat, value = rater_target_psi_val, , )

## 
## 'psi' matrix does not exist and will be added.
##  Added `psi` matrix.

# Target -> A_B link matrix:
zz <- xxmBetweenMatrix(model = zz, parent = "target", child = "a_b", type = "lambda", 
    pattern = lambda_pat, value = lambda_val, label = )

## 
## 'lambda' matrix does not exist and will be added.
##  Added `lambda` matrix.

# Target -> A_B link matrix:
zz <- xxmBetweenMatrix(model = zz, parent = "target", child = "b_a", type = "lambda", 
    pattern = lambda_pat, value = lambda_val, label = )

## 
## 'lambda' matrix does not exist and will be added.
##  Added `lambda` matrix.

# Rater -> A_B link matrix:
zz <- xxmBetweenMatrix(model = zz, parent = "rater", child = "a_b", type = "lambda", 
    pattern = lambda_pat, value = lambda_val, label = )

## 
## 'lambda' matrix does not exist and will be added.
##  Added `lambda` matrix.

# Rater -> B_A link matrix:
zz <- xxmBetweenMatrix(model = zz, parent = "rater", child = "b_a", type = "lambda", 
    pattern = lambda_pat, value = lambda_val, label = )

## 
## 'lambda' matrix does not exist and will be added.
##  Added `lambda` matrix.

Add within-level matrices

For each declared level xxmWithinMatrix() is used to add within-level parameter matrices. For each parameter matrix, the function adds the three matrices constructed earlier:

pattern
value
label (optional)

Add across-level matrices

Pairs of levels that share parent-child relationship have regression relationships. xxmBetweenMatrix() is used to add corresponding rergession matrices connecting the two levels.

Level with the independent variable is the parent level.
Level with the dependent variable is the child level.

For each parameter matrix, the function adds the three matrices constructed earlier:

pattern
value
label (optional)

Estimate model parameters

Estimation process is initiated by xxmRun(). If all goes well, a q&d summary of the results is printed.

zz <- xxmRun(zz)

## ------------------------------------------------------------------------------
## Estimating model parameters
## ------------------------------------------------------------------------------
##                3845.6787733309 
## 1010
##                3827.0022965414 
##                3813.2807817324 
##                3800.1973631633 
##                3792.8027509368 
##                3787.2429488352 
##                3781.1339685640 
##                3773.3712242050 
##                3754.7344587277 
## 1010
##                3745.7106123263 
##                3743.0787783140 
##                3741.5194034727 
##                3732.1782471290 
##                3728.1121445111 
##                3727.2437463610 
##                3724.7609317541 
##                3718.9189004660 
##                3717.5244086705 
##                3717.4465498261 
##                3717.6208968386 
##                3716.8867202835 
##                3717.5992659136 
##                3718.5415793043 
##                3722.2117551244 
##                3722.0321341161 
##                3721.9343251739 
##                3721.8510434386 
##                3721.7369256222 
##                3721.6958881212 
##                3721.6649608229 
##                3721.6546033034 
##                3721.6416413801 
##                3721.6309393732 
##                3721.6287124801 
##                3721.6263509909 
##                3721.6253391056 
##                3721.6249232333 
##                3721.6243861740 
##                3721.6237942327 
##                3721.6217367473 
##                3721.6195869373 
##                3721.6172747980 
##                3721.6130437022 
##                3721.6097309518 
##                3721.6087287044 
##                3721.6080373175 
##                3721.6078041279 
##                3721.6069571707 
##                3721.6060237656 
##                3721.6056525807 
##                3721.6047551826 
##                3721.6046911642 
##                3721.6044512253 
##                3721.6043859237 
##                3721.6042990222 
##                3721.6042344755 
##                3721.6042182606 
##                3721.6042180248 
##                3721.6042180269 
##                3721.6042180280 
##                3721.6042180166 
## Model converged normally
## nParms: 18
## ------------------------------------------------------------------------------
## *
##  1:                                     like_like ::      2.705 [     0.000,      0.000]
## 
##  2:                                     like_warm ::      0.741 [     0.000,      0.000]
## 
##  3:                                     warm_warm ::      3.160 [     0.000,      0.000]
## 
##  4:                                     like_mean ::      6.683 [     0.000,      0.000]
## 
##  5:                                     warm_mean ::      5.677 [     0.000,      0.000]
## 
##  6:                                   like1_like2 ::      1.158 [     0.000,      0.000]
## 
##  7:                                   like1_warm2 ::      0.678 [     0.000,      0.000]
## 
##  8:                                   warm1_warm2 ::      0.592 [     0.000,      0.000]
## 
##  9:                                target_psi_1_1 ::      0.406 [     0.000,      0.000]
## 
## 10:                                target_psi_1_2 ::      0.614 [     0.000,      0.000]
## 
## 11:                                target_psi_2_2 ::      0.977 [     0.000,      0.000]
## 
## 12:                          target_rater_psi_1_1 ::      0.477 [     0.000,      0.000]
## 
## 13:                          target_rater_psi_2_1 ::      0.626 [     0.000,      0.000]
## 
## 14:                          target_rater_psi_1_2 ::      0.224 [     0.000,      0.000]
## 
## 15:                          target_rater_psi_2_2 ::      0.355 [     0.000,      0.000]
## 
## 16:                                 rater_psi_1_1 ::      0.785 [     0.000,      0.000]
## 
## 17:                                 rater_psi_1_2 ::      0.360 [     0.000,      0.000]
## 
## 18:                                 rater_psi_2_2 ::      0.533 [     0.000,      0.000]
## 
## ------------------------------------------------------------------------------

Estimate profile-likelihood confidence intervals

Once parameters are estimated, confidence intervals are estimated by invoking xxmCI(). Depending on the the number of observations and the complexity of the model, xxmCI() may take a long time to compute. xxmCI() also prints a summary of parameter estimates and CIS.

zz <- xxmCI(zz)

View results

A summary of results may be retrived as an R list by a call to xxmSummary(). The returned list has two elements:

fit is a list with five elements:
- deviance is \( -2 Log Likelihood \) for the maximum likelihood fit function.
- nParameters is the total number of unique parameters.
- nObservations is the total number of observations across all levels.
- aic is Akaike's Information Criterion or AIC computed as \( -2ll + 2*p \).
- bic is Bayesian Information Criterion or BIC computed as \( -2ll + p*\log(n) \).
estimates is a single table of free parameter estimates All xxM parameters have superscripts {child, parent} and subscripts {to, from}. xxM adds a descriptive parameter label if one is not already provided by the user.

bivariate.srm.results <- xxmSummary(zz)
bivariate.srm.results

## $fit
## $fit$deviance
## [1] 3722
## 
## $fit$nParameters
## [1] 18
## 
## $fit$nObservations
## [1] 907
## 
## $fit$aic
## [1] 3758
## 
## $fit$bic
## [1] 3844
## 
## 
## $estimates
##     child parent          to        from                label estimate
## 1     a_b    a_b        like        like            like_like   2.7051
## 2     a_b    a_b        like        warm            like_warm   0.7412
## 3     a_b    a_b        warm        warm            warm_warm   3.1601
## 4     a_b    a_b        like         One            like_mean   6.6832
## 5     a_b    a_b        warm         One            warm_mean   5.6766
## 6     a_b    b_a        like        like          like1_like2   1.1579
## 7     a_b    b_a        warm        like          like1_warm2   0.6781
## 8     a_b    b_a        like        warm          like1_warm2   0.6781
## 9     a_b    b_a        warm        warm          warm1_warm2   0.5920
## 10    b_a    b_a        like        like            like_like   2.7051
## 11    b_a    b_a        like        warm            like_warm   0.7412
## 12    b_a    b_a        warm        warm            warm_warm   3.1601
## 13    b_a    b_a        like         One            like_mean   6.6832
## 14    b_a    b_a        warm         One            warm_mean   5.6766
## 23 target target like-target like-target       target_psi_1_1   0.4059
## 24 target target like-target warm-target       target_psi_1_2   0.6136
## 25 target target warm-target warm-target       target_psi_2_2   0.9769
## 34 target  rater like-target  like-rater target_rater_psi_1_1   0.4769
## 35 target  rater warm-target  like-rater target_rater_psi_2_1   0.6257
## 36 target  rater like-target  warm-rater target_rater_psi_1_2   0.2240
## 37 target  rater warm-target  warm-rater target_rater_psi_2_2   0.3547
## 38  rater  rater  like-rater  like-rater        rater_psi_1_1   0.7851
## 39  rater  rater  like-rater  warm-rater        rater_psi_1_2   0.3598
## 40  rater  rater  warm-rater  warm-rater        rater_psi_2_2   0.5329
##          low   high
## 1   2.287256 3.2109
## 2   0.379996 1.1559
## 3   2.652929 3.7913
## 4   6.386750 6.9747
## 5   5.371974 5.9772
## 6   0.501311 1.7819
## 7   0.056552 1.2619
## 8   0.056552 1.2619
## 9  -0.541477 1.6542
## 10  2.287256 3.2109
## 11  0.379996 1.1559
## 12  2.652929 3.7913
## 13  6.386750 6.9747
## 14  5.371974 5.9772
## 23  0.171238 0.7465
## 24  0.336852 0.9695
## 25  0.539157 1.5608
## 34  0.197031 0.8120
## 35  0.268629 1.0393
## 36 -0.065961 0.5347
## 37 -0.007319 0.7394
## 38  0.436826 1.2729
## 39  0.078493 0.7133
## 40  0.188153 1.0057

Free model object

xxM model object may hog a significant amount of RAM outside of R's workspace. This memory will automatically be released, when the workspace is cleared by a call to rm(list=ls()) or at the end of the R session. Alternatively, it is recommended that xxmFree() may be called to release the memory.

zz <- xxmFree(zz)
rm(list = ls())

alt bv.srm.results