Overview
This vignette shows the bivariate workflow implemented in
dyadicMarkov. In the bivariate setting, two categorical
variables are observed repeatedly for the two members of a dyad. The
bivariate method follows the global-and-local procedure described in
Böllenrücher et al. (in press).
The current bivariate functions support states = 2. With
two binary variables observed for two members, the previous state is
described by four binary components. The empirical bivariate count
matrix therefore has 16 rows and 2 columns.
Although the data are synthetic, the four columns can be read like
real repeated observations from a dyad. For example, V1
could represent one coded behavior or response, and V2
could represent a second coded behavior or response observed at the same
measurement occasions. The columns FM_V1 and
SM_V1 then describe the two members on the main variable,
while FM_V2 and SM_V2 describe the same two
members on the second variable.
The 16 rows arise because the previous state combines four binary
lagged components: first member on V1, second member on
V1, first member on V2, and second member on
V2. With two possible states for each component, this gives
previous-state combinations. The 2 columns represent the possible next
states of the member sequence analyzed on the current main variable.
Data
The example data set dyadic_bivariate_example contains
two categorical variables for the first member and the second member of
a dyad. Each row corresponds to one measurement occasion.
utils::data("dyadic_bivariate_example", package = "dyadicMarkov")
head(dyadic_bivariate_example)
#> time FM_V1 SM_V1 FM_V2 SM_V2
#> 1 1 2 1 1 2
#> 2 2 2 1 2 1
#> 3 3 2 2 2 1
#> 4 4 2 2 2 2
#> 5 5 2 2 1 2
#> 6 6 2 2 1 1
dim(dyadic_bivariate_example)
#> [1] 90 5The four chains are first-member and second-member sequences for the
main variable (V1) and the second variable
(V2).
Empirical bivariate transition counts
The first step is to construct the empirical bivariate transition
count matrix with countEmpBivariate(). Rows represent the
16 possible previous-state combinations of the two dyadic variables.
Columns represent the next state of the first member on the main
variable.
emp_bi <- dyadicMarkov::countEmpBivariate(
chainFM_V1 = dyadic_bivariate_example$FM_V1,
chainSM_V1 = dyadic_bivariate_example$SM_V1,
chainFM_V2 = dyadic_bivariate_example$FM_V2,
chainSM_V2 = dyadic_bivariate_example$SM_V2,
states = 2L
)
emp_bi
#> next_1 next_2
#> mainFM1_mainSM1_secondFM1_secondSM1 7 0
#> mainFM1_mainSM1_secondFM1_secondSM2 0 2
#> mainFM1_mainSM1_secondFM2_secondSM1 0 0
#> mainFM1_mainSM1_secondFM2_secondSM2 1 0
#> mainFM1_mainSM2_secondFM1_secondSM1 0 1
#> mainFM1_mainSM2_secondFM1_secondSM2 0 1
#> mainFM1_mainSM2_secondFM2_secondSM1 2 0
#> mainFM1_mainSM2_secondFM2_secondSM2 0 2
#> mainFM2_mainSM1_secondFM1_secondSM1 0 0
#> mainFM2_mainSM1_secondFM1_secondSM2 2 1
#> mainFM2_mainSM1_secondFM2_secondSM1 1 1
#> mainFM2_mainSM1_secondFM2_secondSM2 0 3
#> mainFM2_mainSM2_secondFM1_secondSM1 1 1
#> mainFM2_mainSM2_secondFM1_secondSM2 1 4
#> mainFM2_mainSM2_secondFM2_secondSM1 1 5
#> mainFM2_mainSM2_secondFM2_secondSM2 0 52
class(emp_bi)
#> [1] "dyadic_counts" "matrix" "array"
dim(emp_bi)
#> [1] 16 2Global bivariate case
The function bivariateCase() performs the global step of
the bivariate method. It applies the package chi-squared comparisons to
classify the dependence structure of the member sequence analyzed as a
trivial, univariate, partial bivariate or complete bivariate case.
case_bi <- dyadicMarkov::bivariateCase(emp_bi, alpha = 0.05)
case_bi
#> Bivariate dyadic case
#> Case: complete
#> Alpha: 0.05
case_bi$case
#> [1] "complete"
summary(case_bi)
#> $case
#> [1] "complete"
#>
#> $alpha
#> [1] 0.05
#>
#> $call
#> dyadicMarkov::bivariateCase(empirical = emp_bi, alpha = 0.05)
#>
#> attr(,"class")
#> [1] "summary_dyadic_case" "list"This example is identified as a complete bivariate case. The appropriate local step is therefore to compare complete bivariate candidate patterns.
Local pattern identification for a complete case
For a complete bivariate case, completePattern()
compares the complete bivariate candidate structures by AIC and returns
the selected pattern.
complete_bi <- dyadicMarkov::completePattern(emp_bi)
complete_bi
#> Dyadic interaction pattern
#> Pattern: complete actor on the main, actor partner on the second (D2)
complete_bi$pattern
#> [1] "complete actor on the main, actor partner on the second (D2)"
complete_bi$aic
#> pattern matrix
#> 1 complete actor partner on both (C) C
#> 2 complete partner on the main, actor partner on the second (D1) D1
#> 3 complete actor on the main, actor partner on the second (D2) D2
#> 4 complete actor partner on the main, partner on the second (D3) D3
#> 5 complete actor partner on the main, actor on the second (D4) D4
#> 6 complete partner on both (E1) E1
#> 7 complete partner on the main, actor on the second (E2) E2
#> 8 complete actor on the main, partner on the second (E3) E3
#> 9 complete actor on both (E4) E4
#> aic
#> 1 32.00000
#> 2 30.03720
#> 3 28.42735
#> 4 33.33973
#> 5 38.60730
#> 6 30.55374
#> 7 42.91748
#> 8 36.47261
#> 9 40.57170In this example, the selected complete bivariate pattern is
D2, labelled by the package as complete actor on the main,
actor-partner on the second.
Reassigning member and main-variable roles
The previous analysis used FM_V1 as the member-variable
sequence analyzed, V1 as the main variable and
V2 as the second variable. In the bivariate method, these
are analysis roles rather than fixed identities. To describe the dyad
more completely, the roles can be reassigned so that each
member-variable sequence is analyzed in turn.
The same data set gives different branches of the global-and-local procedure depending on the member-variable sequence analyzed and the main variable.
analyze_bivariate <- function(label, fm_v1, sm_v1, fm_v2, sm_v2) {
emp <- dyadicMarkov::countEmpBivariate(
chainFM_V1 = fm_v1,
chainSM_V1 = sm_v1,
chainFM_V2 = fm_v2,
chainSM_V2 = sm_v2,
states = 2L
)
case <- dyadicMarkov::bivariateCase(emp, alpha = 0.05)
cat("\n", label, "\n", sep = "")
print(case)
if (identical(case$case, "complete")) {
print(dyadicMarkov::completePattern(emp))
}
if (identical(case$case, "partial")) {
print(dyadicMarkov::partialPattern(emp))
}
if (identical(case$case, "univariate")) {
print(dyadicMarkov::univariatePattern(fm_v1, sm_v1, states = 2L, alpha = 0.05))
}
}
d <- dyadic_bivariate_example
analyze_bivariate(
"FM_V1 as analyzed sequence, V1 as main variable",
d$FM_V1, d$SM_V1, d$FM_V2, d$SM_V2
)
#>
#> FM_V1 as analyzed sequence, V1 as main variable
#> Bivariate dyadic case
#> Case: complete
#> Alpha: 0.05
#> Dyadic interaction pattern
#> Pattern: complete actor on the main, actor partner on the second (D2)
analyze_bivariate(
"SM_V1 as analyzed sequence, V1 as main variable",
d$SM_V1, d$FM_V1, d$SM_V2, d$FM_V2
)
#>
#> SM_V1 as analyzed sequence, V1 as main variable
#> Bivariate dyadic case
#> Case: complete
#> Alpha: 0.05
#> Dyadic interaction pattern
#> Pattern: complete actor partner on the main, partner on the second (D3)
analyze_bivariate(
"FM_V2 as analyzed sequence, V2 as main variable",
d$FM_V2, d$SM_V2, d$FM_V1, d$SM_V1
)
#>
#> FM_V2 as analyzed sequence, V2 as main variable
#> Bivariate dyadic case
#> Case: partial
#> Alpha: 0.05
#> Dyadic interaction pattern
#> Pattern: partial actor (B2)
analyze_bivariate(
"SM_V2 as analyzed sequence, V2 as main variable",
d$SM_V2, d$FM_V2, d$SM_V1, d$FM_V1
)
#>
#> SM_V2 as analyzed sequence, V2 as main variable
#> Bivariate dyadic case
#> Case: univariate
#> Alpha: 0.05
#> Dyadic interaction pattern
#> Pattern: APM (A1)
#> Alpha: 0.05
#> States: 2For this example, the four role assignments illustrate three branches
of the procedure: complete bivariate cases for the two V1
member-variable sequences, a partial bivariate case for
FM_V2, and a univariate case for SM_V2. The
partial branch is therefore demonstrated with the same bivariate example
data rather than with an artificial seeded example.
Reading the global and local steps together
The global and local steps should be read together. If
bivariateCase() returns trivial, no subsequent
local pattern is selected. If it returns univariate, the
bivariate workflow returns to univariatePattern() using the
first- and second-member sequences of the current main variable. If it
returns partial, the local step is
partialPattern(). If it returns complete, the
local step is completePattern().