Fits latent profiles to repeated observations of the same group and
estimates the probabilities of moving between them from one occasion to the
next. The measurement model is that of multilpa() – Gaussian, categorical
or mixed indicators, invariant across occasions – so profiles keep the same
meaning at every occasion and a change of profile is a change of state
rather than a change of definition. With more than one group class, each
class carries its own initial distribution and its own transition matrix,
which is how groups that differ in their dynamics are separated from groups
that differ only in where they start.
Usage
lta(
data,
vars,
id,
n_profiles,
time,
n_group_classes = 1L,
variance_model = c("varying", "equal"),
n_starts = 10L,
max_iter = 1000L,
tol = 1e-08,
min_variance = 1e-06,
seed = NULL,
missing = c("error", "fiml"),
covariance_model = c("diagonal", "full"),
categorical = character(),
min_probability = 1e-10,
occasions = c("observed", "grid"),
select_start = c("likelihood", "converged"),
transitions = c("homogeneous", "occasion"),
transition_covariates = character(),
initial_covariates = character(),
measurement = c("invariant", "occasion"),
order = 1L,
model = NULL,
mover_stayer = FALSE,
weights = NULL,
ordinal = character(),
count = character(),
count_model = c("poisson", "negative_binomial"),
count_dispersion = c("varying", "equal")
)Arguments
- data
A data frame containing indicators and a group identifier.
- vars
Unique character vector of continuous indicator column names.
- id
Name of the observed group identifier column. Character, factor, or numeric identifiers are supported; missing identifiers are not. Required here: this model has a second level by construction, so it has no single-level form and does not take
multilpa()'sid = NULL.- n_profiles
Integer number of individual profiles, at least two.
- time
Name of the column giving each observation's occasion within its group. Required: this model is defined by the ordering. Values must be complete and unique within each group.
- n_group_classes
Positive integer number of latent group classes. One gives an ordinary single-level latent transition model; more than one fits a mixture of transition models over groups.
- variance_model
Either
"varying"(profile-specific variances or covariance matrices) or"equal"(shared across profiles).- n_starts
Positive integer number of EM starts. The first is deterministic: Ward's hierarchical clustering of the standardized indicators (on at most 2000 evenly spaced rows), each cluster described by its own means and spread, which is the start mclust's name refers to and finds solutions whose profiles differ in shape. The rest are k-means from random centres. When
startis supplied, it supplies the first start instead; remaining starts are random initializations. It is ignored whenmax_iter = 0andstartis supplied: that call evaluates the supplied parameters and nothing else, so exactly one start is run whatevern_startssays.- max_iter
Nonnegative integer maximum number of EM updates per start.
max_iter = 0performs no update. Withstart, it returns the model evaluated at exactly those values, whatevern_startsis, so thatlogLik()scores a parameter set supplied from elsewhere rather than the best of some random initializations that were never asked for. Withoutstartthere is nothing to evaluate at, so the random initializations are scored and the highest is returned.- tol
Positive relative log-likelihood tolerance. Convergence requires absolute change no greater than
tol * (1 + abs(previous log likelihood)).- min_variance
Positive lower bound on each variance, or each covariance eigenvalue for full covariance, in squared input units. This defines a constrained maximum-likelihood problem. Bound-active estimates are explicitly reported and generate a warning.
- seed
Optional random seed: any whole number
set.seed()accepts, negative ones included. With a supplied seed, the caller's random-number state is restored on exit.- missing
"error"rejects missing indicators;"fiml"maximizes the observed-data likelihood under an ignorable missingness mechanism (MAR). Missing indicators are integrated out, not filled in for likelihood fitting. This holds with membership covariates too; the covariates themselves must be complete (latents_bad_data).- covariance_model
"diagonal"assumes conditional independence;"full"estimates within-profile residual covariances.- categorical
Character vector naming indicators to treat as categorical. Each is modelled by unrestricted, profile-specific response probabilities over its observed categories, which is the latent class measurement model. Binary, ordinal and unordered indicators are all handled by the same unrestricted parameterization; numeric, integer, logical, character and factor columns are accepted. Indicators not named here stay Gaussian, so naming a subset fits a mixed-mode model.
- min_probability
Positive lower bound on every categorical response probability, defining a constrained maximum-likelihood problem in the same way
min_variancedoes for Gaussian indicators.- occasions
How a group's occasions are placed on the transition grid.
"observed"numbers each group's own observations consecutively, so a transition always links two adjacent observations."grid"places them on the grid of every position seen in the data, so a group that skips a position still consumes a transition across the gap and contributes no measurement information at it. The two agree whenever every group is observed at every position.- select_start
Which start the fit reports.
"likelihood", the default, takes the highest log likelihood across starts, preferring a converged start only among those tied with it to within a relative1e-10."converged"takes the highest log likelihood among the converged starts whenever at least one converged, and falls back to every start when none did. Use it when an unconverged start edges out a converged one by a negligible amount, which happens when a membership logit creeps along a flat ridge of the likelihood: the converged start is then a maximum that the standard errors can work with, and the other is not.summary()andget_results(fit, "starts")show every start either way.- transitions
"homogeneous"(the default) uses one transition matrix for every occasion;"occasion"estimates a separate one for each move (occasion 1 to 2, 2 to 3, ...).- transition_covariates
Names of columns whose values shift the transition probabilities through a multinomial logit per origin profile: the log odds of moving to each other profile rather than staying. A column may change over occasions; the value at the destination occasion is used. Factors are expanded to indicator columns.
- initial_covariates
Names of columns that shift the initial profile distribution (a multinomial logit, reference: the last profile), read at each group's first occasion.
- measurement
"invariant"(the default) keeps the profiles' means, variances and response probabilities equal across occasions;"occasion"estimates them separately at each occasion, so a profile is defined by its position in the sequence of transitions rather than by one fixed measurement model.- order
1(the default) or2. With2, the profile at each occasion from the third on depends on the profiles at the two previous occasions (a second-order Markov chain); the first move stays first order.- model
A covariance structure as
multilpa()takes it ("EEI","VVI","VEI", ... ; mclust's codes). Standard errors are given for EEI, VVI, EEE and VVV in the homogeneous model and for the diagonal EEI and VVI with the extensions; other structures are refused (latents_unsupported_inference).- mover_stayer
TRUEadds a class of stayers: groups that never change profile (identity transitions) and have their own initial profile distribution (Goodman's mover-stayer model).n_group_classesthen counts the mover classes; the stayer class is reported as"stayers".- weights
NULL, or the name of a column ofdataholding one sampling weight perid(constant over its occasions). As inmultilpa(): pseudo maximum likelihood with the weights scaled to sum to the number ofidunits, sandwich standard errors by default, and integer weights equal to repeating each sequence. A weighted fit uses the general transition engine and returns amultilpa_lta.- ordinal, count, count_model, count_dispersion
Ordinal and count indicators, as in
multilpa(); withmeasurement = "occasion"their parameters are estimated per occasion. Read them withget_results(fit, "ordinal")andget_results(fit, "count_means"). A fit with them uses the general transition engine.
Value
A multilpa_transitions object containing means, variances,
optional covariances and response_probabilities,
initial_probabilities (group classes by profiles),
transition_probabilities (profiles by profiles by group classes, rows
indexing the profile moved from), group_probabilities, posterior
matrices, classifications, log likelihood, information criteria, restart
diagnostics and convergence history. Use get_results() for the tidy
transition table and for the other
tables. No standard
errors, likelihood-ratio tests or guarantees of global optimality are
given for this model family.
With transitions = "occasion", transition_covariates,
initial_covariates, measurement = "occasion" or order = 2 the
result is a multilpa_lta object instead, read with
get_results.multilpa_lta(): transition and initial logit coefficients
with Wald standard errors (observed, robust or OPG, from analytic
scores), model-implied transition probabilities per occasion, profiles
per occasion. Missing indicators (missing = "fiml") and covariance
structures are supported; diagonal fits are finished by a quasi-Newton
search on the exact likelihood, others by EM.
Agreement with Mplus (User's Guide examples 8.13, 8.14; occasion-specific
thresholds), depmixS4 and LMest is recorded in
equivalence/lta-extensions/ of the source repository.
Details
This is latent transition analysis (LTA); with more than one group class it
is a mixture over transition patterns, so the classes are trajectories rather
than states. multilpa() is the cross-sectional counterpart.
By default transitions are first order and homogeneous over
occasions: the probability of moving from one profile to another does not
depend on the occasion or on earlier profiles. Measurement parameters are
shared across occasions and across group classes. A profile that no group occupies before
its final occasion leaves its transition row without information; the row
is then uniform by construction rather than estimated. Such a row is warned
about when the model is fitted, is flagged by the estimated column of
the transition table and is listed by
get_results(fit, "transitions", estimated = FALSE).
Profile labels are arbitrary and
are not comparable across fits without alignment.
max_iter = 0 updates nothing: the starting values are evaluated and
returned with the expectation they imply, and the fit reports
converged = FALSE after zero iterations. Every reported quantity,
including expected_count and the implied profile prevalence, is then
read off that single expectation.
References
Collins, L. M., & Lanza, S. T. (2010). Latent class and latent transition analysis. Wiley.
Vermunt, J. K. (2003). Multilevel latent class models. Sociological Methodology, 33, 213–239. doi:10.1111/j.0081-1750.2003.t01-1-00131.x.
See also
get_results() for the fitted transition probabilities and for the
assignments in order, and multilpa() for the
cross-sectional model this one shares its measurement parameters with.
Examples
# Students take their courses in their own order, which `sequence` records,
# so the same enrolments that fit a cross-sectional model fit a transition
# one. Engagement mostly persists from one course to the next.
fit <- lta(
course_engagement,
vars = c("browse", "lectures", "forum_read", "forum_post", "attendance"),
id = "student", n_profiles = 2, time = "sequence", n_starts = 2, seed = 1
)
get_results(fit, what = "transitions")
#> group_class from to probability expected_count stable estimated
#> 1 1 1 1 0.8614039 670.56162 TRUE TRUE
#> 2 1 1 2 0.1385961 107.89439 FALSE TRUE
#> 3 1 2 1 0.1850203 99.45941 FALSE TRUE
#> 4 1 2 2 0.8149797 438.08458 TRUE TRUE
#> group_class_probability
#> 1 1
#> 2 1
#> 3 1
#> 4 1
summary(fit)
#> Latent transition model: 2 profiles and 1 group class
#> Observations: 1422; groups: 106; up to 15 occasions (unbalanced, observed grid)
#> Parameters: 23; converged: TRUE
#> Log likelihood: -8335.287419; AIC: 16716.575
#> BIC (groups): 16777.834; BIC (individuals): 16837.551
#> Best likelihood replicated in 2/2 starts (absolute tolerance 0.00834).
#>
#> -- profiles --------------------------------------------------------
#> profile indicator mean variance standard_deviation mean_standard_error
#> 1 browse 0.5405 0.5929 0.7700 0.02688
#> 1 lectures 0.4266 0.8376 0.9152 0.03197
#> 1 forum_read 0.6173 0.4860 0.6972 0.02454
#> 1 forum_post 0.5297 0.6877 0.8293 0.02902
#> 1 attendance 0.6436 0.3916 0.6258 0.02207
#> 2 browse -0.7720 0.5139 0.7168 0.03027
#> 2 lectures -0.6093 0.5456 0.7387 0.03095
#> 2 forum_read -0.8814 0.3582 0.5985 0.02531
#> 2 forum_post -0.7565 0.4181 0.6466 0.02731
#> 2 attendance -0.9189 0.3780 0.6148 0.02598
#> variance_standard_error
#> 0.02930
#> 0.04135
#> 0.02453
#> 0.03411
#> 0.01971
#> 0.03088
#> 0.03266
#> 0.02152
#> 0.02552
#> 0.02257
#>
#> -- responses -------------------------------------------------------
#> (no rows)
#>
#> -- covariances -----------------------------------------------------
#> profile indicator indicator_2 covariance
#> 1 browse browse 0.5929
#> 1 lectures browse 0.0000
#> 1 forum_read browse 0.0000
#> 1 forum_post browse 0.0000
#> 1 attendance browse 0.0000
#> 1 browse lectures 0.0000
#> 1 lectures lectures 0.8376
#> 1 forum_read lectures 0.0000
#> 1 forum_post lectures 0.0000
#> 1 attendance lectures 0.0000
#> ... 40 more rows. get_results(x, what = "covariances")
#>
#> -- transitions -----------------------------------------------------
#> group_class from to probability expected_count stable estimated
#> 1 1 1 0.8614 670.56 TRUE TRUE
#> 1 1 2 0.1386 107.89 FALSE TRUE
#> 1 2 1 0.1850 99.46 FALSE TRUE
#> 1 2 2 0.8150 438.08 TRUE TRUE
#> group_class_probability
#> 1
#> 1
#> 1
#> 1
#>
#> -- initial ---------------------------------------------------------
#> group_class profile probability prevalence group_class_probability
#> 1 1 0.6258 0.5882 1
#> 1 2 0.3742 0.4118 1
#>
#> -- counts ----------------------------------------------------------
#> level class effective_count effective_proportion
#> individuals 1 836.4 0.5882
#> individuals 2 585.6 0.4118
#> groups 1 106.0 1.0000
#>
#> -- posteriors ------------------------------------------------------
#> row group profile posterior modal
#> 1 1 1 1.000e+00 TRUE
#> 2 1 1 9.971e-01 TRUE
#> 3 1 1 1.387e-05 FALSE
#> 4 1 1 8.982e-07 FALSE
#> 5 1 1 1.263e-06 FALSE
#> 6 1 1 4.927e-06 FALSE
#> 7 1 1 4.797e-06 FALSE
#> 8 1 1 2.279e-06 FALSE
#> 9 1 1 4.709e-07 FALSE
#> 10 1 1 2.102e-06 FALSE
#> ... 2834 more rows. get_results(x, what = "posteriors")
#>
#> -- group_posteriors ------------------------------------------------
#> group group_size log_likelihood group_class posterior modal
#> 1 13 -59.47 1 1 TRUE
#> 2 13 -71.93 1 1 TRUE
#> 3 14 -75.38 1 1 TRUE
#> 4 13 -66.30 1 1 TRUE
#> 5 14 -79.40 1 1 TRUE
#> 6 12 -80.17 1 1 TRUE
#> 7 15 -88.32 1 1 TRUE
#> 8 14 -81.68 1 1 TRUE
#> 9 14 -82.52 1 1 TRUE
#> 10 15 -82.01 1 1 TRUE
#> ... 96 more rows. get_results(x, what = "group_posteriors")
#>
#> -- assignments -----------------------------------------------------
#> student sequence browse lectures forum_read forum_post attendance profile
#> 1 1 0.73 -0.08 0.30 0.67 0.72 1
#> 1 2 0.68 0.53 -0.02 -0.55 0.70 1
#> 1 3 -1.51 -0.51 -0.91 -0.80 -0.97 2
#> 1 4 0.64 -1.02 -1.62 -1.20 -1.26 2
#> 1 5 -0.94 -0.37 -0.71 -0.28 -1.52 2
#> 1 6 -0.36 -0.46 -0.77 -0.43 -1.34 2
#> 1 7 0.38 -1.85 -0.84 -1.09 -1.12 2
#> 1 8 -1.44 0.66 -0.67 -1.67 -1.00 2
#> 1 9 -0.76 -0.43 -1.04 -0.69 -1.37 2
#> 1 10 0.26 -0.95 -0.51 -1.89 -1.45 2
#> group_class uncertainty posterior_profile_1 posterior_profile_2
#> 1 1.174e-05 1.000e+00 1.174e-05
#> 1 2.864e-03 9.971e-01 2.864e-03
#> 1 1.387e-05 1.387e-05 1.000e+00
#> 1 8.982e-07 8.982e-07 1.000e+00
#> 1 1.263e-06 1.263e-06 1.000e+00
#> 1 4.927e-06 4.927e-06 1.000e+00
#> 1 4.797e-06 4.797e-06 1.000e+00
#> 1 2.279e-06 2.279e-06 1.000e+00
#> 1 4.709e-07 4.709e-07 1.000e+00
#> 1 2.102e-06 2.102e-06 1.000e+00
#> ... 1412 more rows. get_results(x, what = "assignments")
#>
#> -- classification --------------------------------------------------
#> level class n_modal proportion_modal estimated_n estimated_proportion
#> individuals 1 837 0.5886 836.4 0.5882
#> individuals 2 585 0.4114 585.6 0.4118
#> groups 1 106 1.0000 106.0 1.0000
#> average_posterior odds_correct_classification
#> 0.9915 81.98
#> 0.9890 128.22
#> 1.0000 NA
#>
#> -- average_posteriors ----------------------------------------------
#> level assigned_class class n_assigned average_posterior
#> individuals 1 1 837 0.991531
#> individuals 1 2 837 0.008469
#> individuals 2 1 585 0.011015
#> individuals 2 2 585 0.988985
#> groups 1 1 106 1.000000
#>
#> -- classification_errors -------------------------------------------
#> level true_class assigned_class probability
#> individuals 1 1 0.992295
#> individuals 1 2 0.007705
#> individuals 2 1 0.012104
#> individuals 2 2 0.987896
#> groups 1 1 1.000000
#>
#> -- bch_weights -----------------------------------------------------
#> level unit assigned_class class weight
#> individuals 1 1 1 1.00786
#> individuals 1 1 2 -0.00786
#> individuals 2 1 1 1.00786
#> individuals 2 1 2 -0.00786
#> individuals 3 2 1 -0.01235
#> individuals 3 2 2 1.01235
#> individuals 4 2 1 -0.01235
#> individuals 4 2 2 1.01235
#> individuals 5 2 1 -0.01235
#> individuals 5 2 2 1.01235
#> ... 2940 more rows. get_results(x, what = "bch_weights")
#>
#> -- entropy ---------------------------------------------------------
#> level n_classes n_units entropy_sum relative_entropy
#> individuals 2 1422 38.3 0.9611
#> groups 1 106 0.0 NA
#>
#> -- residuals -------------------------------------------------------
#> profile indicator_1 indicator_2 kind observed expected residual
#> profile_1 lectures attendance gaussian 0.23623 0 0.23623
#> profile_2 forum_read attendance gaussian 0.23302 0 0.23302
#> profile_1 forum_read attendance gaussian 0.21662 0 0.21662
#> profile_1 forum_post attendance gaussian 0.20818 0 0.20818
#> profile_2 browse attendance gaussian 0.19679 0 0.19679
#> profile_1 browse attendance gaussian 0.16968 0 0.16968
#> profile_2 lectures attendance gaussian 0.11963 0 0.11963
#> profile_2 forum_post attendance gaussian 0.11548 0 0.11548
#> profile_2 browse forum_read gaussian 0.10229 0 0.10229
#> profile_2 forum_read forum_post gaussian 0.06942 0 0.06942
#> effective_n statistic df p_value p_adjusted
#> 836.4 6.951 NA 3.637e-12 3.637e-12
#> 585.6 5.730 NA 1.005e-08 1.005e-08
#> 836.4 6.354 NA 2.096e-10 2.096e-10
#> 836.4 6.099 NA 1.068e-09 1.068e-09
#> 585.6 4.813 NA 1.487e-06 1.487e-06
#> 836.4 4.946 NA 7.573e-07 7.573e-07
#> 585.6 2.901 NA 3.714e-03 3.714e-03
#> 585.6 2.800 NA 5.113e-03 5.113e-03
#> 585.6 2.478 NA 1.322e-02 1.322e-02
#> 585.6 1.678 NA 9.329e-02 9.329e-02
#> ... 10 more rows. get_results(x, what = "residuals")
#>
#> -- information_criteria --------------------------------------------
#> log_likelihood n_parameters aic kic bic_groups bic_individual sabic_groups
#> -8335 23 16717 16743 16778 16838 16705
#> sabic_individual caic_groups caic_individual awe_groups awe_individual
#> 16764 16801 16861 16954 17150
#> icl_groups icl_individual clc_groups clc_individual
#> 16778 16914 16671 16747
#>
#> -- model -----------------------------------------------------------
#> n_observations n_informative n_groups n_profiles n_group_classes centering
#> 1422 1422 106 2 1 none
#> covariance_structure n_parameters n_parameters_with_measurement log_likelihood
#> <NA> 23 23 -8335
#> aic bic_groups bic_individual converged iterations boundary small_classes
#> 16717 16778 16838 TRUE 6 FALSE FALSE
#> best_start n_best_replicated weights
#> 2 2 <NA>
#>
#> -- sequences -------------------------------------------------------
#> group group_class time profile
#> 1 1 1 1
#> 1 1 2 1
#> 1 1 3 2
#> 1 1 4 2
#> 1 1 5 2
#> 1 1 6 2
#> 1 1 7 2
#> 1 1 8 2
#> 1 1 9 2
#> 1 1 10 2
#> ... 1412 more rows. get_results(x, what = "sequences")
#>
#> -- sequence_summary ------------------------------------------------
#> group_class groups observations mean_length median_length shortest longest
#> 1 106 1422 13.42 14 10 15
#> complete gaps
#> 24 0
#>
#> -- sequence_lengths ------------------------------------------------
#> group group_class occasions observations complete
#> 1 1 13 13 FALSE
#> 2 1 13 13 FALSE
#> 3 1 14 14 FALSE
#> 4 1 13 13 FALSE
#> 5 1 14 14 FALSE
#> 6 1 12 12 FALSE
#> 7 1 15 15 TRUE
#> 8 1 14 14 FALSE
#> 9 1 14 14 FALSE
#> 10 1 15 15 TRUE
#> ... 96 more rows. get_results(x, what = "sequence_lengths")
#>
#> -- starts ----------------------------------------------------------
#> start log_likelihood converged iterations error
#> 1 -8335 TRUE 4 <NA>
#> 2 -8335 TRUE 6 <NA>
#>
#> -- data ------------------------------------------------------------
#> student sequence browse lectures forum_read forum_post attendance
#> 1 1 0.73 -0.08 0.30 0.67 0.72
#> 1 2 0.68 0.53 -0.02 -0.55 0.70
#> 1 3 -1.51 -0.51 -0.91 -0.80 -0.97
#> 1 4 0.64 -1.02 -1.62 -1.20 -1.26
#> 1 5 -0.94 -0.37 -0.71 -0.28 -1.52
#> 1 6 -0.36 -0.46 -0.77 -0.43 -1.34
#> 1 7 0.38 -1.85 -0.84 -1.09 -1.12
#> 1 8 -1.44 0.66 -0.67 -1.67 -1.00
#> 1 9 -0.76 -0.43 -1.04 -0.69 -1.37
#> 1 10 0.26 -0.95 -0.51 -1.89 -1.45
#> ... 1412 more rows. get_results(x, what = "data")
#>
#> 22 tables above, truncated to fit. get_results(x, what = ) returns any
#> of them whole, and get_results(x, what = "all") returns every one.