Introduction to LMest
Bartolucci, F., Pandolfi, S., Pennoni, F., Serafini, A.
2024-08-30
Introduction
The package LMest allows users
to specify and fit Latent (or Hidden) Markov (LM) models for the analysis of longitudinal continuous data,
categorical data and time-series.
It
includes
functions for parameter estimation, via the Expectation-Maximization (EM) algorithm,
covering
the basic LM model and its extensions with individual covariates
through
suitable parameterizations.
Additionally, it provides
functions for
simulated data
from these models,
performing model selection
(including searching
for the global maximum likelihood), and
conducting
local and global decoding.
The package also offer
standard errors for model parameters which,
can be obtained using
numerical or exact methods
as well as
parametric bootstrap
techniques.
This document introduces the LMest’s basic set of tools, and
demonstrates how to apply them for the different model specifications,
using data frames in
various formats.
LMest contains the following main functions
each with specific tasks:
lmestData() |
to manipulate data in long format |
lmestFormula() |
to build the formulas specifying the model to be estimated |
lmest() |
to estimate LM models for categorical responses with or without covariates |
se() |
to obtain standard errors for the estimated LM model |
lmestCont() |
to estimate LM models for continuous outcomes with or without covariates |
lmestMixed() |
to estimate LM models for categorical responses with initial and transition probabilities of the latent process allowed to vary across different latent sub-populations |
lmestSearch() |
to search for the global maximum of the model log-likelihood and to select the optimal number of latent states |
lmestDecoding() |
to perform local and global decoding (Viterbi algorithm) |
bootstrap() |
to perform bootstrap parametric resampling to compute standard errors for the parameter estimates |
draw() |
to draw samples from the LM models |
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|
Data available are the following:
data_criminal_sim |
Simulated dataset about crimes committed by a cohort of subjects |
data_drug |
Longitudinal dataset about marijuana consumption deriving from the National Youth Survey |
data_SRHS_long |
Dataset about self-reported health status deriving from the Health and Retirement Study conducted by the University of Michigan in long format |
PSIDlong |
Longitudinal dataset deriving from the Panel Study of Income Dynamics |
RLMSdat |
Longitudinal dataset deriving from the Russia Longitudinal Monitoring Survey (RLMS) about job satisfaction |
RLMSlong |
Longitudinal dataset deriving from the Russia Longitudinal Monitoring Survey (RLMS) about job satisfaction in long format |
NLSYlong |
Longitudinal dataset deriving from the National Longitudinal Survey of Youth (NLSY) about antisocial behavior and self-esteem |
data_employment_sim |
Simulated dataset assuming interviews conducted among a nationally representative sample of graduates to investigate their employment status after graduation |
data_market_sim |
Simulated dataset assuming observations of customers of four different brands along with the prices of each transaction. |
data_heart_sim |
Simulated dataset assuming an observational retrospective study to assess the health state progression of individuals after treatment. |
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|
See help(package="LMest") for further details and citation("LMest") for main references.
library(LMest)
#> LMest: Generalized Latent Markov Models (Version 3.2.0)
#> Type 'citation("LMest")' for citing this package.
Data: RLMSlong
This dataset,
included
in the LMest package, concerns the evaluation of job satisfaction of \(n\) = 1,718 individuals followed
over
\(T\) = 7 years from 2008 to 2014.
The data come from the Russia Longitudinal Monitoring Survey, and are documented in ?RLMSlong.
The response variable (named value), corresponding to the reported job satisfaction at different time occasions, has five ordered categories from 1: absolutely satisfied to 5: absolutely not satisfied.
The longitudinal dataset is in long format:
data("RLMSlong")
dim(RLMSlong)
#> [1] 12026 4
str(RLMSlong)
#> 'data.frame': 12026 obs. of 4 variables:
#> $ time : int 1 1 1 1 1 1 1 1 1 1 ...
#> $ id : int 1 2 3 4 5 6 7 8 9 10 ...
#> $ rlms : Factor w/ 7 levels "IKSJQ","IKSJR",..: 1 1 1 1 1 1 1 1 1 1 ...
#> $ value: int 2 2 2 2 2 3 1 3 1 1 ...
Data: PSIDlong
This dataset
is derived from the Panel Study of Income Dynamics (https://psidonline.isr.umich.edu).
The data used in the following application concern \(n\) = 1,446 women who were followed from 1987 to 1993.
There are two binary response variables:
Y1Fertility (indicating whether a woman had given birth to a child in a certain year) and
Y2Employment (indicating whether she was employed).
The covariates are: X1Race (dummy variable equal to 1 for a black woman); X2Age (in 1986, rescaled by its maximum value); X3Age2 (squared age); X4Education (number of years of schooling); X5Child1_2 (number of children in the family aged between 1 and 2 years, referred to the previous year); X6Child3_5; X7Child6_13; X8Child14; X9Income of the husband (in dollars, referred to the previous year, divided by 1,000).
data("PSIDlong")
dim(PSIDlong)
#> [1] 10122 13
str(PSIDlong)
#> 'data.frame': 10122 obs. of 13 variables:
#> $ id : num 1 1 1 1 1 1 1 2 2 2 ...
#> $ time : num 1 2 3 4 5 6 7 1 2 3 ...
#> $ Y1Fertility : num 0 0 0 0 0 0 0 1 0 0 ...
#> $ Y2Employment: num 1 1 1 1 1 1 1 1 1 1 ...
#> $ X1Race : num 0 0 0 0 0 0 0 0 0 0 ...
#> $ X2Age : num -8 -8 -8 -8 -8 -8 -8 -15 -15 -15 ...
#> $ X3Age2 : num 0.64 0.64 0.64 0.64 0.64 0.64 0.64 2.25 2.25 2.25 ...
#> $ X4Education : num 12 12 12 12 12 12 12 12 12 12 ...
#> $ X5Child1_2 : num 0 0 0 0 0 0 0 0 0 1 ...
#> $ X6Child3_5 : num 0 0 0 0 0 0 0 0 0 0 ...
#> $ X7Child6_13 : num 0 2 2 2 2 1 1 0 1 1 ...
#> $ X8Child14 : num 0 0 0 0 0 1 1 0 0 0 ...
#> $ X9Income : num 35.3 28.2 33.2 37 39.9 ...
Data: data_criminal_sim
This simulated dataset
contains \(n\) = 60,000 observations related to the complete conviction histories of a cohort of offenders followed from the age of criminal responsibility, 10 years, until 40 years. It also includes the proportion of non-offenders.
We consider \(T=6\) age bands
each of five years in length.
For every age band, we
have a binary variable equal to 1 if the subject has been convicted for a crime of one of the following ten typologies and to 0 otherwise.
The typologies of crime are: y1 violence against the person, y2 sexual offenses, y3 burglary, y4 robbery, y5 theft and handling stolen goods, y6 fraud and forgery, y7 criminal damage, y8 drug offenses, y9 motoring offenses, y10 other offenses.
Gender is a covariate coded equal to 1 for male and 2 for female:
data(data_criminal_sim)
dim(data_criminal_sim)
#> [1] 60000 13
str(data_criminal_sim)
#> num [1:60000, 1:13] 1 1 1 1 1 1 2 2 2 2 ...
#> - attr(*, "dimnames")=List of 2
#> ..$ : NULL
#> ..$ : chr [1:13] "id" "sex" "time" "y1" ...
Data: NLSYlong
This dataset in the LMest package, also downloadable from the package panelr, is derived from the National Longitudinal Survey of Youth (https://www.nlsinfo.org/content/cohorts/nlsy79).
The data are a subset concerning \(n\) = 581 individuals who were followed from 1990 to 1994.
We consider two response variables: anti, providing a measure of antisocial behavior (measured on a scale ranging from 0 to 6), self, which is a measure of self-esteem (measured on a scale ranging from 6 to 24).
The covariates are the following: momage, a continuous variable indicating the age of the mother at birth; gender, a dummy variable equal to 1 for females; childage, a continuous variable indicating the age of the child at the first interview; momwork, a dummy variable equal to 1 if mother works; married, a dummy variable equal to 1 if parents are married; hispanic and black, two dummy variables for ethnicity; pov, a time-varying binary variable indicating the poverty status of the family in the years 1990, 1992 and 1994.
data("NLSYlong")
dim(NLSYlong)
#> [1] 1743 12
Prepare and explore data
Function lmestData() allows us to check and prepare the data. For example, for the NLSYlong dataset we have:
dt <- lmestData(data = NLSYlong, id = "id", time="time",
responsesFormula= anti+self ~NULL)
summary(dt, dataSummary="responses", varType=rep("c",ncol(dt$Y)))
#>
#> Data Info:
#> ----------
#>
#> Observations: 581
#> Time occasions: 3
#> Variables: 2
#>
#>
#> Summary:
#> ----------
#>
#> anti self
#> Min. :0.000000 Min. : 6.0000
#> 1st Qu.:0.000000 1st Qu.:18.0000
#> Median :1.000000 Median :21.0000
#> Mean :1.636833 Mean :20.3502
#> 3rd Qu.:3.000000 3rd Qu.:23.0000
#> Max. :6.000000 Max. :24.0000
#>
#> Summary by year:
#> ----------
#>
#>
#> Time = 0
#>
#> anti self
#> Min. :0.000000 Min. : 9.00000
#> 1st Qu.:0.000000 1st Qu.:18.00000
#> Median :1.000000 Median :21.00000
#> Mean :1.567986 Mean :20.07057
#> 3rd Qu.:2.000000 3rd Qu.:23.00000
#> Max. :6.000000 Max. :24.00000
#>
#> Time = 2
#>
#> anti self
#> Min. :0.000000 Min. : 6.00000
#> 1st Qu.:0.000000 1st Qu.:18.00000
#> Median :1.000000 Median :21.00000
#> Mean :1.595525 Mean :20.36213
#> 3rd Qu.:3.000000 3rd Qu.:23.00000
#> Max. :6.000000 Max. :24.00000
#>
#> Time = 4
#>
#> anti self
#> Min. :0.000000 Min. : 9.0000
#> 1st Qu.:0.000000 1st Qu.:19.0000
#> Median :1.000000 Median :21.0000
#> Mean :1.746988 Mean :20.6179
#> 3rd Qu.:3.000000 3rd Qu.:23.0000
#> Max. :6.000000 Max. :24.0000
We can display the summary of each response variable for every time occasion.
For the criminal data, if we select only females, we have:
data_criminal_sim<-data.frame(data_criminal_sim)
crimf <- data_criminal_sim[data_criminal_sim$sex == 2,]
dt1 <- lmestData(data = crimf, id = "id", time = "time")
summary(dt1, varType = rep("d",ncol(dt1$Y)))
#>
#> Data Info:
#> ----------
#>
#> Observations: 5200
#> Time occasions: 6
#> Variables: 11
#>
#>
#> Proportion:
#> ----------
#>
#> time sex y1 y2 y3 y4 y5
#> 1:5200 2:1 0:0.9838782 0:0.9971474 0:0.9796154 0:0.9981731 0:0.9427564
#> 2:5200 1:0.0161218 1:0.0028526 1:0.0203846 1:0.0018269 1:0.0572436
#> 3:5200
#> 4:5200
#> 5:5200
#> 6:5200
#> y6 y7 y8 y9 y10
#> 0:0.9909615 0:0.9839103 0:0.991859 0:0.9987179 0:0.9849679
#> 1:0.0090385 1:0.0160897 1:0.008141 1:0.0012821 1:0.0150321
#>
#>
#>
#>
#>
#> Proportion by year:
#> ----------
#>
#>
#> Time = 1
#>
#> time sex y1 y2 y3 y4 y5
#> 1:5200 2:1 0:0.9886538 0:0.9982692 0:0.9696154 0:0.9990385 0:0.9328846
#> 1:0.0113462 1:0.0017308 1:0.0303846 1:0.0009615 1:0.0671154
#> y6 y7 y8 y9 y10
#> 0:0.9957692 0:0.9830769 0:0.9963462 0:0.9990385 0:0.9884615
#> 1:0.0042308 1:0.0169231 1:0.0036538 1:0.0009615 1:0.0115385
#>
#> Time = 2
#>
#> time sex y1 y2 y3 y4 y5
#> 2:5200 2:1 0:0.9755769 0:0.9942308 0:0.9598077 0:0.9965385 0:0.8901923
#> 1:0.0244231 1:0.0057692 1:0.0401923 1:0.0034615 1:0.1098077
#> y6 y7 y8 y9 y10
#> 0:0.9832692 0:0.9696154 0:0.9880769 0:0.9973077 0:0.9707692
#> 1:0.0167308 1:0.0303846 1:0.0119231 1:0.0026923 1:0.0292308
#>
#> Time = 3
#>
#> time sex y1 y2 y3 y4 y5 y6
#> 3:5200 2:1 0:0.975 0:0.9963462 0:0.975 0:0.9973077 0:0.9290385 0:0.9871154
#> 1:0.025 1:0.0036538 1:0.025 1:0.0026923 1:0.0709615 1:0.0128846
#> y7 y8 y9 y10
#> 0:0.9778846 0:0.9859615 0:0.9975 0:0.9773077
#> 1:0.0221154 1:0.0140385 1:0.0025 1:0.0226923
#>
#> Time = 4
#>
#> time sex y1 y2 y3 y4 y5
#> 4:5200 2:1 0:0.9825 0:0.9965385 0:0.9861538 0:0.9980769 0:0.9523077
#> 1:0.0175 1:0.0034615 1:0.0138462 1:0.0019231 1:0.0476923
#> y6 y7 y8 y9 y10
#> 0:0.9901923 0:0.9867308 0:0.9907692 0:0.9994231 0:0.9876923
#> 1:0.0098077 1:0.0132692 1:0.0092308 1:0.0005769 1:0.0123077
#>
#> Time = 5
#>
#> time sex y1 y2 y3 y4 y5
#> 5:5200 2:1 0:0.9873077 0:0.9982692 0:0.9919231 0:0.9984615 0:0.97
#> 1:0.0126923 1:0.0017308 1:0.0080769 1:0.0015385 1:0.03
#> y6 y7 y8 y9 y10
#> 0:0.9938462 0:0.9909615 0:0.9936538 0:0.9994231 0:0.9909615
#> 1:0.0061538 1:0.0090385 1:0.0063462 1:0.0005769 1:0.0090385
#>
#> Time = 6
#>
#> time sex y1 y2 y3 y4 y5
#> 6:5200 2:1 0:0.9942308 0:0.9992308 0:0.9951923 0:0.9996154 0:0.9821154
#> 1:0.0057692 1:0.0007692 1:0.0048077 1:0.0003846 1:0.0178846
#> y6 y7 y8 y9 y10
#> 0:0.9955769 0:0.9951923 0:0.9963462 0:0.9996154 0:0.9946154
#> 1:0.0044231 1:0.0048077 1:0.0036538 1:0.0003846 1:0.0053846
Latent Markov models for categorical responses
Function lmest() allows us to estimate LM models for categorical responses with different model specifications,
both
with and without covariates.
These models rely on
the
homogeneous
first-order Markov chain
with a finite number of states.
Maximum likelihood estimation of model parameters is performed through the EM algorithm.
Standard errors for the parameter estimates are obtained by exact computation of the information matrix or through reliable numerical approximations of this matrix.
This can be done by using option out_se=TRUE or by using the suitable function
se().
Using PSIDlong, the basic LM model with time heterogenous transition probabilities can be estimated as follows:
mod <- lmest(responsesFormula = fmLatent$responsesFormula,
index = c("id","time"),
data = PSIDlong, k = 2)
The function requires a data in long format, so
“id” column and a “time” column must be specified in the argument “index”. The number of latent state may be specified as a single
value
or a vector of integer values as follows:
mod <- lmest(responsesFormula = fmLatent$responsesFormula,
index = c("id","time"),
data = PSIDlong, k = 1:3)
The suitable number of latent states is selected using the BIC or AIC criterion and returned.
Print method shows the main results:
print(mod)
#>
#> Basic Latent Markov model
#> Call:
#> lmest(responsesFormula = fmLatent$responsesFormula, data = PSIDlong,
#> index = c("id", "time"), k = 1:3)
#>
#> Available objects:
#> [1] "lk" "piv" "Pi" "Psi" "np" "k"
#> [7] "aic" "bic" "lkv" "n" "TT" "modBasic"
#> [13] "yv" "ns" "Lk" "Bic" "Aic" "call"
#> [19] "data"
#>
#> Convergence info:
#> LogLik np k AIC BIC n TT
#> -6900.447 17 2 13834.89 13924.6 1446 7
Standard errors can be obtained with function se() as:
se(mod)
#> $sepiv
#> [1] 0.01541024 0.01541024
#>
#> $sePi
#> , , time = 1
#>
#> state
#> state 1 2
#> 1 0 0
#> 2 0 0
#>
#> , , time = 2
#>
#> state
#> state 1 2
#> 1 0.02212323 0.02212323
#> 2 0.01311762 0.01311762
#>
#> , , time = 3
#>
#> state
#> state 1 2
#> 1 0.02149914 0.02149914
#> 2 0.01074772 0.01074772
#>
#> , , time = 4
#>
#> state
#> state 1 2
#> 1 0.023346423 0.023346423
#> 2 0.009465333 0.009465333
#>
#> , , time = 5
#>
#> state
#> state 1 2
#> 1 0.02084359 0.02084359
#> 2 0.01037142 0.01037142
#>
#> , , time = 6
#>
#> state
#> state 1 2
#> 1 0.02017732 0.02017732
#> 2 0.01002753 0.01002753
#>
#> , , time = 7
#>
#> state
#> state 1 2
#> 1 0.02561638 0.02561638
#> 2 0.01156591 0.01156591
#>
#>
#> $sePsi
#> , , item = 1
#>
#> state
#> category 1 2
#> 0 0.005557113 0.00282196
#> 1 0.005557113 0.00282196
#>
#> , , item = 2
#>
#> state
#> category 1 2
#> 0 0.01066309 0.004149002
#> 1 0.01066309 0.004149002
For the data PSIDlong, we can estimate an LM model with covariates affecting the distribution of the latent process by fixing \(k\) = 2 latent states as follows:
mod2 <- lmest(responsesFormula = fmLatent$responsesFormula,
latentFormula = fmLatent$latentFormula,
index = c("id","time"),
data = PSIDlong, k = 2,
paramLatent = "multilogit",
start = 0, out_se=TRUE)
#> ------------|-------------|-------------|-------------|-------------|-------------|
#> k | start | step | lk | lk-lko | discrepancy |
#> ------------|-------------|-------------|-------------|-------------|-------------|
#> 2 | 0 | 0 | -8957.56 |
#> 2 | 0 | 10 | -6782.67 | 0.320199 | 0.00644032 |
#> 2 | 0 | 20 | -6782.14 | 0.00309348 | 0.000645718 |
#> 2 | 0 | 29 | -6782.13 | 5.10208e-05 | 8.22408e-05 |
#> ------------|-------------|-------------|-------------|-------------|-------------|
Every 10 iterations of the EM algorithm,
the function displays the following information:
- the chosen model specification
- number of latent states
- the type of starting values used
- the number of iterations
- the value of the log-likelihood at the end of the current iteration
- the difference
in log-likelihood
from the previous iteration
- the discrepancy between the corresponding parameter vectors.
The summary() method returns the estimation results:
summary(mod2)
#> Call:
#> lmest(responsesFormula = fmLatent$responsesFormula, latentFormula = fmLatent$latentFormula,
#> data = PSIDlong, index = c("id", "time"), k = 2, start = 0,
#> paramLatent = "multilogit", out_se = TRUE)
#>
#> Coefficients:
#>
#> Be - Parameters affecting the logit for the initial probabilities:
#> logit
#> 2
#> (Intercept) -1.4473
#> X1Race 0.0783
#> X2Age -0.0952
#> X3Age2 -0.5027
#> X4Education 0.2170
#> X5Child1_2 -0.8882
#> X6Child3_5 -0.5797
#> X7Child6_13 0.1647
#> X8Child14 0.1014
#> X9Income -0.0189
#>
#> Standard errors for Be:
#> logit
#> 2
#> (Intercept) 0.6170
#> X1Race 0.1625
#> X2Age 0.0678
#> X3Age2 0.2916
#> X4Education 0.0376
#> X5Child1_2 0.1171
#> X6Child3_5 0.1116
#> X7Child6_13 0.7246
#> X8Child14 0.1638
#> X9Income 0.0044
#>
#> p-values for Be:
#> logit
#> 2
#> (Intercept) 0.019
#> X1Race 0.630
#> X2Age 0.160
#> X3Age2 0.085
#> X4Education 0.000
#> X5Child1_2 0.000
#> X6Child3_5 0.000
#> X7Child6_13 0.820
#> X8Child14 0.536
#> X9Income 0.000
#>
#> Ga - Parameters affecting the logit for the transition probabilities:
#> logit
#> 1 2
#> (Intercept) -3.5582 -3.7495
#> X1Race -0.1057 -0.4382
#> X2Age 0.0298 -0.0497
#> X3Age2 0.2579 0.2224
#> X4Education 0.1658 -0.0061
#> X5Child1_2 -0.1673 -0.2244
#> X6Child3_5 -0.2238 0.1116
#> X7Child6_13 0.0874 -0.0968
#> X8Child14 -0.0671 0.1715
#> X9Income -0.0113 0.0088
#>
#> Standard errors for Ga:
#> logit
#> 1 2
#> (Intercept) 0.7166 0.7512
#> X1Race 0.1920 0.2026
#> X2Age 0.0728 0.0777
#> X3Age2 0.3135 0.3174
#> X4Education 0.0407 0.0421
#> X5Child1_2 0.1373 0.1780
#> X6Child3_5 0.1250 0.1444
#> X7Child6_13 0.0751 0.0996
#> X8Child14 0.1392 0.1326
#> X9Income 0.0033 0.0028
#>
#> p-values for Ga:
#> logit
#> 1 2
#> (Intercept) 0.000 0.000
#> X1Race 0.582 0.031
#> X2Age 0.683 0.522
#> X3Age2 0.411 0.483
#> X4Education 0.000 0.885
#> X5Child1_2 0.223 0.207
#> X6Child3_5 0.073 0.440
#> X7Child6_13 0.244 0.331
#> X8Child14 0.630 0.196
#> X9Income 0.001 0.002
#>
#> Psi - Conditional response probabilities:
#> , , item = 1
#>
#> state
#> category 1 2
#> 0 0.8992 0.9485
#> 1 0.1008 0.0515
#>
#> , , item = 2
#>
#> state
#> category 1 2
#> 0 0.9068 0.0347
#> 1 0.0932 0.9653
#>
#>
#> Standard errors for the conditional response probability matrix:
#> , , item = 1
#>
#> state
#> category 1 2
#> 0 0.0056 0.0028
#> 1 0.0056 0.0028
#>
#> , , item = 2
#>
#> state
#> category 1 2
#> 0 0.0099 0.0041
#> 1 0.0099 0.0041
A plot of the conditional response probabilities referred to the categories of the multivariate response is obtained as:
plot(mod2, what = "CondProb")
To
gain insight the results, we observe that the second latent state corresponds to women with the lowest propensity
for fertility and the highest propensity
for employment.
The first state corresponds to women with both low propensity
for
fertility and
a low likelihood of having a job.
A path diagram of the estimated transition probabilities is shown below:
plot(mod2, what="transitions")
The averaged estimated transition matrix
reveals
a high level of persistence
within
the same latent state.
Specifically,
the probability of transitioning from the first state to the second state is approximately 0.06.
The estimated marginal distribution of the latent states for each time occasion can be represented in the following plot:
plot(mod2, what="marginal")
Latent Markov model for continuous outcomes
The LM model for continuous outcomes may be estimated by using function lmestCont(), assuming a Gaussian distribution for the response variables given the latent process.
For the data NLSYlong, we estimate a multivariate LM model with covariates in the latent process. The selection of the number of latent states can be performed by setting option k
appropriately:
dt$data$id = as.numeric(dt$data$id)
dt$data$time = as.numeric(dt$data$time)
modc <- lmestCont(responsesFormula = anti + self ~ NULL,
latentFormula = ~ gender + childage + hispanic + black + pov +
momwork + married| gender + childage + hispanic + black+ pov+
momwork + married,
index = c("id", "time"),
data = dt$data,
k = 1:3,
modBasic=1,
output = TRUE,
tol=10^-1)
We can display the estimation results with a plot of the indices based on the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC):
plot(modc,what="modSel")
A plot of the ellipse of the estimated overall density, weighted by the estimated marginal distribution of the latent states, is obtained as follows:
plot(modc, what="density")
A density plot for each component (latent state) is obtained as:
plot(modc,what="density",components=c(1,2))
The latent states are ordered according to increasing values of antisocial behavior.
The path diagram of the estimated transition probabilities is obtained as follows:
plot(modc,what="transitions")
Other results and asymptotic tests can be obtained using the estimated standard errors
semodc<-se(modc)
TabBe <-cbind(modc$Be, semodc$seBe, modc$Be/semodc$seBe)
colnames(TabBe) <- c("estBe", "s.e.Be","t-test")
round(TabBe,3)
#> estBe s.e.Be t-test
#> (Intercept) -0.223 2.056 -0.108
#> gender -0.505 0.280 -1.803
#> childage 0.045 0.231 0.196
#> hispanic -0.392 0.374 -1.048
#> black 0.043 0.351 0.122
#> pov 0.403 0.352 1.145
#> momwork 0.091 0.293 0.311
#> married -0.258 0.361 -0.716
The argument Be, returned by the function, contains the estimated regression parameters affecting the distribution of the initial probabilities. The gender log-odds (second row of Be) is negative and significant, indicating that females tend to be allocated to the first latent state at the beginning of the survey.
TabGa1 <- cbind(modc$Ga,semodc$seGa,modc$Ga/semodc$seGa)
colnames(TabGa1) <- c("estGa(1)","estGa(2)", "s.e.Ga(1)","s.e.Ga(2)", "t-test(1)","t-test(2)")
round(TabGa1,3)
#> estGa(1) estGa(2) s.e.Ga(1) s.e.Ga(2) t-test(1) t-test(2)
#> (Intercept) -2.388 -1.494 6.285 5.247 -0.380 -0.285
#> gender -0.272 0.388 0.853 1.073 -0.319 0.362
#> childage 0.015 -0.125 0.736 0.573 0.021 -0.218
#> hispanic -0.025 -0.063 1.008 1.260 -0.024 -0.050
#> black 0.186 -0.232 1.285 0.987 0.145 -0.235
#> pov 0.141 -0.125 1.842 0.768 0.077 -0.163
#> momwork -0.045 -0.083 0.957 0.755 -0.047 -0.110
#> married -0.120 0.115 1.376 1.049 -0.087 0.110
Output Ga contains the estimated parameters affecting the distribution of the transition probabilities.
These parameters
measure
the influence of each covariate on the transition between states.
Mixed Latent Markov model
Function lmestMixed() allows us to estimate mixed LM models for categorical responses,
taking
into account additional sources of (time-fixed) dependence in the data.
For the data data_criminal_sim we are interested to evaluate the patterns of criminal behavior among individuals.
To this end,
we estimate a model with \(k_1\) = 2 latent classes and \(k_2\) = 2 latent states, restricting the analysis to females.
responsesFormula <- lmestFormula(data = crimf,response = "y")$responsesFormula
modm <- lmestMixed(responsesFormula =responsesFormula,
index = c("id","time"),
k1 = 2, k2 = 2,
tol = 10^-3,
data = crimf)
Results:
summary(modm)
#> Call:
#> lmestMixed(responsesFormula = responsesFormula, data = crimf,
#> index = c("id", "time"), k1 = 2, k2 = 2, tol = 10^-3)
#>
#> Coefficients:
#>
#> Mass probabilities:
#> [1] 0.5929 0.4071
#>
#> Initial probabilities:
#> u
#> v 1 2
#> 1 0.9186 0.7904
#> 2 0.0814 0.2096
#>
#> Transition probabilities:
#> , , u = 1
#>
#> v1
#> v0 1 2
#> 1 0.9667 0.0333
#> 2 0.4983 0.5017
#>
#> , , u = 2
#>
#> v1
#> v0 1 2
#> 1 0.9762 0.0238
#> 2 0.3826 0.6174
#>
#>
#> Conditional response probabilities:
#> , , j = 1
#>
#> v
#> y 1 2
#> 0 0.9962 0.8637
#> 1 0.0038 0.1363
#>
#> , , j = 2
#>
#> v
#> y 1 2
#> 0 0.9985 0.9838
#> 1 0.0015 0.0162
#>
#> , , j = 3
#>
#> v
#> y 1 2
#> 0 0.9988 0.7935
#> 1 0.0012 0.2065
#>
#> , , j = 4
#>
#> v
#> y 1 2
#> 0 1 0.9807
#> 1 0 0.0193
#>
#> , , j = 5
#>
#> v
#> y 1 2
#> 0 0.9821 0.5601
#> 1 0.0179 0.4399
#>
#> , , j = 6
#>
#> v
#> y 1 2
#> 0 0.9985 0.9176
#> 1 0.0015 0.0824
#>
#> , , j = 7
#>
#> v
#> y 1 2
#> 0 0.9975 0.8521
#> 1 0.0025 0.1479
#>
#> , , j = 8
#>
#> v
#> y 1 2
#> 0 0.9981 0.931
#> 1 0.0019 0.069
#>
#> , , j = 9
#>
#> v
#> y 1 2
#> 0 1 0.9863
#> 1 0 0.0137
#>
#> , , j = 10
#>
#> v
#> y 1 2
#> 0 0.9993 0.8454
#> 1 0.0007 0.1546
round(modm$Psi[2, , ], 3)
#> j
#> v 1 2 3 4 5 6 7 8 9 10
#> 1 0.004 0.001 0.001 0.000 0.018 0.001 0.003 0.002 0.000 0.001
#> 2 0.136 0.016 0.207 0.019 0.440 0.082 0.148 0.069 0.014 0.155
We can identify the first latent state as
representing
females with null or very low tendency to commit crimes, whereas the second latent state corresponds to criminals
primarily engaged in
theft, burglary, and other offences
According to the estimated transition matrix, females classified in the first cluster present a higher probability (around 0.5)
of moving
from the second to the first latent state
compared to
those
in
the second cluster (of around 0.4).
This indicates
a more pronounced tendency
for individuals in the first cluster
to commit less crimes over time.
Search for the global maximum of the log-likelihood
Function lmestSearch() addresses both model selection and the multimodality of the likelihood function. It employs different initializations to search for the global maximum of the log-likelihood function.
Two main criteria are provided to select the number of latent states: AIC and BIC.
For example, for the RLMSlong dataset we can estimate the basic LM model for increasing values of the latent states \(k\) ranging from 1 to 4:
out <- lmestSearch(responsesFormula = fmBasic$responsesFormula,
index = c("id","time"),
data = RLMSlong,version ="categorical", k = 1:4,
modBasic = 1, seed = 123)
We can display the results of the model selection using:
summary(out)
#> Call:
#> lmestSearch(responsesFormula = fmBasic$responsesFormula, data = RLMSlong,
#> index = c("id", "time"), k = 1:4, version = "categorical",
#> seed = 123, modBasic = 1)
#>
#> states lk np AIC BIC
#> 1 -14943.73 4 29895.45 29917.25
#> 2 -13921.09 11 27864.18 27924.12
#> 3 -13557.20 20 27154.41 27263.39
#> 4 -13392.93 31 26847.86 27016.78
The minimum BIC index corresponds to the model with \(k\)=4 latent states, and the model has 31 free parameters.
The estimation results for the selected number of states can be displayed as follows:
mod4 <- out$out.single[[4]]
summary(mod4)
#> Call:
#> NULL
#>
#> Coefficients:
#>
#> Initial probabilities:
#> est_piv
#> [1,] 0.1717
#> [2,] 0.4400
#> [3,] 0.2021
#> [4,] 0.1863
#>
#> Transition probabilities:
#> , , time = 2
#>
#> state
#> state 1 2 3 4
#> 1 0.8809 0.0675 0.0250 0.0266
#> 2 0.0173 0.9219 0.0520 0.0088
#> 3 0.0165 0.0925 0.8727 0.0183
#> 4 0.0145 0.0642 0.1568 0.7645
#>
#> , , time = 3
#>
#> state
#> state 1 2 3 4
#> 1 0.8809 0.0675 0.0250 0.0266
#> 2 0.0173 0.9219 0.0520 0.0088
#> 3 0.0165 0.0925 0.8727 0.0183
#> 4 0.0145 0.0642 0.1568 0.7645
#>
#> , , time = 4
#>
#> state
#> state 1 2 3 4
#> 1 0.8809 0.0675 0.0250 0.0266
#> 2 0.0173 0.9219 0.0520 0.0088
#> 3 0.0165 0.0925 0.8727 0.0183
#> 4 0.0145 0.0642 0.1568 0.7645
#>
#> , , time = 5
#>
#> state
#> state 1 2 3 4
#> 1 0.8809 0.0675 0.0250 0.0266
#> 2 0.0173 0.9219 0.0520 0.0088
#> 3 0.0165 0.0925 0.8727 0.0183
#> 4 0.0145 0.0642 0.1568 0.7645
#>
#> , , time = 6
#>
#> state
#> state 1 2 3 4
#> 1 0.8809 0.0675 0.0250 0.0266
#> 2 0.0173 0.9219 0.0520 0.0088
#> 3 0.0165 0.0925 0.8727 0.0183
#> 4 0.0145 0.0642 0.1568 0.7645
#>
#> , , time = 7
#>
#> state
#> state 1 2 3 4
#> 1 0.8809 0.0675 0.0250 0.0266
#> 2 0.0173 0.9219 0.0520 0.0088
#> 3 0.0165 0.0925 0.8727 0.0183
#> 4 0.0145 0.0642 0.1568 0.7645
#>
#>
#> Conditional response probabilities:
#> , , item = 1
#>
#> state
#> category 1 2 3 4
#> 0 0.5793 0.0694 0.0239 0.0181
#> 1 0.3463 0.8413 0.3040 0.1679
#> 2 0.0493 0.0635 0.5576 0.1851
#> 3 0.0222 0.0248 0.1035 0.4541
#> 4 0.0030 0.0009 0.0111 0.1748
A plot of the conditional response probabilities referred to the categories of the univariate response is obtained with:
plot(mod4, what="CondProb")
Local and global decoding
Function lmestDecoding() allows us to predict the sequence of latent states for the sample units on the basis of the output of the main estimation functions, thus enabling
“dynamic pattern recognition”.
For the basic LM model estimated
by using data PSIDlong the local (Ul) and global (Ug) decoding (using the Viterbi algorithm) are given by:
dec <- lmestDecoding(mod)
head(dec$Ul)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 2 2 2 2 2 2 2
#> [2,] 2 2 2 1 1 1 2
#> [3,] 2 2 2 2 2 2 2
#> [4,] 1 1 2 2 2 2 2
#> [5,] 2 2 2 2 2 2 2
#> [6,] 2 2 2 2 2 2 2
head(dec$Ug)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 2 2 2 2 2 2 2
#> [2,] 2 2 2 1 1 1 2
#> [3,] 2 2 2 2 2 2 2
#> [4,] 1 1 2 2 2 2 2
#> [5,] 2 2 2 2 2 2 2
#> [6,] 2 2 2 2 2 2 2
Bootstrapping
Function bootstrap() allows us to obtain standard errors through parametric bootstrap. A reasonable number of bootstrap samples is \(B=200\).
For illustrative purposes, we show how to obtain \(B=2\) samples from the output modc of the model with covariates for the NLSYlong dataset, estimated
using the
function lmestCont() as follows:
mboot <- bootstrap(modc, n = 581, B = 2, seed = 172)
Object seMu contains the estimated standard errors for the conditional means:
mboot$seMu
#> state
#> 1 2
#> [1,] 0.01857819 0.07561814
#> [2,] 0.09538772 0.18128943
Draw samples
Function draw()
allows us to draw samples from the estimated basic LM model.
For example, considering the basic LM model illustrated with the RLMSlong dataset, we can obtain a sample of responses of size \(n\) = 100 as follows:
draw3 <- draw(est = mod4, format = "matrices", seed = 4321, n = 100)
#> ------------|
#> sample unit|
#> ------------|
#> 100 |
#> ------------|
head(draw3$Y)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 1 0 1 1 1 1 2
#> [2,] 1 0 1 1 1 1 1
#> [3,] 0 2 1 0 1 2 1
#> [4,] 1 1 1 1 1 1 1
#> [5,] 1 2 1 1 1 1 1
#> [6,] 1 1 1 1 3 1 2
Each line of \(Y\) contains the sample responses of each unit for the seven time occasions.
The package also provides functions to draw samples from other LM models.