# binaryMM: Fitting Flexible Marginalized Models for Binary Correlated Outcomes

The binaryMM package allows users to fit marginalized transition and latent variables (mTLV) models for binary longitudinal data. The aim of this vignette is to provide an overview of the models together with example code and analyses.

# The Marginalized Transition and Latent Variables Model

Let $$N$$ be the total number of subjects, $$\boldsymbol{Y}_{i}$$ be the $$n_i-$$vector of binary responses for subject $$i$$, $$\boldsymbol{X}_i$$ be the $$n_i \times p$$ matrix of covariates and $$U_i \sim N(0, 1)$$. The marginalized transition and latent variable (mTLV) model described in Schildcrout and Heagerty (2007) can be defined by two equations:

$logit\left(\mu_{ij}^m\right) = \boldsymbol{\beta}^T\boldsymbol{X}_i$ $logit\left(\mu_{ij}^c\right) = \Delta_{ij}(\boldsymbol{X}_i) + \gamma(\boldsymbol{X}_i)Y_{i(j-1)} + \sigma(\boldsymbol{X}_i) U_i$

where $$\mu_{ij}^m = E[Y_{ij} | \boldsymbol{X}_i]$$ and $$\mu_{ij}^c = E[Y_{ij} | \boldsymbol{X}_i, Y_{i(j-1)}, U_i]$$.

The first equation describes the marginal mean model and the relationship between the outcome $$\boldsymbol{Y}_{i}$$ and the covariates $$\boldsymbol{X}_i$$. The second equation describes the conditional mean model (also named the dependence model) and the relationship between the outcome $$\boldsymbol{Y}_{i}$$ measured over time for each subject $$i$$. In particular, the conditional model includes a short-term transition component $$\gamma(\boldsymbol{X}_i)Y_{i(j-1)}$$, and a random intercept term, $$\sigma(\boldsymbol{X}_i) U_i$$, describing long-term non-diminishing dependence.

$$\Delta_{ij}(\boldsymbol{X}_i)$$ is a function of the marginal mean, $$\mu_{ij}^m$$, and the conditional mean, $$\mu_{ij}^c$$, such that the two model above are cohesive. In particular, $$\Delta_{ij}(\boldsymbol{X}_i)$$ is the value that satisfies the convolution equation:

$\mu_{ij}^m = E_{U_i, Y_{i(j-1)}}(\mu_{ij}^c) = E_{Z_i}[E_{Y_{i(j-1)}}[logit^{-1}(\Delta_{ij}(\boldsymbol{X}_i) + \gamma(\boldsymbol{X}_i)Y_{i(j-1)} + \sigma(\boldsymbol{X}_i) U_i)]]$ $$\Delta_{ij}(\boldsymbol{X}_i)$$ in mTLV is analytically intractable and its value is computed iteratively with a Newton-Raphson method.

Detailed information on marginalized models with transition and/or latent terms can be found in Heagerty (2002), Heagerty and Zeger (1999) and Schildcrout and Heagerty (2007).

# Basic Examples

The next two sections explain how different specifications of mTLV models can be fitted using the binaryMM package. The data used are part of the package.

library(binaryMM)

## The Madras Longitudinal Schizophrenia Study

madras contains a subset of the data from the Madras Longitudinal Schizophrenia Study Diggle et al. (2002), which collected monthly symptom data on 86 schizophrenia patients after their initial hospitalization. The dataframe has 922 observations on 86 patients and includes the variables:

• though. An indicator for thought disorders

• age. An indicator for age-at-onset $$\geq$$ 20 years

• gender. An indicator for female gender

• month. Months since hospitalization

• id. A unique patient identifiers

The primary question of interest is whether subjects with an older age-at-onset tend to recover more or less quickly, and whether female patients recover more or less quickly. Recovery is measured by a reduction in the presentation of symptoms.

data(madras)
#> 'data.frame':    922 obs. of  5 variables:
#>  $id : int 1 1 1 1 1 1 1 1 1 1 ... #>$ thought: int  1 1 1 1 1 0 0 0 0 0 ...
#>  $month : int 0 1 2 3 4 5 6 7 8 9 ... #>$ gender : int  0 0 0 0 0 0 0 0 0 0 ...
#>  $age : num 1 1 1 1 1 1 1 1 1 1 ... The marginal mean model is defined as: $logit(\mu_{ij}^m) = \beta_0 + \beta_1month_{ij} + \beta_2age_i + \beta_3gender_i + \beta_4 age_i \times month_{ij} + \beta_5 gender_i \times month_{ij}$ Multiple dependence models are explored to demonstrate how the mm function can be used. The different dependence models are declared by changing the t.formula and lv.formula arguments. Note that by default formula both are initially assigned NULL and if neither association models are specified, then an error is returned. • Case 1. A dependence model with a transition term only: $logit(\mu_{ij}^c) = \Delta_{ij}(\boldsymbol{X}_i) + \gamma Y_{i(j-1)}$ mod.mt <- mm(thought ~ month*gender + month*age, t.formula = ~1, data = madras, id = id) summary(mod.mt) #> #> Class: #> MMLong #> #> Call: #> mm(mean.formula = thought ~ month * gender + month * age, t.formula = ~1, #> id = id, data = madras) #> #> Information Criterion: #> AIC BIC logLik Deviance #> 688.3789 705.5594 -337.1895 674.3789 #> #> Marginal Mean Parameters: #> Estimate Model SE Chi Square Pr(>Chi) #> (Intercept) 1.183683 0.444318 7.0971 0.007721 #> month -0.342857 0.081841 17.5501 2.798e-05 #> gender -0.141884 0.416152 0.1162 0.733147 #> age -0.649770 0.449183 2.0925 0.148021 #> month:gender -0.143788 0.081853 3.0859 0.078975 #> month:age 0.111555 0.085896 1.6867 0.194040 #> #> Dependence Model Parameters: #> Estimate Model SE Chi Square Pr(>Chi) #> gamma:(Intercept) 3.16583 0.23014 189.23 < 2.2e-16 #> #> Number of clusters: 86 #> Maximum cluster size: 12 #> Convergence status (nlm code): 1 #> Number of iterations: 22 • Case 2. A dependence model with a latent term only: $logit(\mu_{ij}^c) = \Delta_{ij}(\boldsymbol{X}_i) + \sigma U_i$ mod.mlv <- mm(thought ~ month*gender + month*age, lv.formula = ~1, data = madras, id = id) summary(mod.mlv) #> #> Class: #> MMLong #> #> Call: #> mm(mean.formula = thought ~ month * gender + month * age, lv.formula = ~1, #> id = id, data = madras) #> #> Information Criterion: #> AIC BIC logLik Deviance #> 750.5767 767.7571 -368.2883 736.5767 #> #> Marginal Mean Parameters: #> Estimate Model SE Chi Square Pr(>Chi) #> (Intercept) 1.569015 0.399400 15.4326 8.550e-05 #> month -0.399007 0.062845 40.3107 2.166e-10 #> gender -0.539932 0.355893 2.3016 0.12924 #> age -0.911165 0.393487 5.3621 0.02058 #> month:gender -0.081899 0.060179 1.8521 0.17354 #> month:age 0.140215 0.063107 4.9366 0.02629 #> #> Dependence Model Parameters: #> Estimate Model SE Chi Square Pr(>Chi) #> log(sigma):(Intercept) 0.81289 0.12764 40.559 1.908e-10 #> #> Number of clusters: 86 #> Maximum cluster size: 12 #> Convergence status (nlm code): 1 #> Number of iterations: 42 • Case 3. A dependence model with a transition and a latent term: $logit(\mu_{ij}^c) = \Delta_{ij}(\boldsymbol{X}_i) + \gamma Y_{i(j-1)} + \sigma U_i$ mod.mtlv <- mm(thought ~ month*gender + month*age, t.formula = ~1, lv.formula = ~1, data = madras, id = id) summary(mod.mtlv) #> #> Class: #> MMLong #> #> Call: #> mm(mean.formula = thought ~ month * gender + month * age, lv.formula = ~1, #> t.formula = ~1, id = id, data = madras) #> #> Information Criterion: #> AIC BIC logLik Deviance #> 680.1283 699.7631 -332.0642 664.1283 #> #> Marginal Mean Parameters: #> Estimate Model SE Chi Square Pr(>Chi) #> (Intercept) 1.327440 0.434588 9.3299 0.002255 #> month -0.367564 0.077443 22.5268 2.072e-06 #> gender -0.282497 0.402015 0.4938 0.482241 #> age -0.732176 0.436180 2.8177 0.093228 #> month:gender -0.111740 0.078306 2.0362 0.153591 #> month:age 0.117705 0.080870 2.1184 0.145536 #> #> Dependence Model Parameters: #> Estimate Model SE Chi Square Pr(>Chi) #> gamma:(Intercept) 2.511119 0.303963 68.2486 <2e-16 #> log(sigma):(Intercept) 0.074494 0.244870 0.0925 0.761 #> #> Number of clusters: 86 #> Maximum cluster size: 12 #> Convergence status (nlm code): 1 #> Number of iterations: 50 • Case 4. A dependence model with a transition term that is modified by gender. $logit(\mu_{ij}^c) = \Delta_{ij}(\boldsymbol{X}_i) + (\gamma_0 + \gamma_1 gender_i) Y_{i(j-1)}$ mod.mtgender <- mm(thought ~ month*gender + month*age, t.formula = ~gender, data = madras, id = id) summary(mod.mtgender) #> #> Class: #> MMLong #> #> Call: #> mm(mean.formula = thought ~ month * gender + month * age, t.formula = ~gender, #> id = id, data = madras) #> #> Information Criterion: #> AIC BIC logLik Deviance #> 690.2915 709.9263 -337.1458 674.2915 #> #> Marginal Mean Parameters: #> Estimate Model SE Chi Square Pr(>Chi) #> (Intercept) 1.175157 0.444200 6.9990 0.008156 #> month -0.342431 0.081702 17.5662 2.775e-05 #> gender -0.137202 0.416448 0.1085 0.741809 #> age -0.635415 0.452122 1.9752 0.159901 #> month:gender -0.143961 0.082080 3.0763 0.079443 #> month:age 0.110312 0.085973 1.6464 0.199456 #> #> Dependence Model Parameters: #> Estimate Model SE Chi Square Pr(>Chi) #> gamma:(Intercept) 3.11501 0.28599 118.634 <2e-16 #> gamma:gender 0.14321 0.48550 0.087 0.768 #> #> Number of clusters: 86 #> Maximum cluster size: 12 #> Convergence status (nlm code): 1 #> Number of iterations: 34 • Case 5. A dependence model with a latent term that is modified by gender. Note that because $$\sigma$$ is a positive quantity, to fit a mTLV model where the latent term is modified by gender, we need to specify two indicator variables: I0 for gender = 0, and I1 for gender = 1. The model to be specified in lv.fomula will take the form: ~0+I0+I1. $logit(\mu_{ij}^c) = \Delta_{ij}(\boldsymbol{X}_i) + [\sigma_0I(gender_i == 0) + \sigma_1I(gender_i == 1)]U_i$ # set-up two new indicator variables for gender madras$g0    <- ifelse(madras$gender == 0, 1, 0) madras$g1    <- ifelse(madras$gender == 1, 1, 0) mod.mlvgender <- mm(thought ~ month*gender + month*age, lv.formula = ~0+g0+g1, data = madras, id = id) summary(mod.mlvgender) #> #> Class: #> MMLong #> #> Call: #> mm(mean.formula = thought ~ month * gender + month * age, lv.formula = ~0 + #> g0 + g1, id = id, data = madras) #> #> Information Criterion: #> AIC BIC logLik Deviance #> 752.4992 772.1340 -368.2496 736.4992 #> #> Marginal Mean Parameters: #> Estimate Model SE Chi Square Pr(>Chi) #> (Intercept) 1.566764 0.400938 15.2705 9.316e-05 #> month -0.395507 0.064395 37.7227 8.155e-10 #> gender -0.515176 0.366086 1.9804 0.15935 #> age -0.921331 0.395394 5.4296 0.01980 #> month:gender -0.091526 0.069640 1.7273 0.18875 #> month:age 0.140228 0.063187 4.9251 0.02647 #> #> Dependence Model Parameters: #> Estimate Model SE Chi Square Pr(>Chi) #> log(sigma):g0 0.84429 0.17128 24.297 8.257e-07 #> log(sigma):g1 0.77199 0.19523 15.636 7.678e-05 #> #> Number of clusters: 86 #> Maximum cluster size: 12 #> Convergence status (nlm code): 1 #> Number of iterations: 50 The parameters from the marginal mean model have the same interpretation regardless of the dependence model used. Overall, older individuals tend to have slower recovery time than younger subjects, while females recover quicker than males. ## Weighted Likelihood The binaryMM package allows user to add sampling weights and estimates the parameters of interest in those cases where the available sample might not be representative of the target population (i.e., survey data). This section shows how the sampling weights can be added in the mm syntax using the datarand dataframe. The dataframe has 24,999 observation on 2,500 subjects and includes the variables: • id. A unique patient identifier • Y. A binary longitudinal outcome • time. A continuous time-varying covariate indicating time of each follow-up • binary. A binary time-fixed covariate indicating whether a patient was assigned to a treatment arm (1) or a control arm (0) data(datrand) str(datrand) #> 'data.frame': 24999 obs. of 4 variables: #>$ id    : int  1 1 1 1 1 1 1 1 1 2 ...
#>  $Y : int 0 0 1 1 0 0 0 0 0 0 ... #>$ time  : num  0 1 2 3 4 5 6 7 8 0 ...
#>  $binary: num 0 0 0 0 0 0 0 0 0 0 ... From datarand a biased sampled can be created by assuming that complete data are available only for 1) every one who experienced the event Y at least once, and 2) 20% of the subjects who never experienced the event Y. # create the sampling scheme Ymean <- tapply(datrand$Y, FUN = mean, INDEX = datrand$id) some.id <- names(Ymean[Ymean != 0]) none.id <- names(Ymean)[!(names(Ymean) %in% some.id)] samp.some <- some.id[rbinom(length(none.id), 1, 1) == 1] samp.none <- none.id[rbinom(length(none.id), 1, 0.20) == 1] # sample subjects and create a weight vector datrand$sampled <- ifelse(datrand$id %in% c(samp.none, samp.some), 1, 0) dat.small <- subset(datrand, sampled == 1) wt <- ifelse(dat.small$id %in% samp.none, 1/1, 1/0.2)

# fit the mTLV model
mod.wt          <- mm(Y ~ time*binary, t.formula = ~1, data = dat.small,
id = id, weight = wt)
summary(mod.wt)
#> Warning in summary.MMLong(mod.wt): When performing a weighted likelihood
#> analysis (by specifying the weight argument), robust standard errors are
#> reported. Model based standard errors will not be correct and should not be
#> used.
#>
#> Class:
#> MMLong
#>
#> Call:
#> mm(mean.formula = Y ~ time * binary, t.formula = ~1, id = id,
#>     data = dat.small, weight = wt)
#>
#> Information Criterion:
#>       AIC        BIC     logLik   Deviance
#>  74768.35   74795.57  -37379.17   74758.35
#>
#> Marginal Mean Parameters:
#>              Estimate Robust SE Chi Square  Pr(>Chi)
#> (Intercept) -1.014456  0.045303    501.435 < 2.2e-16
#> time        -0.161408  0.010246    248.177 < 2.2e-16
#> binary       0.320099  0.072732     19.369 1.077e-05
#> time:binary  0.158889  0.013982    129.130 < 2.2e-16
#>
#> Dependence Model Parameters:
#>                   Estimate Robust SE Chi Square  Pr(>Chi)
#> gamma:(Intercept) 1.043764  0.042677     598.17 < 2.2e-16
#>
#> Number of clusters:             1712
#> Maximum cluster size:           15
#> Convergence status (nlm code):  1
#> Number of iterations:           27

Note that when the weight argument is specified, model-based standard error will not be correct and should not be reported. Thus, the software will return robust standard errors only together with a warning message.

# Functions Available in the Package

The two examples above showed how different mTLV model can be used using simulated data as well as data from the Madras Longitudinal Schizophrenia Study. The table below summarizes the functions in mm available to the user.

Function Description
GenBinaryY Generate binary response variable under a user-specified mTLV model. The outcome is generated from a Bernoulli distribution where the probability of success is computed as the inverse-logit of the conditional mean. The function requires the user to specify the mean model formula (mean.formula) in which a binary covariate is regressed on covariates, one or both components of the dependence model (the latent variable component lv.formula or the transition term component t.formula), the vector of cluster identifiers (id), a vector of values for the parameters of the mean model (beta), a vector of values for the parameters of the transition component of the dependence model (gamma), a vector of values for the latent component of the dependence model (sigma), a dataframe (data) with the mean model covariates (ordered by id and time) and a string of the mane of the new binary variable (Yname). The function returns the entire data object with an additional column Yname of the binary longitudinal outcome
mm Fit mTLV model. The function requires the user to specify the mean model formula (mean.formula) in which a binary covariate is regressed on covariates, one or both components of the dependence model (the latent variable component lv.formula or the transition term component t.formula), the vector of cluster identifiers (id), and the dataframe to use (data). Users can additionally specify the sampling weights (weight) to estimate the parameters using weighted likelihood.
summary Summarize the results of a class MMLong generated using mm. Tables with estimated parameters, standard errors and p-value are printed for both the mean model and the dependence model parameters
anova Allows to compare two nested models of class MMLong generated using mm. fits using mTLV. Currently comparison can be made for two models only

# Reference

Diggle, P, PJ Heagerty, KY Liang, and Zeger SL. 2002. Analysis of Longitudinal Data. Oxford University Press.
Heagerty, PJ. 2002. “Marginalized Transition Models and Likelihood Inference for Longitudinal Categorical Data.” Biometrics 58: 342–51.
Heagerty, PJ, and SL Zeger. 1999. “Marginalized Multilevel Models and Likelihood Inference.” Statistical Science 15 (1): 1–19.
Schildcrout, JS, and PJ Heagerty. 2007. “Marginalized Models for Moderate to Long Series of Longitudinal Binary Response Data.” Biometrics 63 (2): 322–31.