Methods for object of class "MixMod" for standard generic functions.

coef(object, ...)

# S3 method for MixMod
coef(object, sub_model = c("main", "zero_part"),
...)

fixef(object, ...)

# S3 method for MixMod
fixef(object, sub_model = c("main", "zero_part"), ...)

ranef(object, ...)

# S3 method for MixMod
ranef(object, post_vars = FALSE, ...)

confint(object, parm, level = 0.95, ...)

# S3 method for MixMod
confint(object,
parm = c("fixed-effects", "var-cov","extra", "zero_part"),
level = 0.95, sandwich = FALSE, ...)

anova(object, ...)

# S3 method for MixMod
anova(object, object2, test = TRUE,
L = NULL, sandwich = FALSE, ...)

fitted(object, ...)

# S3 method for MixMod
fitted(object,
type = c("mean_subject", "subject_specific", "marginal"),

residuals(object, ...)

# S3 method for MixMod
residuals(object,
type = c("mean_subject", "subject_specific", "marginal"),
link_fun = NULL, tasnf_y = function (x) x, ...)

predict(object, ...)

# S3 method for MixMod
predict(object, newdata, newdata2 = NULL,
type = c("mean_subject", "subject_specific", "marginal", "zero_part"),
se.fit = FALSE, M = 300, df = 10, scale = 0.3, level = 0.95,
seed = 1, return_newdata = FALSE, sandwich = FALSE, ...)

simulate(object, nsim = 1, seed = NULL, ...)

# S3 method for MixMod
simulate(object, nsim = 1, seed = NULL,
type = c("subject_specific", "mean_subject"), new_RE = FALSE,
acount_MLEs_var = FALSE, sim_fun = NULL,
sandwich = FALSE, ...)

terms(x, ...)

# S3 method for MixMod
terms(x, type = c("fixed", "random", "zi_fixed", "zi_random"), ...)

formula(x, ...)

# S3 method for MixMod
formula(x, type = c("fixed", "random", "zi_fixed", "zi_random"), ...)

model.frame(formula, ...)

# S3 method for MixMod
model.frame(formula, type = c("fixed", "random", "zi_fixed", "zi_random"), ...)

model.matrix(object, ...)

# S3 method for MixMod
model.matrix(object, type = c("fixed", "random", "zi_fixed", "zi_random"), ...)

nobs(object, ...)

# S3 method for MixMod
nobs(object, level = 1, ...)

VIF(object, ...)

# S3 method for MixMod
VIF(object, type = c("fixed", "zi_fixed"), ...)

cooks.distance(model, ...)

# S3 method for MixMod
cooks.distance(model, cores = max(parallel::detectCores() - 1, 1), ...)

## Arguments

object, object2, x, formula, model objects inheriting from class "MixMod". When object2 is also provided, then the model behind object must be nested within the model behind object2. character string indicating for which sub-model to extract the estimated coefficients; it is only relevant for zero-inflated models. logical; if TRUE the posterior variances of the random effects are returned as an extra attribute of the numeric matrix produced by ranef(). character string; for which type of parameters to calculate confidence intervals. Option "var-cov" corresponds to the variance-covariance matrix of the random effects. Option extra corresponds to extra (shape/dispersion) parameters in the distribution of the outcome (e.g., the $$\theta$$ parameter in the negative binomial family). Option zero_inflated corresponds to the coefficients of the zero-inflated sub-model. numeric scalar between 0 and 1 denoting the level of the confidence interval. In the nobs() method it denotes the level at which the number of observations is counted. The value 0 corresponds to the number of independent sample units determined by the number of levels of the grouping variable. If set to a value greater than zero, it returns the total number of observations. logical; should a p-value be calculated. a numeric matrix representing a contrasts matrix. This is only used when in anova() only object is provided, and it can only be specified for the fixed effects. When L is used, a Wald test is performed. logical; if TRUE the sandwich estimator is used in the calculation of standard errors. character string indicating the type of fitted values / residuals / predictions / variance inflation factors to calculate. Option "mean_subject" corresponds to only using the fixed-effects part; option "subject_specific" corresponds to using both the fixed- and random-effects parts; option "marginal" is based in multiplying the fixed effects design matrix with the marginal coefficients obtained by marginal_coefs. the link_fun of marginal_coefs. a function to transform the grouped / repeated measurements outcome before calculating the residuals; for example, relevant in two-part models for semi-continuous data, in which it is assumed that the log outcome follows a normal distribution. a data frame based on which predictions are to be calculated. newdata2 is only relevant when level = "subject_specific"; see Details for more information. character string indicating at which scale to calculate predictions. Options are "link" indicating to calculate predictions at the link function / linear predictor scale, and "response" indicating to calculate predictions at the scale of the response variable. logical, if TRUE standard errors of predictions are returned. numeric scalar denoting the number of Monte Carlo samples; see Details for more information. numeric scalar denoting the degrees of freedom for the Student's t proposal distribution; see Details for more information. numeric scalar or vector denoting the scaling applied to the subject-specific covariance matrices of the random effects; see Details for more information. numerical scalar giving the seed to be used in the Monte Carlo scheme. logical; if TRUE the predict() method returns a copy of the newdata and of newdata2 if the corresponding argument was not NULL, with extra columns the predictions, and the lower and upper limits of the cofidence intervals when type = "subject_specific". numeric scalar giving the number of times to simulate the response variable. logical; if TRUE, new random effects will be simulated, and new outcome data will be simulated by simulate() using these new random effect. Otherwise, the empirical Bayes estimates of the random effects from the fitted model will be used. logical; if TRUE it accounts for the variability of the maximum likelihood estimates (MLEs) by simulating a new value for the parameters from a multivariate normal distribution with mean the MLEs and covariance matrix the covariance matrix of the MLEs. a function based on which to simulate the response variable. This is relevant for non-standard models. The simulate() function also tries to extract this function from the family component of object. The function should have the following four arguments: n a numeric scalar denoting the number of observations to simulate, mu a numeric vector of means, phis a numeric vector of extra dispersion/scale parameters, and eta_zi a numeric vector for the zero-part of the model, if this is relevant. the number of cores to use in the computation. further arguments; currently none is used.

## Details

In generic terms, we assume that the mean of the outcome $$y_i$$ ($$i = 1, ..., n$$ denotes the subjects) conditional on the random effects is given by the equation: $$g{E(y_i | b_i)} = \eta_i = X_i \beta + Z_i b_i,$$ where $$g(.)$$ denotes the link function, $$b_i$$ the vector of random effects, $$\beta$$ the vector of fixed effects, and $$X_i$$ and $$Z_i$$ the design matrices for the fixed and random effects, respectively.

Argument type_pred of predict() specifies whether predictions will be calculated in the link / linear predictor scale, i.e., $$\eta_i$$ or in the response scale, i.e., $$g{E(y_i | b_i)}$$.

When type = "mean_subject", predictions are calculated using only the fixed effects, i.e., the $$X_i \beta$$ part, where $$X_i$$ is evaluated in newdata. This corresponds to predictions for the 'mean' subjects, i.e., subjects who have random effects value equal to 0. Note, that in the case of nonlinear link functions this does not correspond to the averaged over the population predictions (i.e., marginal predictions).

When type = "marginal", predictions are calculated using only the fixed effects, i.e., the $$X_i \beta$$ part, where $$X_i$$ is evaluated in newdata, but with $$\beta$$ coefficients the marginalized coefficients obtain from marginal_coefs. These predictions will be marginal / population averaged predictions.

When type = "zero_part", predictions are calculated for the logistic regression of the extra zero-part of the model (i.e., applicable for zero-inflated and hurdle models).

When type = "subject_specific", predictions are calculated using both the fixed- and random-effects parts, i.e., $$X_i \beta + Z_i b_i$$, where $$X_i$$ and $$Z_i$$ are evaluated in newdata. Estimates for the random effects of each subject are obtained as modes from the posterior distribution $$[b_i | y_i; \theta]$$ evaluated in newdata and with $$theta$$ (denoting the parameters of the model, fixed effects and variance components) replaced by their maximum likelihood estimates.

Notes: (i) When se.fit = TRUE and type_pred = "response", the standard errors returned are on the linear predictor scale, not the response scale. (ii) When se.fit = TRUE and the model contains an extra zero-part, no standard errors are computed when type = "mean_subject". (iii) When the model contains an extra zero-part, type = "marginal" predictions are not yet implemented.

When se.fit = TRUE and type = "subject_specific", standard errors and confidence intervals for the subject-specific predictions are obtained by a Monte Carlo scheme entailing three steps repeated M times, namely

Step I

Account for the variability of maximum likelihood estimates (MLES) by simulating a new value $$\theta^*$$ for the parameters $$\theta$$ from a multivariate normal distribution with mean the MLEs and covariance matrix the covariance matrix of the MLEs.

Step II

Account for the variability in the random effects estimates by simulating a new value $$b_i^*$$ for the random effects $$b_i$$ from the posterior distribution $$[b_i | y_i; \theta^*]$$. Because the posterior distribution does not have a closed-form, a Metropolis-Hastings algorithm is used to sample the new value $$b_i^*$$ using as proposal distribution a multivariate Student's-t distribution with degrees of freedom df, centered at the mode of the posterior distribution $$[b_i | y_i; \theta]$$ with $$\theta$$ the MLEs, and scale matrix the inverse Hessian matrix of the log density of $$[b_i | y_i; \theta]$$ evaluated at the modes, but multiplied by scale. The scale and df parameters can be used to adjust the acceptance rate.

Step III

The predictions are calculated using $$X_i \beta^* + Z_i b_i^*$$.

Argument newdata2 can be used to calculate dynamic subject-specific predictions. I.e., using the observed responses $$y_i$$ in newdata, estimates of the random effects of each subject are obtained. For the same subjects we want to obtain predictions in new covariates settings for which no response data are yet available. For example, in a longitudinal study, for a subject we have responses up to a follow-up $$t$$ (newdata) and we want the prediction at $$t + \Delta t$$ (newdata2).

## Value

The estimated fixed and random effects, coefficients (this is similar as in package nlme), confidence intervals fitted values (on the scale on the response) and residuals.

mixed_model, marginal_coefs

## Examples

# \donttest{
# simulate some data
set.seed(123L)
n <- 500
K <- 15
t.max <- 25

betas <- c(-2.13, -0.25, 0.24, -0.05)
D <- matrix(0, 2, 2)
D[1:2, 1:2] <- c(0.48, -0.08, -0.08, 0.18)

times <- c(replicate(n, c(0, sort(runif(K-1, 0, t.max)))))
group <- sample(rep(0:1, each = n/2))
DF <- data.frame(year = times, group = factor(rep(group, each = K)))
X <- model.matrix(~ group * year, data = DF)
Z <- model.matrix(~ year, data = DF)

b <- cbind(rnorm(n, sd = sqrt(D[1, 1])), rnorm(n, sd = sqrt(D[2, 2])))
id <- rep(1:n, each = K)
eta.y <- as.vector(X %*% betas + rowSums(Z * b[id, ]))
DF$y <- rbinom(n * K, 1, plogis(eta.y)) DF$id <- factor(id)

################################################

fm1 <- mixed_model(fixed = y ~ year + group, random = ~ year | id, data = DF,
family = binomial())

#> 1   -2.305583  0.54222964 -0.265744
#> 2   -1.905620 -0.12525835 -0.265744
#> 3   -2.395862 -0.14287663 -0.265744
#> 4   -2.379605  0.42518796 -0.265744
#> 5   -2.348230  0.01936688 -0.265744
#> 6   -2.056225  0.69488984 -0.265744fixef(fm1)#> (Intercept)        year      group1
#>  -2.2391690   0.1968593  -0.2657440 head(ranef(fm1))#>   (Intercept)       year
#> 1 -0.06641381  0.3453703
#> 2  0.33354914 -0.3221176
#> 3 -0.15669315 -0.3397359
#> 4 -0.14043607  0.2283287
#> 5 -0.10906060 -0.1774924
#> 6  0.18294398  0.4980305

confint(fm1)#>                  2.5 %   Estimate      97.5 %
#> (Intercept) -2.4728845 -2.2391690 -2.00545354
#> year         0.1529535  0.1968593  0.24076514
#> group1      -0.5673166 -0.2657440  0.03582867confint(fm1, "var-cov")#>                       2.5 %    Estimate      97.5 %
#> var.(Intercept)  0.22607344  0.50791169  0.06600001
#> cov.(Int)_year  -0.08018454 -0.03554827  0.06888253
#> var.year         0.15508238  0.16033127 -0.81295806
head(fitted(fm1, "subject_specific"))#>          1          2          3          4          5          6
#> 0.09066165 0.15603499 0.83100175 0.96225413 0.97895242 0.97984947 head(residuals(fm1, "marginal"))#>          1          2          3          4          5          6
#> -0.1805435 -0.1980669  0.6881459  0.6194025  0.5928527  0.5908654
fm2 <- mixed_model(fixed = y ~ year * group, random = ~ year | id, data = DF,
family = binomial())

# likelihood ratio test between fm1 and fm2
anova(fm1, fm2)#>
#>         AIC     BIC  log.Lik  LRT df p.value
#> fm1 5203.04 5228.33 -2595.52
#> fm2 5201.62 5231.12 -2593.81 3.43  1  0.0642
#>
# the same but with a Wald test
anova(fm2, L = rbind(c(0, 0, 0, 1)))#>
#> Marginal Wald Tests Table
#>
#> User-defined contrasts matrix:
#>  (Intr) year group1 yr:gr1
#>       0    0      0      1
#>
#>    Chisq df Pr(>|Chi|)
#>   3.7927  1     0.0515
#> # }