fit_LME_longitudinal {Landmarking}

Fit a landmarking model using a linear mixed effects (LME) model for the longitudinal submodel

Description

This function is a helper function for fit_LME_landmark.

Usage

fit_LME_longitudinal(
  data_long,
  x_L,
  fixed_effects,
  random_effects,
  fixed_effects_time,
  random_effects_time,
  standardise_time = FALSE,
  random_slope_in_LME = TRUE,
  random_slope_as_covariate = FALSE,
  cv_name = NA,
  individual_id,
  lme_control = nlme::lmeControl()
)

Arguments

data_long

Data frame containing repeat measurement data and time-to-event data in long format.

x_L

Numeric specifying the landmark time(s)

fixed_effects

Vector of character strings specifying the column names in data_long which correspond to the fixed effects

random_effects

Vector of character strings specifying the column names in data_long which correspond to the random effects

fixed_effects_time

Character string specifying the column name in data_long which contains the time at which the fixed effects were recorded

random_effects_time

Vector of character strings specifying the column names in data_long which contain the times at which repeat measures were recorded. This should either be length 1 or the same length as random_effects. In the latter case the order of elements must correspond to the order of elements in random_effects.

standardise_time

Boolean indicating whether to standardise the time variable by subtracting the mean and dividing by the standard deviation (see Details section for more information)

random_slope_in_LME

Boolean indicating whether to include a random slope in the LME model

random_slope_as_covariate

Boolean indicating whether to include the random slope estimate from the LME model as a covariate in the survival submodel.

cv_name

Character string specifying the column name in data_long that indicates cross-validation fold

individual_id

Character string specifying the column name in data_long which contains the individual identifiers

lme_control

Object created using nlme::lmeControl(), which will be passed to the control argument of the lme function

Details

For an individual i, the LME model can be written as

Y_i = X_i \beta + Z_i U_i + \epsilon_i

where

• Y_i is the vector of outcomes at different time points for the individual

• X_i is the matrix of covariates for the fixed effects at these time points

• \beta is the vector of coefficients for the fixed effects

• Z_i is the matrix of covariates for the random effects

• U_i is the matrix of coefficients for the random effects

• \epsilon_i is the error term, typically from N(0, \sigma)

By using an LME model to fit repeat measures data we can allow measurements from the same individuals to be more similar than measurements from different individuals. This is done through the random intercept and/or random slope.

Extending this model to the case where there are multiple random effects, denoted k, we have

Y_{ik} = X_{ik} \beta_k + Z_{ik} U_{ik} + \epsilon_{ik}

Using this model we can allow a certain covariance structure within the random effects term U_{ik}, for example a sample from the multivariate normal (MVN) distribution MVN(0,\Sigma_u). This covariance structure means the value of one random effects variable informs about the value of the other random effects variables, leading to more accurate predictions and allowing there to be missing data in the random effects variables.

The function fit_LME_landmark uses a unstructured covariance for the random effects when fitting the LME model (i.e. no constraints are imposed on the values). To fit the LME model the function lme from the package nlme is used. The fixed effects are calculated as the LOCF for the variables fixed_effects at the landmark age x_L and the random effects are those stated in random_effects and at times random_effects_time. The random intercept is always included in the LME model. Additionally, the random slope can be included in the LME model using the parameter random_slope_in_LME=TRUE. The model is used to predict the values of the random effects at the landmark time x_L, and these are used as predictors in the survival model along with the LOCF values of the fixed effects. Additionally, the estimated value of the random slope can be included as predictors in the survival model using the parameter random_slope_as_covariate=TRUE.

It is important to distinguish between the validation set and the development set for fitting the LME model. The development set includes all the repeat measurements (including those after the landmark age x_L). Conversely, the validation set only includes the repeat measurements recorded up until and including the landmark age x_L.

There is an important consideration about fitting the linear mixed effects model. As the variable random_effects_time gets further from 0, the random effects coefficients get closer to 0. This causes computational issues as the elements in the covariance matrix of the random effects, \Sigma_u, are constrained to be greater than 0. Using parameter standard_time=TRUE can prevent this issue by standardising the time variables to ensure that the random_effects_time values are not too close to 0.

The LOCF values for the fixed effects and the prediction of the random effects at the landmark age are used as the covariates for the survival submodel, in addition to the estimated random slopes if option random_effects_as_covariate is selected.

Value

List containing elements: data_longitudinal, model_longitudinal, model_LME, and model_LME_standardise_time.

data_longitudinal has one row for each individual in the risk set at x_L and contains the value of the covariates at the landmark time x_L of the fixed_effects using the LOCF model and random_effects using the LME model.

model_longitudinal indicates that the LME approach is used.

model_LME contains the output from the lme function from package nlme. For a model using cross-validation, model_LME contains a list of outputs with each element in the list corresponds to a different cross-validation fold.

model_LME_standardise_time contains a list of two objects mean_response_time and sd_response_time if the parameter standardise_time=TRUE is used. This is the mean and standard deviation used to normalise times when fitting the LME model.