# Half Kernel Estimation with Backward Lagged Covariates

### Description

A kernel weighting scheme to evaluate the effects of longitudinal covariates on the occurrence of events when the time-dependent covariates are measured intermittently. Regression parameter estimation using half kernel imputation of missing values with backward lagged covariates.

### Usage

1 2 | ```
halfKernel(X, Z, tau, kType = "epan", bw = NULL, tol = 0.001,
maxiter = 100)
``` |

### Arguments

`X ` |
an object of class data.frame. The structure of the data.frame must be {patient ID, event time, event indicator}. Patient IDs must be of class integer or be able to be coerced to class integer without loss of information. Missing values must be indicated as NA. The event indicator is 1 if the event occurred; 0 if censored. |

`Z ` |
an object of class data.frame. The structure of the data.frame must be {patient ID, time of measurement, measurement(s)}. Patient IDs must be of class integer or be able to be coerced to class integer without loss of information. Missing values must be indicated as NA. |

`tau ` |
an object of class numeric. The desired time point. |

`kType` |
An object of class character indicating the type of smoothing kernel to use in the estimating equation. Must be one of {"epan", "uniform", "gauss"}, where "epan" is the Epanechnikov kernel and "gauss" is the Gaussian kernel. |

`bw ` |
If provided, bw is an object of class numeric or a numeric vector containing the bandwidths for which parameter estimates are to be obtained. If NULL, an optimal bandwidth will be determined using an adaptive selection procedure. The range of the bandwidth search space is taken to be 2*(Q3 - Q1)*n^-0.7 to 2*(Q3 - Q1)*n^-0.3, where Q3 is the 0.75 quantile and Q1 is the 0.25 quantile of the measurement times for the covariate and n is the effective number of patients, taken as the total number of patients that experienced an event. |

`tol ` |
An object of class numeric. The minimum change in the regression parameters deemed to indicate convergence of the Newton-Raphson method. |

`maxiter ` |
An object of class numeric. The maximum number of iterations used to estimate regression parameters. |

### Value

A list is returned. If bandwidths are provided, each element of the list is a matrix, where the ith row corresponds to the ith bandwidth of argument “bw," and the columns correspond to the model parameters. If the bandwidth is determined automatically, each element is a named vector calculated at the optimal bandwidth.

`betaHat ` |
The estimated model coefficients. |

`stdErr ` |
The standard error for each coefficient. |

`zValue ` |
The estimated z-value for each coefficient. |

`pValue ` |
The p-value for each coefficient. |

If the bandwidth is determined automatically, three additional list elements are returned:

`optBW ` |
The estimated optimal bandwidth. |

`minMSE ` |
The mean squared error at the optimal bandwidth. |

`MSE ` |
The vector of MSE for each bandwidth. |

### Author(s)

Hongyuan Cao, Mathew M. Churpek, Donglin Zeng, Jason P. Fine, and Shannon T. Holloway

### References

Cao, H., Churpek, M. M., Zeng, D., and Fine, J. P. (2015). Journal of the American Statistical Association, in press.

### See Also

`fullKernel`

, `lastValue`

, `nearValue`

### Examples

1 2 3 | ```
data(SurvLongData)
exp <- halfKernel(X = X, Z = Z, tau = 1.0, bw = 0.015)
``` |