# vcPB: Varying Coefficient Peters-Belson Method for Longitudinal Data

A package for estimating the disparity between a majority group and minority group based on the extended model of the Peters-Belson method. Our model is the first extension of Peters-Belson method to the longitudinal data.

Furthermore, our method can set a modifiable variable to find the complicated association between other variables and the modifiable variable.

### Installation

The current version can be installed from source using the package devtools

devtools::install_github("SangkyuStat/vcPB")

It can also be found on CRAN

install.packages("vcPB")

### Usage Examples

#### - vc.pb function

vc.pb function provides three different types of models based on the different input arguments: modifier and time varying coefficients.

If modifier is NULL (the default setting is NULL) and at least a time-varying variable exists, then the simple varying-coefficient Peters-Belson method using a gaussian kernel regression can be performed as below:

vc.pb(formula = response ~ (time varying variable | time variable) +
variable,
id,
data = input_data,
group = disparity_group)

If modifier is not NULL and is a discrete variable, and at least a time-varying variable exists, then the modifiable varying-coefficient Peters-Belson method using a gaussian kernel regression can be performed as below:

vc.pb(formula = response ~ (time varying variable | time variable) +
variable + discrete modifier,
id,
data = input_data,
group = disparity_group,
modifier = "discrete modifier")

If modifier is not NULL and is a continuous variable, and at least a time-varying variable exists, then the simple varying-coefficient Peters-Belson method using a gaussian kernel regression can be performed as below:

vc.pb(formula = response ~ (time varying variable | time variable) +
variable + continuous modifier,
id,
data = input_data,
group = disparity_group,
modifier = "continuous modifier")

The type of modifier returns the different results. If there are more than one time-varying variables, the user can perform the function as below:

vc.pb(formula = response ~ (time varying variable1 | time variable) +
(time varying variable2 | time variable) + variable,
id,
data = input_data,
group = disparity_group)

If there is no modifier and time-varying variable, then the model is just the naive PB model. For this case, the user can use pb function instead.

The user needs to define group properly to measure the disparity between two groups in group variable, there should be 2 levels for this variable.

The user needs to define id properly to have the exact identification on observations whether they are measured repeatedly across the time.

The selection of bandwidths is essential and important for the kernel regression. If there is nothing given as initial values, we get and use the default marginal bandwidth from the function KernSmooth::dpill. For all models, bandwidth_M, bandwidth_m, bandwidth_xM and bandwidth_xm are essential. If modifier is not NULL and is a continuous variable, then bandwidth_Z_M, bandwidth_Z_m, bandwidth_Z_xM and bandwidth_Z_xm are needed more.

Also, use needs to specify local time points (local_time) for the time-varying kernel regression. The function will automatically give the time points if there is nothing given. The local time points will be returned in the fitted object.

#### - pb function

pb function provides the original Peters-Belson method of Peters (1941) and Belson (1956). The usage is as similar as the vc.pb but the user should not put the time varying coefficients and a modifier variable.

### Developing

• The conditional version will be uploaded very soon.
• The cross-validation function for choosing the bandwidths will be developed.
• We are trying to develop other methods as well.

### References

Peters, C. C. (1941) A method of matching groups for experiment with no loss of population. Journal of Educational Research, 34, 606-612.

Belson, W. A. (1956) A Technique for Studying the Effects of a Television Broadcast. JRSSC, 5(3), 195-202.