# New-TestStatistics

The package offers several test statistics that can be used for goodness-of-fit tests of conditional distribution families. However, there might be other statistics that the user wishes to use in their analysis. In the following, we will describe how custom test statistics can be defined and used along with the package.

## The abstract base class `TestStatistic`

Whenever the user wants to define a new test statistic, they have to create an R6 class inheriting from the abstract base class `TestStatistic` implementing the method `calc_stat()`. Given some data and a model to test for, the method calculates the value of the test statistic as well as two vectors (`plot.x` and `plot.y`) that can be used to plot the corresponding process.

## Example

In the following example, we will define a new test statistic, specifically the marked empirical process (MEP) as defined in Stute (1997). It is given by \[R_n^1(x) = n^{-1/2} \sum_{i=1}^n 1_{\{X_i \le x\}} (Y_i - m(X_i, \hat{\vartheta}_n)),\] where \(m(x, \vartheta)\) denotes the regression function corresponding to a given model with parameter \(\vartheta\). In order to be able to plot the process in a two-dimensional grid, we will assume the covariates \(X\) to be one-dimensional.

    MEP_Stute97 <- R6::R6Class(
      classname = "MEP_Stute97",
      inherit = TestStatistic,
      public = list(
        calc_stat = function(data, model) {
          # check for correct shape of data and definedness of model params
          checkmate::assert_data_frame(data)
          checkmate::assert_names(names(data), must.include = c("x", "y"))
          checkmate::assert_matrix(as.matrix(x), ncols = 1)
          checkmate::assert_class(model, "ParamRegrModel")
          params <- model$get_params()
          if (anyNA(params)) {
            stop("Model first needs to be fitted to the data.")
          }

          # compute residuals and order them according to X
          res <- data$y - model$mean_yx(data$x)
          ord.id <- order(c(data$x))
          res.ord <- res[ord.id]

          # compute MEP (cumulative sum of the ordered residuals)
          proc <- cumsum(res.ord) / sqrt(n)

          # set private fields accordingly
          private$value <- max(abs(proc))
          private$plot.x <- c(data$x)[ord.id]
          private$plot.y <- proc
          invisible(self)
        }
      )
    )

Now, let us create an artificial dataset to use the test statistic with.

    set.seed(123)
    n  <- 100
    x <- rnorm(n)
    model <- NormalGLM$new()
    params_true <- list(beta = 3, sd = 0.5)
    y <- model$sample_yx(x^2, params_true)
    data <- dplyr::tibble(x = x, y = y)
    head(data)

    #> # A tibble: 6 × 2
    #>         x      y
    #>     <dbl>  <dbl>
    #> 1 -0.560   0.587
    #> 2 -0.230   0.287
    #> 3  1.56    7.17
    #> 4  0.0705 -0.159
    #> 5  0.129  -0.426
    #> 6  1.72    8.80

To evaluate the goodness-of-fit test using the new test statistic, we to fit two different models to the data: the correct model (using x^2) and a wrong model (using x).

    model$fit(data, params_init = list(beta = 1, sd = 5), inplace = TRUE)
    model$get_params()

    #> $beta
    #> [1] 0.9404959
    #>
    #> $sd
    #> [1] 4.094303

    gt <- GOFTest$new(data = data, model_fitted = model, test_stat = MEP_Stute97$new(), nboot = 100)
    gt$get_pvalue()

    #> [1] 0

As expected, the p-value for the wrong model is very low, so it would get rejected. To visualize the results, we can plot the process corresponding to the given data together with the bootstrap processes.

    gt$plot_procs()

Now, let us consider the correct model with covariates \(X^2\).

    data_x2 <- dplyr::tibble(x = data$x^2, y = data$y)
    model$fit(data_x2, params_init = list(beta = 1, sd = 5), inplace = TRUE)
    model$get_params()

    #> $beta
    #> [1] 2.946964
    #>
    #> $sd
    #> [1] 0.4783823

    gt <- GOFTest$new(data = data_x2, model_fitted = model, test_stat = MEP_Stute97$new(), nboot = 100)
    gt$get_pvalue()

    #> [1] 0.95

The p-value for the correct model is rather high and would thus be accepted. We can again plot the corresponding processes.

    gt$plot_procs()