New-TestStatistics

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()