# Non-parametric Two-sample test

Let \(x_1, x_2, \ldots, x_{n_1} \sim F\) and \(y_1, y_2, \ldots, y_{n_2} \sim G\) be random samples from the distributions \(F\) and \(G\), respectively. We test the null hypothesis that the two samples are generated from the same unknown distribution \(\bar{F}\), that is:

\[ H_0: F = G = \bar{F} \]

versus the alternative hypothesis that the two distributions are different, that is

\[ H_1: F \not = G. \] We compute the kernel-based quadratic distance (KBQD) tests

\[ \mathrm{trace}_n = \frac{1}{n_1(n_1-1)}\sum_{i=1}^{n_1} \sum_{j \not=i}^{n_1} K_{\bar{F}}(\mathbf{x}_i,\mathbf{x}_j) + \frac{1}{n_2(n_2-1)}\sum_{i=1}^{n_2} \sum_{j \not=i}^{n_2} K_{\bar{F}}(\mathbf{y}_i,\mathbf{y}_j), \] and \[ D_{n} = \frac{1}{n_1(n_1-1)}\sum_{i=1}^{n_1} \sum_{j \not=i}^{n_2} K_{\bar{F}}(\mathbf{x}_i,\mathbf{x}_j) - \frac{2}{n_1 n_2}\sum_{i=1}^{n_1} \sum_{j =1}^{n_2} K_{\bar{F}}(\mathbf{x}_i,\mathbf{y}_j) + \frac{1}{n_2(n_2-1)}\sum_{i=1}^{n_2} \sum_{j \not=i}^{n_2} K_{\bar{F}}(\mathbf{y}_i,\mathbf{y}_j). \nonumber \] where \(K_{\bar{F}}\) denotes the Normal kernel \(K\) defined as

\[ K(\mathbf{s}, \mathbf{t}) = (2 \pi)^{-d/2} \left(\det{\mathbf{\Sigma}_h}\right)^{-\frac{1}{2}} \exp\left\{-\frac{1}{2}(\mathbf{s} - \mathbf{t})^\top \mathbf{\Sigma}_h^{-1}(\mathbf{s} - \mathbf{t})\right\}, \]

for every \(\mathbf{s}, \mathbf{t} \in \mathbb{R}^d \times \mathbb{R}^d\), with covariance matrix \(\mathbf{\Sigma}_h = h^2 I\) and tuning parameter \(h\), centered with respect to \(\bar{F} = \frac{n_1F + n_2G}{n_1 + n_2}\). For more information about the centering of the kernel, see the documentation of the `kb.test()` function.

    help(kb.test)

The KBQD tests exhibit high power against asymmetric alternatives that are close to the null hypothesis and with small sample size. We consider an example of this scenario. We generate the samples \(x = (x_1, \ldots,x_n)\) from a standard normal distribution \(N_d(0,I_d)\) and \(y = (y_1, \ldots,y_n)\) from a skew-normal distribution \(SN_d(0,I_d, \lambda)\), where \(d=4\), \(n=100\) and \(\lambda= (0.5,\ldots,0.5)\).

    library(sn)
    library(mvtnorm)
    library(QuadratiK)
    n <- 100
    d <- 4
    skewness_y <- 0.5
    set.seed(2468)
    x_2 <- rmvnorm(n, mean = rep(0, d))
    y_2 <- rmsn(n = n, xi = 0, Omega = diag(d), alpha = rep(skewness_y, d))

The two-sample test can be performed by providing the two samples to be compared as `x` and `y` to the `kb.test()` function. If a value of \(h\) is not provided, the function automatically performs the function `select_h`.

    set.seed(2468)
    two_test <- kb.test(x = x_2, y = y_2)
    two_test

    ##
    ##  Kernel-based quadratic distance two-sample test
    ## U-statistic   Dn          Trace
    ## ------------------------------------------------
    ## Test Statistic:   1.679763    2.312427
    ## Critical Value:   1.087455    1.497868
    ## H0 is rejected:   TRUE        TRUE
    ## CV method:  subsampling
    ## Selected tuning parameter h:  1.6

We can display the chosen optimal value of \(h\) together with the power plot obtained versus the considered \(h\), for the alternatives \(\delta\) in the `select_h()` function.

    two_test@h$h_sel

    ## [1] 1.6

    two_test@h$power.plot

For more details visit the help documentation of the `select_h()` function.

    help(select_h)

For the two-sample case, the `summary` function provides the results from the test and a list tables of the standard descriptive statistics for each variable, computed per group and overall. Additionally, it generates the qq-plots comparing the quantiles of the two groups for each variable.

    summary_two <- summary(two_test)

    ##
    ##  Kernel-based quadratic distance two-sample test
    ##   Statistic    Value Critical_Value Reject_H0
    ## 1        Dn 1.679763       1.087455      TRUE
    ## 2     Trace 2.312427       1.497868      TRUE

    summary_two$summary_tables

    ## [[1]]
    ##             Group 1    Group 2    Overall
    ## mean   -0.005393522  0.3197861  0.1571963
    ## sd      1.039119207  0.9094193  0.9875137
    ## median -0.019317321  0.4448058  0.1601955
    ## IQR     1.562613453  1.3612937  1.4292426
    ## min    -2.675477796 -1.4256211 -2.6754778
    ## max     2.151784802  2.9375947  2.9375947
    ##
    ## [[2]]
    ##            Group 1    Group 2     Overall
    ## mean   -0.10005083  0.1936138  0.04678149
    ## sd      1.10476260  1.0556439  1.08777010
    ## median -0.07955849  0.2235325  0.10130247
    ## IQR     1.48816630  1.4716179  1.41498342
    ## min    -3.22222061 -2.4336333 -3.22222061
    ## max     2.50192633  2.6879362  2.68793623
    ##
    ## [[3]]
    ##             Group 1    Group 2     Overall
    ## mean   -0.006524772  0.1701261  0.08180065
    ## sd      0.958942739  0.9524916  0.95742170
    ## median -0.039301279  0.1887394  0.11877637
    ## IQR     1.329868172  1.4657107  1.40312077
    ## min    -2.860006689 -2.1762183 -2.86000669
    ## max     1.923763114  2.4237195  2.42371949
    ##
    ## [[4]]
    ##            Group 1    Group 2     Overall
    ## mean   -0.06757686  0.2236458  0.07803449
    ## sd      0.98684958  0.9862135  0.99481815
    ## median -0.03258747  0.1097711  0.05517931
    ## IQR     1.30933016  1.4088334  1.39890664
    ## min    -2.29625537 -2.1827156 -2.29625537
    ## max     2.40795077  2.9929942  2.99299420

#### Select h

The search for the optimal value of the tuning parameter \(h\) can be performed independently from the test computation using the `select_h` function. It requires the two samples, provided as `x` and `y`, and the considered family of alternatives.

    set.seed(2468)
    two_test_h <- select_h(x = x_2, y = y_2, alternative = "skewness")

The code is not evaluated since we would obtain the same results.

#### Note

Notice that the test statistics for two-sample testing coincide with the \(k\)-sample test statistics when \(k=2\). Hence, alternatively the two sample tests can be performed providing the two samples together as `x` and indicating the membership to the groups with the argument `y`.

    x_pool <- rbind(x_2, y_2)
    y_memb <- rep(c(1, 2), each = n)
    h <- two_test@h$h_sel
    set.seed(2468)
    kb.test(x = x_pool, y = y_memb, h = h)

    ##
    ##  Kernel-based quadratic distance k-sample test
    ## U-statistic   Dn          Trace
    ## ------------------------------------------------
    ## Test Statistic:   1.679763    2.312427
    ## Critical Value:   1.087455    1.497868
    ## H0 is rejected:   TRUE        TRUE
    ## CV method:  subsampling
    ## Selected tuning parameter h:  1.6

See the k-sample test vignette for more details.

In the `kb.test()` function, the critical value can be computed with the subsampling, bootstrap or permutation algorithm. The default method is set to subsampling since it needs less computational time. For details on the sampling algorithm see the documentation of the `kb.test()` function.

For more details on the level and power performance of the considered two-sample tests, see the extensive simulation study reported in the following reference.

## References

Markatou, M. and Saraceno, G. (2024). “A Unified Framework for Multivariate Two- and k-Sample Kernel-based Quadratic Distance Goodness-of-Fit Tests.”
https://doi.org/10.48550/arXiv.2407.16374