Author: Stéphane Laurent
Date: 2023-04-11

# Owen cumulative functions

#### Stéphane Laurent

#### 2023-04-11

# The Owen distribution

Let \(Z \sim {\cal N}(0,1)\) and \(X \sim \chi^2_\nu\) be two independent random variables. For real numbers \(\delta_1\) and \(\delta_2\), define the two random variables \[ T_1 = \frac{Z+\delta_1}{\sqrt{X/\nu}} \quad \text{and} \quad\; T_2 = \frac{Z+\delta_2}{\sqrt{X/\nu}}. \]

Both \(T_1\) and \(T_2\) follow a non-central Student distribution. The number of degrees of freedom is \(\nu\) for each of them, and their respective non-centrality parameters are \(\delta_1\) and \(\delta_2\) respectively.

Owen (1965) studied the distribution of the pair \((T_1, T_2)\).

The four Owen cumulative functions are \[ \begin{align} O_1(\nu, t_1, t_2, \delta_1, \delta_2) & = \Pr(T_1 \leq t_1, T_2 \leq t_2), \\ O_2(\nu, t_1, t_2, \delta_1, \delta_2) & = \Pr(T_1 \leq t_1, T_2 \geq t_2), \\ O_3(\nu, t_1, t_2, \delta_1, \delta_2) & = \Pr(T_1 \geq t_1, T_2 \geq t_2), \\ O_4(\nu, t_1, t_2, \delta_1, \delta_2) & = \Pr(T_1 \geq t_1, T_2 \leq t_2). \end{align} \]

Owen provided an efficient way to evaluate these functions when \(\nu\) is an integer number. Owen’s algorithms are implemented in the `OwenQ` package.

For \(\delta_1 > \delta_2\), these four functions are implemented in the `OwenQ` package under the respective names `powen1`, `powen2`, `powen3` and `powen4`. For general values of \(\delta_1\) and \(\delta_2\), they are implemented under the respective names `psbt1`, `psbt2`, `psbt3` and `psbt4`.

# Non-central Student distribution

Owen (1965) also provided an algorithm to evaluate the cumulative distribution function of a univariate non-central Student distribution with an integer number of degrees of freedom. This evaluation is performed by the function `ptOwen` of the `OwenQ` package.

    ptOwen(q=1, nu=3, delta=2)
    ## [1] 0.1573494
    pt(q=1, df=3, ncp=2)
    ## [1] 0.1573494

It is known that the `pt` function is not reliable when the non-centrality parameter `ncp` is large. Below we compare the values given by `ptOwen` and `pt` to the value given by Wolfram|Alpha (returned by the command `N[CDF[NoncentralStudentTDistribution[4,70],80]]`):

    p1 <- pt(q=80, df=4, ncp=70)
    p2 <- ptOwen(q=80, nu=4, delta=70)
    wolfram <- 0.54742763380700947685
    p1 - wolfram
    ## [1] 0.02268711
    p2 - wolfram
    ## [1] 3.330669e-16

# Limitations

When `q`\(=\)`delta`, the value of `ptOwen(q, nu, delta)` should go to `0.5` as `nu` increases to infinity. The examples below show the failure of this expectation when `nu` is too large.

    ptOwen(q=50, nu=3500, delta=50)
    ## [1] 0.4986697
    ptOwen(q=50, nu=3600, delta=50)
    ## [1] 0.4989289
    ptOwen(q=50, nu=3650, delta=50)
    ## [1] 0.4522949
    ptOwen(q=50, nu=3660, delta=50)
    ## [1] 0.3349762
    ptOwen(q=50, nu=3670, delta=50)
    ## [1] 0.7607795
    ptOwen(q=50, nu=3680, delta=50)
    ## [1] 0

Since all Owen’s algorithms are somehow similar to the algorithm evaluating `ptOwen`, we can suspect that the other ones also suffer from certain limitations. In the other vignette, we investigate the reliability and the limitations of `OwenQ`.

In order to do some comparisons, and thanks to the `BH` package, we have ported the `boost` implementation of the cumulative Student distribution to `OwenQ`. It is evaluated by the internal function `pt_boost`. We concluded that this function is highly reliable. In particular it does not suffer from the failures of `ptOwen` we have just shown:

    OwenQ:::pt_boost(q=50, nu=3500, delta=50)
    ## [1] 0.4986697
    OwenQ:::pt_boost(q=50, nu=3600, delta=50)
    ## [1] 0.4987041
    OwenQ:::pt_boost(q=50, nu=3650, delta=50)
    ## [1] 0.4987206
    OwenQ:::pt_boost(q=50, nu=3660, delta=50)
    ## [1] 0.4987238
    OwenQ:::pt_boost(q=50, nu=3670, delta=50)
    ## [1] 0.4987271
    OwenQ:::pt_boost(q=50, nu=3680, delta=50)
    ## [1] 0.4987303

# Application to equivalence testing

The Owen distribution intervenes in the calculation of the power of equivalence tests.

## One sample

Assume a statistical model given by a sample \(y_i \sim_{\text{iid}} {\cal N}(\mu, \sigma^2)\) for \(i=1, \ldots, n\). We want to demonstrate that, up to a given confidence level, the mean \(\mu\) belongs to a certain interval \([\Delta_1, \Delta_2]\). In other words, we are interested in the alternative hypothesis \(H_1\colon\{\Delta_1 \leq \mu \leq \Delta_2\}\).

Consider the usual \(100(1-2\alpha)\%\)-confidence interval about \(\mu\): \[ \left[\bar y - t^\ast_{n-1}(\alpha)\frac{\hat\sigma}{\sqrt{n}}, \, \bar y + t^\ast_{n-1}(\alpha)\frac{\hat\sigma}{\sqrt{n}} \right]. \]

The \(H_1\) hypothesis is accepted at level \(\alpha\) when this interval falls into the interval \([\Delta_1, \Delta_2]\).

This can be written as follows: \[ T_1 := \frac{\bar y - \Delta_1}{\hat\sigma/\sqrt{n}} \geq t^\ast_{n-1}(\alpha) \quad \text{and} \quad T_2 := \frac{\bar y - \Delta_2}{\hat\sigma/\sqrt{n}} \leq - t^\ast_{n-1}(\alpha). \]

Observe that \[ T_1 = \frac{z - \delta_1}{\dfrac{\sqrt{n-1}\hat\sigma/\sigma}{\sqrt{n-1}}} \] where \[ z = \frac{\sqrt{n}}{\sigma}(\bar y - \mu) \sim {\cal N}(0,1) \quad \text{and} \quad \delta_1 = \frac{\mu - \Delta_1}{\sigma/\sqrt{n}}. \]

By reasoning in the same way for \(T_2\), we find that the pair \((T_1, T_2)\) follows the Owen distribution with degrees of freedom \(\nu = n-1\), and non-centrality parameters \(\delta_1\) given above and \(\delta_2 = \tfrac{\mu - \Delta_2}{\sigma/\sqrt{n}}\).

Therefore the power of the test - i.e. the probability to accept \(H_1\) - is given by \[ O_4\bigl(n-1, t^\ast_{n-1}(\alpha), -t^\ast_{n-1}(\alpha), \delta_1, \delta_2\bigr), \] and then can be evaluated thanks to `powen4`.

## Inconclusive equivalence test

The result of the equivalence test is said to be inconclusive when only one of the bounds of the confidence interval falls into the interval \([\Delta_1, \Delta_2]\).

The probability to get an inconclusive result can be obtained with the `OwenQ` package. We show and check that below with the help of simulations.

    Delta1 <- -2; Delta2 <- 2
    mu <- 1; sigma <- 6; n <- 30L
    alpha <- 0.05
    nsims <- 1e6L
    equivalence <- inconclusive <- numeric(nsims)
    for (i in 1L:nsims) {
      y <- rnorm(n, mu, sigma)
      CI <- t.test(x = y, conf.level = 1-2*alpha)$conf.int
      equivalence[i] <- (CI[1] > Delta1) && (CI[2] < Delta2)
      inconclusive[i] <- ((CI[1] < Delta1) && (CI[2] > Delta1)) ||
        ((CI[1] < Delta2) && (CI[2] > Delta2))
    }

    dof <- n-1
    q <- qt(1-alpha, dof)
    se <- sqrt(1/n)*sigma
    delta1 <- (mu-Delta1)/se; delta2 <- (mu-Delta2)/se
    # probability to get equivalence
    mean(equivalence)
    ## [1] 0.092532
    powen4(dof, q, -q, delta1, delta2)
    ## [1] 0.09300963
    # probability to get inconclusive
    mean(inconclusive)
    ## [1] 0.901865
    ptOwen(q, dof, delta2) - ptOwen(-q, dof, delta1) - powen4(dof, q, -q, delta1, delta2)
    ## [1] 0.90139

## Parallel design

Now consider two independent samples \(x_i \sim_{\text{iid}} {\cal N}(\mu, \sigma^2)\) for \(i=1, \ldots, n_1\). and \(y_i \sim_{\text{iid}} {\cal N}(\nu, \sigma^2)\) for \(i=1, \ldots, n_2\) and the alternative hypothesis \(H_1\colon\bigl\{|\mu-\nu| \leq \Delta\bigr\}\). The power of this test is evaluated by the function `powerTOST` below. The parameter `delta0` corresponds to the difference \(\mu-\nu\).

    powerTOST <- function(alpha, delta0, Delta, sigma, n1, n2) {
      se <- sqrt(1/n1 + 1/n2) * sigma
      delta1 <- (delta0 + Delta) / se
      delta2 <- (delta0 - Delta) / se
      dof <- n1 + n2 - 2
      q <- qt(1 - alpha, dof)
      powen4(dof, q, -q, delta1, delta2)
    }

# The Owen \(T\)-function

The `OwenQ` package also provides an implementation of the Owen \(T\)-function, under the name `OwenT`. This is a port of the function `owens_t` of `boost`, the peer-reviewed collection of C++ libraries.

# The Owen \(Q\)-functions

The Owen cumulative functions are based on the two Owen \(Q\)-functions \[ Q_1(\nu, t, \delta, R) = \frac{1}{\Gamma\left(\frac{\nu}{2}\right)2^{\frac12(\nu-2)}} \int_0^R \Phi\left(\frac{tx}{\sqrt{\nu}}-\delta\right) x^{\nu-1} e^{-\frac{x^2}{2}} \mathrm{d}x \] and \[ Q_2(\nu, t, \delta, R) = \frac{1}{\Gamma\left(\frac{\nu}{2}\right)2^{\frac12(\nu-2)}} \int_R^\infty \Phi\left(\frac{tx}{\sqrt{\nu}}-\delta\right) x^{\nu-1} e^{-\frac{x^2}{2}} \mathrm{d}x. \]

They are implemented in the `OwenQ` package under the respective names `OwenQ1` and `OwenQ2` (for integer values of \(\nu\)).

# Application: Equal-tailed tolerance interval

Consider the statistical model given by a sample \(y_i \sim_{\text{iid}} {\cal N}(\mu, \sigma^2)\) for \(i=1, \ldots, n\).

An equal-tailed \((p,\alpha)\)-tolerance interval is an interval containing at least \(100p\%\) of the “center data” with \(100(1-\alpha)\%\) confidence.

The natural choice for such an interval is, as shown by Owen (1965), \[ \bigl[\bar y - k_e \hat\sigma, \bar y + k_e \hat\sigma] \] where \(k_e\) satisfies \[ O_2(n-1, k_e\sqrt{n}, -k_e\sqrt{n}, \delta, -\delta) = 1-\alpha \] where \(\delta = \sqrt{n}z_{(1+p)/2}\).

Therefore, the tolerance factor \(k_e\) can be determined with the help of the `powen2` function of the `OwenQ` package. But it is more efficient to use the function `spowen2` for this purpose; `spowen2(nu, t, delta)` has the same value as `powen2(nu, t, -t, delta, -delta)` but it is evaluated more efficiently.

    p <- 0.9; alpha <- 0.05
    n <- 100
    delta <- sqrt(n)*qnorm((1+p)/2)
    uniroot(function(ke) spowen2(n-1, ke*sqrt(n), delta) - (1-alpha),
            lower=qnorm(1-alpha), upper=5, extendInt = "upX", tol=1e-9)$root
    ## [1] 1.981513

The \(k_e\) factors are tabulated in Krishnamoorthy & Mathew’s book.

# Internal functions: numerical integration

The `OwenQ` package provides three internal functions, `iOwenQ1`, `iOwenQ2` and `ipowen4`, which respectively perform the evaluation of \(Q_1\), \(Q_2\) and \(O_4\) by numerical integration using the `RcppNumerical` package. They can be called with a positive non-integer value of \(\nu\). The evaluation of \(O_4\) with `ipowen4` is correct only for \(t_1 > t_2\) and \(\delta_1 > \delta_2\).

# References

- Owen, D. B. (1965). A special case of a bivariate noncentral t-distribution. Biometrika 52, 437-446.

- Krishnamoorthy & Mathew (2009). Statistical Tolerance Regions. Wiley.