Examples

The following examples illustrate the use of the numerical optimization-integration algorithms described in the previous section. Work is measured in terms of the number of density function evaluations needed for the integration to compute $P(t)$. The basic starting interval $[t_a, t_b]$ was computed using the Pegasus method; the cost for this is minimal because only univariate and bivariate ${\bf T}$ values are required. The initial bracketing interval for $t_\alpha$ is given in the first row in each table, and the work for that row is the work to compute the initial $h(t_a)$, $h(t_c)$ and $h(t_b)$ values (with only $h(t_c)$ required to start the Newton iteration). Subsequent rows show results of the iterations necessary to determine $t_\alpha$ to the specified accuracy.

Our first example used data for starch thickness taken from Hsu and Nelson (1998). The desired CI's were for two-sided comparisons versus a control, with $P(t) = {\bf T}(-{\bf t},{\bf t}, \mbox{\boldmath$\Sigma$}, 86)$, and

$$\mbox{\boldmath $\Sigma$}= \left[ \begin{array}{cccccc} 1&-&-... ...&-\\ .5505& .4922& .8651& .7738& .7915&1 \end{array} \right].$$

For $\alpha = .1$, we computed $[t'_a,t'_b] = [1.663,2.442]$ and $[t_a,t_b] = [2.116,2.324]$. The following numerical optimization-integration results were obtained.

Pegasus Method Results for $\tau = 0.001$

$t_a, t_c, t_b$	$h(t_a),h(t_c),h(t_b)$	$\hat{h'}$, Work
2.220, 2.264, 2.324	-.01, .0005, .01	.22, 39952
2.220, 2.262, 2.264	-.01, .00001, .0005	.22, 51152

Newton Method Results for $\tau = 0.001$

$t_a, t_c, t_b$	$h(t_a),h(t_c),h(t_b)$	$\hat{h'}$, Work
2.116, 2.220, 2.324	, -.01,	.26, 11200
2.220, 2.261, 2.324	-.01, -.0002,	.24, 28752

For $\alpha = .05$, we computed $[t'_a,t'_b] = [1.988, 2.701]$, $[t_a,t_b] = [2.429, 2.606]$ and:

Pegasus Method Results for $\tau = 0.001$

$t_a, t_c, t_b$	$h(t_a),h(t_c),h(t_b)$	$\hat{h'}$, Work
2.518,2.561,2.606	-.005, .0002, .006	.12, 146432
2.518,2.559,2.561	-.005,.00004,.0002	.12, 187984

Newton Method Results for $\tau = 0.001$

$t_a, t_c, t_b$	$h(t_a),h(t_c),h(t_b)$	$\hat{h'}$, Work
2.429,2.518,2.606	, -.005,	.14, 41552
2.518,2.557,2.606	-.005,-.0002,	.13, 104880
2.557,2.559,2.606	-.0002,.00001,	.12, 168208

Our second example is an all-pairwise comparisons example based on a one-way ANOVA design with sample sizes $n_i = 20, 3, 3, 15$ (see Westfall et.al., 1999). We need two sided CI's with $P(t) = {\bf T}(-{\bf t},{\bf t}, \mbox{\boldmath$\Sigma$}, 37)$, given $\mbox{\boldmath$\Sigma$}=$

$$\left[\begin{array}{rrrrrr} 1&-&-&-&-&-\\ .1304&1&-&-&-&-\... ...1&-\\ 0&-.8513& .3086&-.6455& .1667&1\\ \end{array} \right].$$

For $\alpha = .1$, we computed $[t'_a,t'_b] = [1.687, 2.508]$, $[t_a,t_b] = [2.265, 2.383]$ and:

Pegasus Method Results for $\tau = 0.001$

$t_a, t_c, t_b$	$h(t_a),h(t_c),h(t_b)$	$\hat{h'}$, Work
2.324,2.339,2.383	-.003,.0003,.009	.21, 86144
2.324,2.337,2.339	-.003,-.0002,.0003	.21, 127696

Newton Method Results for $\tau = 0.001$

$t_a, t_c, t_b$	$h(t_a),h(t_c),h(t_b)$	$\hat{h'}$, Work
2.265,2.324,2.383	, -.003,	.25, 41552
2.324,2.338,2.383	-.003,-.00006,	.21, 68592

For $\alpha = .05$, we computed $[t'_a,t'_b] = [2.050, 2.788]$, $[t_a,t_b] = [2.026, 2.684]$ and:

Pegasus Method Results for $\tau = 0.001$

$t_a, t_c, t_b$	$h(t_a),h(t_c),h(t_b)$	$\hat{h'}$, Work
2.648,2.654,2.684	-.0006,-.00001,.003	.11, 255680

Newton Method Results for $\tau = 0.001$

$t_a, t_c, t_b$	$h(t_a),h(t_c),h(t_b)$	$\hat{h'}$, Work
2.612,2.648,2.684	, -.0006,	.13, 63328
2.648,2.654,2.684	-.0006,.00003,	.12, 159504

The results from these example tests demonstrate that the algorithms described in this paper provide feasible methods for computing $t_\alpha$ values for confidence intervals. Given good starting intervals determined from bivariate distribution values, the numerical optimzation based on the use of the Newton method is more efficient than optimization based on the Pegasus method. These conclusions are also supported by a variety of other examples that we have considered.

REFERENCES

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