Computing optimal sampling designs for two-stage studies

Stata Technical Bulletin

Koopman’s method

Let X and Y be two independent binomial variates based on sample sizes m and n and parameters p₁ and p₂, respectively.
Let θ = pι∕p₂∙ Koopman (1984) proposed a method for constructing confidence intervals for θ based on a chi-squared test.
oopman’s method has been widely used in medical research for evaluating drug efficacy and treatment effects.

Assume we test for H₀ : θ = Θq against Ha : O ≠ 0o∙ For this problem there is no uniformly most powerful test as extreme
values may occur in the sample, but a chi-squared test seems a reasonable choice. The test statistic Ug₀ is then given by

_{ττ t} ч {x - τnp₁}² (y - np₂}²

^{θo x,y} _mp₁ (I-Pi) np₂(1 - p₂)

where pɪ and p₂ are the maximum likelihood estimates under the restriction θ = 0_o, It can be proved that

ʌ _ θ₀(m + y) + x + n- [{0_o(m + ?/) + x + n}² - ⅛θ₀(m + n)(x + y)]¹/²

2(m + n)

andp₂ = pi∕0_o.

For 0 = 1, the statistic Ug₀(x,y) is the traditional Pearson chi-square. Rearranging Ug₀(x,y) results in

rr ( ʌ (ж-Wi)² ʃ m(0_θ-Pι)l

This shows that under H₀, Ug₀{x, y) has asymptotically for m —> ∞ and n —> ∞, a chi-squared distribution with 1 degree
of freedom independent of Θq (Bishop et al. 1977). Hence, an approximate 1 — α two-sided confidence region for θ is given by

{U_θo(x,y) < X1,1-_α}

where χ² _1-α is the 1 — α fractile of the chi-squared distribution with 1 degree of freedom. Since U is a convex function of θ,
this is an asymmetric interval (0∕,0∙_u), where

Uθ_l(x,y) = U_θJx,y} = χ² ι-_o.

and

θι < θ_u

As Ug_l (ж, ye reduces to the usual chi-squared when 0 = 1, this interval will always agree with the chi-squared test.

Because there is no explicit expression for the inverse function of U, the values of 0/ and θ_u have to be solved by numerical
procedures. The main concern of the command koopman is to obtain 0/ and θ_u by using repeated bisection as suggested by
Koopman (1984).

Syntax

koopman vαr_xevent var.group [weihht∖ [if ixp∖ [in range [, level (#) ]

koopmani #_x #_m #_y #_n [, level (#) ]

koopman allows fweights.

Description

koopman computes confidence intervals for the ratio of two binomial proportions based on two independent binomially
distributed random variables using Koopman’s method. Point estimates and confidence intervals for the odds ratio are calculated.
ivent-var contains a one if the observation represents an event and 0 otherwise. group~var indicates the group to which each
observation belongs. The variable must have only two values. Observations with missing values are not used.

koopmani is the immediate form of koopman.

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