Boschloo's test

Boschloo's test is a statistical hypothesis test for analysing 2x2 contingency tables. It examines the association of two Bernoulli distributed random variables and is a uniformly more powerful alternative than Fisher's exact test. It was published in 1970 by R. D. Boschloo.

Setting

A 2x2 contingency table visualizes $n$ independent observations of two binary variables $A$ and $B$:

${\begin{array}{c|cc|c}&B=1&B=0&{\mbox{Total}}\\\hline A=1&x_{11}&x_{10}&n_{1}\\A=0&x_{01}&x_{00}&n_{0}\\\hline {\mbox{Total}}&s_{1}&s_{0}&n\\\end{array}}$

The probability distribution of such tables can be classified in three distinct cases.

1. The row sums $n_{1},n_{0}$ and column sums $s_{1},s_{0}$ are fixed in advance and not random.
   Then all $x_{ij}$ are determined by $x_{11}$. If $A$ and $B$ are independent, $x_{11}$ follows a hypergeometric distribution with parameters $n,n_{1},s_{1}$:
   $x_{11}\sim {\mbox{Hypergeometric}}(n,n_{1},s_{1})$.
2. The row sums $n_{1},n_{0}$ are fixed in advance but the column sums $s_{1},s_{0}$ are not.
   Then all random parameters are determined by $x_{11}$ and $x_{01}$ and $x_{11},x_{01}$ follow a binomial distribution with probabilities $p_{1},p_{0}$:
   $x_{11}\sim B(n_{1},p_{1})$
   $x_{01}\sim B(n_{0},p_{0})$
3. Only the total number $n$ is fixed but the row sums $n_{1},n_{0}$ and the column sums $s_{1},s_{0}$ are not.
   Then the random vector $(x_{11},x_{10},x_{01},x_{00})$ follows a multinomial distribution with probability vector $(p_{11},p_{10},p_{01},p_{00})$.

Fisher's exact test is designed for the first case and therefore an exact conditional test (because it conditions on the column sums). The classical example of such a case is the Lady tasting tea: A lady tastes 8 cups of tea with milk. In 4 of those cups the milk is poured in before the tea. In the other 4 cups the tea is poured in first. The lady tries to assign the cups to the two categories. Following our notation, the random variable $A$ represents the used method (1 = milk first, 0 = milk last) and $B$ represents the lady's guesses (1 = milk first guessed, 0 = milk last guessed). Then the row sums are the fixed numbers of cups prepared with each method: $n_{1}=4,n_{0}=4$. The lady will assign 4 cups to each method, so also the column sums are fixed in advance: $s_{1}=4,s_{0}=4$. If she is not able to tell the difference, $A$ and $B$ are independent and the number $x_{11}$ of correctly classified cups with milk first follows the hypergeometric distribution ${\mbox{Hypergeometric}}(8,4,4)$.

Boschloo's test is designed for the second case and therefore an exact unconditional test. Examples of such a case are often found in medical research, where a binary endpoint is compared between two patient groups. Following our notation, $A=1$ represents the first group that receives some medication of interest. $A=0$ represents the second group that receives a placebo. $B$ indicates the cure of a patient (1 = cure, 0 = no cure). Then the row sums equal the group sizes and usually are fixed in advance. The column sums are the total number of cures respectively disease continuations and not fixed in advance.

An example for the third case can be constructed as follows: Simultaneously flip two distinguishable coins $A$ and $B$ and do this $n$ times. If we count the number of results in our 2x2 table (1 = head, 0 = tail), we neither know in advance how often coin $A$ shows head or tail (row sums random), nor do we know how often coin $B$ shows head or tail (column sums random).

Test hypothesis

The null hypothesis of Boschloo's one-tailed test is (high values of $x_{1}$ favor the alternative hypothesis):

$H_{0}:p_{1}\leq p_{0}$

The null hypothesis of the one-tailed test can also be formulated in the other direction (small values of $x_{1}$ favor the alternative hypothesis):

$H_{0}:p_{1}\geq p_{0}$

The null hypothesis of the two-tailed test is:

$H_{0}:p_{1}=p_{0}$

There is no unique definition of the two-tailed version of Fisher's exact test. This is also true for Boschloo's test as it is based on Fisher's exact test. In the following we deal with the one-tailed test and $H_{0}:p_{1}\leq p_{0}$.

Boschloo's Idea

We denote the desired significance level by $\alpha$. Fisher's exact test is a conditional test and appropriate for the first above mentioned case. But if one pretends that the observed column sum $s_{1}$ would have been fixed in advance, Fisher's exact test can also be applied to the second case. The true size of the test then depends on the nuisance parameters $p_{1},p_{0}$. It can be shown that the size maximum $\max \limits _{p_{1}\leq p_{0}}{\big (}{\mbox{size}}(p_{1},p_{0}){\big )}$ is taken for equal proportions $p=p_{1}=p_{0}$ and is still controlled by $\alpha$. However, Boschloo stated that for small sample sizes, the maximal size is often considerably smaller than $\alpha$. This leads to an undesirable loss of power.

Boschloo's Idea was to use Fisher's exact test with a greater nominal level $\alpha ^{*}>\alpha$. Thereby, $\alpha ^{*}$ should be chosen maximal such that the maximal size is still controlled by $\alpha$: $\max \limits _{p\in [0,1]}{\big (}{\mbox{size}}(p){\big )}\leq \alpha$. This method was especially advantageous at that time because $\alpha ^{*}$ could be looked up for common values of $\alpha ,n_{1}$ and $n_{0}$. This made performing Boschloo's test computationally easy.

Test statistic

The decision rule of Boschloo's approach is based on Fisher's exact test. An equivalent way of formulating the test is to use the p-value of Fisher's exact test as test statistic. Fisher's p-value is calculated from the hypergeometric distribution (for ease of notation we write $x_{1},x_{0}$ instead of $x_{11},x_{01}$:

$p_{F}=1-F_{{\mbox{Hypergeometric}}(n,n_{1},x_{1}+x_{0})}(x_{1}-1)$

The distribution of $p_{F}$ is determined by the binomial distributions of $x_{1}$ and $x_{0}$ and depends on the unknown nuisance parameters $p_{1},p_{0}$. For a specified significance level $\alpha$, the critical value of $p_{F}$ is the maximal value $\alpha ^{*}$ that fulfilles $\max \limits _{p\in [0,1]}P(p_{F}\leq \alpha ^{*})\leq \alpha$. The critical value $\alpha ^{*}$ equals the nominal level of Boschloo's original approach.

Modification

Boschloo's test deals with the unknown nuisance parameter $p$ by taking the maximum over the whole parameter space $[0,1]$. The Berger & Boos procedure takes a different approach by maximizing $P(p_{F}\leq \alpha ^{*})$ over a $(1-\gamma )$ confidence interval and adding $\gamma$. $\gamma$ is usually a small value such as 0.001 or 0.0001. This results in a modified Boschloo's test which is also exact.

Comparison to other exact tests

All exact tests adhere to the specified significance level but can have varying power in different situations. Mehrotra et al. compared the power of some exact tests in different situations. The results regarding Boschloo's test are summarized in the following.

Modified Boschloo's test

Boschloo's test und the modified Boschloo's test have similar power in all considered scenarios. In some cases Boschloo's test had slightly more power and in some cases vice versa.

= Fisher's exact test

Boschloo's test is by construction uniformly more powerful than Fisher's exact test. For small sample sizes (e.g. 10 per group) the power difference is significant, ranging from 16 to 20 percentage points in the regarded cases. This difference is smaller for greater sample sizes.

$Z$-Pooled test

This test is based on the test statistic

$Z_{P}(x_{1},x_{0})={\frac {{\hat {p}}_{1}-{\hat {p}}_{0}}{\sqrt {{\tilde {p}}(1-{\tilde {p}})({\frac {1}{n_{1}}}+{\frac {1}{n_{0}}})}}},$

where $p_{i}={\frac {x_{i}}{n_{i}}}$ are the group event rates and ${\tilde {p}}={\frac {x_{1}+x_{0}}{n_{1}+n_{0}}}$ is the pooled event rate.

The power of this test is similar to that of Boschloo's test in most scenarios. In some cases, the $Z$-Pooled test has greater power, with differences mostly ranging from 1 to 5 percentage points. In very few cases, the difference goes up to 9 percentage points.

This test can also be modified by the Berger & Boos procedure. However, the resulting test has very similar power to the unmodified test in all scenarios.

$Z$-Unpooled test

This test is based on the test statistic

$Z_{U}(x_{1},x_{0})={\frac {{\hat {p}}_{1}-{\hat {p}}_{0}}{\sqrt {{\frac {{\hat {p}}_{1}(1-{\hat {p}}_{1})}{n_{1}}}+{\frac {{\hat {p}}_{0}(1-{\hat {p}}_{0})}{n_{0}}}}}},$

where $p_{i}={\frac {x_{i}}{n_{i}}}$ are the group event rates.

The power of this test is similar to that of Boschloo's test in many scenarios. In some cases, the $Z$-Unpooled test has greater power, with differences ranging from 1 to 5 percentage points. However, in some other cases, Boschloo's test has significantly more power, with differences up to 68 percentage points.

This test can also be modified by the Berger & Boos procedure. The resulting test has similar power to the unmodified test in most scenarios. In some cases, the power is considerably improved by the modification but the overall power comparison to Boschloo's test remains unchanged.

See also

• Fisher's exact test
• Barnard's test