# On two-sided p-values for non-symmetric distributions

###### Abstract

Two-sided statistical tests and p-values are well defined only
when the test statistic in question has a symmetric distribution.
A new two-sided p-value called conditional p-value is introduced here.
It is closely related
to the doubled p-value and has an intuitive appeal. Its use is advocated
for both continuous and discrete distributions. An important advantage of this p-value
is that equivalent 1-sided tests are transformed into -equivalent
2-sided tests. It is compared to the widely used doubled and minimum likelihood p-values.
Examples include the variance test, the binomial and the Fisher’s exact test.

keywords: two-sided tests, Fisher’s exact test, variance test, binomial test, test,
minimum likelihood

## 1 Introduction

Two-sided statistical tests are widely used and misused in numerous applications of statistics. In fact, some applied journals do not accept papers quoting 1-sided p-values anymore. Examples include The New England Journal of Medicine, Journal of the National Cancer Institute and Journal of Clinical Oncology among others.

Unfortunately, two-sided statistical tests and p-values are well defined only when the test statistic in question has a symmetric distribution. The difficulties with two-sided p-values arise in a general case of a non-symmetric distribution, though they are more often commented on for discrete distributions.

The most famous example is an ongoing discussion about how 2-sided p-values
should be constructed for the Fisher’s exact test .
This discussion was started in 1935 by Fisher (1935) and Irwin (1935).
Numerous developments of the next 50 years are summarised in Yates (1984) and
discussion thereof. The more recent contributions include
several proposals based on an a modified UMPU test
Lloyd (1988), Dunne et al. (1996), Meulepas (1998). See also Agresti and Wackerly (1977), Dupont (1986), Davis (1986),
Agresti (1992).
A (far from exhausting) list of 9 different proposals is given
in Meulepas (1998). The problem is still not resolved.

Fisher advocated doubling the 1-sided p-value in his
letter to Finney in 1946 (Yates, 1984, p.444). This doubled p-value is denoted by .
Fisher’s motivation was an equal prior weight of departure in either direction. Other arguments for doubling include
invariance under transformation of the distribution to a normal scale, and ease of approximation by the chi-square distribution
(Yates, 1984). One of the evident drawbacks of the doubling rule is that it may result in a p-value greater than 1. The doubled p-value is used in the majority of statistical software in the case of continuously distributed statistics and often in the discrete case.

The primary contribution of this article is the introduction of a new method of
defining two-sided p-values to be called ‘conditional two-sided p-values’ denoted by .
The conditional p-value is closely related
to the doubled p-value and has an intuitive appeal. It is demonstrated that this new two-sided p-value has properties which make it a
definite improvement on currently used two-sided p-values for both
discrete and continuous non-symmetric distributions.

Another popular two-sided p-value for non-symmetric discrete distributions implemented in computer packages, R (R Development Core Team, 2004) in particular, is adding the probabilities
of the points less probable than the observed (at both tails). This p-value is denoted by . This method
was introduced in Freeman and Halton (1951), and is based on Neyman and Pearson (1931) idea of ordering multinomial
probabilities; this is called ‘the principle of minimum likelihood’ by Gibbons and Pratt (1975), see also George and Mudholkar (1990).
Hill and Pike (1965) were the first to use this p-value for Fisher’s exact test.
Many statisticians objected to this principle. Gibbons and Pratt (1975)
commented that ‘The minimum likelihood method can also lead to
absurdities, especially when the distribution is U-shaped, J-shaped, or simply not unimodal.’ Radlow and Alf (1975) pointed out ‘This procedure is justified only if events of lower probability
are necessarily more discrepant from the null hypothesis. Unfortunately, this is frequently not true.’

The following example clearly demonstrates another unfortunate feature of this p-value. When a value of density is associated with a high 1-sided p-value at one tail,
the value at the opposite tail cannot be rejected even though it may have a very low 1-sided p-value.

Example: Two-sided variance test based on the Chi-square distribution Suppose we have 6 observations from a perfectly normal population and wish to test the null hypothesis that the variance against a two-sided alternative. The test statistic , where is the sample variance. For (or ) the 1-sided p-value on the left tail is 0.0374, the density is 0.0807, the symmetric value on the right tail is , the 1-sided p-value is 0.2431, see dotted lines on the left plot of Figure 1; similarly for () the density is 0.0366, the p-value on the left tail is 0.0079, the symmetric value is 9.255, p-value=0.0993 (dashed lines on the same plot). It is very difficult to reject the null for small observed values.

Given critical values on the left and right tail and , such that , the power of a two-sided variance test at level is calculated as , where .
The power of four -level tests is plotted at the right plot of Figure 1. The test based on the is evidently biased,
with
very low power for , i.e. when . The minimum value of the power is .

The uniformly most powerfull unbiased (UMPU) test for this example has the critical region defined by and corresponding to critical levels and at the left and right tail, respectively. Finally, the generalized likelihood ratio (GLR) test is based on the statistic , and it is biased (Stuart and Ord, 1991, Example 23.5, p.882). This is not exceptional; Bar-Lev et al. (2002) showed that for a continuous exponential family on the real line, the GLR and UMPU tests coincide if and only if, up to an affine transformation, is either a normal, inverse Gaussian or gamma family.

The new conditional 2-sided p-value is formally defined in the next section.
The power of the tests based on the doubled and conditional p-value
for the chi-square example is also plotted in
Figure 1. They are much less biased, with minimum power of and respectively.

The formal definition of the conditional 2-sided p-value and the comparison of its properties to those of the doubled p-value and the for a case of continuous distributions is given in Section 2, and for discrete distributions (binomial and hypergeometric) in Section 3. Discussion is in Section 4. The use of the conditional 2-sided p-value is advocated for both continuous and discrete distributions. An important advantage of this p-value is that equivalent 1-sided tests are transformed into -equivalent 2-sided tests.

## 2 Two-sided p-values for continuous asymmetric distributions

Consider a general case of a statistic with a strictly increasing continuous
null distribution with continuous density . For an observed value of , one-sided p-value on the left tail is defined as , where independent from . Similarly, on the right tail the p-value is
. Denote by a generic location parameter chosen to separate the two tails of the distribution . Particular examples include the mean , the mode ,
or the median . Which parameter should be used to separate the two tails depends on the context;
the mean seems to be the most appropriate when a test statistic is based on an estimate of a natural parameter in an
exponential family, as is the case with binomial or Fisher’s exact test.
General theory below is applicable regardless of the parameter chosen, though the details of examples may differ.
Interestingly, it does not matter much
for the most important non-symmetric discrete distributions: the mean when attainable coincides with the mode (or one of the two modes) for Poisson, binomial and hypergeometric distributions. The latter two distributions are discussed in Section 3.

###### Definition 1

Weighted two-tailed p-value centered at with weights satisfying is defined as

(1) |

Doubled p-value denoted by has weights . Without loss of generality assume that . Then the doubled p-value is equal to for ,
1 for , and for . Thus the doubled p-value is not continuous at unless , its derivative is also discontinuous at .

Similarly, a weighted p-value is continuous at iff and an additional requirement of results in and arriving at the next definition.

###### Definition 2

Conditional 2-sided p-value centered at is defined as

(2) |

This is a smooth function of (but at ),
with a maximum of at . It strictly increases for and decreases for . The conditional p-value is conceptually close to the doubled p-value, the only difference being that the two tails are
weighted inversely
proportionate to their probabilities. This results in inflated p-values on the thin tail, and
deflated p-values on the thick tail when compared to the doubled p-value. When the tails are defined in respect to
the median, the two p-values coincide: . Thus conditional p-value is equal to the usual doubled
p-value for a symmetric distribution.
It is easy to see that under the null hypothesis is uniformly distributed on given a particular tail, i.e.
,
similar to a 1-sided p-value.

There is an evident connection between a choice of a two-sided p-value and a critical
region (CR) for a two-sided test at level . A CR is defined through critical values corresponding to probabilities
and , with the weights of the two tails
. It can equivalently be defined through a weighted p-value as . The doubled p-value corresponds to . The conditional p-value is equivalent to the choice , .

For a two-sided test, critical values and satisfy and . Thus . Define . Then the CR is . Therefore any 2-sided test, a UMPU test inclusive, is a test based on conditional p-value centered at some . Conversely, if the value is chosen to be independent of , the resulting test is, in general, biased. Since an independence from level is a natural requirement for a p-value, some bias cannot be escaped.

###### Lemma 1

For a one-parameter exponential family , a two-sided level- test based on the conditional p-value is less biased in the neighborhood of the null value than the standard equal tails test based on the doubled p-value whenever , where is the weight at the left tail of the UMPU test.

Proof Denote test critical function by . This is an indicator function of the CR, so , and the power is . Without loss of generality is the sufficient statistic. The derivative of the power is (Lehmann, 1959, p. 127)

(3) |

For an UMPU test . For a test with weight at the left tail,

For , this is strictly decreasing function of equal zero at . When , any provides positive values of , and when , the values of are negative; in any case the gradient is the steepest and the bias is the largest at , as required.

Lemma 1 provides a sufficient condition for the -based test to be less biased than the equal tails test, but this condition is not necessary. This condition holds for the distribution, and the variance test
based on is uniformly (in ) less biased then the test based on the doubled p-value, left plot of Figure 2.
The doubled p-value based test is asymptotically UMPU, Shao (1999), and so is the -based test. In the two-sample case, the equal-tails -test of the equality of variances is UMPU for equal sample sizes, and the -based test is less biased when the ratio of sample sizes is larger than 1.7 (starting from ), whereas lemma 1 holds for even more unbalanced sample sizes with the ratio of 2.5 or above, right plot of Figure 2.

Finally, consider the minimum likelihood p-value.

###### Definition 3

Minimum likelihood p-value is .

reaches 1 at the mode, and whenever . It is not a unimodal function of when the density is not unimodal. In a case of a unimodal distribution, for a pair of conjugate points , it is calculated as . It has a distribution under the null.

When used to define a test, the acceptance region defined as contains the points
with the highest density, and is therefore of minimum length.
Inverting this test results in the shortest
confidence intervals, see Sterne (1954) for the binomial and Baptista and Pike (1977) for the hypergeometric distribution.
It is also related to Bayes shortest posterior confidence intervals,
see Wilson and Tonascia (1971) for the intervals for the standard deviation and
the ratio of variances in normal populations, based on
inverse chi and distribution, respectively.

The next three examples clarify the properties of the conditional p-value in comparison to .

Example: Triangular distribution

Suppose that the null density is given by for and for . The mode , and . Then for , and for , , George and Mudholkar (1990). Thus, . This is the only unimodal distribution for which this equality holds as it requires the linearity of the density .

Example: Uniform distribution

Consider a distribution. This is a symmetric distribution with , and for ,
and
for , whereas for all values of .

This example shows the cardinal difference between
the two p-values. acknowledges unusual values of at the ends of the interval, and the does not.
This is a somewhat extreme example, because the uniform distribution has a whole interval of modes. The next example
deals with a unimodal distribution but shows exactly the same properties of the respective p-values.

Example: Left-truncated normal distribution.

Denote the standard normal distribution function and density by and , respectively.
Consider a left-truncated at normal distribution defined for .
The mode is at zero. Then for , and for .
reaches 1 at ,
and is continuous at , but its derivative is not continuous at .
The mean is , and the conditional p-value reaches 1 at .
An example for is plotted in the right plot in Figure 3.
For this example and the weight of the left tail is . The main difference between the two p-values is that at the left tail, so even the low values of
in the vicinity of have rather high p-values. On the other hand, is very close to zero for these values,
recognizing that it is rather unusual to get close to . It seems that a small two-sided p-value at the left tail
makes more sense.

The above two examples show the properties of the which are perhaps clear from its definition: it compares a value
to other values at the same tail. On the other hand, depends on the values at both tails.
The same circumstances arise in the variance test example which was introduced in the Introduction.

Example: variance test based on the Chi-square distribution (continued) Recall, that for () the 1-sided p-value is .0079, and the value of with equal density is with the 1-sided p-value of .0993. The mean , and the conditional p-values are , and , the weight of the left tail is . The value with the same conditional p-value on the opposite tail is with the 1-sided p-value of 0.0056. Clearly, is more comparable to than the value . The three p-values are plotted at the left plot in Figure 3.

The power of the three tests and of the UMPU test (all at 5% level) is shown in the right plot in Figure 1. The UMPU test is the conditional test with , corresponding to the weight . All three tests are biased, with the bias defined as the minimum difference between the power and level
being for the doubled and for the conditional test. This agrees with Lemma 1.
The doubled test is slightly less powerful on the right, and slightly more on the left.
The test based on has very large bias and such low power for the alternatives , that it
does not deserve to be called a two-sided test.

The main difficulty associated with the two-sided tests is that two equivalent 1-sided tests may result in distinct 2-sided tests. For the variance test example the tests based on and on are not equivalent. Let be a measure of distance from . It imposes an equivalence of points at two sides of : each value has an equidistant value . Two-sided tests based on and are not equivalent, generally speaking, because for , the equidistant value . This results in different rejection regions for the two tests. The main advantages of the conditional p-value are given in the next Lemma.

###### Lemma 2

(i) For a strictly increasing function , the conditional p-value
.

(ii) Suppose strictly decreases for and strictly
increases for , and . Define the conditional p-value for the distance as
.
Then .

The first statement of the lemma easily follows from the definition of , and for the second statement take . This is a strictly increasing function of , and the proof follows from part (i).

The first part of the lemma ensures that equivalent 1-sided tests are transformed into -equivalent
2-sided tests. The second part states that the 2-sided tests based on any measure of distance from are -equivalent.
This is true because the conditional p-value ignores
any equivalence between the points at different tails.

## 3 Discrete distributions

In this section the 2-sided conditional p-value is defined for a discrete distribution.
It is also compared to and for two important cases: binomial and hypergeometric
distributions.

The definition of the conditional p-value (2) is also applicable in a discrete case, but it may require a modification when the value is attainable. Since the value belongs to both tails, the previously defined weights of the tails and add up to . The modified weights of the tails are and . This modification is akin to continuity correction. The formal definition of is

###### Definition 4

Conditional two-sided p-value for a discrete distribution is

(4) |

where the weights are and . Modified conditional p-value is defined with weights and in equation (4).

Two definitions coincide when the value is not attainable. In a discrete symmetric case when is an attainable value the values of are doubled 1-sided values, and the values of are times smaller, and the -based test is therefore more liberal. The conditional p-value has a mode of 1 at when this value is attainable, and two modes of 1 at the attainable values above and below when is not an attainable value. It has discrete uniform distribution when restricted to values at a particular tail, though not overall. In what follows we consider the case of , and use the notation .

### 3.1 Binomial distribution

For distribution the mode is .
When is an integer, and are both modes,
and the mean is unattainable. When is an integer, . In all cases the distance
. The median is one of or .

Consider first the symmetric case . For odd , the value is an integer, both tails of the distribution have weight 0.5
and . For even , the mean is an integer, , but .
Unmodified version is symmetric
at with values for . The modified version .

Statistical packages differ in regards to the 2-sided p-values for the binomial test: R (R Development Core Team, 2004) uses and StatXact (www.cytel.com) uses the doubled p-value.

The three p-values, , , and are plotted in Figure 4
for and two values of , and .
In the first case is an attainable value.
It can be seen that on the left plot. The weight of the left tail
is vs . Consequently, for all but .
Modified conditional p-value is considerably
closer to at the left tail, and at the right (thin) tail.
In fact, in this example for , provides exact 1-sided p-values for ,
and for . So , , , and
. The two-sided binomial test as programmed in R uses and would reject the null
hypothesis of at 5% level given an observed value of 5, whereas a test based on the doubled or conditional p-value would not reject.
The same thing may happen for much larger values of . For example, for ,
and the observed value of the values are
, and .

For (right plot) is not attainable. has two modes at 2 and 3. Here and for , whereas for .

Typically, at the thick tail, and at the thin tail. Even for large the difference between and is rather large. For example, for and the values are and in comparison to the 1-sided p-value of .

For the binomial distribution the weight of the tails converges to 0.5 rather slowly, and , see Table 1. Whenever the mean is attainable, the weight of the thin right tail is also more than 0.5, and . If E is not attainable, the weight of the thin tail is less than 0.5, and then . This is always true for . The distribution is more symmetric when the mean is attainable. Otherwise even for , the weight of the left tail is for .

10 | 0.736 | 1.130 | 0.531 | 0.678 | 1.086 | 0.521 |
---|---|---|---|---|---|---|

11 | 0.697 | 2.304 | 0.697 | 0.617 | 1.614 | 0.617 |

20 | 0.677 | 1.113 | 0.527 | 0.630 | 1.070 | 0.517 |

21 | 0.648 | 1.844 | 0.648 | 0.586 | 1.416 | 0.586 |

50 | 0.616 | 1.083 | 0.520 | 0.584 | 1.049 | 0.512 |

51 | 0.598 | 1.485 | 0.598 | 0.556 | 1.250 | 0.556 |

100 | 0.583 | 1.063 | 0.515 | 0.559 | 1.036 | 0.509 |

101 | 0.570 | 1.325 | 0.570 | 0.540 | 1.172 | 0.540 |

200 | 0.559 | 1.046 | 0.511 | 0.542 | 1.026 | 0.507 |

201 | 0.550 | 1.221 | 0.550 | 0.528 | 1.119 | 0.528 |

500 | 0.538 | 1.030 | 0.507 | 0.527 | 1.017 | 0.504 |

501 | 0.532 | 1.135 | 0.532 | 0.518 | 1.074 | 0.518 |

1000 | 0.527 | 1.022 | 0.505 | 0.519 | 1.012 | 0.503 |

1001 | 0.522 | 1.094 | 0.522 | 0.513 | 1.052 | 0.513 |

### 3.2 Hypergeometric distribution

Consider a crosstabulation of two binary variables and . We shall refer to numbers of observations in the cell and respective probabilities as and , . The value is the statistic of Fisher’s exact test used to test for association of and given fixed margins . The value defines all the other entries in a table with given margins. A parameter of primary importance is the odds ratio estimated by . The case of no association is equivalent to . Denote the expected values , with . The number iff . Fisher (1935) derived the distribution of as

The null distribution (standard hypergeometric) is for . For testing vs the p-value is .
For the p-value is . For a two-sided test, seems to be the p-value of choice, implemented both in R and in StatExact.

Sometimes other one-sided test statistics are used to test for association; they may be based on the differences of proportions in rows
or columns (e.g. ) or on the . Nevertheless, all other possible 1-sided tests are equivalent to
Fisher’s exact test since their statistics are strictly increasing functions of , as shown by Davis (1986).
The Fisher’s exact test is also the UMPU test if the randomization is allowed (Tocher, 1950).

For distribution, the full range of values for fixed margins
is , where and .
The mode is . Therefore .
When is an integer, and are both modes
and the mean is unattainable. When is an integer, .
In all cases the distance .

Exact 2-sided tests for association are used when both
positive and negative associations are of interest. However, there
is ongoing controversy about how 2-sided p-values should be
constructed for the hypergeometric distribution (Yates, 1984; Agresti and Wackerly, 1977; Meulepas, 1998; Dunne et al., 1996).

Davis (1986) compares the p-values associated with the following 6 statistics:

Consider the table with margins used as an example in Davis (1986). The possible values are 0 through 5, , so the left tail has two tables only, for and 1, with the total probability of . Tables with are on the right tail, the total probability is . Two tails are rather close in probability. Davis (1986) looks at the orderings of tables according to the increasing values of test statistics, as follows:

Due to monotonicity of all statistics at both sides of the mean , the
conditional p-values for all 6 statistics do not differ (Lemma 2). Therefore all 6 2-sided tests are equivalent. This is the main advantage
of the conditional p-value for hypergeometric distribution. Fisher’s exact test is usually superseded by the chi-square test for large cell numbers.
Equivalence of these two tests is of practical importance, for example when testing for linkage disequilibrium in genetics.

The probabilities of the 6 tables along with their one-sided p-values, and values are given in columns 2-5 of Table 2.Conditional p-values are very close to doubled 1-sided p-values. The second set of tables in Table 2 corresponds to margins . Here the left tail probability is 0.689, and the thin right tail has probability 0.311. The probabilities and the p-values are given in columns 6-9. Here the inflation of the conditional p-values on the right tail is more prominent.

0 | .143 | .143 | .286 | .274 | .258 | .258 | .570 | .374 |

1 | .378 | .521 | 1 | 1 | .430 | .689 | 1 | 1 |

2 | .336 | .479 | .622 | 1 | .246 | .311 | .311 | 1 |

3 | .124 | .143 | .143 | .299 | .059 | .065 | .065 | .209 |

4 | .019 | .019 | .019 | .040 | .006 | .006 | .006 | .028 |

5 | .001 | .001 | .001 | .002 | .0002 | .0002 | .0002 | .0006 |

## 4 Discussion

Two-sided testing in non-symmetric distributions is not straightforward. The UMPU tests are not implemented in the mainstream software packages even for continuous problems, and require randomization in the discrete case. The non-asymptotic GLR tests are also not implemented, and are, in general, biased, Bar-Lev et al. (2002). At the same time the two-sided tests are the staple in all applications. An importance of a conceptually and computationally simple approach to two-sided testing is self-evident.

The conditional 2-sided p-value introduced in Section 1 is closely related
to doubled p-value and has an intuitive appeal. Its use is advocated
for both continuous and discrete distributions. An important advantage of this p-value
is that equivalent 1-sided tests are transformed into -equivalent
2-sided tests. This helps to resolve the ongoing controversy about which 2-sided tests should be used for the association
in 2 by 2 tables.

The properties of this p-value compare favorably to the doubled p-value and to the minimum likelihood p-value , the main two implemented options in statistical tests for non-symmetric distributions. For the variance test, the bias of the -based test is smaller than the bias of the standard equal tails test based on the doubled p-value, and much smaller than the bias of the -based test. For the considerably unbalanced sample sizes, the -based test is also less biased than the equal tails F-test of the equality of variances.

We did not compare the power and the bias of the resulting tests for the binomial and the hypergeometric cases. This is difficult to do for tests at different levels without recourse to randomisation. For asymptotically normal tests, both p-values should result in asymptotically UMPU tests, though the minimum likelihood p-value may require more stringent conditions to ensure the convergence of the density to normal density. The proof of these statements is a matter for further research.

Another open question is which version or should be used for an attainable value of . Motivation for is less clear, it also results in a more conservative test on top of the inescapable conservativeness due to discrete distribution.

Gibbons and Pratt (1975) consider a large number of 2-sided p-values and find them lacking. They recommend reporting one-tailed p-value with the direction of the observed departure from the null hypothesis. In this spirit, the conditional p-value conditions on this direction.

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