CLs method (particle physics)

In particle physics, CLs represents a statistical method for setting "upper limits" (also called "exclusion limits") on model parameters, a particular form of interval estimation used for parameters that can take only non-negative values. Although CLs are said to refer to Confidence Levels, "The method's name is ... misleading, as the CLs exclusion region is not a confidence interval." It was first introduced by physicists working at the LEP experiment at CERN and has since been used by many high energy physics experiments. It is a frequentist method in the sense that the properties of the limit are defined by means of error probabilities, however it differs from standard confidence intervals in that the stated confidence level of the interval is not equal to its coverage probability. The reason for this deviation is that standard upper limits based on a most powerful test necessarily produce empty intervals with some fixed probability when the parameter value is zero, and this property is considered undesirable by most physicists and statisticians.
Upper limits derived with the CLs method always contain the zero value of the parameter and hence the coverage probability at this point is always 100%. The definition of CLs does not follow from any precise theoretical framework of statistical inference and is therefore described sometimes as "ad hoc". It has however close resemblance to concepts of "statistical evidence"
proposed by the statistician Allan Birnbaum.
Definition.
Let "X" be a random sample from a probability distribution with a real non-negative parameter formula_1. A "CLs" upper limit for the parameter "θ", with confidence level formula_2, is a statistic (i.e., observable random variable) formula_3 which has the property:
The inequality is used in the definition to account for cases where the distribution of "X" is discrete and an equality can not be achieved precisely. If the distribution of "X" is continuous then this should be replaced by an equality. Note that the definition implies that the coverage probability formula_4 is always larger than formula_2.
An equivalent definition can be made by considering a hypothesis test of the null hypothesis formula_6 against the alternative formula_7. Then the numerator in (), when evaluated at formula_8, correspond to the type-I error probability (formula_9) of the test (i.e., formula_8 is rejected when formula_11) and the denominator to the power (formula_12). The criterion for rejecting formula_13 thus requires that the ratio formula_14 will be smaller than formula_15. This can be interpreted intuitively as saying that formula_8 is excluded because it is formula_15 less likely to observe such an extreme outcome as "X" when formula_8 is true than it is when the alternative formula_19 is true.
The calculation of the upper limit is usually done by constructing a test statistic formula_20 and finding the value of formula_21 for which
where formula_23 is the observed outcome of the experiment.
Usage in high energy physics.
Upper limits based on the CLs method were used in numerous publications of experimental results obtained at particle accelerator experiments such as LEP, the Tevatron and the LHC, most notable in the searches for new particles.
Origin.
The original motivation for CLs was based on a conditional probability calculation suggested by physicist G. Zech for an event counting experiment. Suppose an experiment consists of measuring formula_24 events coming from signal and background processes, both described by Poisson distributions with respective rates formula_25 and formula_26, namely formula_27. formula_26 is assumed to be known and formula_25 is the parameter to be estimated by the experiment. The standard procedure for setting an upper limit on formula_25 given an experimental outcome formula_31 consists of excluding values of formula_25 for which formula_33, which guarantees at least formula_34 coverage. Consider, for example, a case where formula_35 and formula_36 events are observed, then one finds that formula_37 is excluded at 95% confidence level. But this implies that formula_38 is excluded, namely all possible values of formula_25. Such a result is difficult to interpret because the experiment cannot essentially distinguish very small values of formula_25 from the background-only hypothesis, and thus declaring that such small values are excluded (in favor of the background-only hypothesis) seems inappropriate. To overcome this difficulty Zech suggested conditioning the probability that formula_41 on the observation that formula_42, where formula_43 is the (unmeasurable) number of background events. The reasoning behind this is that when formula_43 is small the procedure is more likely to produce an error (i.e., an interval that does not cover the true value) than when formula_43 is large, and the distribution of formula_43 itself is independent of formula_25. That is, not the over-all error probability should be reported but the conditional probability given the knowledge one has on the number of background events in the sample. This conditional probability is
which correspond to the above definition of CLs. The first equality just uses the definition of Conditional probability, and the second equality comes from the fact that if formula_49 and the number of background events is by definition independent of the signal strength.
Generalization of the conditional argument.
Zech's conditional argument can be formally extended to the general case. Suppose that formula_50 is a test statistic from which the confidence interval is derived, and let
where formula_52 is the outcome observed by the experiment. Then formula_53 can be regarded as an unmeasurable (since formula_21 is unknown) random variable, whose distribution is uniform between 0 and 1 independent of formula_21. If the test is unbiased then the outcome formula_52 implies
from which, similarly to conditioning on formula_43 in the previous case, one obtains
Relation to foundational principles.
The arguments given above can be viewed as following the spirit of the conditionality principle of statistical inference, although they express a more generalized notion of conditionality which do not require the existence of an ancillary statistic. The conditionality principle however, already in its original more restricted version, formally implies the likelihood principle, a result famously shown by Birnbaum. CLs does not obey the likelihood principle, and thus such considerations may only be used to suggest plausibility, but not theoretical completeness from the foundational point of view. (The same however can be said on any frequentist method if the conditionality principle is regarded as necessary).
Birnbaum himself suggested in his 1962 paper that the CLs ratio formula_14 should be used as a measure of the strength of "statistical evidence" provided by significance tests, rather than formula_9 alone. This followed from a simple application of the likelihood principle: if the outcome of an experiment is to be only reported in a form of a "accept"/"reject" decision, then the overall procedure is equivalent to an experiment that has only two possible outcomes, with probabilities formula_9,formula_63 and formula_34,formula_65 under formula_66. The likelihood ratio associated with the outcome "reject formula_67" is therefore formula_14 and hence should determine the evidential interpretation of this result. (Since, for a test of two simple hypotheses, the likelihood ratio is a compact representation of the likelihood function). On the other hand, if the likelihood principle is to be followed consistently, then the likelihood ratio of the original outcome should be used and not formula_14, making the basis of such an interpretation questionable. Birnbaum later described this as having "at most heuristic, but not substantial, value for evidential interpretation".
A more direct approach leading to a similar conclusion can be found in Birnbaum's formulation of the "Confidence principle", which, unlike the more common version, refers to error probabilities of both kinds. This is stated as follows:
"A concept of statistical evidence is not plausible unless it finds 'strong evidence for formula_70 as against formula_67' with small probability (formula_9) when formula_67 is true, and with much larger probability (1 -formula_74) when formula_70 is true. " 
Such definition of confidence can naturally seem to be satisfied by the definition of CLs. It remains true that
both this and the more common (as associated with the Neyman-Pearson theory) versions of the confidence principle are incompatible with the likelihood principle, and therefore no frequentist method can be regarded as a truly complete solution to the problems raised by considering conditional properties of confidence intervals.
Calculation in the large sample limit.
If certain regularity conditions are met, then a general likelihood function will become a Gaussian function in the large sample limit. In such case the CLs upper limit at confidence level formula_2 (derived from the uniformly most powerful test) is given by
where formula_78 is the standard normal cumulative distribution, formula_79 is the maximum likelihood estimator of formula_21 and formula_81 is its standard deviation; the latter might be estimated from the inverse of the Fisher information matrix or by using the "Asimov" data set. This result happens to be equivalent to a Bayesian credible interval if a uniform prior for formula_21 is used.