Brit. J. Phil. Sci. 53 (2002), 69±76

Resolving Neyman's Paradox Max Albert ABSTRACT According to Fisher, a hypothesis specifying a density function for X is falsi®ed (at the level of signi®cance a) if the realization of X is in the size-a region of lowest densities. However, non-linear transformations of X can map low-density into high-density regions. Apparently, then, falsi®cations can always be turned into corroborations (and vice versa) by looking at suitable transformations of X (Neyman's Paradox). The present paper shows that, contrary to the view taken in the literature, this provides no argument against a theory of statistical falsi®cation.

1 2 3 4

The problems of statistical falsi®cation Redhead's version of Neyman's Paradox A counterargument Dierent measurement processes

1 The problems of statistical falsi®cation From a theoretical perspective, statistical inference is a serious problem for falsi®cationism. Actual tests of scienti®c hypotheses in most cases involve statistical arguments. Falsi®cationism is therefore plausible only if it can deal with statistical hypotheses. However, there is no falsi®cationist theory of statistical inference that is accepted even by falsi®cationists.1 Except for the trivial case where a zero-probability event is observed, a falsi®cation of a statistical hypothesis presupposes the choice of a rejection region, that is, an event that is possible under the hypothesis in question but the observation of which is nevertheless taken as a falsi®cation.2 Following Popper ([1984]) and Gillies ([1971], [1973]), the rejection region is to be 1

2

Forster ([1995], p. 402) explains the attraction of Bayesianism in philosophy of science by the fact that the logic-based accounts of scienti®c inference (which include falsi®cationism) have nothing to say about statistical problems. Nevertheless, statistical practice is often quite falsi®cationist in spirit; cf. Gillies ([forthcoming]) on Neyman's use of the w2 test. This causes problems to non-falsi®cationists, who have to argue that standard and successful practices are in fact illegitimate. `Rejection' and `falsi®cation' are used as synonyms throughout, although this is not always in accordance with the use of the term `rejection' in the statistical literature.

& British Society for the Philosophy of Science 2002

70

Max Albert

chosen according to a methodological rule, a falsifying rule for statistical inference (FRSI). According to Neyman and Pearson's ([1933]) theory of statistical inference (NPT), such a rule should take two kinds of errors into account, usually called error of the ®rst and error of the second kind. A ®rst-kind error is an erroneous falsi®cation of a true hypothesis, while a second-kind error is the failure to falsify a false hypothesis (erroneous corroboration). The risk of a ®rst-kind error can be controlled by choosing a rejection region that has a suciently small probability a, also called level of signi®cance, under the hypothesis. The risk of a second-kind error has to be controlled by considering at least one explicit alternative to the hypothesis under test. The optimal test in the case of only two alternatives minimizes the probability b of not falsifying the original hypothesis if the alternative is true. Equivalently, it maximizes the power 1 ÿ b. The NPT is acceptable to falsi®cationists if and only if the complete set of alternative hypotheses considered is part of the background knowledge, i.e. can be taken for granted in the context of the inquiry. If, for example, it is known that a certain procedure guarantees random sampling from a ®nite population (in the sense of every element of the population having the same probability to be included in the sample), then there is a ®xed set of hypotheses concerning the population average of any variable, and these hypotheses give rise to a ®xed set of statistical hypotheses. The randomsampling hypothesis is part of the background knowledge, and the set of hypotheses in which we can use an NP test is determined by the logical consequences of this basic hypothesis together with certain non-statistical hypotheses about the population. In such cases, a falsi®cationist can agree with Neyman ([1965], p. 448) that there is no dierence between estimation (choosing the in some sense best hypothesis from a given set) and testing.3 When, however, the question arises (as it must at some stage) as to whether the basic hypothesis in the background is true, the situation is dierent. One may be able to ®nd some test based on a still more general basic hypothesis, but if the hypothesis under scrutiny becomes more and more general, the NPT quickly runs out of tests. In practice, therefore, some assumptions are tested with the help of tests that are quite general but questionable from the standpoint of the NPT, like the w2 test. Thus, a falsi®cationist interpretation of the NPT must be based on some other, more fundamental approach answering the question of how to assess, 3

For these cases, the interpretation and extension of the NPT by Mayo ([1996]) ®ts into a falsi®cationist framework. See also Mayo and Spanos ([2000]) for further progress in rendering the idea of the severity of a test more precise. Note that the falsi®cationist idea of severity of tests relies on background knowledge that can be (or already has been) tested independently; cf. Musgrave's ([1974]) discussion of Hempel's raven paradox.

Resolving Neyman's Paradox

71

as econometricians are wont to say, the assumptions of the statistical model. At this level, at least, tests should be independent from alternative hypotheses (Gillies [1971], [1973], [forthcoming]; Albert [1992]). The obvious candidate for a theory of statistical falsi®cation, Gillies' ([1971], [1973]) theory, is open to three important objections that have already been raised against Fisher's theory of signi®cance testing (cf. Fisher [1990], Spielman [1974]). Consider a statistical hypothesis stating a distribution function for a one-dimensional random variable (rv). Let us assume that we have an FRSI specifying, without recourse to alternative hypotheses, a rejection region for a single observation of the rv. Then the following three problems must be solved.4 1. Selection of a test statistic. It is not obvious how to extend the FRSI to the case of n > 1 observations. The solution is to select a one-dimensional test statistic; however, there are many candidates. 2. Optional stopping. The rule determining the number of observations, the stopping rule, has a potential in¯uence on the level of signi®cance if the decision to stop is not independent from the observations (optional stopping). For any experiment, there are always several interpretations yielding dierent signi®cance levels: the conventional interpretation that assumes stopping to be independent from observations, and many alternative interpretations specifying dierent ways of optional stopping. If one concludes that optional stopping makes a dierence, the experimenter's intentions matter for the evaluation of otherwise indistinguishable observations. 3. Neyman's Paradox. Assume that the rv is continuously distributed with a density. Fisher's and Gillies' theories select the lowest-density regions as rejection regions. Non-linear transformations of the rv can map low-density into high-density regions and vice versa, leading to dierent decisions on the basis of the same FRSI and the same observations. The NPT solves the ®rst and the third problem by appealing to alternative hypotheses. As already argued, this is not always an option for falsi®cationists. The second problem also arises for the NPT. However, these problems pose no insurmountable obstacles for a theory of statistical falsi®cation. Just adopting some test by convention (e.g. the w2 4

The problem of selecting a test statistic originally motivated Neyman and Pearson ([1930], [1933]) to modify Fisher's theory; see also Hacking ([1965], pp. 75±81). On optional stopping, see Hacking ([1965], pp. 107±9) and Berger and Berry ([1988]). Neyman's Paradox, originally formulated in connection with the t test, goes back to a 1929 paper of Neyman; cf. Neyman and Pearson ([1930], p. 101n) and Neyman ([1952], pp. 45±51). The general formulation is due to Redhead ([1974]).

72

Max Albert

goodness-of-®t test for discrete hypotheses) goes far in solving the ®rst problem.5 Mayo ([1996]) argues that one actually should worry about optional stopping, thus attacking the position of adherents of the likelihood principle (Bayesians and others) that the insensitivity of likelihood-based inferences to optional stopping is a point in their favor. The present note argues that Neyman's Paradox does not arise if one accepts a reasonable condition concerning the scope of an FRSI.

2 Redhead's version of Neyman's Paradox The very general nature of Neyman's Paradox has been argued most forcefully by Redhead ([1974]) in his critique of Gillies' rule. Consider the hypothesis X N, meaning that the rv X is distributed according to the cumulative distribution function N of the standard normal distribution, and restrict the analysis to the case of a single observation. The so-called Gauss test at a level of signi®cance 0.05 then requires rejection if jxj51:96. This test is accepted by Fisher and Gillies, at least in principle. Consider the rv Y t X, where ÿ1 3 def N 2 ÿ N X , X 4 0 t X Nÿ1 12 ÿ N X , X 5 0 and where therefore Y N.6 The same Gauss test can be applied to Y as well. If X is in the rejection region, Y is not, and vice versa. Thus one can always reformulate the hypothesis in such a way that it is not rejected by mathematically the same test. Redhead's argument is completely general. Transformations that map the tails of the original distribution to the centre of the new distribution can always be found. Moreover, Redhead's transformation guarantees for any unimodal distribution with mode 0 that X and t X are distributed identically.

3 A Counterargument The present paper argues that Neyman's Paradox, taken as a general criticism of a theory of statistical falsi®cation, misses its target. An FRSI has only one purpose, namely to render statistical hypotheses falsi®able. Thus, I suggest that a theory of statistical falsi®cation is based on the following adequacy condition. 5

6

Albert ([1992]) proposes an FRSI for discrete rvs similar in spirit to Gillies' rule but covering the test-selection problem. This rule selects the multinomial goodness-of-®t test, which can be approximated by the w2 test under certain conditions. On continuous distributions, see below. t 0 is not de®ned. This is not important, however, since distributions diering only w.r.t. zeroprobability events are taken to be equivalent.

Resolving Neyman's Paradox

73

Scope of an FRSI. An FRSI only applies to hypotheses that can immediately be confronted with data (low-level hypotheses). Real-world measurement processes have ®nite precision. For this reason, continuous distributions, as far as they are part of the low-level hypotheses under test, are approximations. This means that Neyman's Paradox does not arise since it depends on the assumption that real variables can be observed with in®nite precision. The adequacy condition destroys the basis for the paradox. Let us make this clear by reference to an example. Since we consider just Neyman's Paradox and not the other objections against statistical falsi®cations, assume that there is an FRSI selecting a test for discrete rvs. Assume that the theory from which the low-level hypothesis under test is derived postulates a continuous distribution. For example, one might argue that the height of humans at age 25 (X) is a continuous rv. We want to test the hypothesis that this rv obeys a continuous distribution with density f. Since we focus on Neyman's Paradox, assume that we test the hypothesis on the basis of a single observation; the argument is completely analogous if we consider, for example, the mean value of a sample. The actual process of measurement yields intervals on a speci®c scale, and these intervals have a certain probability according to the hypothesis. We are back to a test of a discrete hypothesis where, by assumption, the FRSI unambiguously picks a rejection region.

4 Dierent measurement processes In a presentation of this argument, the following point was raised. It is physically possible to measure Y a=X instead of X (where a is some constant). Let us make this clear by an example. Assume that for some obscure reason a person's height is measured in the following way. The person stands upright against a wall. A sliding bar that projects 20 cm from the wall at a right angle is moved down until it touches the top of the person's head. The tip of the bar (point A), the point exactly below it on the ground (point B), and a third point C on the ground 320 cm from point B form a right-angled triangle (see Figure 1). The length of AB is the height of the person, the length of BC is known to be 300 cm. From A to C, a string is stretched. We mark the point D where the height of the string is exactly 100 cm; the distance of D from C is the rv Y. We have Y a=X where a 30,000. With Y, we have a transformed variable with density f a=Y. Actual measurements again yield intervals on a certain scale, resulting in a dierent discrete hypothesis that can be tested by using the FRSI. It is possible that the

74

Max Albert

Figure 1. Measuring Y a=X, where X is the height of a person.

same observation gives rise to a falsi®cation if X is measured, while a corroboration occurs if Y is measured. At ®rst glance, then, it seems that Neyman's Paradox again raises its ugly head. However, a closer examination shows that this is not true. While there is one observation, there are two dierent low-level hypotheses since two dierent measurement processes are used. These hypotheses dier not only w.r.t. the rv and the density function, but also w.r.t. the if-clause describing the experimental set-up. It is even possible that one of the hypotheses is true and the other is false since they are necessarily derived from dierent auxiliary assumptions. For instance, the more complicated process of observing Y may introduce errors of measurement that are absent when X is observed. The hypothesis that the density of Y is f a=Y is derived under the implicit assumption that no such measurement errors are introduced by observing Y instead of X, an assumption which might be false. But let us grant the truth of all auxiliary assumptions. The situation is nevertheless dierent from Neyman's Paradox. In the case of Neyman's Paradox, we assume that measurements of X and Y result in real values. We then have to decide how to interpret these results. We do not gain anything by combining both measurements: we can compute one from the other. With ®nite precision, the situation is quite dierent. If we can subject the same person to both measurement processes, we get a more precise result in

Resolving Neyman's Paradox

75

all those cases where the observed X interval and the observed Y interval are not mapped onto each other by taking inverse values. Assume that, in our example, the realization of X is found to be in the interval 110, 111. Measuring Y with the same precision, we ®nd an interval 270, 271. Combining the information, we learn that the true height is in the interval 110.70, 111. Hence, we can test a discrete hypothesis that gives the probabilities for the smaller intervals for the height of a person ascertainable by combining both measurements. Thus, the cases where the results of the tests dier are just the cases where we get more precise results from using both methods of measurement. The combined measurement obviously allows for a more severe test than any of the single measurements. Of course, if we have to decide between the two experiments because only one of them is feasible, it may depend on our decision whether we get a falsi®cation or a corroboration. But this is only reasonable. The combined experiment delivers maximum precision. If there are several second-best methods of measurement, it is plausible that there are cases where these measurements taken in isolation come to dierent results. This is not dierent from testing a deterministic hypothesis. If you want to test the hypothesis that swans are white, and if you can only aord a trip either to Germany or to New Zealand, it depends on your decision whether you falsify or corroborate the hypothesis. There is only one strange feature of the statistical setting: we are better o (in that we can avoid Neyman's Paradox) because we know less (namely, intervals instead of exact values). But since no problem results from this, we should be grateful. The working properties of the tests are not aected. There is no ambiguity. Thus, Neyman's Paradox does not exist after all if one accepts the very reasonable adequacy condition for the scope of an FRSI. It comes up only in thought experiments where one confronts hypotheses with actually unobservable events.

Acknowledgements For useful hints and discussions, I am grateful to Donald Gillies and the participants, especially Joseph Berkovitz, Jeremy Butter®eld and Michael Redhead, of a Cambridge University S Club session. Department of Economics University of Koblenz±Landau August-Croissant-Str. 5 D-76829 Landau, Germany [email protected]

76

Max Albert

References Albert, M. [1992]: `Die Falsi®kation statistischer Hypothesen', Journal for General Philosophy of Science, 23, pp. 1±32. Berger, J. O. and Berry, D. A. [1988]: `The Relevance of Stopping Rules in Statistical Inference (with discussion)', in S. S. Gupta and J. O. Berger (eds), 1988, Statistical Decision Theory and Related Topics IV, Vol. I, New York: Springer, pp. 29±72. Fisher, R. A. [1990]: Statistical Methods, Experimental Design and Scienti®c Inference, Oxford: Oxford University Press. Forster, M. R. [1995]: `Bayes and Bust: Simplicity as a Problem for a Probabilist's Approach to Con®rmation', British Journal for the Philosophy of Science, 46, pp. 399±424. Gillies, D. A. [1971]: `A Falsifying Rule for Probability Statements', British Journal for the Philosophy of Science, 22, pp. 231±61. Gillies, D. A. [1973]: An Objective Theory of Probability, London: Methuen. Gillies, D. A. [forthcoming]: `Bayesianism and the Fixity of the Theoretical Framework', in D. Cor®eld and J. Williamson (eds), Foundations of Bayesianism, Kluwer. Hacking, I. [1965]: Logic of Statistical Inference, Cambridge: Cambridge University Press. Mayo, D. G. [1996]: Error and the Growth of Experimental Knowledge, Chicago and London: University of Chicago Press. Mayo, D. G., and Spanos, A. [2000]: `A Post-Data Interpretation of Neyman±Pearson Methods Based on a Conception of Severe Testing', unpublished, Blacksburgh, VA: Virginia Polytechnic Institute and State University. Musgrave, A. [1974]: `Logical versus Historical Theories of Con®rmation', British Journal for the Philosophy of Science, 25, pp. 1±23. Neyman, J. [1952]: Lectures and Conferences on Mathematical Statistics, Washington: Graduate School of the U.S. Department of Agriculture. Neyman, J. [1965]: `Behavioristic Points of View on Mathematical Statistics', in (no ed.), 1965, On Political Economy and EconometricsÐEssays in Honor of Oskar Lange, Oxford: Pergamon Press, pp. 445±62. Neyman, J., and Pearson, E. S. [1930]: `On the Problem of Two Samples', in J. Neyman and E. S. Pearson, 1967, Joint Statistical Papers, Cambridge: Cambridge University Press, pp. 99±115. Neyman, J., and Pearson, E. S. [1933]: `On the Problem of the Most Ecient Tests of Statistical Hypotheses', in J. Neyman and E. S. Pearson, 1967, Joint Statistical Papers, Cambridge: Cambridge University Press, pp. 140±85. Popper, K. R. [1984]: Die Logik der Forschung, 8th impr. and enh. edn, TuÈbingen: Mohr (Siebeck). Redhead, M. L. G. [1974]: `On Neyman's Paradox and the Theory of Statistical Tests', British Journal for the Philosophy of Science, 25, pp. 265±71. Spielman, S. [1974]: `The Logic of Tests of Signi®cance', Philosophy of Science, 41, pp. 211±26.

Resolving Neyman's Paradox Max Albert ABSTRACT According to Fisher, a hypothesis specifying a density function for X is falsi®ed (at the level of signi®cance a) if the realization of X is in the size-a region of lowest densities. However, non-linear transformations of X can map low-density into high-density regions. Apparently, then, falsi®cations can always be turned into corroborations (and vice versa) by looking at suitable transformations of X (Neyman's Paradox). The present paper shows that, contrary to the view taken in the literature, this provides no argument against a theory of statistical falsi®cation.

1 2 3 4

The problems of statistical falsi®cation Redhead's version of Neyman's Paradox A counterargument Dierent measurement processes

1 The problems of statistical falsi®cation From a theoretical perspective, statistical inference is a serious problem for falsi®cationism. Actual tests of scienti®c hypotheses in most cases involve statistical arguments. Falsi®cationism is therefore plausible only if it can deal with statistical hypotheses. However, there is no falsi®cationist theory of statistical inference that is accepted even by falsi®cationists.1 Except for the trivial case where a zero-probability event is observed, a falsi®cation of a statistical hypothesis presupposes the choice of a rejection region, that is, an event that is possible under the hypothesis in question but the observation of which is nevertheless taken as a falsi®cation.2 Following Popper ([1984]) and Gillies ([1971], [1973]), the rejection region is to be 1

2

Forster ([1995], p. 402) explains the attraction of Bayesianism in philosophy of science by the fact that the logic-based accounts of scienti®c inference (which include falsi®cationism) have nothing to say about statistical problems. Nevertheless, statistical practice is often quite falsi®cationist in spirit; cf. Gillies ([forthcoming]) on Neyman's use of the w2 test. This causes problems to non-falsi®cationists, who have to argue that standard and successful practices are in fact illegitimate. `Rejection' and `falsi®cation' are used as synonyms throughout, although this is not always in accordance with the use of the term `rejection' in the statistical literature.

& British Society for the Philosophy of Science 2002

70

Max Albert

chosen according to a methodological rule, a falsifying rule for statistical inference (FRSI). According to Neyman and Pearson's ([1933]) theory of statistical inference (NPT), such a rule should take two kinds of errors into account, usually called error of the ®rst and error of the second kind. A ®rst-kind error is an erroneous falsi®cation of a true hypothesis, while a second-kind error is the failure to falsify a false hypothesis (erroneous corroboration). The risk of a ®rst-kind error can be controlled by choosing a rejection region that has a suciently small probability a, also called level of signi®cance, under the hypothesis. The risk of a second-kind error has to be controlled by considering at least one explicit alternative to the hypothesis under test. The optimal test in the case of only two alternatives minimizes the probability b of not falsifying the original hypothesis if the alternative is true. Equivalently, it maximizes the power 1 ÿ b. The NPT is acceptable to falsi®cationists if and only if the complete set of alternative hypotheses considered is part of the background knowledge, i.e. can be taken for granted in the context of the inquiry. If, for example, it is known that a certain procedure guarantees random sampling from a ®nite population (in the sense of every element of the population having the same probability to be included in the sample), then there is a ®xed set of hypotheses concerning the population average of any variable, and these hypotheses give rise to a ®xed set of statistical hypotheses. The randomsampling hypothesis is part of the background knowledge, and the set of hypotheses in which we can use an NP test is determined by the logical consequences of this basic hypothesis together with certain non-statistical hypotheses about the population. In such cases, a falsi®cationist can agree with Neyman ([1965], p. 448) that there is no dierence between estimation (choosing the in some sense best hypothesis from a given set) and testing.3 When, however, the question arises (as it must at some stage) as to whether the basic hypothesis in the background is true, the situation is dierent. One may be able to ®nd some test based on a still more general basic hypothesis, but if the hypothesis under scrutiny becomes more and more general, the NPT quickly runs out of tests. In practice, therefore, some assumptions are tested with the help of tests that are quite general but questionable from the standpoint of the NPT, like the w2 test. Thus, a falsi®cationist interpretation of the NPT must be based on some other, more fundamental approach answering the question of how to assess, 3

For these cases, the interpretation and extension of the NPT by Mayo ([1996]) ®ts into a falsi®cationist framework. See also Mayo and Spanos ([2000]) for further progress in rendering the idea of the severity of a test more precise. Note that the falsi®cationist idea of severity of tests relies on background knowledge that can be (or already has been) tested independently; cf. Musgrave's ([1974]) discussion of Hempel's raven paradox.

Resolving Neyman's Paradox

71

as econometricians are wont to say, the assumptions of the statistical model. At this level, at least, tests should be independent from alternative hypotheses (Gillies [1971], [1973], [forthcoming]; Albert [1992]). The obvious candidate for a theory of statistical falsi®cation, Gillies' ([1971], [1973]) theory, is open to three important objections that have already been raised against Fisher's theory of signi®cance testing (cf. Fisher [1990], Spielman [1974]). Consider a statistical hypothesis stating a distribution function for a one-dimensional random variable (rv). Let us assume that we have an FRSI specifying, without recourse to alternative hypotheses, a rejection region for a single observation of the rv. Then the following three problems must be solved.4 1. Selection of a test statistic. It is not obvious how to extend the FRSI to the case of n > 1 observations. The solution is to select a one-dimensional test statistic; however, there are many candidates. 2. Optional stopping. The rule determining the number of observations, the stopping rule, has a potential in¯uence on the level of signi®cance if the decision to stop is not independent from the observations (optional stopping). For any experiment, there are always several interpretations yielding dierent signi®cance levels: the conventional interpretation that assumes stopping to be independent from observations, and many alternative interpretations specifying dierent ways of optional stopping. If one concludes that optional stopping makes a dierence, the experimenter's intentions matter for the evaluation of otherwise indistinguishable observations. 3. Neyman's Paradox. Assume that the rv is continuously distributed with a density. Fisher's and Gillies' theories select the lowest-density regions as rejection regions. Non-linear transformations of the rv can map low-density into high-density regions and vice versa, leading to dierent decisions on the basis of the same FRSI and the same observations. The NPT solves the ®rst and the third problem by appealing to alternative hypotheses. As already argued, this is not always an option for falsi®cationists. The second problem also arises for the NPT. However, these problems pose no insurmountable obstacles for a theory of statistical falsi®cation. Just adopting some test by convention (e.g. the w2 4

The problem of selecting a test statistic originally motivated Neyman and Pearson ([1930], [1933]) to modify Fisher's theory; see also Hacking ([1965], pp. 75±81). On optional stopping, see Hacking ([1965], pp. 107±9) and Berger and Berry ([1988]). Neyman's Paradox, originally formulated in connection with the t test, goes back to a 1929 paper of Neyman; cf. Neyman and Pearson ([1930], p. 101n) and Neyman ([1952], pp. 45±51). The general formulation is due to Redhead ([1974]).

72

Max Albert

goodness-of-®t test for discrete hypotheses) goes far in solving the ®rst problem.5 Mayo ([1996]) argues that one actually should worry about optional stopping, thus attacking the position of adherents of the likelihood principle (Bayesians and others) that the insensitivity of likelihood-based inferences to optional stopping is a point in their favor. The present note argues that Neyman's Paradox does not arise if one accepts a reasonable condition concerning the scope of an FRSI.

2 Redhead's version of Neyman's Paradox The very general nature of Neyman's Paradox has been argued most forcefully by Redhead ([1974]) in his critique of Gillies' rule. Consider the hypothesis X N, meaning that the rv X is distributed according to the cumulative distribution function N of the standard normal distribution, and restrict the analysis to the case of a single observation. The so-called Gauss test at a level of signi®cance 0.05 then requires rejection if jxj51:96. This test is accepted by Fisher and Gillies, at least in principle. Consider the rv Y t X, where ÿ1 3 def N 2 ÿ N X , X 4 0 t X Nÿ1 12 ÿ N X , X 5 0 and where therefore Y N.6 The same Gauss test can be applied to Y as well. If X is in the rejection region, Y is not, and vice versa. Thus one can always reformulate the hypothesis in such a way that it is not rejected by mathematically the same test. Redhead's argument is completely general. Transformations that map the tails of the original distribution to the centre of the new distribution can always be found. Moreover, Redhead's transformation guarantees for any unimodal distribution with mode 0 that X and t X are distributed identically.

3 A Counterargument The present paper argues that Neyman's Paradox, taken as a general criticism of a theory of statistical falsi®cation, misses its target. An FRSI has only one purpose, namely to render statistical hypotheses falsi®able. Thus, I suggest that a theory of statistical falsi®cation is based on the following adequacy condition. 5

6

Albert ([1992]) proposes an FRSI for discrete rvs similar in spirit to Gillies' rule but covering the test-selection problem. This rule selects the multinomial goodness-of-®t test, which can be approximated by the w2 test under certain conditions. On continuous distributions, see below. t 0 is not de®ned. This is not important, however, since distributions diering only w.r.t. zeroprobability events are taken to be equivalent.

Resolving Neyman's Paradox

73

Scope of an FRSI. An FRSI only applies to hypotheses that can immediately be confronted with data (low-level hypotheses). Real-world measurement processes have ®nite precision. For this reason, continuous distributions, as far as they are part of the low-level hypotheses under test, are approximations. This means that Neyman's Paradox does not arise since it depends on the assumption that real variables can be observed with in®nite precision. The adequacy condition destroys the basis for the paradox. Let us make this clear by reference to an example. Since we consider just Neyman's Paradox and not the other objections against statistical falsi®cations, assume that there is an FRSI selecting a test for discrete rvs. Assume that the theory from which the low-level hypothesis under test is derived postulates a continuous distribution. For example, one might argue that the height of humans at age 25 (X) is a continuous rv. We want to test the hypothesis that this rv obeys a continuous distribution with density f. Since we focus on Neyman's Paradox, assume that we test the hypothesis on the basis of a single observation; the argument is completely analogous if we consider, for example, the mean value of a sample. The actual process of measurement yields intervals on a speci®c scale, and these intervals have a certain probability according to the hypothesis. We are back to a test of a discrete hypothesis where, by assumption, the FRSI unambiguously picks a rejection region.

4 Dierent measurement processes In a presentation of this argument, the following point was raised. It is physically possible to measure Y a=X instead of X (where a is some constant). Let us make this clear by an example. Assume that for some obscure reason a person's height is measured in the following way. The person stands upright against a wall. A sliding bar that projects 20 cm from the wall at a right angle is moved down until it touches the top of the person's head. The tip of the bar (point A), the point exactly below it on the ground (point B), and a third point C on the ground 320 cm from point B form a right-angled triangle (see Figure 1). The length of AB is the height of the person, the length of BC is known to be 300 cm. From A to C, a string is stretched. We mark the point D where the height of the string is exactly 100 cm; the distance of D from C is the rv Y. We have Y a=X where a 30,000. With Y, we have a transformed variable with density f a=Y. Actual measurements again yield intervals on a certain scale, resulting in a dierent discrete hypothesis that can be tested by using the FRSI. It is possible that the

74

Max Albert

Figure 1. Measuring Y a=X, where X is the height of a person.

same observation gives rise to a falsi®cation if X is measured, while a corroboration occurs if Y is measured. At ®rst glance, then, it seems that Neyman's Paradox again raises its ugly head. However, a closer examination shows that this is not true. While there is one observation, there are two dierent low-level hypotheses since two dierent measurement processes are used. These hypotheses dier not only w.r.t. the rv and the density function, but also w.r.t. the if-clause describing the experimental set-up. It is even possible that one of the hypotheses is true and the other is false since they are necessarily derived from dierent auxiliary assumptions. For instance, the more complicated process of observing Y may introduce errors of measurement that are absent when X is observed. The hypothesis that the density of Y is f a=Y is derived under the implicit assumption that no such measurement errors are introduced by observing Y instead of X, an assumption which might be false. But let us grant the truth of all auxiliary assumptions. The situation is nevertheless dierent from Neyman's Paradox. In the case of Neyman's Paradox, we assume that measurements of X and Y result in real values. We then have to decide how to interpret these results. We do not gain anything by combining both measurements: we can compute one from the other. With ®nite precision, the situation is quite dierent. If we can subject the same person to both measurement processes, we get a more precise result in

Resolving Neyman's Paradox

75

all those cases where the observed X interval and the observed Y interval are not mapped onto each other by taking inverse values. Assume that, in our example, the realization of X is found to be in the interval 110, 111. Measuring Y with the same precision, we ®nd an interval 270, 271. Combining the information, we learn that the true height is in the interval 110.70, 111. Hence, we can test a discrete hypothesis that gives the probabilities for the smaller intervals for the height of a person ascertainable by combining both measurements. Thus, the cases where the results of the tests dier are just the cases where we get more precise results from using both methods of measurement. The combined measurement obviously allows for a more severe test than any of the single measurements. Of course, if we have to decide between the two experiments because only one of them is feasible, it may depend on our decision whether we get a falsi®cation or a corroboration. But this is only reasonable. The combined experiment delivers maximum precision. If there are several second-best methods of measurement, it is plausible that there are cases where these measurements taken in isolation come to dierent results. This is not dierent from testing a deterministic hypothesis. If you want to test the hypothesis that swans are white, and if you can only aord a trip either to Germany or to New Zealand, it depends on your decision whether you falsify or corroborate the hypothesis. There is only one strange feature of the statistical setting: we are better o (in that we can avoid Neyman's Paradox) because we know less (namely, intervals instead of exact values). But since no problem results from this, we should be grateful. The working properties of the tests are not aected. There is no ambiguity. Thus, Neyman's Paradox does not exist after all if one accepts the very reasonable adequacy condition for the scope of an FRSI. It comes up only in thought experiments where one confronts hypotheses with actually unobservable events.

Acknowledgements For useful hints and discussions, I am grateful to Donald Gillies and the participants, especially Joseph Berkovitz, Jeremy Butter®eld and Michael Redhead, of a Cambridge University S Club session. Department of Economics University of Koblenz±Landau August-Croissant-Str. 5 D-76829 Landau, Germany [email protected]

76

Max Albert

References Albert, M. [1992]: `Die Falsi®kation statistischer Hypothesen', Journal for General Philosophy of Science, 23, pp. 1±32. Berger, J. O. and Berry, D. A. [1988]: `The Relevance of Stopping Rules in Statistical Inference (with discussion)', in S. S. Gupta and J. O. Berger (eds), 1988, Statistical Decision Theory and Related Topics IV, Vol. I, New York: Springer, pp. 29±72. Fisher, R. A. [1990]: Statistical Methods, Experimental Design and Scienti®c Inference, Oxford: Oxford University Press. Forster, M. R. [1995]: `Bayes and Bust: Simplicity as a Problem for a Probabilist's Approach to Con®rmation', British Journal for the Philosophy of Science, 46, pp. 399±424. Gillies, D. A. [1971]: `A Falsifying Rule for Probability Statements', British Journal for the Philosophy of Science, 22, pp. 231±61. Gillies, D. A. [1973]: An Objective Theory of Probability, London: Methuen. Gillies, D. A. [forthcoming]: `Bayesianism and the Fixity of the Theoretical Framework', in D. Cor®eld and J. Williamson (eds), Foundations of Bayesianism, Kluwer. Hacking, I. [1965]: Logic of Statistical Inference, Cambridge: Cambridge University Press. Mayo, D. G. [1996]: Error and the Growth of Experimental Knowledge, Chicago and London: University of Chicago Press. Mayo, D. G., and Spanos, A. [2000]: `A Post-Data Interpretation of Neyman±Pearson Methods Based on a Conception of Severe Testing', unpublished, Blacksburgh, VA: Virginia Polytechnic Institute and State University. Musgrave, A. [1974]: `Logical versus Historical Theories of Con®rmation', British Journal for the Philosophy of Science, 25, pp. 1±23. Neyman, J. [1952]: Lectures and Conferences on Mathematical Statistics, Washington: Graduate School of the U.S. Department of Agriculture. Neyman, J. [1965]: `Behavioristic Points of View on Mathematical Statistics', in (no ed.), 1965, On Political Economy and EconometricsÐEssays in Honor of Oskar Lange, Oxford: Pergamon Press, pp. 445±62. Neyman, J., and Pearson, E. S. [1930]: `On the Problem of Two Samples', in J. Neyman and E. S. Pearson, 1967, Joint Statistical Papers, Cambridge: Cambridge University Press, pp. 99±115. Neyman, J., and Pearson, E. S. [1933]: `On the Problem of the Most Ecient Tests of Statistical Hypotheses', in J. Neyman and E. S. Pearson, 1967, Joint Statistical Papers, Cambridge: Cambridge University Press, pp. 140±85. Popper, K. R. [1984]: Die Logik der Forschung, 8th impr. and enh. edn, TuÈbingen: Mohr (Siebeck). Redhead, M. L. G. [1974]: `On Neyman's Paradox and the Theory of Statistical Tests', British Journal for the Philosophy of Science, 25, pp. 265±71. Spielman, S. [1974]: `The Logic of Tests of Signi®cance', Philosophy of Science, 41, pp. 211±26.