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No scientifi c worker has a fi xed level of signifi cance at which from year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas. (Ronald A. Fisher, 1956, p. 42) It is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail. One of us once had a student who ran an experiment for his thesis. Let us call him Pogo. Pogo had an experimental group and a control group and found that the means of both groups were exactly the same. He believed it would be unscientifi c to simply state this result; he was anxious to do a signifi cance test. Th e result of the test was that the two means did not diff er signifi cantly, which Pogo reported in his thesis. In 1962, Jacob Cohen reported that the experiments published in a major psychology journal had, on average, only a 50 : 50 chance of detecting a medium-sized eff ect if there was one. Th at is, the statistical power was as low as 50%. Th is result was widely cited, but did it change researchers' practice? Sedlmeier and Gigerenzer (1989) checked the studies in the same journal, 24 years later, a time period that should allow for change. Yet only 2 out of 64 researchers mentioned power, and it was never estimated. Unnoticed, the average power had decreased (researchers now used alpha adjustment, which shrinks power). Th us, if there had been an eff ect of a medium size, the researchers would have had a better chance of fi nding it by throwing a coin rather than conducting their experiments. When we checked the years 2000 to 2002, with some 220 empirical articles, we fi nally found 9 researchers who computed the power of their tests. Forty years after Cohen, there is a fi rst sign of change. Editors of major journals such as A. W. Melton (1962) made null hypothesis testing a neces-sary condition for the acceptance of papers and made small p-values the hallmark of excellent experimentation. Th e Skinnerians found themselves forced to start a new journal, the Journal of the Experimental Analysis of Behavior, to publish their kind of experiments (Skinner, 1984, p. 138). Similarly, one reason for launching the Journal of Mathematical Psychology was to escape the edi-tors' pressure to routinely perform null hypothesis testing. One of its founders, R. D. Luce (1988), called this practice a "wrongheaded view about what constituted scientifi c progress" and "mind-less hypothesis testing in lieu of doing good research: measuring eff ects, constructing substantive theories of some depth, and developing probability models and statistical procedures suited to these theories" (p. 582).
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Published in: D. Kaplan (Ed.). (2004). e Sage handbook of quantitative methodology for the social sciences (pp. 391–408).
ousand Oaks, CA: Sage.
© 2004 Sage Publications.
e Null Ritual
What You Always Wanted to Know About
Signifi cance Testing but Were Afraid to Ask
Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch
No scientifi c worker has a xed level of signifi cance at which from year to year, and in all circumstances, he
rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas.
(Ronald A. Fisher, 1956, p. 42)
It is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail.
(A. H. Maslow, 1966, pp. 15–16)
One of us once had a student who ran an experiment for his thesis. Let us call him Pogo. Pogo had
an experimental group and a control group and found that the means of both groups were exactly
the same. He believed it would be unscientifi c to simply state this result; he was anxious to do a
signifi cance test. e result of the test was that the two means did not di er signifi cantly, which
Pogo reported in his thesis.
In 1962, Jacob Cohen reported that the experiments published in a major psychology journal
had, on average, only a 50 : 50 chance of detecting a medium-sized e ect if there was one. at is,
the statistical power was as low as 50%. is result was widely cited, but did it change researchers’
practice? Sedlmeier and Gigerenzer (1989) checked the studies in the same journal, 24 years later,
a time period that should allow for change. Yet only 2 out of 64 researchers mentioned power,
and it was never estimated. Unnoticed, the average power had decreased (researchers now used
alpha adjustment, which shrinks power). us, if there had been an e ect of a medium size, the
researchers would have had a better chance of nding it by throwing a coin rather than conducting
their experiments. When we checked the years 2000 to 2002, with some 220 empirical articles, we
nally found 9 researchers who computed the power of their tests. Forty years after Cohen, there
is a fi rst sign of change.
Editors of major journals such as A. W. Melton (1962) made null hypothesis testing a neces-
sary condition for the acceptance of papers and made small p-values the hallmark of excellent
experimentation. e Skinnerians found themselves forced to start a new journal, the Journal of
the Experimental Analysis of Behavior, to publish their kind of experiments (Skinner, 1984, p. 138).
Similarly, one reason for launching the Journal of Mathematical Psychology was to escape the edi-
torspressure to routinely perform null hypothesis testing. One of its founders, R. D. Luce (1988),
called this practice a “wrongheaded view about what constituted scientifi c progress” and mind-
less hypothesis testing in lieu of doing good research: measuring e ects, constructing substantive
theories of some depth, and developing probability models and statistical procedures suited to
these theories” (p. 582).
Authors note: We are grateful to David Kaplan and Stanley Mulaik for helpful comments and to Katharina
Petrasch for her support with journal analyses.
GG_Null_2004.indd 1 12.04.2007 10:29:09 Uhr
2 The Null Ritual
e student, the researchers, and the editors had engaged in a statistical ritual rather than sta-
tistical thinking. Pogo believed that one always ought to perform a null hypothesis test, without
exception. e researchers did not notice how small their statistical power was, nor did they seem
to care: Power is not part of the null ritual that dominates experimental psychology. e essence
of the ritual is the following:
(1) Set up a statistical null hypothesis of no mean differenceor “zero correlation.” Dont specify
the predictions of your research hypothesis or of any alternative substantive hypotheses.
(2) Use 5% as a convention for rejecting the null. If significant, accept your research hypothesis.
(3) Always perform this procedure.
e null ritual has sophisticated aspects we will not cover here, such as alpha adjustment and
ANOVA procedures, but these do not change its essence. Typically, it is presented without naming
its originators, as statistics per se. Some suggest that it was authorized by the eminent statistician
Sir Ronald A. Fisher, owing to the emphasis on null hypothesis testing (not to be confused with
the null ritual) in his 1935 book. However, Fisher would have rejected all three ingredients of this
procedure. First, null does not refer to a zero mean di erence or correlation but to the hypothesis
to be nullifi ed,which could postulate a correlation of .3, for instance. Second, as the epigram
illustrates, by 1956, Fisher thought that using a routine 5% level of signifi cance indicated lack of
statistical thinking. ird, for Fisher, null hypothesis testing was the most primitive type in a hier-
archy of statistical analyses and should be used only for problems about which we have very little
knowledge or none at all (Gigerenzer et al., 1989, chap. 3). Statistics o ers a toolbox of methods,
not just a single hammer. In many (if not most) cases, descriptive statistics and exploratory data
analysis are all one needs. As we will see soon, the null ritual originated neither from Fisher nor from
any other renowned statistician and does not exist in statistics proper. It was instead fabricated in
the minds of statistical textbook writers in psychology and education.
Rituals seem to be indispensable for the self-defi nition of social groups and for transitions in
life, and there is nothing wrong about them. However, they should be the subject rather than the
procedure of social sciences. Elements of social rituals include (a) the repetition of the same action,
(b) a focus on special numbers or colors, (c) fears about serious sanctions for rule violations, and (d)
wishful thinking and delusions that virtually eliminate critical thinking (Dulaney & Fiske, 1994).
e null ritual has each of these four characteristics: a repetitive sequence, a fi xation on the 5%
level, fear of sanctions by editors or advisers, and wishful thinking about the outcome (the p-value)
combined with a lack of courage to ask questions.
Pogos counterpart in this chapter is a curious student who wants to understand the ritual rather
than mindlessly perform it. She has the courage to raise questions that seem naive at rst glance
and that others do not care or dare to ask.
Question 1: What Does a Signifi cant Result Mean?
What a simple question! Who would not know the answer? After all, psychology students spend
months sitting through statistics courses, learning about null hypothesis tests (signifi cance tests)
and their featured product, the p-value. Just to be sure, consider the following problem (Haller &
Krauss, 2002; Oakes, 1986):
Suppose you have a treatment that you suspect may alter performance on a certain task. You compare the
means of your control and experimental groups (say, 20 subjects in each sample). Furthermore, suppose you
use a simple independent means t-test and your result is signifi cant (t = 2.7, df = 18, p = .01). Please mark
each of the statements below as “true” or “false.False means that the statement does not follow logically
from the above premises. Also note that several or none of the statements may be correct.
GG_Null_2004.indd 2 12.04.2007 10:29:11 Uhr
Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch 3
(1) You have absolutely disproved the null hypothesis
(i.e., there is no difference between the population means). ! True False !
(2) You have found the probability of the null hypothesis being true. ! True False !
(3) You have absolutely proved your experimental hypothesis
(that there is a difference between the population means). ! True False !
(4) You can deduce the probability of the experimental hypothesis
being true. ! True False !
(5) You know, if you decide to reject the null hypothesis, the
probability that you are making the wrong decision. ! True False !
(6) You have a reliable experimental finding in the sense that if,
hypothetically, the experiment were repeated a great number of
times, you would obtain a significant result on 99% of occasions. ! True False !
Which statements are true? If you want to avoid the I-knew-it-all-along feeling, please answer
the six questions yourself before continuing to read. When you are done, consider what a p-value
actually is: A p-value is the probability of the observed data (or of more extreme data points), given
that the null hypothesis H
is true, defi ned in symbols as p(D |H
). is defi nition can be rephrased
in a more technical form by introducing the statistical model underlying the analysis (Gigerenzer
et al., 1989, chap. 3). Let us now see which of the six answers are correct:
Statements 1 and 3: Statement 1 is easily detected as being false. A signifi cance test can never
disprove the null hypothesis. Signifi cance tests provide probabilities, not defi nite proofs. For the
same reason, Statement 3, which implies that a signifi cant result could prove the experimental hy-
pothesis, is false. Statements 1 and 3 are instances of the illusion of certainty (Gigerenzer, 2002).
Statements 2 and 4: Recall that a p-value is a probability of data, not of a hypothesis. Despite
wishful thinking, p(D |H
) is not the same as p(H
|D), and a signifi cance test does not and cannot
provide a probability for a hypothesis. One cannot conclude from a p-value that a hypothesis has
a probability of 1 (Statements 1 and 3) or that it has any other probability (Statements 2 and 4).
erefore, Statements 2 and 4 are false. e statistical toolbox, of course, contains tools that al-
low estimating probabilities of hypotheses, such as Bayesian statistics (see below). However, null
hypothesis testing does not.
Statement 5: e “probability that you are making the wrong decision” is again a probability
of a hypothesis. is is because if one rejects the null hypothesis, the only possibility of making a
wrong decision is if the null hypothesis is true. In other words, a closer look at Statement 5 reveals
that it is about the probability that you will make the wrong decision, that is, that H
is true. us,
it makes essentially the same claim as Statement 2 does, and both are incorrect
Statement 6: Statement 6 amounts to the replication fallacy. Recall that a p-value is the prob-
ability of the observed data (or of more extreme data points), given that the null hypothesis is true.
Statement 6, however, is about the probability of signifi cant” data per se, not about the probability
of data if the null hypothesis were true. e error in Statement 6 is that p = 1% is taken to imply that
such signifi cant data would reappear in 99% of the repetitions. Statement 6 could be made only if
one knew that the null hypothesis was true. In formal terms, p(D |H
) is confused with 1 p(D).
e replication fallacy is shared by many, including the editors of top journals. For instance, the
former editor of the Journal of Experimental Psychology, A. W. Melton (1962), wrote in his edito-
rial, e level of signifi cance measures the confi dence that the results of the experiment would be
repeatable under the conditions described” (p. 553). A nice fantasy, but false.
To sum up, all six statements are incorrect. Note that all six err in the same direction of wishful
thinking: ey overestimate what one can conclude from a p-value.
GG_Null_2004.indd 3 12.04.2007 10:29:12 Uhr
4 The Null Ritual
Studentsand TeachersDelusions
We posed the question with the six multiple-choice answers to 44 students of psychology, 39 lectur-
ers and professors of psychology, and 30 statistics teachers, who included professors of psychology,
lecturers, and teaching assistants. All students had successfully passed one or more statistics courses
in which signifi cance testing was taught. Furthermore, each of the teachers confi rmed that he or
she taught null hypothesis testing. To get a quasi-representative sample, we drew the participants
from six German universities (Haller & Krauss, 2002).
How many students and teachers noticed that all of the statements were wrong? As Figure 1
shows, none of the students did. Every student endorsed one or more of the illusions about the
meaning of a p-value. One might think that these students lack the right genes for statistical
thinking and are stubbornly resistant to education. A glance at the performance of their teachers,
however, indicates that wishful thinking might not be entirely their fault. Ninety percent of the
professors and lecturers also had illusions, a proportion almost as high as among their students.
Most surprisingly, 80% of the statistics teachers shared illusions with their students. us, the
students’ errors might be a direct consequence of their teachers’ wishful thinking. Note that one
does not need to be a brilliant mathematician to answer the question, “What does a signifi cant
result mean?” One only needs to understand that a p-value is the probability of the data (or more
extreme data), given that the H
is true.
If students “inherited” the illusions from their teachers, where did the teachers acquire them?
e illusions were right there in the rst textbooks introducing psychologists to null hypothesis
testing more than 60 years ago. Guilfords Fundamental Statistics in Psychology and Education, rst
published in 1942, was probably the most widely read textbook in the 1940s and 1950s. Guilford
suggested that hypothesis testing would reveal the probability that the null hypothesis is true.
“If the result comes out one way, the hypothesis is probably correct, if it comes out another way,
the hypothesis is probably wrong.” (p. 156) Guilford’s logic was not consistently misleading but
wavered back and forth between correct and incorrect statements, as well as ambiguous ones that
can be read like Rorschach inkblots. He used phrases such as we obtained directly the probabilities
that the null hypothesis was plausible” and the probability of extreme deviations from chance”
interchangeably for referring to the same thing: the level of signifi cance. Guilford is no exception.
He marked the beginning of a genre of statistical texts that vacillate between the researchersdesire
for probabilities of hypotheses and what signifi cance testing can actually provide. Early authors
promoting the illusion that the level of signifi cance would specify the probability of hypothesis
include Anastasi (1958, p. 11), Ferguson (1959, p. 133), and Lindquist (1940, p. 14). But the
belief has persisted over decades: for instance, in Miller and Buckhout (1973; statistical appendix
by Brown, p. 523), Nunally (1975, pp. 194–196), and in the examples collected by Bakan (1966),
Pollard and Richardson (1987), Gigerenzer (1993), Nickerson (2000), and Mulaik, Raju, and
Harshman (1997).
Which of the illusions were most often endorsed, and which relatively seldom? Table 1 shows
that Statements 1 and 3 were most frequently detected as being false. ese claim certainty rather
than probability. Still, up to a third of the students and an embarrassing 10% to 15% of the
group of teachers held this illusion of certainty. Statements 4, 5, and 6 lead the hit list of the most
widespread illusions. ese errors are about equally prominent in all groups, a collective fantasy
that seems to travel by cultural transmission from teacher to student. e last column shows that
these three illusions were also prevalent among British academic psychologists who answered the
same question (Oakes, 1986). Just as in the case of statistical power cited in the introduction, in
which little learning was observed after 24 years, knowledge about what a signifi cant result means
GG_Null_2004.indd 4 12.04.2007 10:29:12 Uhr
Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch 5
does not seem to have improved since Oakes. Yet a persistent blind spot for power and a lack of
comprehension of signifi cance are consistent with the null ritual.
Statements 2 and 4, which put forward the same type of error, were given di erent endorse-
ments. When a statement concerns the probability of the experimental hypothesis, it is much more
accepted by students and teachers as a valid conclusion than one that concerns the probability of the
null hypothesis. e same pattern can be seen for British psychologists (see Table 1). Why are re-
searchers and students more likely to believe that the level of signifi cance determines the probability
of H
rather than that of H
? A possible reason is that the researchers’ focus is on the experimental
hypothesis H
and that the desire to fi nd the probability of H
drives the phenomenon.
Did the students produce more illusions than their teachers? Surprisingly, the di erence was only
slight. On average, students endorsed 2.5 illusions, their professors and lecturers who did not teach
statistics approved of 2.0 illusions, and those who taught signi cance testing endorsed 1.9 illusions.
Could it be that these collective illusions are specifi c to German psychologists and students?
No, the evidence points to a global phenomenon. As mentioned above, Oakes (1986) reported that
97% of British academic psychologists produced at least one illusion. Using a similar test question,
Falk and Greenbaum (1995) found comparable results for Israeli students, despite having taken
measures for debiasing students. Falk and Greenbaum had explicitly added the right alternative
(“None of the statements is correct”), whereas we had merely pointed out that more than one or
none of the statements might be correct. As a further measure, they had made their students read
Bakans (1966) classic article, which explicitly warns against wrong conclusions. Nevertheless,
only 13% of their participants opted for the right alternative. Falk and Greenbaum concluded that
unless strong measures in teaching statistics are taken, the chances of overcoming this misconcep-
tion appear low at present” (p. 93). Warning and reading by itself does not seem to foster much
insight. So what to do?
(n = 44)
& lecturers
not teaching
(n = 39)
& lecturers
(n = 30)
Note. The percentage refer to the participants in
each group who endorsed one or more of the six
false statements (based on Haller & Krauss, 2002).
Figure 1. e Amount of Delusions About the
Meaning of “p = .01”.
GG_Null_2004.indd 5 12.04.2007 10:29:12 Uhr
6 The Null Ritual
Question 2: How Can Students Get Rid of Illusions?
e collective illusions about the meaning of a signifi cant result are embarrassing to our profession.
is state of a airs is particularly painful because psychologists—unlike natural scientists—heav-
ily use signifi cance testing yet do not understand what its product, the p-value, means. Is there a
Yes. e cure is to open the statistical toolbox. In statistical textbooks written by psychologists
and educational researchers, signifi cance testing is typically presented as if it were an all-purpose
tool. In statistics proper, however, an entire toolbox exists, of which null hypothesis testing is only
one tool among many. As a therapy, even a small glance into the contents of the toolbox can be suf-
cient. One quick way to overcome some of the illusions is to introduce students to Bayes’ rule.
Bayes’ rule deals with the probability of hypotheses, and by introducing it alongside null hy-
pothesis testing, one can easily see what the strengths and limits of each tool are. Unfortunately,
Bayesrule is rarely mentioned in statistical textbooks for psychologists. Hays (1963) had a chapter
on Bayesian statistics in the second edition of his widely read textbook but dropped it in the sub-
sequent editions. As he explained to one of us (GG) he dropped the chapter upon pressure from his
publisher to produce a statistical cookbook that did not hint at the existence of alternative tools for
statistical inference. Furthermore, he believed that many researchers are not interested in statistical
thinking in the rst place but solely in getting their papers published (Gigerenzer, 2000).
Here is a short comparative look at two tools:
(1) Null hypothesis testing computes the probability p(D |H
). The form of conditional prob-
abilities makes it clear that with null hypothesis testing, (a) only statements concerning the
probability of data D can be obtained, and (b) the null hypothesis H
functions as the refer-
ence point for the conditional statement. In other words, any correct answer to the question of
what a significant result means must include the conditional phrase “… given H
is true or an
equivalent expression.
Table 1
Percentages of False Answers (i.e., Statements Marked as True)
in the ree Groups of Figure 1
Germany 2000
United Kingdom
Statement (abbreviated)
Professors and lec-
turers: not teaching
Professors and
teaching statistics
Professors and
1. H
is absolutely disproved
34 15 10 1
2. Probability of H
is found
32 26 17 36
3. H
is absolutely proved
20 13 10 6
4. Probability of H
is found
59 33 33 66
5. Probability of wrong decision 68 67 73 86
6. Probability of replication 41 49 37 60
Note. For comparison, the results of Oakes’ (1986) study with academic psychologists in the United Kingdom are shown
in the right column.
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Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch 7
(2) Bayes’ rule computes the probability p(H
|D). In the simple case of two hypotheses, H
, which are mutually exclusive and exhaustive, Bayesrule is the following:
|D) =
) + p(H
For instance, consider HIV screening for people who are in no known risk group (Gigerenzer,
2002). In this population, the a priori probability p(H
) of being infected by HIV is about 1 in
10,000, or .0001. e probability p(D |H
) that the test is positive (D) if the person is infected is
.999, and the probability p(D |H
) that the test is positive if the person is not infected is .0001. What
is the probability p(H
|D) that a person with a positive HIV test actually has the virus? Inserting
these values into Bayesrule results in p(H
|D) = .5. Unlike null hypothesis testing, Bayesrule can
actually provide a probability of a hypothesis.
Now let us approach the same problem with null hypothesis testing. e null is that the person
is not infected. e observation is a positive test, and the probability of a positive test given that
the null is true is p = .0001, which is the exact level of signifi cance. erefore, the null hypothesis
of no infection is rejected with high confi dence, and the alternative hypothesis that the person is
infected is accepted. However, as the Bayesian calculation showed, given a positive test, the prob-
ability of a HIV infection is only .5. HIV screening illustrates how one can reach quite di erent
conclusions with null hypothesis testing or Bayesrule. It also clarifi es some of the possibilities and
limits of both tools. e single most important limit of null hypothesis testing is that there is only
one statistical hypothesis—the null, which does not allow for comparative hypotheses testing.
Bayesrule, in contrast, compares the probabilities of the data under two (or more) hypotheses
and also uses prior probability information. Only when one knows extremely little about a topic
(so that one cannot even specify the predictions of competing hypotheses) might a null hypothesis
test be appropriate.
A student who has understood the fact that the products of null hypothesis testing and Bayes
rule are p(D |H
) and p(H
|D), respectively, will note that the Statements 1 through 5 are all about
probabilities of hypotheses and therefore cannot be answered with signifi cance testing. Statement 6,
in contrast, is about the probability of further signifi cant results, that is, about probabilities of data
rather than hypotheses. at this statement is wrong can be seen from the fact that it does not
include the conditional phrase “… if H
is true.
Note that the above two-step course does not require in-depth instruction in Bayesian statistics
(see Edwards, Lindman, & Savage, 1963; Howson & Urbach, 1989). is minimal course can
be readily extended to a few more tools, for instance, by adding Neyman-Pearson testing, which
computes the likelihood ratio p(D |H
)/p(D |H
). Psychologists know Neyman-Pearson testing in
the form of signal detection theory, a cognitive theory that has been inspired by the statistical tool
(Gigerenzer & Murray, 1987). e products of the three tools can be easily compared:
(a) p(D |H
) is obtained from null hypothesis testing.
(b) p(D |H
)/p(D |H
) is obtained from Neyman-Pearson hypotheses testing.
(c) p(H
|D) is obtained by Bayesrule.
For null hypothesis testing, only the likelihood p(D |H
) matters; for Neyman-Pearson, the likeli-
hood ratio matters; and for Bayes, the posterior probability matters. By opening the statistical
toolbox and comparing tools, one can easily understand what each tool delivers and what it does
not. For the next question, the fundamental di erence between null hypothesis testing and other
statistical tools such as Bayes’ rule and Neyman-Pearson testing is that in null hypothesis testing,
only one hypothesis—the null—is precisely stated. With this technique, one is not able to compare
GG_Null_2004.indd 7 12.04.2007 10:29:12 Uhr
8 The Null Ritual
two or more hypotheses in a symmetric or fair” way and might draw wrong conclusions from
the data.
Question 3: Can the Null Ritual Hurt?
But it’s just a little ritual. It may be a bit silly, but it cant hurt, can it? Yes, it can. Consider a study
in which the authors had two precisely formulated hypotheses, but instead of specifying the pre-
dictions of both hypotheses for their experimental design, they performed the null ritual. e
question was how young children judge the area of rectangles, and the two hypotheses were the
following: Children add height plus width, or children multiply height times width (Anderson &
Cuneo, 1978). In one experiment, 5- to 6-year-old children rated the joint area of two rectangles
(not an easy task). e reason for having them rate the area of two rectangles rather than one was
to disentangle the integration rule (adding vs. multiplying) from the response function (linear vs.
logarithmic). Su ce to say that the idea for the experiment was ingenious. e Height + Width
rule was identifi ed with the null hypothesis of no linear interaction in a two-factorial analysis of
variance. e prediction of the second hypothesis, the Height × Width rule, was never specifi ed,
as it never is with null hypothesis testing. e authors found that the curves are nearly parallel
and the interaction did not approach signifi cance, F(4, 56) = 1.20” (p. 352). ey concluded that
this and similar results would support the Height + Width rule and disconfi rm the multiplying
rule. In Andersons (1981) words, “Five-year-olds judge area of rectangles by an adding, Height +
Width rule” (p. 33).
Testing a null, however, is a weak argument if one has some ideas about the subject matter, as
Anderson and Cuneo (1978) did. So let us derive the actual predictions of both of their hypotheses
for their experimental design (for details, see Gigerenzer & Murray, 1987). Figure 2 shows the
prediction of the Height + Width rule and that of the Height × Width rule. ere were eight pairs
of rectangles, shown by the two curves. Note that the middle segment (the parallel lines) does not
di erentiate between the two hypotheses, as the left and the right segments do. us, only these
two segments are relevant. Here, the Height + Width rule predicts parallel curves, whereas the
Height × Width rule predicts converging curves (from left to right). One can see that the data (top
panel) actually show the pattern predicted by the multiplying rule and that the curves converge
even more than predicted. If either of the two hypotheses is supported by the data, then it is the
multiplying rule (this was supported by subsequent experimental research in which the predictions
of half a dozen hypotheses were tested; see Gigerenzer & Richter, 1990). Nevertheless, the null ritual
misled the researchers into concluding that the data would support the Height + Width rule.
Why was the considerable deviation from the prediction of the Height + Width rule not statis-
tically signifi cant? One reason was the large amount of error in the data: Asking young children to
rate the joint area of two rectangles produced highly unreliable responses. is contributed to the
low power of the statistical tests, which was consistently below 10% (Gigerenzer & Richter, 1990)!
at is, the experiments were set up so that the chance of accepting the Height × Width rule if it
is true was less than 1 in 10.
But doesnt the alternative hypothesis always predict a signifi cant result? As Figure 2 illustrates,
this is not the case. Even if the data had coincided exactly with the prediction of the multiplying
rule, the result would not have been signifi cant (because the even larger deviation of the actual data
was not signifi cant either). In general, a hypothesis predicts a value or a curve but not signifi cance
or nonsignifi cance. e latter is the joint product of several factors that have little to do with the
hypothesis, including the number of participants, the error in the data, and the statistical power.
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Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch 9
is example is not meant as a critique of specifi c authors but as an illustration of how routine
null hypothesis testing can hurt. It teaches two aspects of statistical thinking that are alien to the
null ritual. First, it is important to specify the predictions of more than one hypothesis. In the pres-
ent case, descriptive statistics and mere eyeballing would have been better than the null ritual and
analysis of variance. Second, good statistical thinking is concerned with minimizing the real error
in the data, and this is more important than a small p-value. In the present case, a small error can
be achieved by asking children for paired comparisons—which of two rectangles (chocolate bars)
is larger? Unlike ratings, comparative judgments generate highly reliable responses, clear individual
di erences, and allow researchers to test hypotheses that cannot be easily expressed in the main-
e ect plus interactionlanguage of analysis of variance (Gigerenzer & Richter, 1990).
Question 4: Is the Level of Signifi cance the Same ing as Alpha?
Let us introduce Dr. Publish-Perish. He is the average researcher, a devoted consumer of statistical
methods. His superego tells him that he ought to set the level of signifi cance before an experiment
is performed. A level of 1% would be impressive, wouldnt it? Yes, but … there is a dilemma. He
fears that the p-value calculated from the data could turn out slightly higher, such as 1.1%, and he
would then have to report a nonsignifi cant result. He does not want to take that risk. en there
is the option of setting the level at a less impressive 5%. But what if the p-value turned out to be
smaller than 1% or even .1%? en he would regret his decision deeply because he would have to
Size of second
Children’s rating of joint area
10 × 8
6 × 5
10 × 8
6 × 5
10 × 8
6 × 5
Actual data
Height × Width
Height + Width
7 × 7
7 × 11 11 × 7 11 × 11
curve segment
Size of first rectangle
Note. Anderson and Cuneo (1978) asked which of
two hypotheses, Height + Width or Height × Width,
describes young children’s judgments of the joint
area of rectangle pairs. Following null hypothesis
testing, they identified the Height + Width rule with
nonsignificance of the linear interaction in an analysis
of variance and the Height × Width rule with a
significant interaction. The result was not significant;
the Height × Width rule was rejected and the
Height + Width rule accepted. When one inste
specifies the predictions of both hypotheses
(Gigerenzer & Murray, 1987), the Height + Width rule
predicts the parallel curves, and the Height × Width
rule predicts the converging curves. One can see
that the data are actually closer to the pattern predicted
by the Height × Width rule (see text).
Figure 2. How to Draw the Wrong Conclusions
by Using Null Hypothesis Testing.
GG_Null_2004.indd 9 12.04.2007 10:29:13 Uhr
10 The Null Ritual
report this result as p < .05. He does not like that either. So he thinks the only choice left is to cheat
a little and disobey his superego. He waits until he has seen the data, rounds the p-value up to the
next conventional level, and reports that the result is signifi cant at p < .001, .01, or .05, whatever is
next. at smells of deception, and his superego leaves him with feelings of guilt. But what should
he do when everyone else seems to play this little cheating game?
Dr. Publish-Perish does not know that his moral dilemma is caused by a mere confusion, a
product of textbook writers who failed to distinguish the three main interpretations of the level of
signifi cance and mixed them all up.
Interpretation 1: Mere Convention
So far, we have mentioned only in passing the statisticians who have created and shaped the ideas
we are talking about. Similarly, most statistical textbooks for psychology and education are generally
mute about these eminent people and their ideas, which is remarkable for a eld where authors are
cited compulsively, and no shortage of competing theories exists.
e rst person to introduce is Sir Ronald A. Fisher (1890–1962), one of the most infl uen-
tial statisticians ever, who also made rst-rate contributions to genetics and was knighted for his
achievements. Fisher spent most of his career at University College, London, where he held the
chair of eugenics. His publications include three books on statistics. For psychology, the most
infl uential of these was the second one, The Design of Experiments, rst published in 1935. In the
Design, Fisher suggested that we think of the level of signifi cance as a convention: “It is usual and
convenient for experimenters to take 5 per cent as a standard level of signifi cance, in the sense that
they are prepared to ignore all results which fail to reach this standard” (p. 13). Fisher’s assertion
that 5% (in some cases, 1%) is a convention to be adopted by all experimenters and in all experi-
ments, whereas nonsignifi cant results are to be ignored, became part of the null ritual. For instance,
the 1974 Publication Manual of the American Psychological Association instructed experimenters to
make mechanical decisions using a conventional level of signifi cance:
Caution: Do not infer trends from data that fail by a small margin to meet the usual levels of signifi cance.
Such results are best interpreted as caused by chance and are best reported as such. Treat the result section
like an income tax return. Take what’s coming to you, but no more. (p. 19; this passage was deleted in the
3rd edition [American Psychological Association, 1983])
In a recent defense of what he calls NHSTP (null hypothesis signifi cance testing procedure), Chow
(1998) still proclaims that null hypothesis tests should be interpreted mechanically, using the con-
ventional 5% level of signifi cance. is view reminds us of a maxim regarding the critical ratio, the
predecessor of the signifi cance level: “A critical ratio of three, or no Ph.D.
Interpretation 2: Alpha
e second eminent person we would like to introduce is the Polish mathematician Jerzy Neyman,
who worked with Egon S. Pearson (the son of Karl Pearson) at University College in London and
later, when the tensions between Fisher and himself grew too heated, moved to Berkeley, California.
Neyman and Pearson criticized Fisher’s null hypothesis testing for several reasons, including that
no alternative hypothesis is specifi ed, which in turn does not allow computation of the probability
! of wrongly rejecting the alternative hypothesis (Type II error) or of the power of the test (1 – !)
(Gigerenzer et al., 1989, chap. 3). In Neyman-Pearson theory, the meaning of a level of signifi cance
GG_Null_2004.indd 10 12.04.2007 10:29:13 Uhr
Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch 11
such as 3% is the following: If the hypothesis H
is correct, and the experiment is repeated many
times, the experimenter will wrongly reject H
in 3% of the cases. Rejecting the hypothesis H
if it
is correct is called a Type I error, and the probability of rejecting H
if it is correct is called alpha (").
Neyman and Pearson insisted that one must specify the level of signifi cance before the experiment
to be able to interpret it as ". e same holds for !, which is the rate of rejecting the alternative
hypothesis H
if it is correct (Type II error). Here we get the second classical interpretation of the
level of signifi cance: the error rate ", which is determined before the experiment, albeit not by
mere convention but by cost-benefi t calculations that strike a balance between ", !, and sample
size n (Cohen, 1994).
Interpretation 3: e Exact Level of Signifi cance
Fisher had second thoughts about his proposal of a conventional level and stated these most clearly
in the mid-1950s. In his last book, Statistical Methods and Scientifi c Inference (1956, p. 42), Fisher
rejected the use of a conventional level of signifi cance and ridiculed this practice as absurdly aca-
demic” (see epigram). Fishers primary target, however, was the interpretation of the level of signifi -
cance as ", which he rejected as unscientifi c. In science, Fisher argued, unlike in industrial quality
control, one does not repeat the same experiment again and again, as is assumed in Neyman and
Pearsons interpretation of the level of signifi cance as an error rate in the long run. What researchers
should do instead, according to Fisher’s second thoughts, is publish the exact level of signifi cance,
say, p = .02 (not p < .05), and communicate this result to their fellow researchers.
us, the phrase level of signifi cance has three meanings:
(1) the conventional level of significance, a common standard for all researchers (early Fisher);
(2) the " level, that is, the relative frequency of wrongly rejecting a hypothesis in the long run if it is
true, to be decided jointly with ! and the sample size before the experiment and independently
of the data (Neyman & Pearson);
(3) the exact level of significance, calculated from the data after the experiment (late Fisher).
e basic di erence is this: For Fisher, the exact level of signifi cance is a property of the data, that
is, a relation between a body of data and a theory; for Neyman and Pearson, " is a property of the
test, not of the data. Level of signifi cance and " are not the same thing. e practical consequences
are straightforward:
(1) Conventional level: You specify only one statistical hypothesis, the null. You always use the
5% level and report whether the result is signifi cant or not; that is, you report p < .05 or p > .05,
just like in the null ritual. If the result is signifi cant, you reject the null; otherwise, you do not draw
any conclusion. ere is no way to confi rm the null hypothesis. e decision is asymmetric.
(2) Alpha level: You specify two statistical hypotheses, H
and H
, to be able to calculate the
desired balance between ", !, and the sample size n. If the result is signifi cant (i.e., if it falls within
the alpha region), the decision is to reject H
and to act as if H
were true; otherwise, the decision
is to reject H
and to act as if H
were true. (We ignore here, for simplicity, the option of a region
of indecision.) For instance, if " = ! = .10, then it does not matter whether the exact level of sig-
nifi cance is .06 or .001. e level of signifi cance has no infl uence on ". Unlike in null hypothesis
testing with a conventional level, the decision is symmetric.
(3) Exact level of signifi cance: You calculate the exact level of signifi cance from the data. You
report, say, p = .051 or p = .048. You do not use statements of the type p < .05” but report the
exact (or rounded) value. ere is no decision involved. You communicate information; you do
not make yes-no decisions.
GG_Null_2004.indd 11 12.04.2007 10:29:13 Uhr
12 The Null Ritual
ese three interpretations of the level of signifi cance are confl ated in most textbooks used in
psychology and education. is confusion is a direct consequence of the sour fact that these text-
books do not teach the toolbox and competing statistical theories but instead only one apparently
monolithic form of “statistics”—a mishmash that does not exist in statistics proper (Gigerenzer,
1993, 2000).
Now let us go back to Dr. Publish-Perish and his moral confl ict. His superego demands that
he specify the level of signifi cance before the experiment. We now understand that this doctrine is
part of the Neyman-Pearson theory. His ego personifi es Fisher’s theory of calculating the exact level
of signifi cance from the data but is confl ated with Fishers earlier idea of making a yes-no decision
based on a conventional level of signifi cance. e confl ict between his superego and his ego is the
source of his guilt feelings, but he does not know that. Never having heard that there are di erent
theories, he has a vague feeling of shame for doing something wrong. Dr. Publish-Perish does not
follow any of the three di erent conceptions. Unknowingly, he tries to satisfy all of them and ends
up presenting an exact level of signifi cance as if it were an alpha level, yet rst rounding it up to one
of the conventional levels of signifi cance, p < .05, p < .01, or p < .001. e result is not ", nor an
exact level of signifi cance, nor a conventional level. It is an emotional and intellectual confusion.
Question 5: What Emotional Structure Sustains the Null Ritual?
Dr. Publish-Perish is likely to share some of the illusions demonstrated in the fi rst section. Recall
that most of these illusions involve the confusion of the level of signifi cance with the probability
of a hypothesis. Yet every person of average intelligence can understand the di erence between
p(D | H) and p(H | D), suggesting that the issue is not an intellectual but a social and emotional one.
Following Gigerenzer (1993; see also Acree, 1978), we will continue to use the Freudian language
of unconscious confl icts as an analogy to analyze why intelligent people surrender to statistical
rituals rather than engage in statistical thinking.
e Neyman-Pearson theory serves as the superego of Dr. Publish-Perishs statistical thinking,
demanding in advance the specifi cation of precise alternative hypotheses, signifi cance levels, and
power to calculate the sample size necessary, as well as teaching the doctrine of repeated random
sampling (Neyman, 1950, 1957). Moreover, the frequentist superego forbids the interpretation
of levels of signifi cance as the degree of confi dence that a particular hypothesis is true or false.
Hypothesis testing, in its view, is about decision making (i.e., acting as if a hypothesis were true or
false) but not about epistemic statements (i.e., believing in a hypothesis).
e Fisherian theory of signifi cance testing functions as the ego. e ego gets things done in the
laboratory and papers published. e ego determines the level of signifi cance after the experiment,
and it does not specify power or calculate the sample size necessary. e ego avoids precise predic-
tions from its research hypothesis and instead claims support for it by rejecting a null hypothesis.
e ego makes abundant epistemic statements about particular results and hypotheses. But it is
left with feelings of guilt and shame for having violated the rules.
e Bayesian posterior probabilities form the id of this hybrid logic. ese probabilities of
hypotheses are censored by both the frequentist superego and the pragmatic ego. However, they
are exactly what the Bayesian id wants, and it gets its way by wishful thinking and blocking the
intellect from understanding what a level of signifi cance really is.
e Freudian analogy (see Figure 3) illustrates the unconscious confl icts in the minds of the
average student, researcher, and editor and provides a way to understanding why many psycholo-
gists cling to null hypothesis testing like a ritual and why they do not seem to want to understand
GG_Null_2004.indd 12 12.04.2007 10:29:13 Uhr
Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch 13
what they easily could. e analogy brings the anxiety and guilt, the compulsive behavior, and
the intellectual blindness associated with the hybrid logic into the foreground. It is as if the raging
personal and intellectual confl icts between Fisher and Neyman and Pearson, as well as between
these frequentists and the Bayesians, were projected into an “intra-psychic” confl ict in the minds
of researchers. In Freudian theory, ritual is a way of resolving unconscious confl ict.
Textbook writers, in turn, have tried to resolve the conscious confl ict between statisticians
by collective silence. You will rarely nd a textbook for psychologists that points out even a few
issues in the heated debate about what is good hypotheses testing, which is covered in detail in
Gigerenzer et al. (1989, chaps. 3, 6). e textbook method of denial includes omitting the names
of the parents of the various ideas—that is, Fisher, Neyman, and Pearson—except in connection
with trivialities such as an acknowledgment for permission to reproduce tables. One of the few
exceptions is Hays (1963), who mentioned in one sentence in the second edition that statistical
theory made cumulative progress from Fisher to Neyman and Pearson, although he did not hint at
their di ering ideas or confl icts. In the third edition, however, this sentence was deleted, and Hays
fell back to common standards. When one of us (GG) asked him why he deleted this sentence, he
gave the same reason as for having removed the chapter on Bayesian statistics: e publisher wanted
a single-recipe cookbook, not names of statisticians whose theories might confl ict. e fear seems
to be that a statistical toolbox would not sell as well as one truth or one hammer.
Many textbook writers in psychology continue to spread confusion about statistical theories,
even after they have learned otherwise. For instance, in response to Gigerenzer (1993), Chow
(1998) acknowledges that di erent logics of statistical inference exist. But a few lines later, he falls
back into the “its-all-the-same” fable when he asserts, “To K. Pearson, R. Fisher, J. Neyman, and
E. S. Pearson, NHSTP was what the empirical research was all about” (p. xi). Calling the heroes
of the past to justify the null ritual (to which NHSTP seems to amount) is bewildering. Each of
these statisticians would have rejected NHSTP. Neyman and Pearson spent their careers arguing
against null hypothesis testing, against a magical 5% level, and for the concept of Type II error
(which Chow declares not germane to NHSTP). Chow’s confusion is not an exception. NHSTP is
the symptom of the unconscious confl ict illustrated in Figure 3. Laying open the confl icts between
major approaches rather than denying them would be a rst step to understanding the underlying
issues, a prerequisite for statistical thinking.
The Unconscious Conflict
Two or more hypotheses; alpha and beta determined
before the experiment; compute sample size; no
statements about the truth of hypotheses …
Null hypothesis only; significance level computed
after the experiment; beta ignored; sample size by
rule of thumb; gets papers published but left with
feelings of guilt
Desire for probabilities of hypotheses
Figure 3. A Freudian Analogy for the
Unconscious Confl icts in the Minds of
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14 The Null Ritual
Question 6: Who Keeps Psychologists Performing the Null Ritual?
Ask graduate students, and they likely point to their advisers. e students do not want problems
with their thesis. When we meet them again as post-docs, the answer is that they need a job. After
getting their rst job, they still feel restricted because there is a tenure decision in a couple of years.
When they are safe as associate or full professors, it is still not their fault because they believe the
editors of the major journals will not publish their papers without the null ritual. ere is always
someone else to blame, rather than one’s own lack of having the courage to know. But fears about
punishment for rule violations are not entirely unfounded. For instance, Melton (1962) insisted
on the null ritual and also made it clear in his editorial that he wants to see p < .01, not just p <
.05. e reasons he gave were two of the illusions listed in Question 1. He misleadingly asserted
that the lower the p-value, the higher the confi dence that the alternative hypothesis is true and the
higher the probability that a replication will fi nd a signifi cant result. Nothing beyond p-values is
mentioned in the editorial: Precise hypotheses, good descriptive statistics, confi dence intervals,
e ect sizes, and power do not appear in his statement about good research. us, the null ritual
seems to be enforced by editors.
e story of a recent editor, however, reveals that the truth is not as simple as that. In his
“On the Tyranny of Hypothesis Testing in the Social Sciences,” Geo rey Loftus (1991) reviewed
The Empire of Chance (Gigerenzer et al., 1989), which presented one of the fi rst analyses of how
psychologists mishmashed ideas of Fisher and also Neyman and Pearson into one hybrid logic.
When Loftus (1993) became the editor of Memory & Cognition, he made it clear in his editorial
that he did not want authors to submit papers in which p-, t-, or F-values are mindlessly being
calculated and reported. Rather, he asked researchers to keep it simple and report gures with error
bars, following the proverb that “a picture is worth more than a thousand p-values.” We admire
Loftus for having had the courage to take this step. Years after, one of us (GG) asked Loftus about
the success of his crusade against thoughtless signifi cance testing. Loftus bitterly complained that
most researchers actually refused the opportunity to escape the ritual. Even when he asked in his
editorial letter to get rid of dozens of p-values, the authors insisted on keeping them in. ere is
something deeply engrained in the minds of many researchers that makes them repeat the same
action over and over again.
Question 7: How Can We Advance Statistical inking?
There is no single recipe for promoting statistical thinking, but there are several good heu-
ristics. We sketch a few of these, which the readers can use to construct their own program
or curriculum.
Hypotheses Is in the Plural
If there is one single severe problem with the null ritual, then it is the fact that hypothesis is in the
singular. Hypotheses testing should always be competitive; that is, the predictions of several hypoth-
eses should be specifi ed. Figure 2 gives an example of how the predictions of two hypotheses can be
specifi ed graphically. Rieskamp and Ho rage (1999), for instance, test eight competing hypotheses
about how people predict the profi t of companies, and Gigerenzer and Ho rage (1995) test the
predictions of six cognitive strategies in problem solving. One advantage of multiple hypotheses is
GG_Null_2004.indd 14 12.04.2007 10:29:13 Uhr
Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch 15
the analysis of individual di erences: For instance, one can show that people systematically follow
di erent problem-solving strategies.
Minimize the True Error
Statistical thinking does not simply involve measuring the error and inserting the value into the
denominator of the t-ratio. Good statistical thinking is about how to minimize the real error. By
real error, we refer to the true variability of measurements or observations, not the variance divided
by the square root of the number of observations. W. S. Gosset, who published the t-test in 1908
under the pseudonym “Student” wrote, “Obviously the important thing … is to have a low real
error, not to have a signifi cant’ result at a particular station. e latter seems to me to be nearly
valueless in itself(quoted in Pearson, 1939, p. 247). Methods of minimizing the real error include
proper choice of task (e.g., paired comparison instead of rating) (see Gigerenzer & Richter, 1990),
proper choice of experimental environment (e.g., testing participants individually rather than in
large classrooms), proper motivation (e.g., by performance-contingent payment rather than fl at
sums), instructions that are unambiguous rather than vague, and the avoidance of unnecessary
deception of participants about the purpose of the experiment, which can lead to second-guessing
and increased variability of responses (Hertwig & Ortmann, 2001).
ink of a Toolbox, Not of a Hammer
Recall that the problem of inductive inference has no single best solution—it has many good
solutions. Statistical thinking involves analyzing the problem at hand and then selecting the best
tool in the statistical toolbox or even constructing such a tool. No tool is best for all problems. For
instance, there is no single best method of representing a central tendency: Whether to report the
mean, the median, the mode, or all three of these needs to be decided by the problem at hand.
e toolbox includes, among others, descriptive statistics, methods of exploratory data analysis,
confi dence intervals, Fisher’s null hypothesis testing, Neyman-Pearson hypotheses testing, Wald’s
sequential analysis, and Bayesian statistics.
e concept of a toolbox has an important consequence for teaching statistics. Stop teaching the
null ritual or what is called NHSTP (see, e.g., Chow, 1998; Harlow, 1997). Teach statistics in the
plural: the major statistical tools together with good examples of problems they can solve. For in-
stance, the logic of Fishers (1956) null hypothesis testing can easily be made clear in three steps:
(1) Set up a statistical null hypothesis. The null need not be a nil hypothesis (zero difference).
(2) Report the exact level of significance (e.g., p = .011 or .051). Do not use a conventional 5%
level (e.g., p < .05), and do not talk about accepting or rejecting hypotheses.
(3) Use this procedure only if you know very little about the problem at hand.
Note that Fishers null hypothesis testing is, at each step, unlike the null ritual (see introduction).
One can see that statistical power has no place in Fishers framework—one needs a specifi ed alterna-
tive hypothesis to compute power. In the same way, one can explain the logic of Neyman-Pearson
hypotheses testing, which we illustrate for the case of two hypotheses and a binary decision criterion
as follows:
(1) Set up two statistical hypotheses, H
and H
, and decide about ", !, and sample size before the
experiment, based on subjective cost-benefit considerations. These define a rejection region for
each hypothesis.
GG_Null_2004.indd 15 12.04.2007 10:29:13 Uhr
16 The Null Ritual
(2) If the data falls into the rejection region of H
, accept H
; otherwise, accept H
. Note that accepting
a hypothesis does not imply that you believe in it; it only means that you act as if it were true.
(3) The usefulness of the procedure is limited to situations in which you have a disjunction of
hypotheses (e.g., either µ = 8 or µ = 10 is true) and in which the scientific context can provide
the utilities that enter the choice of " and !.
A typical application of Neyman-Pearson testing is in quality control. Imagine a manufacturer of
metal plates that are used in medical instruments. She considers a mean diameter of 8 mm (H
) as
optimal and 10 mm (H
) as dangerous to the patients and hence unacceptable. From past experi-
ence, she knows that the random uctuations of diameters are approximately normally distributed
and that the standard deviations do not depend on the mean. is allows her to determine the
sampling distributions of the mean for both hypotheses. She considers accepting H
while H
true (Type II error) to be the most serious error because it may cause harm to patients and to the
rms reputation. She sets its probability as ! = 0.1% and " = 10%. Now she calculates the required
sample size n of plates that must be sampled every day to test the quality of the production. When
she accepts H
, she acts as if there were a malfunction and stops production, but this does not mean
that she believes that H
is true. She knows that she must expect a false alarm in 1 out of 10 days
in which there is no malfunction (Gigerenzer et al., 1989, chap. 3).
e basic logic of other statistical tools can be taught in the same way, and examples for their
usefulness and limits can be provided.
Know and Show Your Data
Descriptive statistics and exploratory data analysis are typically more informative than the null
ritual, specifi cally in the presence of multiple hypotheses. For instance, the plot of the three curves
shown in Figure 2 is more informative than the result of the analysis of variance that the data do not
deviate signifi cantly from the predictions of the null. Showing in addition the individual data points
around the means of the data curve, or at least the error bars, would be even more informative.
Similarly, a scatter plot showing the data points is more informative than a correlation coe cient,
for each scatter plot corresponds to one correlation, whereas a correlation of .5, for example, cor-
responds to many and strikingly di erent scatter plots. Wilkinson and the Task Force on Statistical
Inference (1999) give examples for informative graphs.
Keep It Simple
A statistical analysis should be transparent to its author and the readership. Each statistical method
consists of a sequence of mathematical operations, and to understand what the end product (factor
scores, regression weights, nonsignifi cant interactions) means, one needs to check the meaning of
each operation at each step. Transparency allows the reader to follow each step and to understand
or criticize the analysis. e best vehicle for transparency is simplicity. If a point can be made by a
simple analysis, such as plotting the means and standard deviations, one should stick with it rather
than using a less transparent method, such as factor analysis or path analysis. e purpose of a
statistical analysis is not to impress others with a complex method they do not fully understand.
We have witnessed painful talks whereby the audience actually insisted on clarifi cation, only to
learn that the author did not understand his fancy method either. Never use a statistical method
that is not entirely transparent to you.
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Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch 17
p-Values Want Company
If you wish to report a p-value, remember that it conveys very limited information. us, report
p-values together with information about e ect sizes, or power, or confi dence intervals. Recall that
the null hypothesis that defi nes the p-value need not be a nil hypothesis (e.g., zero di erence); any
hypothesis can be a null, and many di erent nulls can be tested simultaneously (e.g., Gigerenzer
& Richter, 1990).
Question 8: How Can We Have More Fun With Statistics?
Many students experience statistics as dry, dull, and dreary. It certainly need not be; real-world
examples (as in Gigerenzer, 2002) can make statistical thinking exciting. Here are several other
ways of turning students into statistics addicts, or at least of making them think. e rst heuristic
is to draw a red thread from the past to the present. We understand the aspirations and fears of a
person better if we know his or her history. Knowing the history of a statistical concept can create
a similar feeling of intimacy.
Connecting to the Past
e rst test of a null hypothesis was by John Arbuthnot in 1710. His aim was to give an empirical
proof of divine providence, that is, of an active God. Arbuthnot observed that “the external ac-
cidents to which males are subject (who must seek their food with danger) do make a great havock
of them, and that this loss exceeds far that of the other sex” (p. 188). To repair this loss, he argued,
God brings forth more males than females, year after year. He tested this hypothesis of divine
purpose against the null hypothesis of mere chance, using 82 years of birth records in London. In
every year, the number of male births was larger than that of female births. Arbuthnot calculated
the “expectation” of these data if the hypothesis of blind chance were true. In modern terms, the
probability of these data if the null hypothesis were true was
p(D | H
) = (1/2)
Because this probability was so small, he concluded that it is divine providence, not chance, that
Scholium. From hence it follows, that Polygamy is contrary to the Law of Nature and Justice, and to the
Propagation of the human Race; for where Males and Females are in equal number, if one Man takes
Twenty Wifes, Nineteen Men must live in Celibacy, which is repugnant to the Design of Nature; nor is it
probable that Twenty Women will be so well impregnated by one Man as by Twenty. (qtd. in Gigerenzer
& Murray, 1987, pp. 4–5)
Arbuthnot’s proof of God highlights the limitations of null hypothesis testing. e research hy-
pothesis (God’s divine intervention) is not stated in statistical terms. Nor is a substantial alternative
hypothesis stated in statistical terms (e.g., 3% of female newborns are abandoned immediately
after birth). Only the null hypothesis (“chance”) is stated in statistical terms—a nil hypothesis.
A result that is unlikely if the null were true (a low p-value) is taken as proofof the unspecifi ed
research hypothesis.
Arbuthnot’s test was soon forgotten. e specifi c techniques of null hypothesis testing, such as
the t-test (devised by Gosset in 1908) or the F-test (F for Fisher, e.g., in analysis of variance), were
GG_Null_2004.indd 17 12.04.2007 10:29:14 Uhr
18 The Null Ritual
rst applied in the context of agriculture. e examples in Fisher’s fi rst book on statistics (1925)
smelled of manure, potatoes, and pigs. In his second book (1935), Fisher had cleaned out this odor,
as well as much of the mathematics, so that social scientists could bond with the new statistics. e
rst applications of these tests in psychology were mostly in parapsychology and education.
A striking change in research practice, which was named the inference revolution in psychology
(Gigerenzer & Murray, 1987), happened from approximately 1940 to 1955 in the United States.
It led to the institutionalization of the null ritual as the method of scientifi c inference in university
curricula, textbooks, and the editorials of major journals. Before 1940, null hypothesis testing
using analysis of variance or the t-test was practically nonexistent: Rucci and Tweney (1980) found
a total of only 17 articles published from 1934 to 1940 that used it. By the early 1950s, half of
the psychology departments in leading U.S. universities had made inferential statistics a graduate
program requirement (Rucci & Tweney, 1980). By 1955, more than 80% of the empirical articles
in four leading journals used null hypothesis testing (Sterling, 1959). Today, the gure is close to
100%. Despite decades of critique of the null ritual, it is still practiced and defended by the ma-
jority of psychologists. For instance, it is often argued that if we can strip routine null hypothesis
testing of the mental confusion associated with it, something of limited but important use is left:
deciding whether or not research data can be explained in terms of chance infl uences” (Chow,
1998, p. 188). We are back to Arbuthnot: e focus is on chance; to test substantive alternative
hypotheses is not an issue. Arbuthnot, it should be said to his defense, was a step ahead—he did
not recommend his procedure as a routine.
Materials to connect with the past can be drawn from two seminal books by Stephen Stigler
(1986, 1999). His writing is so clear and entertaining that it feels as though one had grown up with
statistical thinking. Danziger (1987), Gigerenzer (1987, 2000), and Gigerenzer et al. (1989) tell
the story of the institutionalization of the null ritual in psychology.
Controversies and Polemics
Statistics has plenty of controversies. ese stories of confl ict can provide highly motivating material
for students, who learn that—unlike in their textbooks—statistics is about real people and their
struggles with ideas and with one another. Because of Fisher’s remarkable talent for polemics, his
writings can serve as a starting point. Here are a few highlights.
Fisher once congratulated the Reverend omas Bayes for his insight to withhold his treatise
from publication (it was published posthumously in 1763/1963). Why did Fisher say that? Bayes
rule presupposes the availability of a prior probability distribution over the possible hypotheses,
and Fisher insisted that such a distribution is only meaningful when it can be verifi ed by sampling
from a population. Such distributional data are available in the case of HIV testing (see Question 2)
but obviously uncommon for scientifi c hypotheses. Fisher believed that the Bayesians are wrong
in assuming that all uncertainties can be expressed in terms of probabilities (see Gigerenzer et al.,
1989, pp. 92–93).
Bayesrule and subjective probabilities were not the only target for Fisher. He branded Neymans
position as childishand “horrifying [for] the intellectual freedom of the west.Indeed, he likened
Neyman to
Russians [who] are made familiar with the ideal that research in pure science can and should be geared
to technological performance, in the comprehensive organized e ort of a fi ve-year plan for the nation …
[whereas] in the U.S. also the great importance of organized technology has I think made it easy to confuse
the process appropriate for drawing correct conclusions, with those aimed rather at, let us say, speeding
production, or saving money. (Fisher, 1955, p. 70)
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Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch 19
Why did Fisher link the Neyman-Pearson theory to Stalins 5-year plans? Why did Fisher also
compare them to the Americans, who confuse the process of gaining knowledge with speeding
up production and saving money? It is probably not an accident that Neyman was born in Russia
and, at the time of Fisher’s comment, had moved to the United States. What Fisher believed was
that cost-benefi t calculations, Type I error rates, Type II error rates, and accept-reject decisions had
nothing to do with gaining knowledge but instead with technology and making money, as in qual-
ity control in industry. Researchers do not accept or reject hypotheses; rather, they communicate
the exact level of signifi cance to fellow researchers, so that others can freely make up their minds.
In Fisher’s eyes, free communication was a sign of the freedom of the West, whereas being told a
decision was a sign of communism. For him, the concepts of ", !, and power (1 !) have nothing
to do with testing scientifi c hypotheses.
ey are defi ned as long-run frequencies of errors in repeated experiments, whereas in science,
there are no experiments repeated again and again.
Fisher (1956) drew a bold line between his null hypothesis tests and Neyman-Pearsons tests,
which he ridiculed as originating from the phantasy of circles [i.e., mathematicians] rather remote
from scientifi c research(p. 100). Neyman, for his part, responded that some of Fisher’s tests
are in a mathematically specifi able sense ‘worse than useless’” (Hacking, 1965, p. 99). What did
Neyman have in mind with this verdict? Neyman had estimated the power of some of Fishers tests,
including the famous Lady-tea-tasting experiment in Fisher (1935), and found that the power was
sometimes smaller than ".
Polemics can motivate students to ask questions and to understand the competing ideas un-
derlying the tools in the toolbox. For useful material, see Fisher (1955, 1956), Gigerenzer (1993),
Gigerenzer et al. (1989, chap. 3), Hacking (1965), and Neyman (1950).
Playing Detective
Aside from motivating examples, history, and polemics, a further way to engage students is to
challenge them to fi nd the errors of others. For instance, assign your students the task of looking
up the section on the logic of hypothesis testing in textbooks for statistics in psychology and check-
ing for wishful thinking, as in Table 1. Table 2 shows the result for a widely read textbook whose
author, as usual, did not spell out the di erences between Fisher, Neyman and Pearson, and the
Bayesians but mixed them all up. e price for this was confusion and wishful thinking about the
omnipotence of the level of signifi cance. Table 2 shows quotes from three pages of the textbook,
in which the author tries to explain to the reader what a level of signifi cance means. For instance,
the rst three assertions are unintelligible or plainly wrong and suggest that a level of signifi cance
would provide information about the probability of hypotheses, and the fourth amounts to the
replication fallacy.
Over the years, textbooks writers in psychology have learned to avoid obvious errors but still
continue to teach the null ritual. For instance, the 16th edition of a very infl uential textbook,
Gerrig and Zimbardos (2002) Psychology and Life, contains sections on “inferential statisticsand
“becoming a wise consumer of statistics” (pp. 37–46), which are pure guidelines for the null ritual.
e ritual is portrayed as statistics per se and named the “backbone of psychological research
(p. 46). Our detective student will nd that the names of Fisher, Bayes, Neyman, and Pearson are
not mentioned, nor are concepts such as power, e ect size, or confi dence intervals. She may also
stumble upon the prevailing oracular language: “Inferential statistics indicate the probability that
the particular sample of scores obtained are actually related to whatever you are attempting to
GG_Null_2004.indd 19 12.04.2007 10:29:14 Uhr
20 The Null Ritual
measure or whether they could have occurred by chance” (p. 44). Yet in the midst of unintelligible
and nonsensical explanations such as these appear moments of deep insight: “Statistics can also be
used poorly or deceptively, misleading those who do not understand them” (p. 46).
Question 9: What if ere Were No Signifi cance Tests?
is question has been asked in a series of articles in Harlow, Mulaik, and Steiger (1997) and in
similar debates, which are summarized in the superb review by Nickerson (2000). However, there
are actually two di erent questions: What if there were no null hypothesis testing (signifi cance
testing), as advocated by Fisher? What if there were no null ritual (or NHSTP)?
If eminent psychologists have anything in common, it is their distaste for mindless null hypoth-
esis testing—which contrasts with the taste of the masses. You will not catch Jean Piaget testing a
null hypothesis. Piaget worked out his logical theory of cognitive development, Wolfgang Köhler
the Gestalt laws of perception, I. P. Pavlov the principles of classical conditioning, B. F. Skinner
those of operant conditioning, and Sir Frederick Bartlett his theory of remembering and sche-
mata—all without rejecting a null hypothesis. Moreover, F. Bartlett, R. Duncan Luce, Herbert A.
Simon, B. F. Skinner, and S. S. Stevens explicitly protested in their writings against the null ritual
(Gigerenzer, 1987, 1993; Gigerenzer & Murray, 1987).
So what if there were no null ritual or NHST? Nothing would be lost, except confusion,
anxiety, and a platform for lazy theoretical thinking. Much could be gained, such as knowledge
about di erent statistical tools, training in statistical thinking, and a motivation to deduce precise
predictions from one’s hypotheses. Should we ban the null ritual? Certainly—it is a matter of
intellectual integrity. Every researcher should have the courage not to surrender to the ritual, and
every editor, textbook writer, and adviser should feel obliged to promote statistical thinking and
reject mindless rituals.
What if there were no null hypothesis testing, as advocated by Fisher? Not much would be lost,
except in situations in which we know very little, where a p-value by itself can contribute something.
Note that this question is a di erent one: Fishers null hypothesis testing is one tool in the statistical
Table 2
What Does “Signifi cant at the 5 % Level” Mean?
“If the probability is low, the null hypothesis is improbable
e improbability of observed results being due to error”
e probability that an observed di erence is real”
e statistical confi dencewith odds of 95 out of 100 that the observed di erence will hold up in investigations
Degree to which experimental results are taken seriously”
e danger of accepting a statistical result as real when it is actually due only to error’
Degree of “faith [that] can be placed in the reality of the nding”
e investigator can have 95 percent confi dence that the sample mean actually di ers from the population mean
All of these are di erent ways to say the same thing
Note. Within three pages of text, the author of a widely read textbook explained to the reader that “level of signifi cance”
means all of the above (Nunally, 1975, pp. 194–196). Smart students will be confused, but they may misattribute their
confusion to their own lack of understanding.
Source: Nunally (1975).
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Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch 21
toolbox, not a ritual. Should we ban null hypothesis testing? No, there is no reason to do so; it is
just one small tool among many. What we need is to educate the next generation to dare to think
and free themselves from compulsive hand-washing, anxiety, and feelings of guilt.
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In this thesis I explore the extent to which researchers of animal cognition should be concerned about the reliability of its scientific results and the presence of theoretical biases across research programmes. To do so I apply and develop arguments borne in human psychology’s “replication crisis” to animal cognition research and assess a range of secondary data analysis methods to detect bias across heterogeneous research programmes. After introducing these topics in Chapter 1, Chapter 2 makes the argument that areas of animal cognition research likely contain many findings that will struggle to replicate in direct replication studies. In Chapter 3, I combine two definitions of replication to outline the relationship between replication and theory testing, generalisability, representative sampling, and between-group comparisons in animal cognition. Chapter 4 then explores deeper issue in animal cognition research, examining how the academic systems that might select for research with low replicability might also select for theoretical bias across the research process. I use this argument to suggest that much of the vociferous methodological criticism in animal cognition research will be ineffective without considering how the academic incentive structure shapes animal cognition research. Chapter 5 then beings my attempt to develop methods to detect bias and critically and quantitatively synthesise evidence in animal cognition research. In Chapter 5, I led a team examining publication bias and the robustness of statistical inference in studies of animal physical cognition. Chapter 6 was a systematic review and a quantitative risk-of-bias assessment of the entire corvid social cognition literature. And in Chapter 7, I led a team assessing how researchers in animal cognition report and interpret non-significant statistical results, as well as the p-value distributions of non-significant results across a manually extracted dataset and an automatically extracted dataset from the animal cognition literature. Chapter 8 then reflects on the difficulties of synthesising evidence and detecting bias in animal cognition research. In Chapter 9, I present survey data of over 200 animal cognition researchers who I questioned on the topics of this thesis. Finally, Chapter 10 summarises the findings of this thesis, and discusses potential next steps for research in animal cognition.
... Concernant la significativité des tests (chi2), modèles (régressions logistiques) et variables représentatives dans les ACM, exprimée par la « p value », le seuil de significativité retenu est généralement 0,05 ou 5 %. Ce seuil est relatif à une convention, mais dont la pertinence est controversée (Lambdin, 2012 ;Gigerenzer et al., 2004 ;Poitevineau, 2004 ;Krueger, 2001). Gigerenzer et al. (2004) l'assimilent ainsi à un « rituel » et une « routine ». ...
La transition agroécologique prônée par les politiques publiques en France constitue le point de départ du questionnement développé dans la thèse. Nous focalisons notre attention sur un dispositif public qui vise à stimuler les projets agroécologiques collectifs mis en œuvre par des agriculteurs et des organisations partenaires à l’échelle territoriale, le Groupe environnemental d’intérêt économique et environnemental (GIEE). Le cas d’étude concerne une coopérative qui a eu recours à un tel dispositif dans le cadre du développement de légumes secs, dont les intérêts agronomiques, environnementaux et alimentaires les inscrivent pleinement dans les perspectives de transition agroécologique. À travers ce cas, nous examinons le rôle des interactions sociales dans l’actualisation des modes d’agir et de penser d’acteurs requise dans la transition agroécologique, appréhendée ici comme un changement institutionnel. Dans le prolongement des travaux développés dans le champ de l’économie délibérative, nous nous intéressons aux potentialités de la délibération collective pour provoquer cette actualisation en suscitant l’expression de valeurs. L’originalité de la thèse est de recourir aux apports de la psychologie historico-culturelle, qui montrent que les êtres humains disposent de la capacité à surmonter les perturbations rencontrées dans leur activité et à entretenir un nouveau rapport avec elle en faisant intervenir un artefact médiateur. Un artefact spécifique, le concept de qualité, a ainsi été mobilisé dans des délibérations collectives menées avec des agriculteurs de trois GIEE de la coopérative et des consommateurs.
... The p-value might also be one of the culprits of the reproducibility crisis in science (Nuzzo 2014), including computer science (Berrar and Dubitzky 2019;Cockburn et al. 2020). Criticisms against the p-value have been raised for many decades, notably in psychology and sociology (Rozeboom 1960;Carver 1978;Schmidt 1996;Wagenmakers 2007;Schmidt and Hunter 2016;Gigerenzer 1998;Gigerenzer et al. 2004), medicine and epidemiology (Goodman 1993;Rothman et al. 2008;Stang et al. 2010). Over the last decades, the logic of null hypothesis significance testing has been challenged (Cohen 1990(Cohen , 1994, but also defended (Hagen 1997;Mulaik et al. 2016;Abelson 2016). ...
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The statistical comparison of machine learning classifiers is frequently underpinned by null hypothesis significance testing. Here, we provide a survey and analysis of underrated problems that significance testing entails for classification benchmark studies. The p-value has become deeply entrenched in machine learning, but it is substantially less objective and less informative than commonly assumed. Even very small p-values can drastically overstate the evidence against the null hypothesis. Moreover, the p-value depends on the experimenter’s intentions, irrespective of whether these were actually realized or not. We show how such intentions can lead to experimental designs with more than one stage, and how to calculate a valid p-value for such designs. We discuss two widely used statistical tests for the comparison of classifiers, the Friedman test and the Wilcoxon signed rank test. Some improvements to the use of p-values, such as the calibration with the Bayes factor bound, and alternative methods for the evaluation of benchmark studies are discussed as well.
... This approach is being referred to here as an "integration of philosophies" based approach. Similar to a chemical mixture which retains the properties of its constituents, this integration retains the fundamental shortcomings like "data cranking null hypothesis rituals" (Cohen, 1994;Gigerenzer et al., 2004;Sedlmeier, 2009), skewness towards applied research and research designs casted in same dominant worldview. It may also lead to perceptual errors because the perceptual units as viewed by actors in a worldview may be quite different as viewed by observers in a different worldview. ...
... The apical root third thickness was significantly different than middle and coronal thirds (P ˂ 0.001). A suggestive significance difference was found in samples thickness between the coronal and middle root thirds (P = 0.089) (22). Pearson correlation test showed a negative relation between dentin thickness and the transmitted power (r = À0.45, ...
To measure the transmitted laser power in the coronal, middle and apical root thirds during vertical and horizontal irradiation of laser. 14 mm length whole roots and longitudinal root sections were irradiated using a 940 nm diode laser with 0.5 W in continuous mode for 5 s. A power detector was placed in front of the root apex for the vertical transmitted power measurement and placed laterally to root surface for the horizontal transmission experiment. The transmitted power from the root apex vertically was 53% of the irradiated power, while for the horizontal transmission experiment for the coronal, middle and apical root thirds were 25.6%, 40.4% and 41.3%, respectively. Irradiation of root canals with 940 nm laser power can be transmitted during vertical irradiation in more than 50% of the samples, whereas less transmission happens with horizontal irradiation at apical and middle root third than coronal.
The field of obstetrics and gynecology is constantly replenished with the newest research findings. In an era of rapidly available study publications, there are a number of challenges to interpreting the obstetrics and gynecology literature. Common pitfalls include the over reliance on the dichotomized P-value, lack of transparency, bias in study reporting, limitations of resources, absence of standardized practices and outcomes in study design, and the rare concerns for data integrity. We review these predominant challenges and their potential solutions, in interpreting the obstetrics and gynecology literature.
After 4 decades of severe criticism, the ritual of null hypothesis significance testing - mechanical dichotomous decisions around a sacred .05 criterion - still persists. This article reviews the problems with this practice, including its near-universal misinterpretation of p as the probability that H0s false, the misinterpretation that its complement is the probability of successful replication, and the mistaken assumption that if one rejects H0 one thereby affirms the theory that led to the test. Exploratory data analysis and the use of graphic methods, a steady improvement in and a movement toward standardization in measurement, an emphasis on estimating effect sizes using confidence intervals, and the informed use of available statistical methods is suggested. For generalization, psychologists must finally rely, as has been done in all the older sciences, on replication.