articulos_pruebas_hipotesis
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Análisis y Diseño de Experimentos
Introducción a las Pruebas de Hipótesis
Ing. Héctor Rincón Arredondo 1
Introducción a las Pruebas de Hipótesis
Este artículo es para reafirmar sus conocimientos sobre un tema que, me permito
suponer, debieron haber cubierto en su curso previo de Matemáticas (Probabilidad
y Estadística), me refiero a las Pruebas de Hipótesis.
Más que realizar algunos cálculos para solución de problemas, a continuación
reproduzco para ustedes un artículo que tiene algunos puntos relativos al sistema
de impartición de justicia.
Ejercicio
Lea el siguiente artículo, dedicando especial atención a la Tabla 1.
Consulte la literatura y complete para cada punto que trata sobre el ‘sistema de
impartición de justicia’ las equivalencias con las ‘Pruebas de Hipótesis’
Ej. En el sistema de impartición de justicia: Amplitud de la evidencia
En las Pruebas de Hipótesis: Tamaño de Muestra
Artículo
After introducing new students to a few chapters of descriptive statistics, most
texts move quickly into the subject of hypothesis testing. Many students find the
somewhat convoluted logic of this subject hard to grasp and often without clear
applicability to the situation at hand. After plowing through a few problems
(usually applications of the popular F and T tests), some students begin to get the
feeling they’ve seen it all before. And they have there is an almost perfect
analogy between statistical hypothesis testing and the criminal justice system.
Because the process of setting up and testing against a null hypothesis can seem
illogical to scientis ts and engineers, this important approach is often lost and,
along with it, a sound statistical basis for product development, optimization, or
problem solving. Usually scientists want to run a test because there is good reason
to believe that certain factors are likely to influence the experimental results. The
statistician, however, usually wants the experiment set up under the seemingly
illogical basic assumption that the effects of the factors tested are nonexistent.
Unfortunately, this often sends the scientist off in search of a more sympathetic
environment in which to prove that his intuition was correct.
Such a path of least resistance is fraught with pitfalls. A testing environment that
leaves a research department overloaded with “real” effects is of little value, often
leading to incorrect and costly decisions.
Table 1 presents an incomplete side-by-side comparison of the methodology of
hypothesis testing with the basic tenets of our criminal justice system. (A refresher
of alpha (α) and beta (β) risks is also offered as Table 2.)
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Análisis y Diseño de Experimentos
Introducción a las Pruebas de Hipótesis
Ing. Héctor Rincón Arredondo 2
It has often been said that even though our adversarial justice system can be
cumbersome and time consuming, it is still the best system yet devised to reach the
correct verdict most often. Similarly, in the field of experimental design, a testing
environment that requires a clear statement of the hypothesis to be tested, along
with the acceptable risks, has the best chance of producing the correct decision.
The final point of comparison has to do with the current philosophical trendsregarding who the system is designed to protect. It is generally recognized that our
criminal justice system has been nurtured on a philosophy that demands a high
degree of protection for a defendant’s rights (innocent until proven guilty), often
with little regard for the fate of the victim. Recently, however, there have been
strong public outcries against this type of system. Indeed, the trend of the future
may well be more protection of the victim and fewer guilty defendants set free.
If we examine the history of statistical experimental design, we also find that much
of the emphasis has been on the "α" risk protecting the producer (or protecting
against detecting an effect that is not significant). Many texts devote little space to
the "β" risk protecting the consumer (or protecting against missing an effect
that is truly significant).
This emphasis can produce a testing environment so overburdened with statistical
rigor that many true effects are declared insignificant a state of affairs that can
be as undesirable as the “sympathetic” testing environment described earlier. It is
interesting to note, however, that the pendulum is swinging in the direction of a
better balance between the producer’s and consumer’s concerns, resulting in a
healthier regard for ensuring high quality in the hands of the customer.
Further reading
In the field of experimental design, two texts are highly recommended:
• William J. Diamond, Practical Experiment Designs for Scientists and Engineers
(Lifetime Learning Publications, 1981, Belmont, CA).
Diamond’s step-by-step approach to designing an experiment (requiring upfront
consideration for "α" and "β" risks, effect size, and sample size) is most refreshing.
Additionally, he focuses on the concept of the "resolution" level of fractional
designs and offers a computer program that lays out the design and completes the
analysis.
• Genichi Taguchi, Quality Engineering Methodology and Application
(American Supplier Institute, 1984, Romulus, MI).
Taguchi demonstrates that through ingenious application of orthogonal arraysalmost any nonstandard design can be accommodated. In data analysis, he
stresses extensive pooling of error to enhance the discriminating power of the
experiment.
Although their styles are quite different, the philosophical approach of both
authors is to apply statistics in a manner that is user-friendly and that most
efficiently identifies real effects.
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Análisis y Diseño de Experimentos
Introducción a las Pruebas de Hipótesis
Ing. Héctor Rincón Arredondo 3
Table 1 Hypothesis Testing and the Criminal Justice System: An Analogy
Criminal Justice
Trial
Defendant
Assumption: the defendant is innocent
Charge: the defendant is guilty
Prosecutor
Prosecutor’s task: Show that the assumption is not true (that the defendant is not
innocent) and that the charge is true
Nature of the Trial
Lenient jury
High confidence that when the defendant is judged to be guilty, he is truly guilty
High risk of judging a guilty defendant to be innocent
Will tend to judge an innocent defendant to be innocent
A poor trial for judging that a truly guilty defendant is guilty
Vengeful jury
Low confidence that when the defendant is judged guilty, he is truly guilty
Low risk of judging a guilty defendant to be innocent
Will tend to judge an innocent defendant to be guilty
A good trial for judging that a truly guilty defendant is guilty
If the evidence is ample and comprehensive, then no matter what the nature of the
trial, the correct verdict is likely to be made
The system should allow for full cross-examination to prevent only one side of the
story to be told
Only evidence, witnesses, and questioning that relate to the charge should be
admitted. Otherwise, findings will carry judgements about issues not relating to the
charge.
Table 2. Statistical Risks/Confidences
α Risk
Risk of rejecting a true null hypothesis
Risk of detecting an unreal difference
Error of the first kind
Producer’s risk
Risk of calling good material bad
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Análisis y Diseño de Experimentos
Introducción a las Pruebas de Hipótesis
Ing. Héctor Rincón Arredondo 4
Risk of saying a process is out of control when it is not
1-α = Probability of accepting a true null hypothesis
= Probability of calling good material good
= Confidence of the test
β Risk
Risk of accepting a false null hypothesis
Risk of not detecting a real difference
Error of the second kind
Consumer’s risk
Risk of calling bad material bad
Risk of saying a process is in of control when it is not
1-β = Probability of rejecting a false null hypothesis
= Probability of calling bad material bad
= Power of the test