articulos_pruebas_hipotesis

8/7/2019 articulos_pruebas_hipotesis

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Análisis y Diseño de Experimentos

Introducción a las Pruebas de Hipótesis

Ing. Héctor Rincón Arredondo 1


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.)






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.






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






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

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