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# Hypothesis, Nested

BIBLIOGRAPHY

The term nested hypothesis sometimes refers to a group of hypothesesthat is, tentative assumptions or hypothetical factors in an experimental modeland other times to a subordinate hypothesis within a group. A nested hypothesis is the same as its non-nested analogue, except that one or more of the parameters in the non-nested hypothesis is constrained or limited in the nested hypothesis. To be considered nested, then, a hypothesis must meet the following conditions. First, it must be one among a group of related hypotheses. Second, all of the hypotheses must have some ordering pattern to them. This ordering pattern causes some hypotheses to be logical subsets of other hypotheses. It is possible for all but one hypothesis to be a subset of another hypothesis. Any hypothesis that is a subset of another hypothesis is nested. Third, when the hypotheses are ordered, each hypothesis must completely contain all the hypotheses preceding it. This condition of containment or nesting is the source of the nested hypothesiss name.

Research programs sometimes are designed so that there are multiple groups of hypotheses. In these cases, each of the groups may consist of nested hypotheses, but none are nested with respect to any other group. This type of situation is sometimes referred to as partially nested. Also some research disciplines reverse the ordering so that the main hypothesis comes first and the nested hypotheses follow it. This situation usually occurs in written material and not when the hypotheses are stated mathematically.

A common situation in which nested hypotheses often occur is traditional hypothesis testing using an alternative null hypothesis. The null hypothesis is an alternative to the hypothesis being tested where the explanatory factors being researched have no impact on the situation under consideration. When the null hypothesis is simply the testing hypothesis with one of the parameters set at any constant, then nesting exists. If, for example, a researcher was looking for factors influencing voting decisions and one of the factors chosen was birth year, the null hypothesis would be that birth year is unimportant (and that parameter would be set to zero). In this case, the hypotheses are nested because the null hypothesis is completely included within the main hypothesis.

Nested hypotheses have been around as long as scientific research. It is both logical and easy to use a null hypothesis in cases where the factor being examined simply becomes unimportant. Nested hypotheses also frequently occur in the social sciences when researchers are investigating a primary hypothesis. Sometimes researchers will also examine secondary hypotheses (dependant on the primary one) as nested hypotheses. There are, however, conditions under which nested hypotheses should only be used with caution. Nested hypotheses are sometimes used solely for ease of computation, without any underlying research basis. Before using nested hypotheses, researchers should make certain that the question being researched provides a substantive logical reason for using nesting.

A significant problem that nested models have is deviance. As the number of parameters in a model is decreased, model deviance automatically increases. In instances where there is a significant amount of nesting, the deviance can rise substantially. The difference between the deviances of two models is the basis for comparing their reliability. Therefore if there are a significant number of parameter changes, there will automatically be a bias in favor of the more complicated model, irrespective of the real accuracy increase. According to some experts in statistics, when analysis techniques such as the log-likelihood are involved, it is impossible for a nested model to have a larger maximum than the associated non-nested model.

A lesser concern is that there are different ways of nesting hypotheses. Population groupings provide an example of this. These groupings can either be superordinated, such that one group is separated out in each nesting, or divided, in which case some of the constituent groups become one nested hypothesis and the remainder become the other hypothesis. These latter subgroups are both nested with respect to the main hypothesis but not with respect to each other. This is a different situation from the former grouping condition, where each new hypothesis is nested with respect to all preceding hypotheses. In order to compare the models and their results, it is necessary to know which approach was used. In cases where nested hypotheses are not appropriate or desirable, there are a number of Bayesian, categorical, generalized linear model, and likelihood data-analysis techniques that can be used in place of the traditional hypothesis/null hypothesis framework.

## BIBLIOGRAPHY

Agresti, Alan. 1996. An Introduction to Categorical Data Analysis. New York: Wiley Inter-Science.

Agresti, Alan. 2002. Categorical Data Analysis. New York: Wiley Inter-Science.

Carlin, Bradley P., and Thomas A. Louis. 2000. Bayes and Empirical Bayes Methods for Data Analysis. 2nd ed. Washington, DC: Chapman and Hall/CRC.

Fox, John. 1997. Applied Regression Analysis, Linear Models, and Related Methods. London: Sage.

Long, J. Scott. 1997. Regression Models for Categorical and Limited Dependent Variables. London: Sage.

Weisstein, Eric W. 2002. CRC Concise Encyclopedia of Mathematics. 2nd ed. Washington, DC: Chapman and Hall/CRC.

David B. Conklin