A PRIMER ON INFERENCE

    Inference is defined as the attempt to generalize on the basis of limited information. Information is always limited because it is impractical, in terms of time and cost, to obtain total knowledge about everything. If everything were known, there would be no need for inference. Since science does not claim to know everything, inference is behind all science, except in those few cases where everything is known about a whole population. It is important to note, also, that inference underlies most thinking, even the unscientific type. In this way, scientific inference is not unlike common sense. What distinguishes scientific inference is that the process is made explicit and follows certain rules, which are the subjects of this lecture.

    Inference demonstrates itself in science at least four main ways: (1) hypothesizing; (2) sampling, (3) designing, and (4) interpreting. These four general areas are sometimes referred to as the "wheel" of science. Hypothesizing usually begins after one has examined the existing knowledge base, reviewed the relevancy of theories, and understood something of the context within which the phenomenon of interest occurs. In other words, you begin research by identifying a problem area (picking a topic), reading the theoretical research (especially the literature review sections), and finding a research question of interest to you (something that has puzzled previous theorists and researchers). Research questions are longer and broader than hypotheses. 

    Hypotheses are simply if-then sentences that can be categorized in certain logical forms, such as no difference (null hypothesis), associated difference, directionality of difference, and magnitude of difference. A good hypothesis implies all these forms in a single sentence, and the trick is to express them as briefly as possible and in simple English. All theories contain hypotheses, but you sometimes have to read them into the theories. There's no need to elaborate all hypotheses capable of being generated by every aspect of a theory, but a single theory can generate many hypotheses with its twists and turns.  In the end, all hypotheses demonstrate inference by concisely reducing extant (existing) knowledge into manageable and meaningful form. Extant knowledge is what you obtain from a literature review.

    Sampling goes to the heart of inference because a sample is what one draws on to test hypotheses and make generalizations. The idea of sampling is drawn from the mathematical discipline of probability theory, and a particular subfield of that discipline called frequentism, which combines inductive (particular to general) and deductive (general to particular) reasoning.  It's the selection of observables to make predictions about unobservables. Sampling, at bottom, is a matter of reducing, of simplifying. Since many phenomena in life tend to follow a normal, or almost normal, distribution (according to the central limits theorem), known mathematical properties of the standard normal curve provide the basis for most predictions, as these are considered estimates of the fit between a sample (observables) and the wider population (unobservables). If the researcher has been thinking inferentially, the method of sampling and the size of the sample will be selected on grounds of parsimony (making do with the fewest numbers as possible). There is no automatic need for large sample sizes, and the type of questions asked or relationships predicted will, in part, help determine the sampling plan. If one is going to infer causality, then random sampling, or some variant, is warranted. There are both probabilistic (making use of advanced features of probability theory) and nonprobabilistic (not making use of advanced features of probability theory) sampling methods that suit different purposes. In general, the more one knows about the wider population or context of the research problem, the easier it is to justify use of nonprobabilistic sampling. Representativeness is what one is after with sampling, which means that each and every person or unit in your sample is a near-average person or unit, not some unusual case that would be called an outlier (too far out on some traits or attributes to be near-average). Measurement is a research step related to sampling and the estimates derived from it. It is important that the sample enable measurement of constructs (unobservables) that are strongly linked to concepts (observables). In general, one should attempt to obtain measures that are meaningful, and this means interval or ratio level, especially if one is going to infer causality. Interval (meaningful distances between points) and ratio (fixed point with meaningful distances) measurement is also related to estimates of validity and reliability of one's research. Validity and reliability refer, respectively, to whether one is measuring what one intends to measure and if one is doing it consistently. These qualities of research, as well as the general idea of sampling, demonstrate inference by streamlining a project into manageable and meaningful proportions.

    Design issues depend, in large part, upon the expertise and creativity of the researcher. What one wants is a good tradeoff between a parsimonious design and one that provides the highest level of confidence. There is no automatic need for the Cadillac of designs, the experimental model (with experimental and control groups), when one can get by with a less grand design. Of course, this depends upon the type of questions asked and relationships predicted. If one is predicting causality, or even hypothesizing correlation (that one thing moves up or down in correspondence with another thing), then the experimental model or a close approximation to it is warranted. Designing with confidence does not refer to the power of statistical estimates, although there is such a thing as statistical correspondence validity which means that the intended analysis is consistent with the design to be used. Confidence, as the term means here, simply means that the prospective design is one the researcher feels comfortable with and is likely to be appreciated by the rest of the scientific community. This is often referred to as the requirement of replication. Sound designs are capable of being replicated; each and every procedure is made explicit so that an outsider could come in, repeat the experiment exactly, and probably get the same results. Replication demonstrates, as design issues in general do, the quality of inference. Nothing is ever demonstrated directly and completely. Only by what seems tedious, rigorous, and systematic does more and more tenable generalizations become possible.

    Interpreting research is perhaps the prime example of inference. Interpretation is made on the basis of data analysis using some sort of statistic. A statistic is a mathematical tool or formula used to test the probability of being right or wrong when we say a hypothesis is true or false. There are about 100 common statistical tests. A test of one's hypothesis can always reach statistical significance by increasing the sample size, and that's just because the way cutoffs are placed in the tables of numbers called significance tables. However, there's a difference between statistical and meaningful significance. Statistical significance is no guarantee of meaningful, or social or psychological significance. Generalizeability is what one is after with interpretation, which means that general conclusions can be made on the basis of successful testing of all your hypotheses. There are two things to be wary of: (1) knowing the limitations of one's research, and (2) knowing the delimitations of one's research. Limitations are specific conclusions that refer to the making of generalizations possible from what your analysis actually shows. It may be nothing more than the discovery of a relationship. You should always know your limitations. Delimitations are general conclusions that refer to the making of generalizations beyond the limitations of your study. You should always be cautious of overgeneralizing to wider populations; you may go beyond your sample, but not beyond related populations.  Be humble and modest in presenting your conclusions. One way of demonstrating how limitations are evidence of inference is to look at the requirements of causality: association, temporality, and nonspuriousness. These three requirements of causality can be said to summarize causal inference. Predicted relationships should vary concomitantly (association, as one goes up or down, the other goes up or down), one variable should precede the other in temporal order, and spurious variables should be reduced to a minimum. Spurious variables are things you haven't thought of that might explain what you've found.

    In the end, there are usually more correlates than causes, and one cannot control everything. Causality is always an inference. This particular type of relationship must be inferred from the observed information, and then related back to known information. Inference demonstrates itself in hypothesizing, sampling, designing, and interpreting. It is the basis for scientific generalization, especially those having to do with the explanation of causality. It is never the final proof, but because final proof is itself never possible, inference is the best substitute. It enables ways to advance science, debunk mistaken beliefs, and is always mindful of its own limitations. Certain safeguards are built into the process which protect against unwarranted generalizations. The process of generalizing in an explicit and scientific manner is inference.

REVIEW QUESTIONS:
1. What does the phrase "extant literature" mean, and what would a researcher probably be doing if they were saying "The extant literature is silent on this issue."?
2. The principle of parsimony is sometimes referred to as Occam's Razor, which is often defined as "don't multiply unobservables." Try your hand at explaining the relevance of Occam's razor to research methodology in 50 words or less.
3. Explain what "correlation" means, but don't use the example given in this lecture as "one thing moves up or down in correspondence with another thing." Use the Internet to find a definition or explanation to help construct your answer.

QUIZ:
1. Concomitant variation, or association, is expected as a requirement of causality. Does this mean that causality requires at least some degree of correlation? Yes or No.
2. Determine what the following statements are using this answer key:
    (A) research question; (B) hypothesis; (C) theory; (D) limited conclusion; (E) delimited conclusion

(a) If more police are hired then more crimes will be reported.
(b) One of the primary determinants of crime rates is how many crimes are reported to police, and this may depend upon the number of police to report crimes to.
(c) This research has shown that for police agencies everywhere, crime reporting is a function of size.
(d) This research has shown that for police agencies like those in the sample, crime reporting is a function of size.
(e) The central role of police in society is to reduce crime, and to do so by expanding the number of officers to catch criminals.

3. Apply the principles and ideas in this lecture to a justice problem of your choice. It can be a problem you've learned about in your studies, read about, seen on TV, or experienced personally. Identify the following, and avoid short answers:

A. ___________________________________ (the problem)
B. ___________________________________ (a theory)
C. ___________________________________ (the hypotheses - identify at least two)
D. ___________________________________ (your conclusions - assuming all your hypotheses are supported)

INTERNET RESOURCES
Characteristics of Quack Theories
Hypothesis Testing and the Principles of Inference
Instructor and Student Tools for Teaching the Concepts behind Statistical Inference
Logic of Inference and the Central Limits Theorem
Sci.Skeptic's FAQ on Scientific Method and Occam's Razor

PRINTED RESOURCES
Casella, G. & R. Berger. (1990). Statistical Inference. Belmont, CA: Wadsworth.
Fisher, R. & J. Bennet. (Eds.) (1990). Statistical Methods, Experimental Design, and Scientific Inference. NY: Oxford Univ. Press.
Hempel, G. & C. (1966). Philosophy of Natural Science. NJ: Prentice Hall.
Kanji, G. (1993). One Hundred Statistical Tests. Beverly Hills: Sage.
Lasley, J. (1999). Essentials of Criminal Justice and Criminological Research. NJ: Prentice Hall.
Manski, C. (1999). Identification Problems in the Social Sciences. Cambridge, MA: Harvard Univ. Press.
Pearl, J. (2000). Causality: Models, Reasoning, and Inference. NY: Cambridge Univ. Press.

Last updated: Oct 09, 2006
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