## Understanding Confidence Intervals

Confidence intervals are constructed from an estimate and its standard error. These intervals can be conceptualized two ways. From one perspective, we can say what proportion of random replications of this interval include the "true, fixed value" for the parameter over the long run (Neyman 1952 p.194f). This Neyman-Pearson approach has the weight of history and reassures determinists. From another perspective, we can say what proportion of possible values of the parameter lies within this interval. This Savage-Bayesian approach (Edwards et al. 1963) accommodates the layman interpretation of confidence intervals and also satisfies probabilists.

The Neyman-Pearson understanding is that the "confidence" interval formed by an estimate plus and minus some standard errors is a statistic which, upon random replication, can be expected to contain the unknowable but fixed "true" parameter a specified proportion of the time. Whether the single example of such an interval which we have in hand has succeeded in doing this is, by definition, unknown. Maybe yes. Maybe no. All we claim to know is the long-run behavior of random replications of intervals formed this way.

Few users of confidence intervals, however, couch their inferences so carefully. Most imply, instead, that they have established such-and- such confidence in the claim that the "truth" they seek is inside their particular interval.

The Savage-Bayesian approach, sometimes employing the terms "credible" or "believable" interval, comes closer to practice. The actual Savage-Bayesian understanding, however, is that, given the observed estimate and its error, such-and-such proportion of all possible values of the parameter fall within the interval.

Something important has happened in this shift of interpretation. The unknown parameter is no longer an unknowable but "fixed truth". Rather it is a "variable" with a range of possible values. The Savage-Bayes interval embraces a specified proportion of these possibilities and therefore serves to define a range of reasonable values for the parameter.

Growing beyond a naive search for "fixed truths" and seeking instead for usefully limited "ranges of possible values" is an important step forward in the evolution of inference.

Neyman J (1952) Lectures and Conferences on Mathematical Statistics and Probability. 2nd Ed. Washington: US Dept of Agriculture.

Edwards W, Lindman H, Savage L.J. (1963) Bayesian statistical inference for psychological research. Psychological Review, 70/3, p.193-242. In: The Writings of Leonard Jimmie Savage - A Memorial Selection. Washington: American Statistical Assoc. 1981.

Confidence intervals. Wright BD. … Rasch Measurement Transactions, 1992, 5:4 p.175

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