Noncentrality Interval Estimation and the Evaluation of Statistical Models - Replacing Traditional Hypothesis Tests with Interval Estimates

The STATISTICA Power Analysis module implements a number of confidence interval procedures that can replace and/or augment the traditional hypothesis tests used in classical testing situations. For a review of these techniques, see Steiger & Fouladi (1997).

Analysis of Variance. One area where confidence intervals have seldom been employed is in assessing strength of effects in the Analysis of Variance (ANOVA).

For example, suppose you are reading a paper, which reports that, in a 1-Way ANOVA, with 4 groups, and N = 60 per group, an F statistic was found that is significant at the .05 level ("F = 2.70, p =.0464"). This result is statistically significant, but how meaningful is it in a practical sense? What have we learned about the size of the experimental effects?

Fleischman (1980) discusses a technique for setting a confidence interval on the overall effect size in the Analysis of Variance. This technique allows one to set a confidence interval on the RMSSE, the root-mean-square standardized effect. Standardized effects are reported in standard deviation units, and hence remain constant when the unit of measurement changes. So, for example, an experimental effect reported in pounds would be different from the same effect reported in kilograms, whereas the standardized effect would be the same in each case. In the case of the data mentioned above, the F statistic that is significant at the .05 level yields a 90% confidence interval for the RMSSE that ranges from .0190 to .3139. The lower limit of this interval stands for a truly mediocre effect, less than 1/50th of a standard deviation. The upper limit of the interval represents effects on the order of 1/3 of a standard deviation, moderate but not overwhelming. It seems, then, that the results from this study need not imply really strong experimental effects, even though the effects are statistically "significant."

Multiple Regression. The squared multiple correlation is reported frequently as an index of the overall strength of a prediction equation. After fitting a regression equation, the most natural questions to ask are, (a) "How effective is the regression equation at predicting the criterion?" and (b) "How precisely has this effectiveness been determined?"

Hence, one very common statistical application that practically cries out for a confidence interval is multiple regression analysis. Publishing an observed squared multiple R together with the result of a hypothesis test that the population squared multiple correlation is zero, conveys little of the available statistical information. A confidence interval on the populations squared multiple correlation is much more informative.

The STATISTICA Power Analysis module computes exact confidence intervals for the population squared multiple correlation, following the approach of Steiger and Fouladi (1992). As an example, suppose a criterion is predicted from 45 independent observations on 5 variables and the observed squared multiple correlation is .40. In this case a 95% confidence interval for the population squared multiple correlation ranges from .095 to .562! A 95% lower confidence limit is at .129. On the other hand the sample multiple correlation value is significant "beyond the .001 level," because the p-value is .0009, and the shrunken estimator is .327. Clearly, it is far more impressive to state that "the squared multiple R value is significant at the .001 level" than it is to state that "we are 95% confident that the population squared multiple correlation is between .095 and .562." But we believe the latter statement conveys the quality and meaning of the statistical result more accurately than the former.

Some writers, like Lee (1972), prefer a lower confidence limit, or "statistical lower bound" on the squared multiple correlation to a confidence interval. The rationale, apparently, is that one is primarily interested in assuring that the percentage of variance "accounted for" in the regression equation exceeds some value. Although we understand the motivation behind this view, we hesitate to accept it. The confidence interval, in fact, contains a lower bound, but also includes an upper bound, and, in the interval width, a measure of precision of estimation. It seems to us that adoption of a lower confidence limit can lead to a false sense of security, and reduces that amount of information available in the model assessment process.