Noncentrality Interval Estimation and the Evaluation of Statistical Models

Select the Quick tab of the Multiple R-squared: Interval Estimation dialog box to access options to perform interval estimation calculations for the multiple R². The squared multiple correlation, R², is often reported in experimental research, but is hardly ever reported with a confidence interval, or estimate of variability, probably because such confidence intervals require complicated, iterative calculations. Steiger and Fouladi (1992) presented techniques for computing exact confidence intervals on the population squared multiple correlation.

Observed R². Enter the observed value of R² in the Observed R² box. This value must be the standard R² value, not the "shrunken" estimator.

C² (Null P²). In the C² (Null P²) box, enter the value for the hypothesized multiple correlation. Besides computing confidence intervals on P², the Power Analysis and Interval Estimation module also computes post hoc statistical bounds on power itself. Since the program allows testing of nonstandard hypotheses about P² (i.e., hypotheses other than P² = 0), power is a function of the specific kind of hypothesis being tested, as well as the value of P² and the sample size. Consequently, if you want the post hoc confidence intervals on power to be accurate, you must adjust the value in this field and the Type of Hypothesis (see below) to reflect the actual null hypothesis that was tested.

Sample Size (N). Enter the sample size for the analysis in the Sample Size (N) box.

Ind. Vars. (k). Enter the number of independent variables in the multiple regression prediction equation in the Ind. Vars. (k) box.

Alpha. Enter the type I error rate for the overall significance test in the Alpha box.

Conf. Level. Under Conf. Level, enter the confidence level for the confidence intervals computed by the program (see below). Note that confidence limits are given for the following quantities for the selected effect:

P². These confidence limits are for the population squared multiple correlation.

Power. These confidence limits
are post hoc statistical bounds on power for the statistical test as currently
defined in this dialog.

Required N. These confidence limits are post hoc statistical bounds on the N required to assure the power specified as the Power Goal.

Power Goal. In the Power Goal box, enter the value for power used by the program when converting a confidence interval on P² into a confidence interval for the required N.

Computing Algorithm. The Power Analysis module allows you to choose from two algorithms:

Faster. Select the Faster option button to employ Lee's (1971, p. 123) non-central F approximation. This approximation is accurate to about four decimal places, and is satisfactory for most applications.

More Accurate. Select the Most Accurate option button to perform a slower, but more accurate calculation, based on theory described in Lee (1972), typically exact to at least five decimal places.

Type of Hypothesis. The choice of option button under Type of Hypothesis determines the type of null hypothesis tested.

2-tailed (P2 = C2). Select the 2-tailed (P2 = C2) option button to test null and alternative hypotheses of the form

H0:
P2

1-tailed (P2 <= C2). Select the 1-tailed (P2 <= C2) option button to test null and alternative hypotheses of the form

H0:
P2

1-tailed (P2 >= C2).

H0:
P2

1-tailed (P2 = 0).

H0:
P2

(This is the hypothesis tested by the standard F-statistic for zero multiple correlation.)