TABLE 29-1

ENROLLMENT GPA AND MONTHS TO COMPLETION IN AN RN TO BSN PROGRAM

Step 1: Calculate b.
From the values in Table 29-1, we know that n = 20, Σx = 20, Σy = 267, Σx² = 30, and Σxy = 238. These values are inserted into the formula for b, as follows:

b=20(238)−(20)(267)20(30)−20 2

b=−2.9

Step 2: Calculate α.
From Step 1, we now know that b = −2.9, and we plug this value into the formula for α.

α=267−(−2.9)(20)20

α=16.25

Step 3: Write the new regression equation:

y=−2.9x+16.25

Step 4: Calculate R.
The multiple R is defined as the correlation between the actual y values and the predicted y values using the new regression equation. The predicted y value using the new equation is represented by the symbol ŷ to differentiate from y, which represents the actual y values in the data set. We can use our new regression equation from Step 3 to compute predicted program completion time in months for each student, using their number of academic degrees prior to enrollment in the RN to BSN Program. For example, Student #1 had earned 1 academic degree prior to enrollment, and the predicted months to completion for Student 1 is calculated as:

y ̂ =−2.9(1)+16.25

y ̂ =13.35

Thus, the predicted ŷ is 13.35 months. This procedure would be continued for the rest of the students, and the Pearson correlation between the actual months to completion (y) and the predicted months to completion (ŷ) would yield the multiple R value. In this example, the R = 0.638. The higher the R, the more likely that the new regression equation accurately predicts y, because the higher the correlation, the closer the actual y values are to the predicted ŷ values. Figure 29-1 displays the regression line where the x axis represents possible numbers of degrees, and the y axis represents the predicted months to program completion (ŷ values).

FIGURE 29-1  REGRESSION LINE REPRESENTED BY NEW REGRESSION EQUATION.

Step 5: Determine whether the predictor significantly predicts y.

t=Rn−21−R 2   ‾ ‾ ‾ ‾  √

To know whether the predictor significantly predicts y, the beta must be tested against zero. In simple regression, this is most easily accomplished by using the R value from Step 4:

t=.638200−21−.407  ‾ ‾ ‾ ‾ ‾  √

t=3.52

The t value is then compared to the t probability distribution table (see Appendix A). The df for this t statistic is n − 2. The critical t value at alpha (α) = 0.05, df = 18 is 2.10 for a two-tailed test. Our obtained t was 3.52, which exceeds the critical value in the table, thereby indicating a significant association between the predictor (x) and outcome (y).

Step 6: Calculate R².
After establishing the statistical significance of the R value, it must subsequently be examined for clinical importance. This is accomplished by obtaining the coefficient of determination for regression—which simply involves squaring the R value. The R² represents the percentage of variance explained in y by the predictor. Cohen describes R² values of 0.02 as small, 0.15 as moderate, and 0.26 or higher as large effect sizes (Cohen, 1988). In our example, the R was 0.638, and, therefore, the R² was 0.407. Multiplying 0.407 × 100% indicates that 40.7% of the variance in months to program completion can be explained by knowing the student’s number of earned academic degrees at admission (Cohen & Cohen, 1983).
The R² can be very helpful in testing more than one predictor in a regression model. Unlike R, the R² for one regression model can be compared with another regression model that contains additional predictors (Cohen & Cohen, 1983). The R² is discussed further in Exercise 30.
The standardized beta (β) is another statistic that represents the magnitude of the association between x and y. β has limits just like a Pearson r, meaning that the standardized β cannot be lower than −1.00 or higher than 1.00. This value can be calculated by hand but is best computed with statistical software. The standardized beta (β) is calculated by converting the x and y values to z scores and then correlating the x and y value using the Pearson r formula. The standardized beta (β) is often reported in literature instead of the unstandardized b, because b does not have lower or upper limits and therefore the magnitude of b cannot be judged. β, on the other hand, is interpreted as a Pearson r and the descriptions of the magnitude of β can be applied, as recommended by Cohen (1988). In this example, the standardized beta (β) is −0.638. Thus, the magnitude of the association between x and y in this example is considered a large predictive association (Cohen, 1988).

Step 1: From the “Analyze” menu, choose “Regression” and “Linear.”

Step 2: Move the predictor, Number of Degrees, to the space labeled “Independent(s).” Move the dependent variable, Number of Months to Completion, to the space labeled “Dependent.” Click “OK.”