Testing Significance And Strength Of Relationship In Simple Regression

Although you can do a test of significance on the Pearson correlation (see p. 58) to determine if there is a relationship between the IVs and DVs, this significance testing commonly comes about in a different way. Frequently, the computer printout will list a table of numbers with headings like those in Table 6-1. The coefficients are the intercept (constant), which equals 98.0 and can be identified on the graph. Similarly, the second line of the table is the slope of the regression line, which equals 1.14. The computer also

Skinfold (cm) Figure 6-1 Relationship between blood sugar and skinfold.

calculates the standard error (SE) of these estimates, using complicated formulas. The t test is the coefficient divided by its SE, with n - 2 degrees of freedom (where n is the sample size), and the significance level follows.

Usually, the computer also prints out an "ANOVA table." But, sez you, "I thought we were doing regression, not ANOVA." We have already drawn the parallel between regression and ANOVA in the previous section. These Sums of Squares end up in the ANOVA table as shown in Table 6-2. If you take the square root of the F ratio, it equals exactly the t value calculated earlier (as it should because it is testing the same relationship).

Table 6-1

Output from Regression Analysis

Table 6-1

Output from Regression Analysis

Variable

Coefficient

SE

t

Significance

Constant

98.00

12.3

7.97

0.0001

Skinfold

1.14

0.046

24.70

0.0001

SE = standard error.

SE = standard error.

Table 6-2

ANOVA Table from Regression Analysis

Source Sum of Degrees of Mean

Squares Freedom Square F p

Table 6-2

ANOVA Table from Regression Analysis

Source Sum of Degrees of Mean

Squares Freedom Square F p

Regression

2,401.5

1

2,401.5

610.1 < 0.0001

Residual

270.0

18

15.0

— —

Finally, the strength of relationship could then be expressed as the ratio of Sum of Squares (SS) (regression) to [SS (regression) + SS (residual)], expressing the proportion of variance accounted for by the IV. In fact, the square root usually is used and is called a Pearson correlation coefficient.

Correlation = /_Sum ofSquares(regression)_

■J Sum of Squares (regression) + Sum of Squares (residual)

So, in the present example, the correlation is:

We also could have tested significance of the relationship directly by looking up significance levels for different values of the correlation coefficient and different sample sizes. This is, of course, unnecessary at this point.

We can interpret all this graphically by referring back to Figure 6-1. In general, the individual data points constitute an ellipse around the fitted line. The correlation coefficient is related to the length and width of the ellipse. A higher correlation is associated with a thinner ellipse and better agreement between actual and predicted values.

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