Partial F-Tests

The partial F-test is similar to the F-test, except that individual or subsets of predictor variables are evaluated for their contribution in the model to increase SSr or, conversely, to decrease SSg. In the current example, we ask, what is the contribution of the individual x and X2 variables  

To determine this, we can evaluate the model, first with X in the model, then with X2. We evaluate Xj in the model, not excluding X2, but holding it constant, and then we measure it with the sum-of-squares regression (or sum-of-squares error), and vice versa. That is, the sum-of-squares regression is explained by adding Xj into the model already containing X2 or SSR( cj t2)- The 

X t s contribution to the model containing x. i variables or various other combinations. 

For the present two-predictor variable model, let us assume that Xi is important, and we want to evaluate the contribution of X2, given Xi is in the model. The general strategy of partial F-tests is to perform the following  

Find the difference between the model containing only Xi and the model containing X2, given Xi is already in the model, (x2 xi) this measures the contribution of X2. 

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Each independent variable in the regression model is then re-examined to see if it is still making a significant contribution. Any variable whose partial F-test value is not significant at the 10 level is dropped from the model. This process continues until no more variables enter the model and none are rejected. 

If Ha is accepted, one does not know if all the b,s are significant or only one or two. That requires a partial F-test, which is discussed later.)... 

To compute the partial F-test, the following formula is used ... 

Table 4.4A presents the full regression model this can be decomposed to Table 4.4B, which presents the partial decomposition of the regression. Let us perform a partial F-test of the data from Example 4.1. 

As an alternative to performing the partial F-test to determine the significance of the Xi predictors, one can perform f-tests for each jS which is automatically done on the MiniTab regression output (see Table 4.2, r-ratio column). Recall... 

At times, a researcher may want to know the relative effects of adding not just one, but several variables to the model at once. For example, suppose a basic regression model is Y = bo + b Xi + and the researcher wants to know the effects of adding X3, X4, and JC5 to the model simultaneously. The procedure is a direct extension of the partial F-test just examined. It is the sum-of-squares that results from the addition of JC3, JC4, and X5 to the model already containing x and X2... 

Now that we have calculated the partial F test, let us discuss the procedure in greater depth, particularly the decomposition of the sum-of-squares. Recall... 

Adding extra predictor x, variables that increase the SSr value and decrease SSe incurs a cost. For each additional predictor x, variable added, one loses 1 degree of freedom. Given the SSr value is increased significantly to offset the loss of 1 degree of freedom (or conversely, the SSe is significantly reduced), as determined by partial F-test, the x, predictor variable stays in the model. This is the basis of the partial F-test. That is, if Fc > Ft, the addition of the extra variable(s) was appropriate. ... 

One can also employ partial correlation coefficient values to determine the contribution to increased r or values. This is analogous to partial F-tests in the ANOVA table evaluation in Chapter 4. The multiple r and values are also related to the sum of squares encountered in Chapter 4, in that, as r or increases, so does SSr, and SSr decreases. 

However, the multiple partial coefficient of determination generally is not as useful as the F-test. If the multiple partial F-test is used to evaluate the multiple contribution of independent predictor values, while holding the others (in this case Xi, X2) constant, the general formula is... 

The researcher s perceived importance of the predictor variables (x,) does not hold, based on partial F tests. 

The partial F tests on other models are constructed exactly as presented in Chapter 4. 

The readers can test the other partial F test on their own. Notice, however, that the spline procedure provides a much better fit of the data than does the original polynomial. For work with splines, it is important first to model the curve and then scrutinize the modeled curve overlaid with the actual data. If the model has some areas that do not fit the data by a proposed knot, try moving the knot to a different jc value and reevaluate the model. If this does not help, change the power of the exponent. As must be obvious by now, this is usually an iterative process requiring patience. 

The use of the partial F-test is also an important tool in interaction determination. If F x2, xi,x2) is significant, for example, then significant interaction is present, and the X and X2 terms are conditional. That is, one cannot talk about the effects of X without taking X2 into account. Suppose there are three predictor variables, Xj, X2, and X3. Then, the model with all possible interactions is ... 

Each of the two-way interactions can be evaluated using partial F-tests, as can the three-way interaction. 

Looking at the regression model, one can see that (Ca level) probably serves no use in the model. That variable should be further evaluated using a partial F test. 

Regression models are then recalculated for each group. In the next step, an F test is conducted for (1) y intercept equivalence, (2) parallel slopes, and (3) coincidence, as previously discussed. If the two regression functions are not different—that is, they are coincidental—the model is considered appropriate for evaluating the individual x, variables. If they differ, one must determine where and in what way, and correct the data model, applying the methods discussed in previous chapters. If the split group regressions are equivalent, the evaluation of the actual X predictor variables can proceed. In previous chapters, we used a partial F test to do this. We use the same process... 

The first selection procedure we discuss is stepwise regression. We have dime this earlier, but not with a software package. Instead, we did a number of partial regression contrasts. Briefly, the F-to-Enter value is set, which can be interpreted as an Ft value minimum for an jc,- variable to be accepted into the final equation. That is, each x,- variable must contribute at least that level to be admitted into the equation. The variable is usually selected in terms of entering one variable at a time with n — k ldf. This would provide an Ft at a = 0.05 of Ft(o.o5,i,ii) = 4.84. The F-to-Enter (sometimes referred to as F in ) is arbitrary. For more than one x,- variable, the test is the partial F test, exactly as we have done earlier. We already know that only X2 would enter this model, because SSr sequential for X2 = 28.580 (Section C, Table 10.2). 

Two different regression models, containing different numbers of variables kj (smaller number) and k2 (larger number), can be eompared by a sequential (partial) F test (eq. 132). The use of the model containing the larger number of variables is justified if the resulting partial F value indicates a 95% significance (Table 19) for the introduction of the new variable/s. 

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