Introducing a Third Variable - Intervening Variables
Introducing a Third Variable - Intervening Variables
An intervening variable is conceptually very different from a variable causing spurious association:
SPURIOUS ASSOCIATION INTERVENING VARIABLE
BETWEEN A AND B BETWEEN A AND B
A<-C->B A->C->B
However, from a data analysis viewpoint the two cases are indistinguishable. In each case, the association between A and B will disappear if the analyst controls for C. In the first case, it is because C caused both A and B or is associated with both A and B. In the second, it is because C intervened between A and B; that is, A causes C, which in turn causes B.
The two cases can be distinguished by determining the direction of causation between A and C. If it runs from A to C, then an intervening variable is involved. In the example of Table 15-1, the interpretation is actually rather clear. It is impossible for the advertising during the test period to influence behavior before that period. Further, it is very possible that past usage did not influence attention to, and readership of, the advertisements and, thus, advertising recall. It is not unusual for the advertising exposure to be higher among those in the audience who use and are thus involved with the advertisement. Thus, it was concluded that the most appropriate model was
A SPURIOUS ASSOCIATION
BETWEEN A AND I
A <-U -> I
If the third variable seemed to intervene between the two variables, the association—even though eliminated by the third variable—would not be deemed spurious, because the link between the two variables still would be meaningful. Suppose that three variables of interest were:
A = advertising expenditures O = attitude of opinion leaders G = attitude of the general population
If the association between A and G were eliminated when the analysis controlled for O, the interpretation might be that advertising influenced general attitudes by influencing opinion leaders. Thus we would have
AN INTERVENING VARIBLE BETWEEN A AND G
A->O->G
and the association of A and G would not be regarded as spurious.
Of course, there could be a causal path directly between the two variables A and B and also an indirect influence from A to B operating through variable C:
BOTH DIRECT AND INDIRECT CAUSAL RELATIONSHIPS BETWEEN A AND B A —> B
V
Additive Causal Relationship
The introduction of a third variable also can have an additive causal effect The diagram depicting such an additive causal relationship would be:
AN ADDITIVE EFFECT ON B A->B
/ C
For example, Figure 15-2 indicates that there is an association between people's concern for the environment in general (£) and their concern about air pollution (A). In addition, the method of commuting (C) also influences their attitudes toward air pollution. Thus, it might be reasonable to hypothesize that there is:
AN ADDITIVE EFFECT ON A E-» A
/ C
Environmental Concern (E)
E^Low E^High
Commute (C) Commute (C)
E1-Low E2-High Ct-Car C2-Other C1-Car C2-Other
Attitude toward air
pollution (A)a 4.1 5.6 3.7 4.8 5.2 6.3
Sample size 75 90 30 45 60 30
aOn a 1-7 (low to high concern) scale.
Commute other ;
| 6 .iitliSv .'2'-:::::::
"3
polor
ro a " 'Commute
towardCO by car [C{,
CD
a 2
§ i
Low High
Environmental Environmental
concern (E{i concern CE2)
FIGURE 15-2
Air pollution attitudes.
The lower line in the graphical representation in Figure 15-2 shows the relationship between A and £ for those who commute by car. The upper line is the relationship between A and E for those who commute without using a car. The effect of the commute method has been added to the effect of E. The fact that the two lines are parallel is characteristic of an additive relationship. Recall the examples in the discussion of Figure 10-1. Unless an interactive term is included, the assumption in an experiment is that the treatment and block effects are additive. Similarly, most of the techniques to be introduced in Part IV also will employ the additive assumption.
An intervening variable is conceptually very different from a variable causing spurious association:
SPURIOUS ASSOCIATION INTERVENING VARIABLE
BETWEEN A AND B BETWEEN A AND B
A<-C->B A->C->B
However, from a data analysis viewpoint the two cases are indistinguishable. In each case, the association between A and B will disappear if the analyst controls for C. In the first case, it is because C caused both A and B or is associated with both A and B. In the second, it is because C intervened between A and B; that is, A causes C, which in turn causes B.
The two cases can be distinguished by determining the direction of causation between A and C. If it runs from A to C, then an intervening variable is involved. In the example of Table 15-1, the interpretation is actually rather clear. It is impossible for the advertising during the test period to influence behavior before that period. Further, it is very possible that past usage did not influence attention to, and readership of, the advertisements and, thus, advertising recall. It is not unusual for the advertising exposure to be higher among those in the audience who use and are thus involved with the advertisement. Thus, it was concluded that the most appropriate model was
A SPURIOUS ASSOCIATION
BETWEEN A AND I
A <-U -> I
If the third variable seemed to intervene between the two variables, the association—even though eliminated by the third variable—would not be deemed spurious, because the link between the two variables still would be meaningful. Suppose that three variables of interest were:
A = advertising expenditures O = attitude of opinion leaders G = attitude of the general population
If the association between A and G were eliminated when the analysis controlled for O, the interpretation might be that advertising influenced general attitudes by influencing opinion leaders. Thus we would have
AN INTERVENING VARIBLE BETWEEN A AND G
A->O->G
and the association of A and G would not be regarded as spurious.
Of course, there could be a causal path directly between the two variables A and B and also an indirect influence from A to B operating through variable C:
BOTH DIRECT AND INDIRECT CAUSAL RELATIONSHIPS BETWEEN A AND B A —> B
V
Additive Causal Relationship
The introduction of a third variable also can have an additive causal effect The diagram depicting such an additive causal relationship would be:
AN ADDITIVE EFFECT ON B A->B
/ C
For example, Figure 15-2 indicates that there is an association between people's concern for the environment in general (£) and their concern about air pollution (A). In addition, the method of commuting (C) also influences their attitudes toward air pollution. Thus, it might be reasonable to hypothesize that there is:
AN ADDITIVE EFFECT ON A E-» A
/ C
Environmental Concern (E)
E^Low E^High
Commute (C) Commute (C)
E1-Low E2-High Ct-Car C2-Other C1-Car C2-Other
Attitude toward air
pollution (A)a 4.1 5.6 3.7 4.8 5.2 6.3
Sample size 75 90 30 45 60 30
aOn a 1-7 (low to high concern) scale.
Commute other ;
| 6 .iitliSv .'2'-:::::::
"3
polor
ro a " 'Commute
towardCO by car [C{,
CD
a 2
§ i
Low High
Environmental Environmental
concern (E{i concern CE2)
FIGURE 15-2
Air pollution attitudes.
The lower line in the graphical representation in Figure 15-2 shows the relationship between A and £ for those who commute by car. The upper line is the relationship between A and E for those who commute without using a car. The effect of the commute method has been added to the effect of E. The fact that the two lines are parallel is characteristic of an additive relationship. Recall the examples in the discussion of Figure 10-1. Unless an interactive term is included, the assumption in an experiment is that the treatment and block effects are additive. Similarly, most of the techniques to be introduced in Part IV also will employ the additive assumption.
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