The Power PC Theory

The power PC Theory (Cheng, 1997; Novick & Cheng, 2004) combines the covariational approach with the notion of causal power (Cartwright, 1989). The causal power p_x of an event denotes its capacity to produce an effect: “Causal Power (…) is the intuitive notion that one thing causes another by virtue of the power or energy that it exerts over the other” (Cheng, 1997, p. 368, her italics). Even though Cheng agrees with

other accounts that covariational information is the key to the process of causal induction she assumes that “(…) people do not simply treat observed covariations as equivalent to causal relations; rather, they interpret and explain their observations of covariations as manifestations of the operation of unobservable causal powers, with the tacit goal of estimating the magnitude of these powers.” (Cheng, 1997, p. 369).

According to Cheng, the general idea of causal powers is not derived from experience but is a priori. This domain-independent knowledge enters the process of causal induction in the form of variables which learners seek to estimate. Causal judgments are assumed to be functions of learners’ estimates of causal power.

The power PC model gives a formal account of how estimates of causal power can be derived from covariational information. Cheng’s analysis applies to situations with a candidate cause C and (known or unknown) alternative cause represented as a composite A. In addition, the events involved must be represented by discrete variables which can be present or absent. In accordance with the probabilistic contrast model (Cheng & Novick, 1990), it is assumed that causal relations are evaluated in a chosen focal set (cf. Section 3.2.2). According to the power PC model, the overall probability of the effect depends on the base rates of its causes and their causal powers. The notion of causal power is formalized by introducing parameters representing causal power to standard probability calculus. The probability of an effect E to occur is then given by:⁶

( ) ( ) _c ( ) _a ( ) _c ( ) _a

P e =P c p⋅ +P a p⋅ −P c p P a p⋅ ⋅ ⋅ (6)

Equation (6) states that the overall probability of the effect is a function of the probability of the causes’ base rates (i.e., P(a) and P(c)) and their causal powers (i.e., p_c and p_a), minus their intersection. Provided the causes occur independently, the probability of the effect conditional on the candidate cause yields

( | ) _c ( | ) _{a c} ( | ) _{a c} ( ) _{a c} ( ) _a

P e c =p +P a c p⋅ −p P a c p⋅ ⋅ =p +P a p⋅ − p P a p⋅ ⋅ and (7) ( | ) ( | ) _a ( ) _a

P e c¬ =P a c p⋅ =P a p⋅ (8)

Equation (7) states that when C is present the probability of the effect is determined by i) the causal power of the candidate cause (i.e., p_c), ii) the probability of the alternative cause to occur (i.e., P(a)), and iii) the causal power of the alternative causes (i.e., p_a). Conversely, when C is observed to be absent, the probability of the effect is

6The following equations apply to generative causes. See Cheng (1997) for details on the computation of causal powers for inhibitors.

PSYCHOLOGICAL THEORIES OF C_AUSAL C_OGNITION 26 determined by the probability and causal power of the alternative cause alone (equation

(8)). According to equations (7) and (8) the computation of the conditional probabilities P(e | c) and P(e | ¬c) includes non-observable parameters representing the causal power of the observed events. Substituting equations (7) and (8) into the standard contingency formula (equation (2)) and simplifying yields

(1 ( | )) P pc P e c

∆ = ⋅ − ¬ (9)

Now, by rearranging formula (9) the theoretical entity of causal power p_c can be estimated from observational data:

( | ) ( | ) 1 ( | ) 1 ( | )

c P P e c P e c

p P e c P e c

∆ − ¬

= =

− ¬ − ¬ (10)

Since all parameters on the right-hand side of equation (10) can be estimated from observable frequency information, the unobservable causal power of an event can be estimated from covariational information. Formula (10) states that the contingency ∆P is only an appropriate estimate of causal power when no alternative causes influence the effect. Thus, if P(e | ¬c) = 0, then p_c = ∆P = P(e | c) holds.

According to the power PC model, causal judgments are not determined by the contingency alone but also by the probability P(e | ¬c). With a fixed contingency causal power increases with the number of instances in which the effect occurs in the absence of the candidate cause, a prediction confirmed in a series of experiments by Buehner, Cheng, and Clifford (2003). This finding is also in accordance with the outcome density bias (i.e., the finding that causal judgments are affected by the overall probability of the effect).

The power PC model also makes predictions about the boundary conditions of causal induction. For example, when the effect is always present in the absence of the cause (i.e., P(e | c) = P(e | ¬c) = 1) causal power is not defined because the denominator is zero. Therefore, the model formalizes the intuition that we cannot evaluate the causal power of a generative cause if the effect is constantly present. Indeed, there is empirical evidence that when the effect is always present learners consider covariational data as insufficient to make judgments about a putative cause (Wu & Cheng, 1999).

Critique of the Power PC Model

The power PC model is proposed as both a normative and descriptive model of causal induction. However, the model has been criticized both on grounds of empirical evidence and theoretical analyses.

As noted, the experiments by Buehner et al. (2003) demonstrate that learners take into account the probability of the effect in the absence of the candidate cause. As anticipated by the power PC model, with a fixed contingency learners’ causal judgments varied depending on the magnitude of P(e | ¬c). However, there is also evidence that causal judgments of non-contingent causes are affected by P(e | ¬c), a finding at variance with the power PC model (Buehner et al., 2003; Lober & Shanks, 2000;

Vallée-Tourangeau et al., 1998). However, this finding also challenges all other rational models of causal induction.

Recently, theoretical aspects of the power PC theory have been criticized (Luhmann

& Ahn, 2005; White, 2005). Luhmann and Ahn (2005) have provided a detailed analysis of the assumptions of the power PC model (cf. Cheng, 1997, p. 373).

According to their analysis, the conditions necessary to derive estimates of causal power from observable information are rarely met and, therefore, the model is too restrictive to provide an adequate account of causal induction. In addition, the power PC model tacitly assumes that causal powers are inherently probabilistic (i.e., the capacity of a cause to produce an effect is not only probabilistic because of unobserved inhibitors), an assumption Luhmann and Ahn claim to be at variance with people’s intuition about causality.

While Luhmann and Ahn focus on the assumptions necessary to derive causal power from regularity information, White (2005) criticizes the claim that the power PC model successfully integrates regularity theories with the notion of causal power. In the power PC theory, causal powers are defined as the probability with which one event, the cause, produces another event, the effect. According to White, Cheng’s definition of causal power is incompatible with traditional power theories which assume causal powers to be stable properties grounded in the physical nature of the entities involved (e.g., Harré & Madden, 1975). Therefore, he argues, the power PC model is incomplete and falls short of reconciling the rivaling regularity and power views.

PSYCHOLOGICAL M_{ODELS OF} C_AUSAL C_OGNITION 28