Abstract
Human behavior can be conceptualized as being influenced by three factors: (1) a person’s prior personal dispositions, which include perceptions, attitudes, values, desires, beliefs, capabilities, and schemas; (2) the impingement of social environments on that person; and (3) the interactions between the predisposing and environmental factors. These factors imply a multilevel analysis of at least two levels, that of the individual (referred to as level-1) and that of the environment (referred to as level-2). A contextual study exemplifies a multilevel analysis because it includes variables on the individual and on the environment. Contextual effects are the cross-level interactions between the personal and environmental variables, and the study of these interactions defines contextual analysis. The latter includes comparative analysis, which links the level-2 variable directly to a level-1 response.
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Notes
- 1.
Pearl’s d-separation criterion suggests that in this chain of relationships (Pearl [2000] 2009, second edition, 16–17, and personal communication of 4/2/2010):
marital status (i) → amount of housework (m) → more absenteeism (j)
1. The amount of housework blocks the effect of marital status on more absenteeism;
2. Marital status has a causal effect on absenteeism via more housework.
Definition 1.2.3 (d-Separation)
A path p is said to be d-separated (or blocked) by a set of nodes Z if and only if
1. p contains a chain i → m → j or a fork i ← m → j such that the middle node m is in Z, or
2. p contains an inverted fork (or collider) i → m ← j such that the middle node m is not in Z and such that no descendant of m is in Z.
A set Z is said to d-separate X from Y if and only if Z blocks every path from a node in X to a node in Y.
“Marital status” and “more absenteeism” are marginally associated but become conditionally independent when “the amount of housework” is held constant. That is, by conditioning on “the amount of housework,” “getting married” has no direct effect on “more absenteeism.” Figuratively, the intervening variable blocks the flow of information along the path (Pearl [2000] 2009, second edition, 17), knowing the amount of housework renders knowing marital status irrelevant to absenteeism.
Wermuth labels the following chain as exemplifying spurious dependence:
successful job placement ← field of qualification ← gender
Here, “field of qualification” blocks every path from “gender” to “successful job placement.” She interprets this model as implying that “field of qualification” causes “successful job placement,” and that “gender” does not cause “successful job placement” via its effect on “field of qualification.” Although Pearl interprets this model as showing that gender causes successful job placement indirectly through field of qualification (personal communication, 4/2/2010), he implies that legal cases support Wermuth’s interpretation ([2000] 2009, second edition, 127):
Another class of examples involves legal disputes over race or sex discrimination in hiring. Here, neither the effect of sex or race on applicants’ qualification nor the effect of qualification on hiring are targets of litigation. Rather, defendants must prove that sex and race do not directly influence hiring decisions, whatever indirect effects they might have on hiring by way of applicant qualification.
Whether such antecedent x variables indirectly cause the response y via their effects on intervening variables t requires further study. For example, focus on t → y and this relationship is found not to be spurious when antecedent propensity scores or test factors x i are controlled. Then, the effect of an extra x that affects y and t is controlled, and x (extra) → t → y. What is gained? The t → y relationship is still not spurious, t is a candidate cause of y. Additionally, t interprets the effect of x (extra) on y.
- 2.
Lazarsfeld’s statement suggests the following path-analytic definitions of causal effects: Let x be prior to y and t be either prior to or on equal footing with x. Then, if β yx.t (the path coefficient) is not equal to zero and t has no direct effect on y, then by marginalizing over t, the bivariate relationship r yx would express the total effect of x on y. However, the path-analytic decomposition r yx = β yx.t + r xt β yt.x (i.e., direct effect of x + shared effect with t) suggests that the estimated causal effect of x = r yx − r xt β yt.x = β yx.t , which is the direct effect of x on y. However, if x is prior to t and x → t, then the putative causal effect of x on y is the correlation: r yx = β yx.t + β tx β yt.x (i.e., direct effect + indirect effect of x through t). The size of the path-analytic causal effect is affected by the type of arrow linking x and t in the path diagram, even though the size of r xt = β tx . As Pearl stresses (Pearl 2002, 208) and exemplifies (Pearl [2000] 2009, second edition, 154–157), a graph of the assumed relationships paired with parameter estimates can clarify notions of causality.
- 3.
Pearl (2002, 208) defines a probabilistic cause in the context of a probability distribution and a directed acyclic graph (i.e., a model) as follows: “X is a probabilistic cause of variable Y if P (y| do (x)) ≠ P (y) for some values x and y.” The agent-based model of Smith (2010b) can be used to clarify this notion of causality in the framework of the model. Let the response variable Y be a measure of the nastiness of the agents. A number of prior variables such as cost-benefit distributions, prejudice, sociometric ties, and so forth are set in advance as parameters of a Monte Carlo run of 20 iterations through the model. These parameters define a context S (for set in advance). Additionally, each agent can have his own unique measure of legitimacy that he attributes to authority; these may range from low to high, say from 0.6 to 1.2. Then, the “naturally occurring” distribution of Y, the output distribution, is referred to as P(Y|S); that is, the naturally occurring probability distribution of Y given the parameters S. Next, a second run of the model is implemented, holding constant the parameters composing S. However, in this run the legitimacy of authority parameter is not allowed to vary naturally; it is set by the experimenter so that each agent now has the value x = 1.3. The model now creates a new output probability distribution. This distribution is referred to as P(y | do(x) ∪ S). The causal effect of the intervention that changed the amount of legitimacy of authority is the difference between the results of the otherwise identical two runs of the model: the causal effect of the intervention is δ = P(y| do (x) ∪ S) – P(y|S).
Given this model and the settings of the runs, this expression is totally consistent with Rubin’s potential outcomes view of causality. The agents in both runs are initially identical. Prior to assignment to one run or the other, the agents have potential outcomes if assigned to either run. After assignment and the running of the Monte Carlo trials, the agents have realized outcomes under the intervention (i.e., the treatment) and under the “naturally occurring” null treatment. The fundamental problem of causal inference is not applicable here because the agents in the trials are identical and fully observable; the only difference between them is random variation due to the random number generator that assigns benefits and costs, and the experimental intervention. Thus, the average causal effect of the intervention is δ = P(y| do (x) ∪ S) – P(y|S). The model produces scores for the variables for each agent after each iteration. Consequently, at the level of the individual agent, the agent’s nastiness at a point in time in one trial could be compared to his nastiness at the same point in time in the other trial. The individual-level causal effect of the intervention at a point in time for agent i can be defined as δ i = y i (1) – y i (0), with y indicating the agent’s response, 1 indicating the intervention group, and 0 the “naturally occurring” null group (Morgan and Winship 2007, 33). Average causal effects would result by averaging such differences for individuals across the agents at various points in time.
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Smith, R.B. (2011). Concepts and Considerations. In: Multilevel Modeling of Social Problems. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9855-9_1
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