For the eight misidentification, we regressed X1 on EX31-EX33, LEN1-LEN4 on X1, and Y1 on X1, LEN1-LEN4, and EX1-EX20 using Lavaan (illustrated in the Figure below). This was replicated 10,000 times – using R-loops – while randomly varying the effects for each causal pathway between the constructs in the network excluding the direct effect of X1 on Y1 (true direct causal effect = 1.00; N = 10,000; R-Code).
Previous to the current example, we focused on the average estimated rather than the modal – the most common – estimate because the previous examples produced normal distributions. As observed below, however, the results of the 10,000 simulations are not normally distributed. More explicitly, approximately 3500 (35%) of the 10,000 simulations produced the true causal relationship of 1.00 between X1 and Y1. The remaining estimations were biased to various degrees, with no single estimate appeared more than 500 times. The reason the true causal estimate was produced so often is because we closed all of the back-door and front-door paths between X1 and Y1 (Pearl, 2009). Nevertheless, bias estimates were produced because the remaining true associations in the structural network were set to randomly vary between .15 and 5.00. The magnitude of the true associations along the back-door and front-door paths dictate if (1) those back door and front-door paths must be closed to estimate the true association and (2) the amount of bias produced when the back-door and front-door are closed. Specifically, a weaker back-door or front-door pathway will increase the bias in the estimated association between X1 and Y1. Given that we estimated the true casual association for the first structural network, next week we will create a new structural network and start all over. The next structural network will permit us to estimate the true casual association using descendants along the back-door and front-door pathways.
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