Your understanding is correct. I’ll just note that this equation is equivalent to the formula for the stability factor provided by Powers in his 1978 Psych Review paper:
In Bill’s version, V.exp (the expected variance of the variable if there is no control) is divided by V.obs (the observed variance of the variable) so that good control is indicated when V.exp>>V.obs and S is a large negative number.
In my version, V.obs is divided by V.exp (rather than vice versa) so good control is again indicated when V.exp>>V.obs making S close to 1.0. In both versions, no control is indicated when V.obs = V.exp and S = 0. I like my version better because it seems more intuitive (to me, anyway) that the measure of control, S, range from 0 (no control) to 1.0 (perfect control). Also, my version eliminates the admittedly remote possibility of getting a divide by 0 (if the variable were perfectly controlled so that V.obs = 0).
It’s there because the variance of a controlled variable, q.i, is determined simultaneously by the disturbance, d, and the output, o, of the system.
q.i = o + d
If the variable, q.i, is really not controlled then the variance of o and d will be independent of each other and, from a basic theorem of statistics – for independent random variables the variance of their sum is equal to the sum of their variance – we get:
var (o+d) = var(qi) = var(o) + var(d)
However, when there is control var(o) and var(d) are highly (negatively) correlated (since o is busy cancelling d) so var(o) and var(d) are not independent and:
var (o+d) = var(q.i) << var(o) +var(d)
Since V.obs = var (q.i) and V.exp = var(o) + var(d), the ratio V.obs/V.exp will be close to 1 when there is no control and close to 0.0 when there is control. Since my stability formula is S = 1 - (V.obs/V.exp)^1/2, the stability measure will be close to 1 when there is control and close to 0 when there is no control.
This is not true on both counts. First, I didn’t include the variance of output in order to get the result I was looking for. As noted above, the variance of output is a fundamental component of the calculation of the stability factor. However, in my spreadsheet demo I do get the same results (in terms of demonstrating the Dead Zone) whether I include the variance in output in the calculation of S or not. In other cases, such as my mindreading demo, which uses S as the basis for doing its “mind reading”, the inclusion of the output variance in the calculation of S is essential in order to correctly determine which of the three variables is being controlled relative to varying references.
And, second, I was not looking to show that “collective control with conflict is less stable than control by an individual controller”. I was testing to determine whether, in a conflict, there is, in fact, a Dead Zone where there is no control of the virtually controlled variable. And, per Powers in B:CP, there is!
As explained above, the amount of output is not what matters; What matters is the variance of the output and the degree to which that variance is negatively correlated with the variance in the disturbance.
Thanks for the compliment but it’s not my stability factor, it’s Bill’s (as noted above). And it appears to me that there is no conceptual difference between us. Like you, I think that the quality of control – or just control – is an objective fact that is seen when the variance of a variable, V.obs, is far less than the variance expected, V.exp, if there were no control.
I think you are confusing my comments about the stability of virtual and actual controlled variables with those I made about the error experienced by the parties to a conflict. Regarding the stability of virtual versus actual control, what I have found is that the stability, S, of virtual controlled variables can be equal to or even greater than that of actual controlled variables as long as the amplitude of the distubance to these variables is well outside the range of the Dead Zone of the virtual controlled variable. When, for example, the amplitude of the disturbance is far outside the range of Dead Zone, the behavior of the virtual controlled varable can be indistinguishable from an actual one.
Regarding your second point the subjective experience of the parties to a conflict, the PCT model tells me that, even in the case where the virtual controlled variable is being kept very stable in a virtual reference state, the conflicted systems are experiencing a great deal more error than an equivalent unconflicted control system controlling an actual controlled variable. I think this is why Bill called control that results from conflict “virtual control”. Virtual control looks like actual control but it’s not experienced as actual control by the participants in the conflict.
This doesn’t mean that virtual control is always a bad thing. For example, you can use my spreadsheet to show that, when the amplitude of the disturbance exceeds the range of the Dead Zone by a sufficiently large amount, the stability of the virtually controlled variable is actually greater than that of the actual controlled variable and the error experienced by the parties controlling the virtually controlled variable is actually less than that experienced by the system controlling the actual controlled variable.
For now, my main point in building the Virtual Control spreadsheet is to show that PCT does, indeed, predict a Dead Zone for a virtually controlled variable. This confirms what Bill said (almost as an aside) in B:CP and shows that my description of the Dead Zone in my book, SLCS, based on what Bill said in B:CP, is correct.
I think this finding of a Dead Zone in conflicts is a phenomenon that should be of considerable interest to social psychologists and sociologists interested in understanding interpersonal conflicts (such as those between people and countries) and to mental health professionals interested in understanding intrapersonal conflicts (such as those that occur between control systems in the same individual).