Modeling for Negative Deviations

I asked Peter for his thoughts on possible explanations for the sometimes quite strong negative deviations of the netvar statistic, for example as seen in a number of large, organized meditations. It is actually a general question that bears on mechanism. This page is a discussion of possibilities or tentative models, which leads to the conclusion that it is at least reasonable to consider bit-level autocorrelation as a fundamental part of the picture. More work is needed, but this is interesting and thought provoking material:

Negative Going Netvar

We can frame our results as follows. We look at the 1-sec Stouffer Z and find that Var[Z] and Var[Z^2] deviate positively over the events. These are the netvar and covar. The underlying stat is the stouffer Z. It is the normalized trial mean. In thinking about what is going on we can take Z to be a normal score, OR we can consider it to be a mean value of the reg trials OR we can consider it to be a bitsum over regs. If we ask how to interpret a pos. or neg. netvar/covar, we need to know what level of data structure we're interested in (Z, regs or bits). We'll answer a bit differently in each case.

Z, the network level

If we just consider the mean output, Z, we're not asking anything about the underlying structure at the reg or bit level. Var[Z] and Var[Z^2] can vary, because they are random variables, and finding a pos or neg trend doesn't tell us much, other than there is a deviation for these stats that has a small pvalue.

Note: I hope your email reads Greek fonts; below I use β for beta and Δ for delta and μ for mu...

z, the reg (trial) level

If we consider that Z is based on the sum of z's from independent devices, then we can ask how a deviation in Var[Z] or Var[Z^2] might affect the individual reg statistics. Different senarios can be examined by simulation. For example, the netvar/covar deviations may be due to contributions from all regs. This is what we currently believe. We can express this by saying that the effect is distributed across the network or by saying that the regs are correlated at the 1-sec level. This is demonstrated by a simple simulation that I've done: Input reg trial values as normal z-scores (taking, say, 50 of them to represent a network of 50 online regs). I can run this to simulate events a few hours long and look at the netvar and covar over ~200 events as we do in the formal experiment. Using standard normal scores, the netvar and covar have zero deviation, and all is copacetic. Next, try this while adding a small, random deviation to the mean or variance of *the individual reg distributions*. That is, instead of drawing the reg trial values from a N[0,1] distribution I draw from a N[0+β, 1+Δ. The β and Δ are small random shifts in the reg mean and variance. I choose the shifts randomly each second, but, each second, the same mean/var shift is applied to all regs online. In the past, you have termed this an average mean shift of the regs when discussing the Stouffer Z^2 increase. Here is what the simulation shows: we can reproduce the netvar/covar effects by taking β and Δ to be randomly selected in the range ± 0.01. That is, the mean(and the variance) of the underlying distribution of reg trial values is allowed to vary slightly, each second, but with an *average* variation of zero. This guarantees a Mean[Z] =0 , as we find for the event data, but yields a positive netvar/covar variation on the scale of what we see for the events. We learn one thing here: β affects the netvar and Δ affects the covar. So we can say our data is *consistent* with a correlated mean shift (from the netvar results) and a correlated variance shift (from the covar results) at the reg level, and that these shifts on average are zero and fluctuate around zero by about 1%.

A bottom line comment is that since we don't see deviations in the Mean[Z] or the devvar for the real event data, any deviations in underlying distributions at the reg level probably are fluctuating with zero mean, that is <β> = <Δ> = 0, as implemented in this simulation. Also, for the β,Δ deviation model, we are only able to simulate positive deviations in the netvar and covar.

There are other possibilities and we will need to do some work to assess them. For example, it could be that only one or a few regs have deviating means and variances, or that the deviations are restricted to regs in a geographical area (which would give a distance effect to the pair-products...). More simulations are needed and I think you can see that it quickly becomes a project. I suspect we have some leverage since we can monitor the devvar for these different situations. That might allow us to control for different senarios. We can also look directly at the distributions of the individual regs, both in simulation and for the event data. This will most likely let us put some limits on possible senarios.

There is another point that needs to be stated. Currently, the netvar/covar results cannot be distinguished from a simulation where reg pseudo-data is generated with N[0,1] (or, equivalently, Z is generated from N[0,1] ) *and* then events are selected by a filter giving an average event Z-score of 0.3. This is an experimenter effect senario. I currently have little *in the data* that distinguishes DAT-like selection from the "imposed" β,Δ mean/variance shifts. The distance correlation and the netvar/covar correlation (which comes in at ~ 1 to 2 hour blocking) may be an argument against selection/experimenter intuition, but it's weak at this point.

010101, the bit level

If we go down to the bit level we can start to address your original question. If we consider Z to be based on a bitsum over all regs, we can ask how the bit distribution must deviate in order to give the netvar/covar results. Since the data are XOR'd, we need to consider that as well. I've only thought about this for the netvar. The covar netvar/covar results cannot be distinguished from a simulation where In what follows, consider that the effect is evenly distributed across regs. Then we can take the network as a single bit generator with 200N bits/second (ie, from an N-reg network). I am aware of two ways to alter the binomial output: alter p, the bit probability, or impose a short lag autocorrelation. The binomial variance (which is the netvar in this case) goes as p-p^2. That is, it is an inverted quadratic with a maximum at p=0.5. Any shift in p, positive or negative, will lower the measured variance. The XOR passes the variance change, that is a negative-going netvar will occur, with the same magnitude, whether there is an XOR in place or not. So we can't obtain a netvar increase, as we see for the events, by a shift in p. simulation? What if p is fluctuating second-by-second along the lines of the β,Δ simulation. Here the XOR will still cancel any mean shift on average and the variance will just be the expectation of the variance over the distribution of p's fluctuations, ie, negative-going. So by this reasoning, we can't increase the netvar by diddling p.

What about imposing an autocorrelation on the bits? Suppose we have an autocorr at a lag of 1 with a strength μ. In this case we find that the variance V changes to (1-μ)V. So the variance decreases for a positive autocorr and is increasing for a negative μ (anti-correlation). A simple XOR will invert this. This means that a positive-going netvar *could* indicate a short-lag anti-correlation at the bit level and a negative netvar trend *could* indicate a positive bit autocorrelation. From this corner of senario-space, we are suddenly talking a talk we haven't done before. There is a suggestion that the netvar result implies an autocorrelation in the regs' bitstreams. [It may also put the PEAR data in a new light - they describe operator experiments as a shift in p; here, we'd say that, since the regs are XOR'd, they should really discuss a real autocorrelation in the devices, not a p-shift of the XOR'd output...]

The bottom line to all this is that we can imagine a positive or a negative netvar by imposing an appropriate autocorrelation on the bitstream. One nice thing is that the sign of the autocorrelation gives a very different character to what happens to the data: up is very different than down. This could be mirrored in characters of global consciousness being very different depending on whether netvar trends are positive or negative.

I hope this fleshes things out a bit (no pun intended). This story is one more strong reason for carrying our analyses down to the reg level where some answers might be teased out of the data.

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