# Correcting data using poisson-regression

I'm new to stats and I was wondering if anyone had any good resources that could explain to me:

How one can correct their data (false-positives) using Poisson-regression. I've been looking for some resources to explain it simply and give a step by step (in R) but I've had trouble.

The reason I ask is because I was trying to reproduce the following:

We perform our analysis at a region level, where all region pairs are separated by more than 17.4 mm, which based on simulations (not shown), leads to negligible bias due to distance-related false positive connections. We also employ a Poisson regression-based statistical adjustment that yields measures of $$SC$$ adjusted for the physical distances between region locations. Specifically, we apply a model that assumes that the number of $$DTT$$ streams $$S_{ij}$$ connecting regions $$i$$ and $$j$$ follows a Poisson distribution with the mean $$\mu(S_{ij}|g_{ij})$$ dependent on the physical distance $$g_{ij}$$ between these regions, i.e. $$S_{ij}|g_{ij}\sim \mathcal{Pois}(\mu(S_{ij}|g_{ij}))$$. Therefore, we estimate and subsequently adjust for the association between the physical distances and the $$DTT$$ counts using the effect $$\alpha_1$$ in the log-linear model $$\log(\mu(S_{ij}|g_{ij}))=\alpha_0+\alpha_1 g_{ij}$$ Henceforth, assume that each $$\pi_{ij}$$ is adjusted for physical (geometric) distance to reduce the potential impact of false structural connections on our awFC method.

source: Bowman et al. 2012

## 1 Answer

DTT streams between proximal nodes in a neural network are not false positives, they produce real phenomena, like the pleasurable sensation of Q-tipping ones ears or making funny faces while playing music. Rather, Bowman is trying to increase the precision to detect distal SCs by adjusting for an expected level proximal, spurious signalling by action potentials. The statistical approach to detection of these SCs is to subtract off the expected potentials under the assumption of an unconnected network. To do this, Bowman creates a Poisson model for the count of firing events where expected firing intensity varies exponentially as a function of the Euclidean distance between two nodes. A combination of nodes will be flagged as a plausible SC if their mutual DTT stream is much more than would be expected under this "unconnected null model" where two close-together nodes are expected to fire more often just as a consequence of being close together.

• Assuming I can do this fit, how would the parameter be adjusted? – hsayya Dec 13 '18 at 18:46
• Do you mean how do you get the $\alpha_0$ and $\alpha_1$? It's a simple Poisson GLM. You merely create a data structure which has one row for each pair of nodes assessed at a particular time. The # of DTT streams is the Y and the X is the Euclidean distance between them, or some other suitable metric. – AdamO Dec 13 '18 at 19:44
• No I meant in the final part of that paragraph "Henceforth, assume that each πijπij is adjusted for physical (geometric) distance ... to reduce the potential impact of false structural connections on our awFC method." how would I adjust the connectivity strength parameter? – hsayya Dec 13 '18 at 20:28
• It's the first time we see $\pi_{ij}$ but I'm assuming it's a latent intensity for the DTT streams. Subtract from the observed streams the predicted streams given by the Poisson model. – AdamO Dec 13 '18 at 20:32
• Okay, I think that makes sense. Do you have any references you can point me to that discuss this approach? "Subtract from the observed streams the predicted streams given by the Poisson model." Thank you so much for your time. – hsayya Dec 17 '18 at 18:44