I’m comparing the difference in firing rates for a population of neurons (i.e., cells) in the brain under two conditions (A and B). For each neuron, we have four data points:

```
- sample_time_A
- spike_count_A
- sample_time_B
- spike_count_B
```

For each neuron in each condition, we counted the number of events (`spike_count`

) occurring within a fixed interval (`sample_time`

in seconds). We are interested in determining whether the rate (number of events per second) differs between the two conditions across the two populations. This is the distribution of the rates (`spike_count_A/sample_time_A`

is blue, `spike_count_B/sample_time_B`

is blue):

Now, this seems like a very straightforward t-test or Wilcoxon sign test to see if the means of `A`

and `B`

are different. I can just compute `rate_A`

and `rate_B`

(i.e. `spike_count/sample_time`

) and use a paired t-test or Wilcoxon sign test. Both return p-values that are significant (<= 0.01).

However, I thought it would be fun to test my rudimentary knowledge of Bayesian statistics. Since `rate_A`

and `rate_B`

can be modeled by a gamma distribution and the observed variable (`spike_count`

) can be a poisson, why not try it out? However, I am getting a result that tells me the difference in rate between the two conditions is not significant (i.e., the 90% credible interval overlaps with 0) as shown by the posterior for the difference in the means of `rate_A`

and `rate_B`

.

I calculated the posterior for the mean of `A`

by dividing the posterior of `sr_alpha_A`

by the posterior of `sr_alpha_B`

(since the expected value of a gamma distribution is alpha divided by beta). For `B`

, the calculation was `(sr_alpha_A + sr_alpha_delta_B) / (sr_beta_A + s r_beta_delta_B)`

. The posterior for the difference in the means of `A`

and `B`

is the difference between the relevant posteriors.

**The lack of significance is fine with me.** If it’s not significant, it’s important to report that result in the paper. However, I thought that the paired t-test had pretty weak assumptions about normality. Further, isn’t the Wilcoxon supposed to be non-parameteric? So, shouldn’t the Bayesian and frequentist approaches agree? This makes me wonder if I set up my model wrong. Here it is:

```
data {
int<lower=1> n_cells;
real<lower=0> sample_time_A[n_cells];
real<lower=0> sample_time_B[n_cells];
int<lower=0> spike_count_A[n_cells];
int<lower=0> spike_count_B[n_cells];
}
parameters {
real<lower=0> sr_alpha_A;
real<lower=0> sr_beta_A;
real sr_alpha_delta_B;
real sr_beta_delta_B;
real<lower=0> sr_cell_A[n_cells];
real<lower=0> sr_cell_B[n_cells];
}
model {
sr_alpha_A ~ gamma(0.5, 0.1);
sr_beta_A ~ gamma(0.1, 0.1);
sr_alpha_delta_B ~ normal(0, 0.01);
sr_beta_delta_B ~ normal(0, 0.01);
sr_cell_A ~ gamma(sr_alpha_A, sr_beta_A);
sr_cell_B ~ gamma(sr_alpha_A + sr_alpha_delta_B, sr_beta_A + sr_beta_delta_B);
for (i in 1:n_cells) {
spike_count_A[i] ~ poisson(sr_cell_A[i] * sample_time_A[i]);
spike_count_B[i] ~ poisson(sr_cell_B[i] * sample_time_B[i]);
}
}
```

I would really appreciate any pointers!