Poisson Model Interrupted Time Series Analysis I've run an Interrupted Time Series Analysis on count data, fitting a Poisson Model as below:
glm(`Subject Total` ~ Quarter + int2 + time_since_intervention2 , df, family = "poisson")

This is intended to show the change in number of application received and needs to be interpretable as such. The plot shows a diminishing trend in the [longer] pre-intervention period with a level change and potentially slight trend change.
The output summary from the model is:
Call:
glm(formula = `Subject Total` ~ Quarter + int2 + time_since_intervention2, 
    family = "poisson", data = df)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.72350  -0.67721   0.06528   0.68167   1.35607  

Coefficients:
                          Estimate Std. Error z value Pr(>|z|)    
(Intercept)               3.681163   0.088844  41.434   <2e-16 ***
Quarter                  -0.028692   0.009733  -2.948   0.0032 ** 
int2                     -0.164817   0.202510  -0.814   0.4157    
time_since_intervention2 -0.024379   0.038969  -0.626   0.5316    

I can't really present the ITS outcomes as above as I need to make such more interpretable and intended to do so by just using exp() on the coefficients. However, whilst exp(Intercept) does reflect the plotted value - the others do not. I am sure this is due to my lack of understanding of the Poisson model, but could do with some advice/help as to how to derive a more usable figure??

For example int2 equates to the level change in the post-intervention period.
Under the summary output it is -0.164817. Exp(int2) = 0.8480492.
However, the "real difference" between the fitted value and relevant counterfactual = -4.2 applications.
Whilst I can report said level change in an interpratble way the trend is much more problemtaic.
Anyone have any ideas/is it just me not really understanding the Poisson model.
Thanks

I think that your problem is in the interpretation of the coefficients. This type of display shows coefficients that represents differences from the intercept associated with differences of predictors from their reference values.
In a Poisson model with a log link, the intercept and coefficients are in units of log of mean expected counts. You need to add/subtract the contributions from the predictor coefficients to the intercept in that scale before you exponentiate.
For example, $\exp(3.681163) \approx 39.7$ counts for the intercept at reference values of other predictors. For the difference associated with int2 with other predictors at reference values, you calculate $\exp(3.681163-0.164817) \approx 33.7$ counts, a difference of 6 counts.
This type of work can be simplified by specifying a set of predictor values of interest to post-modeling analysis software like the emmeans package in R. That can provide estimates in the original count units along with confidence intervals and tests of significance between scenarios.
Reporting model coefficients
Reporting coefficients in the way displayed in the question, with coefficients representing changes in logs of counts associated with the predictors, is standard for Poisson models. If you exponentiate the coefficients as you propose, what you get are multiplicative factors acting on the number of counts.
The model that you show fits:
$$\log(\bar T) = \beta_0  + \beta_1 Q + \beta_2 \text{int2} + \beta_3 \text{tsi2}$$
Where $\bar T$ is the mean "Subject total", $Q$ is "Quarter," and $\text{tsi2}$ is "time_since_intervention2." As noted above, you need to add up all the terms on the right side before you exponentiate to get the estimate of mean "Subject total" $\bar T$. If you do that, you get the following product:
$$\bar T = \exp(\beta_0)  \exp(\beta_1 Q) \exp( \beta_2 \text{int2}) \exp (\beta_3 \text{tsi2})$$
Thus the coefficient $\beta_2$ is the multiplicative factor for "Subject total" counts associated with $\text{int2}$, other things being equal. The coefficient is not itself on a count scale. It's not clear that exponentiating the coefficients will make them any more interpretable.
For a Poisson model, I thus don't see any advantage to going beyond the log-scale coefficients. If you nevertheless want to report exponentiated coefficients, you need to make it clear that the exponentiated coefficients represent represent multiplicative associations with counts. You also need to report corresponding confidence intervals. For 95% confidence intervals of a coefficient with value $\beta$, you report $\exp(\beta \pm 1.96 \text{SE}_{\beta})$, where $\text{SE}_{\beta}$ is the standard error in the original scale. Those will be asymmetric about $\exp(\beta)$.
