GEE with gamma family and log link

I am fitting a generalized estimating equation with gamma family and log
link. I am using GEE (geeglm function) from the R pcakage ?geepack? with
gamma family and log link and unstructured correlation structure.

Here, response variable or outcome is IFN_gamma_protein_pg_mg . Exposure is
intervention or probiotic use. Another covariate is Timepoint.

My code and the output is as mentioned below.

m8<geeglm(IFN_gamma_protein_pg_mg~Intervention+Timepoint,data=B,family=Gamma(link=log),id=
Participant_ID,corstr="exchangeable")

summary(m8)

* IFN_gamma_protein_pg_mg*

*Predictors*

*Estimates*

*p*

(Intercept)

0.01

*<0.001*

Intervention [Probiotics]

0.34

*0.029*

Timepoint [T2]

0.99

0.979

Timepoint [T3]

5.30

0.059

Timepoint [T4]

0.48

*0.039*

Timepoint [T5]

0.11

*<0.001*

I am trying to interpret the coefficients as follows:

For every one-unit increase in the probiotic across the population, the log
average of IFN_gamma_protein increases by 0.34 units.

The exponentiated coefficient ( exp )= (exp(0.34)=1.41) is the factor by
which the arithmetic mean outcome on the original scale multiplied, i.e.,
when intervention is probiotic, for every one-unit increase in the
probiotic across the population, the average of IFN_gamma_protein on the
original scale is 1.41 times higher compared to when intervention is
control within levels of other variable.

Similarly, for timepoint 2, the average of IFN_gamma_protein on the
original scale is exp(  )= exp(0.99)= 2.69 times higher compared to
timepoint 1 within levels of other variable.

For time point 3, the average of IFN_gamma_protein on the original scale is
exp(  )= exp(5.30)= 200.34 times higher compared to timepoint 1 within
levels of other variable.

Can someone confirm me that I am in the right track in the interpretation
of parameters?

I am  posting here as I also want to fit a Gamma GLMM.

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