Your therapy, $T$, is a fixed effect with three levels: long, brief, and standard. I'm assuming that cannabis use, $U$, is a continuous variable (ounces smoked per day / week), which is then a random effect. By these two, I believe your motion to use a mixed effect model is appropriate. Diagnosis length, $D_L$ is also a random effect. Your outcome is the change from baseline, $\Delta_{\text{symp}}$, which is continuous.
Therefore, your model should look like
$$
\Delta_{\text{symp}} = \beta_0 + \beta_1T_{\text{long}} + \beta_2T_{\text{brief}} + \beta_3U + \beta_4D_L + \epsilon,
$$
where $\epsilon$ is some random error (that we hope is normally distributed). Note that $T_{\text{long}}$ and $T_{\text{brief}}$ are indicator vectors for the treatment level, with $T_{\text{standard}}$ as the baseline (its effect is part of $\beta_0$ in our model). I would run this as a multiple regression model, and check the following:
- Check that the errors (model residuals) are normally distributed (with mean of 0). I would plot a density estimate of them and check that it's reasonably symmetric and unimodal. You can also use a Q-Q plot for this. If not, you will probably have to transform your data.
- Check that the predictors are not highly correlated with each other. Build a correlation matrix of the predictors, and look to see if any correlations are greater than 0.8 in absolute value. If so, your model may have issues from Multicollinearity.
After you run this model, check the confidence intervals for each $\beta_i$ value. If the confidence interval for a particular $\beta_i$ value does not contain 0, then that predictor is significantly related to the outcome (if you have confirmed that your predictors are not too strongly correlated with each other).
Here is a good summary of how this all works. If you still have questions, feel free to email or message me.
