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My goal is to quantify dosage impact on cure probability for different patients. Let's suppose I have N patients with their charateristics such as age, gender, weight.... Also let's assume there is a known range for dosage, say from 0 to 1.

Here is what I'm currently doing:

  1. Randomly split all patients into 3 groups
  2. Group 1 is given dosage of 0.1, group 2 dosage of 0.5 and group 3 dosage of 0.9. Please note size of group is different. Presumably group 1 is largest because I need decent number of cured patients and I know low dosage leads to low cure probability. Also note that I can not test more dosage because N is limited.
  3. Get outcome (0/1) for all patients
  4. Train logistic regression model for each group
  5. If I want to get cure probability for any particular patient with untested dosage (say 0.7), I just calculate 3 probabilities based on models in step 4 and do some simple interpolation.

My question is if there is any better way to do this. I'm not feeling very comfortable about the interpolation part.

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  • $\begingroup$ Is there a reason you can't vary dosage along your scale continuously, and treat dosage as a continuous variable? You'd get a far better picture of the relationship between dose and response that way. $\endgroup$ – Fomite Aug 15 '11 at 21:46
  • $\begingroup$ Experiment results are given and I have to find a way to deal with it. Agree with you that treating dosage continuously is a better approach and will definitely suggest it if I'm consulted for the next experiment. $\endgroup$ – FMZ Sep 5 '11 at 6:48
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There are two ways in which you could try to get the possible outcome for the untested dose (say 0.7).

Preferable way

Since you have covariates for each patient (i.e., age, gender, etc., say $\mathbf x$ (a vector)), dosage ($d$), and the response ($0/1, y$), fitting a logistic regression with $y$ as response and $(d, \mathbf x)$ as covariates is the easiest way to model the dependence of $y$ on $d$. Once you have trained your logistic regression model on the available data, you could predict for any patient in the future, given you have their covariate $(d_{\text{new}}, \mathbf x_{\text{new}})$ information by

$$ p_{\text{new}}=\frac{\exp(\beta_0 + \beta_d d_{\text{new}} + \mathbf{\beta}_{\mathbf x}^' \mathbf x_{\text{new}})}{1 + \exp(\beta_0 + \beta_d d_{\text{new}} + \mathbf{\beta}_{\mathbf x}^' \mathbf x_{\text{new}})} $$

Because you have trained your logistic regression before, you should have a cutoff $c$ for deciding the possible outcome such that

$$ p_{\text{new}} > C \implies 1 $$ $$ p_{\text{new}} \le C \implies 0 $$

Your way

Lets say you train 3 separate logistic regression models based on 0.1, 0.5, and 0.9 with unequal group size, and you could predict probabilities $p_{0.1}, p_{0.5},$ and $p_{0.9}$ for a particular dose (say 0.7).

A meaningful way of interpolation to determine the probability (only if you insist on it), could be

$$ p_{0.7} = \frac {\displaystyle \sum_{i \in \{0.1, 0.5, 0.9\}} w_i p_i}{\displaystyle\sum_{i \in \{0.1, 0.5, 0.9\}} w_i} $$

where the weights could be decided in various ways. A few options are:

  1. $w_i = \frac{n_i}{(i-0.7)^2}$, where $i$ is 0.1, 0.5, and 0.9, and $(i-0.7)^2$ reflects the "closeness" of the new dosage to the original 3 groups and $n_i$ is the number of individuals in the ith groups controlling for higher confidence in the weights with higher $n_i$.

  2. $w_i = \frac{n_i}{|i-0.7|}$, where $i$ is 0.1, 0.5, and 0.9, and $|i-0.7|$ reflects the "closeness" of the new dosage to the original 3 groups and $n_i$ is the number of individuals in the ith groups controlling for higher confidence in the weights with higher $n_i$.

However, I would not be very confident in these methods as they are based on a lot of assumptions and "hand-waving".

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  • $\begingroup$ I considered the preferable way too, ie put dosage explicitly in the model as a continuous variable. However given the fact that experiments have already been done for only 3 dosage values (0.1 0.5 and 0.9), which way will generate better predictions, especially for those dosage values not tested? Thanks! $\endgroup$ – FMZ Sep 5 '11 at 6:44
  • $\begingroup$ I think the logistic regression is a better way (preferable way) as long as you are interested in determining the relationship between dosage and response. It doesn't matter if you did the experiment only for 3 doses. $\endgroup$ – suncoolsu Sep 5 '11 at 14:04

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