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I would like to build the relationship between the dose given and two outcomes (one acute toxicity and one late toxicity). The model I used was binary logistic regression. For the acute toxicity outcome, the estimated dosage and its 95% confidence interval was 20 (15~35), while for the late toxicity outcome, the estimated dosage and its 95% confidence interval was 75 (50~80).

We can see that 75>20 and these confidence intervals are not overlapping. It seems to indicate that the late toxicity is caused by a higher dose than the dose for acute toxicity. My question is: Is there any statistical method to test that the late toxicity is caused by a higher dose than acute toxicity? Or is simply claiming that their CIs do not overlap is strong enough?

Thank you

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Sorry I did not state my problem clearly. In radiotherapy, I want to build the association between Vx and acute/late toxicity, where Vx represents a series of dosage variables: Vx, indicating the volume of the organ receiving more than x dosages. For acute toxicity as outcome, I could estimate the best Vx (e.g. x=20), using maximum log-likelihood. I can also estimate the confidence interval of x, using likelihood ratio test, this gives me CI=15~35. Similarly, for late toxicity outcome on the same patient cohort, I ended up with best x=75 and CI=50-80. –  tiantianchen Sep 19 '12 at 23:38
    
Thus, I have two sets of estimated parameters (x) of the variable (Vx) for two different outcomes but on the same patient cohort. And I would like to know if it is possible to compare the estimated x for the two outcomes. –  tiantianchen Sep 19 '12 at 23:42
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2 Answers

This answer is responding to your comment above on my first answer. In it, I assume that what you actually want to do is compare the regression lines for models built to predict two different dependent variables.

This link describes a case similar to yours, where what is required is a maximum likelihood estimate for the distribution of the betas. In the case of that link, the requirements are slightly different, because the two models do not share the same variables.

Because your final dependent variables are different, I would argue that the distribution of the betas will likely be different between the two models anyway. As a result, I would say that this method is probably the best way to go as it delivers a sort of pooled standard error, which you can divide the difference of the two betas by to derive a significance test.

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I may be missing something here but is there any reason you can't use a standard t test? You have two outcome groups, your goal is simply to determine whether they have different mean dosages. It does require that you swap you IV and DV, but in this instance that may be ok.

If you don't need to control for any other variables, the t test will give you the probability that there is a significant difference between the two groups in terms of original dosage. As a simpler model, it will be more easily interpretable and widely understood than logistic regression - important for any peer reviewed work. Further, you can use Cohen's D to get an effect size measure if you need that.

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thank you for the answer. I have re-edited my question. My problem is not comparing means of two independent samples. It is to compare the estimated values of the parameters in a model regarding two different outcomes, but on the sample data sample. –  tiantianchen Sep 19 '12 at 23:51
    
What are the parameters in the model that you are estimating? The actual Vx (organ volume is given or calculated. Maybe what you want is to show that at some dose level the regression curve for probability of late toxicity is higher than for acute toxicity. My confusion which others may have as well is that Vx appears to be a covariate and not a parameter. It is the coefficient of the parameter that is estimated by maximum likelihood. I would think that the magnitude of the parameter is what is estimated by maximum likelihood and the confidence intervals are for teh parameter. –  Michael Chernick Sep 20 '12 at 15:40
    
If it is the parameter that you are estimating and the coefficients of the parameter that you are comparing I do not see that the size of the parameters tells you anything about dose causing late toxicity. What the logistic regression tells you is whether or not the covariates have a significant effect on the probability of the outcome (early toxicity in one model and late toxicity in the other). The p-value and the fact that the confidence interval for the parameter does not contain 0. Which seems to be the case in both models. –  Michael Chernick Sep 20 '12 at 15:47
    
Logistic regression does not tell you the dose, it tells you the probability of the outcome given the covariates. Dose and Vx seem like they might both be covariates and not parameters. Maybe you can clear this up so that we can give you good advice. –  Michael Chernick Sep 20 '12 at 15:49
    
Your comment above makes things even more unclear to me; in your original question you talked about confidence intervals of dosages, however in your comment, you've spoken about testing the difference between two beta parameters... As @MichaelChernick mentions, you may be confusing the coefficients with the dosage required. In actual fact, to predict an outcome, you multiply the coefficient by the value of the variable. As such, the coefficient tells you something (sort of) about the importance of that variable in the outcome, but doesn't correspond to any value which the variable might take. –  analystic Sep 29 '12 at 7:29
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