1
$\begingroup$

I have a question based on the following approach for the analysis of RCT's. The following works well for the outcome (and baseline) being continuous with normal errors. enter image description here

Expanding upon this, I was curious what would be the required form if one was assessing rates. Assume Yt was with respect to some denominator of relevance, N. Also, the baseline Y0 was with respect to some denominator of relevance, N0.

In Generalized Linear Modelling parlance, would either of the following be correct?:

  1. Yt = beta0 + beta1X + beta2*ln(Y0/N0)
  2. Yt = beta0 + beta1X + beta2*ln(Y0) + beta3*ln(N0)

In both cases; link=log, errors=Poisson, offset=ln(N)

My main query is the term Y0/N0. Should it be calculated directly and ln(Y0/N0) be a single variable as an independent (case 1), or, split into two vars as using the log rule: ln(Y0/N0) = ln(Y0) - ln(N0) (case (2))?

Data format is:

enter image description here

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

If you have individual patient data for rates (something like disease episodes in observation time $t_i$ with a baseline for the past 1 time unit e.g. 1 year), you would traditionally do $$\log EY_i = \beta_0 + \beta_1 x_i +\beta_2 \log\text{base}_i + \log t_i$$ (taking the logarithm of the baseline is not totally standard, but makes sense, just make sure to have some rule in case of zeroes; you could also divide by the baseline observation time). Poisson is not used much in practice and Negative Binomial is much more popular (it accounts for unexplained between patient variation).

If you have something with $Y$ number of successes out of $N$ tries, then logistic regression (binomial outcome, logit link function) would seem like a better fit. You'd do much the same as above, but for the link function, you'd probably write it for each individual post-baseline try and for the baseline term $\log (Y_{0i}/(N_{0i}-Y_{0i}))$ (again with some solution needed in case of all zeroes or ones). I.e. for one trial out of the many for a person the logit of getting a 1 in that trial is $$\text{logit} EY_i = \beta_0 + \beta_1 x_i +\log (y_{0i}/(N_{0i}-y_{0i})),$$ i.e. logit link and binomial distribution.

Improve this answer
$\endgroup$
7
  • $\begingroup$ I've included the data layout above. I'm not following your formula Bjorn, where is the baseline info (in full) incorporated? i.e Y0 and N0 $\endgroup$
    – nevermind
    Jan 13, 2019 at 9:07
  • $\begingroup$ I've spelled this out for the binomial case now. $\endgroup$
    – Björn
    Jan 13, 2019 at 9:19
  • $\begingroup$ Thanks, I appreciate it, however, it's not binomial. It's rates; counts per person-year type thing. For example; if the data was with respect to dentists. Dentist 1: no. people requiring fillings (Y0) and the number of hours worked (N0). $\endgroup$
    – nevermind
    Jan 13, 2019 at 9:45
  • 1
    $\begingroup$ In that case the Poison or NegBin version with log(base/baseline hours). $\endgroup$
    – Björn
    Jan 13, 2019 at 9:55
  • 1
    $\begingroup$ Yes, except for using log E Y. $\endgroup$
    – Björn
    Jan 13, 2019 at 11:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.