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I ran a study in which participants were randomized to either a control or an intervention, with outcomes in the form of time-to-event data. While overall time-to-event is shorter in the intervention group than in the control group (log-rank test $p < 0.05$), I am interested in detecting if there was a certain time after the start of the study beyond which the intervention had no effect. More formally, I'm interested in detecting the minimum time $t$ beyond which the intervention's hazard function value was not significantly higher than the hazard function value for the control group.

One idea I had would be to re-compute the log-rank test p-value, ignoring all events that happened before some time $t$ in the intervention and control groups (left-censoring the time-to-event data). Repeating this process for a range of $t$ values, I could determine the minimum value $t$ beyond which there's no significant intervention effect.

I was wondering if there were more standard approaches to determining a time $t$ beyond which an intervention has no detectable effect on the time to event.

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  • $\begingroup$ Was it randomized? $\endgroup$
    – Adam Robinsson
    Feb 24, 2015 at 16:42
  • $\begingroup$ @AdamRobinsson yes, it was randomized $\endgroup$
    – josliber
    Feb 24, 2015 at 21:18
  • $\begingroup$ If it was randomized ("treatment allocation"), and you are measuring time until an event, then I guess a simple Kaplan-Meier plot would provide the best visualization of how survival varies during follow-up in each group. If the favorable effect vanishes after some time (or the effect of any event varies with time), the K-M plot will reveal this. But I might have misunderstood the aim... $\endgroup$
    – Adam Robinsson
    Feb 24, 2015 at 21:21
  • $\begingroup$ @AdamRobinsson yes, an "eyeball test" on a Kaplan-Meier plot could identify where the slopes of the control and intervention survival curves appear to equalize. I am interested in whether there is a statistical test to identify the time at which an equalization of hazards has occurred. $\endgroup$
    – josliber
    Feb 24, 2015 at 21:25
  • $\begingroup$ Thats beyond me but then I realize that this is a very good question, because it will apply for a range of disciplines, so I'm interested as well! $\endgroup$
    – Adam Robinsson
    Feb 24, 2015 at 21:27

2 Answers 2

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Given that you seem to have a panel of individuals who you follow over time of which some are treated and others are not you could run a difference in difference analysis. You could run a regression like $$y_{it} = \beta_1 (\text{treat}_{i}) + \beta_2 (\text{intervention}_t) + \beta_3 (\text{treat}_{i} \cdot \text{intervention}_t) + \epsilon_{it}$$ where $\text{treat}_{i}$ is a dummy for whether individual $i$ is in the treatment group, $\text{intervention}_t$ is a dummy for the post-treatment period, and the interaction between the two captures the treatment effect in $\beta_3$.

If you now want to estimate the fading out time, estimate instead $$y_{it} = \sum^m_{\gamma = 0} \beta_{-\gamma}(\text{treatment}_{it}) + \eta_{it}$$ where $\text{treatment}_{it}$ is a dummy variable which equals one if individual $i$ is in the treatment group AND time $t$ is at or after the treatment date. This estimates the first equation but with $m$ lags of the treatment for which you can choose the number of periods you have from the start of the treatment to the end of the sample period.

Then $\beta_0$ is the treatment effect at the intervention date, $\beta_1$ is the effect of the intervention at the first period after the intervention date, and so on. The nice thing about this approach is that

  • it is easily implemented in any statistical software (you just need to create the dummies and run a regression)
  • the $\beta_0, ..., \beta_m$ coefficients will have standard errors and confidence intervals which you can use to see the time (lag of the treatment) from when the intervention stops to have an effect
  • the $\beta_0, ..., \beta_m$ coefficients will give you an estimate of the magnitude the intervention had in subsequent periods

If you also have additional control variables like characteristics of the study participants $X_{it}$ you can easily include them in the regression, $$y_{it} = \sum^m_{\gamma = 0} \beta_{-\gamma}(\text{treatment}_{it}) + X'_{it}\rho + \eta_{it}$$ this will not affect the estimate of the intervention effect (because identification comes from the group differences between treatment and control groups) but it helps to reduce residual variance and therefore increases precision.

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  • $\begingroup$ Thanks for pointing me to difference-in-difference analysis. A difficulty I'm having in seeing how it applies is that it's not clear what $y_{it}$ is in my case. In your linked answer you mention a continuously varying value (e.g. wages, health) that can be observed before and after the intervention. In my case I'm using time-to-event data and I don't have any events before the intervention. If I chose a $y_{it}$ like number of people who have experienced the event by $t$, then the error terms $\epsilon_{it}$ will be auto-correlated so standard regression assumptions won't hold. $\endgroup$
    – josliber
    Mar 3, 2015 at 8:50
  • $\begingroup$ You would typically cluster the standard errors at the individual level. This corrects for autocorrelation and heteroscedasticity. If you have no real outcome then this is more difficult, yes. It can be a binary one also though, doesn't have to be continuous. $\endgroup$
    – Andy
    Mar 3, 2015 at 8:57
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Intervention Detection http://www.unc.edu/~jbhill/tsay.pdf and elsewhere can be employed with or without a user-suggested intervention variable. In either case one needs to treat any auto-projective process that might be present i.e. the ARIMA structure. Identifying both the ARIMA structure and the response to any user-suggested variable and the "new intervention series" requires some trial and error as the search process/method is to identify if and when this waiting-to-be-discovered intervention arises. This search process can be solved/aided with software but one needs to confirm that the software incorporates any needed impact i.e. user-specified possible predictor series/variables including their lag structures and that the ARIMA process identification phase was not damaged/impacted by the intervention that is waiting-to-be-discovered which of course would have a deleterious effect.

The length would be the difference between the point of the user-specified intervention variable ...if it existed OR the difference between the two identified Level Shift/Step Shift variables that were found/discovered/unmasked .

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