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For 2004-2014, I have monthly measurements of my outcome of interest - some kind of physical exposure - for a collective of many thousand persons. The main determinant for the average exposure level is the intensity of the physical source (correlation ~ 0.7). This intensity was measured 1993-2014 in monthly intervals - it is periodic with a known cycle length.

The collective has three groups (A, B, C) which differ in their behavior such that their average time within proximity of the physical source differs - and thus their average exposure level. Yet, the correlation between their average monthly exposure is very high (~ 0.95). The smoothed monthly group averages look like this:

enter image description here

For group A, I have individual exposure measurements also for 1993-2004, but for B and C, no measurements exist for this period. I would like to retrospectively forecast the missing exposure given a) the intensity of the toxic source and b) the measured exposure for A. How can I do that?

I am only interested in prediction, not in testing. However, I also need a quantification of the forecast error variance. Pointers to relevant R functions / packages would be highly appreciated as well.

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    $\begingroup$ Interesting question. Do you also have the data for the driver in 1993-2004? Could you post the data? An extremely simple approach would of course be regressing B and C on A and then predicting them, using the known values of A, and then you could use simple cross-validation/bagging to estimate the error variance. How certain are you that the correlation of 0.95 you measured in one 10 year period also holds in a different 10 year period? $\endgroup$ – Stephan Kolassa Jul 16 '15 at 12:00
  • $\begingroup$ @StephanKolassa With "driver", do you mean the intensity of the physical source? Yes, that data is available for 1993-2004. Unfortunately, the data is confidential. Yes, I expect a high correlation also in 1993-2004: When members of B/C are exposed, their exposure is the same as for members of A. It's just that they can differ with respect to how much time per month they are exposed (this "time under exposure" is unkown). I have no background in time series, so I have no idea how simple a possible solution might be. I just don't want to make typical beginner's mistakes. $\endgroup$ – caracal Jul 16 '15 at 12:36
  • $\begingroup$ Thanks for the clarification. I +1'ed Gaurav's answer. A simple regression should be quite fine for your use case. Good luck! $\endgroup$ – Stephan Kolassa Jul 16 '15 at 19:43
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I can't comment so I am answering it directly.

For the part b - I agree with Stephan - Use regression. From the graph, it seems a linear regression should work fine without any transformation.

For the part a - Since you have intensity of toxic source from 1993 to 2014, and you have exposure for both B and C, you could directly regress Exposure on intensity for B and C individually. You will need to analyze the graph of intensity/exposure for the best transformation that you might need before doing a linear regression.

Since you will be regressing only on one variable, this link should help you with R functions. http://courses.statistics.com/software/R/R_Ch02.htm.

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  • $\begingroup$ I am concerned about the OLS parameter estimates given correlated errors from the repeated measures. I somewhere read that they loose efficiency. Are there better estimators? $\endgroup$ – caracal Jul 17 '15 at 8:40
  • $\begingroup$ yes, correlated errors cause problems is regression. One easy way to find out is to plot the residuals and check if they look correlated. There are some tests like Durbin Watson test but using this test correctly requires knowledge of time series. I would suggest that "correlation in errors" is a different question, and this question might have been answered already. $\endgroup$ – Gaurav Singhal Jul 20 '15 at 9:25

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