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From |
"Dupont, William" <william.dupont@Vanderbilt.Edu> |

To |
<statalist@hsphsun2.harvard.edu>, "Arbogast, Patrick" <patrick.arbogast@Vanderbilt.Edu>, "LaFleur, Bonnie" <bonnie.lafleur@Vanderbilt.Edu> |

Subject |
st: Interupted time series with variable dispersion |

Date |
Mon, 23 Dec 2002 13:38:15 -0600 |

I am doing an interrupted time-series analysis. A simplified description of my model is as follows: An intervention occurs on a specific date. The data fits two simple linear models of response vs. date both before and after the intervention, (with the slope and intercept coefficients before the intervention being different from these coefficients after the intervention). A first order autoregressive-moving average (arima(1,0,1)) model fits the data quite well. My questions have to do with dispersion. There is an abrupt, large and persistent drop in the response variable that starts at the time of the intervention. The dispersion after the intervention is substantially less then in the baseline interval. My colleagues are interested in assessing both the drop in the response variable as well as the reduction in dispersion. They are used to using the coefficient of variation to assess such changes. Simple coefficients of variation are not ideal in this example because there is some systematic variation in response with time in both the baseline and post intervention intervals. What I have done so far is the following: 1. Fit separate time series models to the baseline and post intervention data. 2. Calculate the mean squared error (MSE) for both models using the structural option of the predict command. 3. Divide the square root of the MSE by the mean response in either the baseline or post intervention period. This is a coefficient of variation like statistic that is adjusted for temporal trends within the interval of analysis (i.e. trends within either the baseline or post intervention interval). Doing this gives a 30% reduction in my "adjusted" coefficient of variation. 4. The next issue is whether this reduction is statistically significant. I have calculated res2, the squared residual for each day divided by the squared mean for the interval (either baseline or post intervention). I then entered res2 into a separate arima model. The outcome shows a significant drop in res2, which I would like to interpret as evidence that there is a significant drop in my "adjusted" coefficient of variation after the intervention. Questions My fundamental question is does the preceding make sense and is there a better way of doing things? My specific questions are 1. Is there a better way of fitting my original model? My original model does a good job at dealing with the temporal correlation of my response variable and produces an expected response curve that fits the data well. However, it also assumes a constant white noise error term that is clearly not correct. I could, of course, transform the data to obtain more homoscedastic errors but I am concerned that this would result is a less satisfactory estimate of the expected response. Is there a way that I can allow for separate residual errors before and after the intervention without giving up a linear model of the response as a function of time? 2. A problem with my approach to assessing the significance of the drop in adjusted coefficient of variation is that my res2 variable is both highly skewed and temporally correlated. If I use an arima model to deal with the temporal correlation, I am assuming a white noise term that is normally distributed. This assumption is clearly incorrect. However, I do not know of a nonparametric method that can account for the autoregressive-moving average nature of my data. What do you recommend? (This problem is made more acute by the fact that my P value comparing adjusted coefficients of variation is only 0.03, and its statistical significance might easily vanish if a more appropriate model was used.) 3. Can anyone suggest references that might be helpful? In particular, can anyone tell me of a reference for calculating a coefficient of variation adjusted for within-group systematic trends? *************** I apologize for writing such a long query. I would be most grateful for any insights on how to analyze this sort of problem. Bill Dupont * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

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