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Regression Model 48. What was the overall approach used in this regression methodology? The following is a summary of the regression model:
49. What is the equation that you used? (Click here for larger view of equation)  The terms in the first line of the equation account for brand-specific differences in sales levels and trends in sales levels. Note that this includes cubic trend to account for possible S-shaped diffusion trends. The second line of the equation is the detailing effect. Note that there are interactions between detailing and SIZE and LAUNCH, so that we obtain different effects depending on brand size and launch date. We also allowed for interactions with PCP% and with the other marketing variables. The last two lines control for competitive spending, price, and generic presence in the market. The SDTC, SDET, SJAD, and SPME variables are "ad stocks" (see Rizzo, Journal of Law and Economics, April 1999). A higher value of l means that advertising stock deteriorates more slowly over time and therefore that the effect of advertising is more long term. The total effect of advertising depends on the l but also on the total effect obtained by substituting median values for the interaction terms in the SDET, etc, terms beginning in line 2 of the equation. 50. The result of a regression is a y-intercept. What are the y-intercepts? There is a separate y-intercept for each brand (see the b0 and the b2j's in the above equation), and so there are 391 separate "brand intercepts." There are also 391 separate trend effects, one for each brand. 51. What are the coefficients for each variable that was regressed? It is extremely difficult to interpret the coefficients by themselves, and there are almost 1,600 coefficients in total because of all the brand-specific intercepts, trend terms, and interactions. What is interpretable is the average effect that was calculated by substituting in median values for the interaction variables, and that is presented in the ROI numbers that we've published. 52. Was the P value for each variable statistically significant (less than .05)? Not for every variable. This is a typical situation in a model with many interactions. The statistical significance of the calculated ROI is what matters, and those are reported in the aggregate ROI results. 53. The ROI for each variable assumes a linear model. Why was this model chosen? The model used is not linear; it is semi-log, which is nonlinear. The log transformation was used because the dependent variable was highly skewed. In practical terms, however, since the total effect of the ad-stock variables was relatively small in absolute terms (a thousand dollar increase in expenditure is not going to increase sales by a very large percentage), the model appeared to be linear within the relevant range of the data. 54. How did you use the parameter estimate in your model to calculate the ROI? The effect of each marketing activity depends on several interactions between that marketing activity and the variables (such as brand size, launch date, PCP%, etc). Median values were substituted in for those variables, multiplied by the corresponding parameter, and then summed up to calculate the effect for a given activity. That effect then represents the % increase in units per dollar spent, which was then multiplied times median units times median price to get ROI. 55. What is a "grid search"? Is it some sort of step-wise regression, or is it just a search which seeks to optimize the lambda by finding the value which minimizes the MSE? We searched to find the set of lambdas (there are four of them, one for each marketing activity) that minimized MSE (mean square error). This was practical to do because of the fact that we knew the lambdas have to be between zero and one. We looked at several combinations of values for the lambdas and it became clear that the high or low values for detailing and journal advertising, and low values for lambda for DTC and PME, would not be optimal. After looking at several additional combinations, the final values that minimize MSE were lambda = .5 for DET and JAD, and lambda = .9 for DTC and PME. Two simplifications were considered: one would be to assume a lambda of .5; another would be to assume the same lambda for each activity. Neither of these alternatives seemed reasonable, so we went with the grid search approach. Note that this is not a step-wise regression. It was estimated using OLS. The reason we did not use step-wise is that this would exclude variables correlated with other variables, and thus would create an omitted variables bias — the included variables would pick up the effects of the excluded variables. An omitted variables bias could be very dangerous for this study. For example, if various detailing variables became excluded, their effect might be picked up by JAD, etc. This would inflate the effectiveness/ROI of JAD. Step-wise is useful for predictive studies where one is just looking to predict sales trends, but not looking to understand what is causing those trends. In this case, our study is about those causes, so we did not want to use step-wise. 56. How did you handle the monthly lags? Did you fit a separate parameter for each month? How did you handle the autocorrelation associated with that kind of analysis? An "ad-stock" variable was used for each of the four marketing activities. The ad-stock variable can be viewed as a Koyk-lag structure. The Koyk parameter (the "Lambda") was allowed to vary for each activity, and a grid search was conducted to find the best set of lambda's, using MSE as a criterion. Autocorrelation was not explicitly controlled for in the estimation, but rather in the model specification itself through the trend variables. Also included were linear, quadratic, and cubic trend terms interacted with each brand dummy. Such a detailed trend structure was used to take into account other systematic changes in sales over time, the type that typically shows up as autocorrelation in a regression model. This was all possible due to the availability of 16,000 observations. 57. The lag effects from DTC are very complex. How did you determine that there is a 2-year effect if the impact of DTC was not significant in the first place? The Koyk-lag was used for DTC as well as the others. Indeed the effect might be more complex, and this would be an avenue for further investigation. The grid search found a lambda of .9 for DTC and, given this lambda, one can obtain the confidence intervals for the effects, and therefore the statistical significance. Note the uncertainty in lambda is not explicitly considered in these confidence intervals which probably makes these intervals conservative. 58. The issue of the lags is complex. I can easily believe that a detail has a lag of up to 3 months, an event has a lag of about 6 months, and a journal ad a lag of about 1 month. But I am unsure about DTC because it may take a considerable amount of time for a patient to see the ad, make an appointment with the physician, talk to the physician about the product, and get a prescription. The amount of time is probably random for a patient in a way that it is not random for a physician. The easiest way around that problem is to use longer time periods (eg, a quarter), but that decreases the number of observations available. In fact, the lambda found for DTC was bigger than the others, meaning that it did take longer for the full effect of DTC to be realized. The findings are in line with your hypothesis. Yes, quarterly level aggregates were not used because one would lose observations. 59. Our company has considerable experience in using regression to model promotion response. Usually, we analyze the response of personal promotion at the individual physician level. Thus we know a great deal about detailing, but not a lot about journals and DTC. Yes, that's another way to go. A lot of this type of analysis has been done with packaged goods panel data (provided by IRI or Nielsen). Logit and nested logit are used in these cases since the dependent variable is categorical. However, these are very similar to regression in the end. |
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