So the log log model is one adaptation of the regulation model that is necessary when you apply it into marketing. Now we are going to look at many more of those adaptations that are necessary. The main thing is regulations are used in marketing for something called marketing mix models. This is where you're trying to find out how does marketing, the different aspects of marketing, affect sales. Now one of the things we saw in multiple regression is, it's not just about what you include in the model, it's also about what you have missed out. So what are the common variable to consider when including in marketing mix models? So I go back to the basics here. I always say remember to include the four Ps, product, price, place, distribution. So what you have up here are different aspects of these 4Ps, product quality and brand lifecycle, whether it's a new product or the first P, product. Now you got distribution which is about place, we got price and promotion. So one of the things about promotion to understand is that promotions have carry over, advertising today has an effect on sales tomorrow or the day after, and so forth. Because people remember the advertisement even when the advertisement stops. So you need to remember to include the carryover, the effect of advertising yesterday on sales today. So here are the four Ps, product, price, place, promotions. A nice little tidbit from recent research is that when you include these four Ps, what's more important? Turns out the first thing is product line. Product is the most important factor determining sales. Second is distribution. How widely you distribute the product. Third is price and fourth is promotion. You get a lot of play about promotion, a lot of thinking about promotion. Mainly because that's the thing you can change much more quickly and the one thing that is more prominent among the consumers. But the fact that it is actually, product, place, and price that are effects on sales. Now let's turn to another thing that is important when you use regression in marketing. This is the difference between statistical and economic significance. Statistical significance is something that we are commonly used to, right? This is the relationship observed in the sample likely to be observed in the population as well. So this is what e saw in p-value. We looked at p-values and said look is the p-value of a coefficient is less than 10%. Then the coefficient in the model is actually significant which means that this coefficient is likely to be observe to have an effect even if you look at another sample of data. Now economic significance is likely different than statistical significance. You can have statistical significance, but still not have economic significance. So what economic significance is, is that does the benefit from a marketing intervention justify the expense of that intervention. So that actually looks at what is called effect size, right? How big is the effect of a coefficient, so that it is worth the investment put into it. So now, let's look at an example here. Let's go back to the first example. Let's look at number of promotions in the x axis and y axis is dollar spend by the consumer, and we saw all those red dots, and the regression line going through, and by now you know what these numbers are, right? This is the intercept. And this is the coefficient of x. You know that by now. And when we looked at the regression you saw that the number of promotions has a p-value less than 0.1, so it is likely that number of promotions will have an effect even when you look at another sample. So you know that number of promotions has statistical significance. Now what you need to know is does number of promotions have economic significance. Let's see how we do that. So, a unit increase in number of increase in number of promotions increases units purchased by 1.42, that you get from the coefficient. Now assume the company making this product has a gross profit of $5. Cost of promotion is $0.50. Now what you want to do is construct an equation for profit. So how do we get profit? Profit is units purchased times gross profit minus cost of promotion times number of promotions. With the numbers we have up here, we can say units purchased is 1.42 that you get from here for a single promotion. If you have promotion of one unit, the coefficient gives you that value is 1.42, that is the predicted sales. Times gross profit, which is $5, minus cost of promotion, which is 50 cents, times you are doing 1 promotion. So 7.1, that's 1.42 times 5 minus 50 cents times 1, that is 0.5 that gives you 6.6. So for a single promotion you make a profit of 6.6. This means that promotions have economic significance. The cost of the promotion is less than the benefit from the promotion. If the coefficient was not 1.42, what if the coefficient was 0.01? What if the coefficient of 0.01 and not 1.42? Then, it is not worth it. The cost is equal to the benefit, you don't do the promotion. So, that's what I meant by saying that even when you have statistical significance, you may not have economic significance. You need to look at the effect size. You need to plug this into a profit equation and see whether it makes sense to invest into the promotion.