 # STOP! – Understand and Apply Growth Curves Correctly

Growth curves are a critical part of many different disciplines including the physical disciplines, however, many business,  with the exclusion of financial companies, tend to neglect growth curves due to the preference of simpler, easy to implement linear lines. They are often misunderstood and when incorrectly applied,  lead to very divergent results. If used properly, however,  they are a great way to understand long term behavior of business activity. This article presents a brief analysis of a few different curves and compares them to a linear approach and further examines their application.

THE LINEAR LINE

One of the key problems for a linear line is that it assumes that as x (or time) increases, the increase in the variable of interest increases in a corresponding manner, using it as a measure of the slope. Using the linear line is easy to implement and interpret, but business activity is rarely linear.  Linear lines will work fairly well, when growth is not clearly understood, or the growth cannot be seen because the visibility of the data observations is too narrow, i.e. a small period of data is analyzed, but the long term data is unavailable or unknown.

In the first chart, we have an estimation line to predict revenue on a weekly basis using 4 periods. Four periods isn’t really enough, however, we are just using this for the purposes of this example. As you can see the straight line somewhat predicts where revenue should be, however, we probably need more data points. In the next chart we are given ten points for prediction, and as we can see a straight line provides us with a pretty good estimate. Unfortunately for the analyst, without visibility into the future, or at least knowledge of how growth occurs, they would never be able to correctly model the prediction line.

As the third chart demonstrates, over time the graph bends, and it would only be through long period observation, theoretical understanding, or clear knowledge of how the pattern increases that the analyst would be able to see this curvature and create a proper model. Even applying the proper growth curves is important because not all growth curves are the same. Understanding the differences between the power curve, logarithmic growth curves and square root curves is critical to more accurate long term prediction.

FINDING THE RIGHT CURVE

Since not all growth curves are the same, how can an analyst tell which one to use. First, the analyst must clearly determine whether or not a linear line is really the best approach to long term prediction. If not , then the analyst must choose which growth curve to use. The first growth curve that should be attempted is a natural log growth curve. Then using forward predictions, you can estimate whether or not the log curve is under fitting or over fitting. If there is overfitting or under fitting then the analyst should look at fitting a power curve where the exponent is less than 1. Recall that an exponent of 1 would mean a linear growth pattern. Further an analyst should recall that a square root curve is a power growth curve with an exponent of 1/2. In a future article we will analyze how to apply these more succinctly.

It should be obvious that not all growth curves are the same. An analyst should be careful of trying to apply a growth curve for one data set to another data set simply because they believe they are two sides of a coin. For example, lets assume a subscription provider is examining the number of interactions they have with a customer, and these interactions diminish over time, e.g. website visits. If you group customers together to form a cohort and you wish to aggregate their visits since their inception, as one would do in an Lifetime Value cohort analysis, the cumulative interactions could continually increase but the increases would diminish over time. For example, month 1 may have 100 interactions, then in month 2 , the same cohort has 50 interactions, generating a total number of interactions for the cohort at 150. The third month can have 25 total interactions and now the total number reaches 175, etc.

Examining the interactions and its growth provides the analyst with insight into the behavior. A natural log model may work well in this cass, of course this is dependent on the data. However, if one wishes to look at the revenue associated with those interactions, the analyst could make a mistake by applying the same curve to those interactions.

Lets assume that each interaction has the same value, then it is likely that the curve might be similar since the units are stable. However, what if the revenue for each interaction is different. The differences in revenue create a brand new problem for the pervious model. In the previous model it assumes that all interactions were equal. In this new example, the interactions have a different value.

NOT ALL GROWTH CURVES ARE CREATED EQUAL

The next chart demonstrates three different revenue levels. The first 10 weeks have an average of \$20 per interaction, the last 10 weeks have an average of 5\$ per interaction, and the ones in between have an average revenue of \$10 per interaction. If the interactions at the beginning are more valuable than the interactions at the end , then the model created could significantly overfit or overstate the revenue toward the end of the model. In the second graph below, if the average revenue per interaction is more valuable at the end, then the model’s prediction will vary widely from the actual values. These examples are an oversimplification, but presents the challenge using a growth model in one instance, and applying it to a second. In general, its true for linear models as well, however, the growth models will exacerbate the issue.

Analysts should take every caution to understand the data and the general pattern, also known as a theoretical pattern, in order to create proper models for predictions. Generally speaking, one of the most critical mistakes made by analysts is not recognizing distributions and growth patterns, attempting to use techniques in a manner not intended, or not understanding the consequences of inappropriate models. We highly recommend that analysts make every effort to understand the models they are using and employ the correct methods, and avoid simply using models from one context to the other without proper testing and validation.

### Alexander Pelaez, Ph.D Alexander Pelaez, Ph.D., is a President of Five Element Analytics, an analytics consulting firm. He has served as a senior executive to a number of firms in healthcare, retail and media. He is also a professor of Information Systems and Business Analytics at Hofstra University.