Can Math Teach Us to Strengthen Systems to Help Children and Families?

By: Scott Orsey

Change is hard, especially where structure creates imbalance in perspectives, power and impact. When years of good intentions yield underwhelming results for children and families, it is time to question the approach. In this four-part blog series, Scott Orsey explores the model used by scientists to measure health and well-being. He arrives at three conditions for change. Might these be the building blocks for the transformation we seek?

I am fascinated by the mathematical models people build to simulate activity observed in the real world. Models, of course, are imperfect. They can only approximate underlying behaviors, and they can have biases or be incomplete. They are built from observation, and can, at best, only emulate universal truths. However, even with these modeling limitations, once a system is expressed in mathematical terms, then the math itself can uncover ever more implications about what is happening in the world. Derived conclusions can be tested against observations and logic. I’ve written previously about one of the most famous physical-world predictions of mathematics that took a century to prove in my article on measurement challenges and Einstein’s “gravitational waves.” It is profound that math can give so much insight.

Unfortunately, there are few “first principles” equivalents to Einstein’s Theory of Relativity in the health and social services work that so many of us engage in to strengthen children and families. We don’t have fundamental E = mc2 types of understanding of human behavior and health. It is far harder to build sophisticated deterministic models (admittedly, physicists have also learned that even our universe is not as deterministic as we once thought). Instead, we model results using probability and we test the likelihood that our hypotheses about what works are true. In so doing, we seek to tease out the influence of a whole host of factors related to our desired outcomes.

I wonder if looking at the mathematical models we use to predict outcomes and evaluate our programs could provide deeper insights into the nature of our work and derive the conditions for our success.

In this blog article, I will frame a common mathematical model in what I hope to be a simple, easy-to-understand form. In future articles in this series, I plan to use that formulation to draw implications for our work. I hope in doing so, we might glean answers to some challenging questions. For example, what must we do to have the impact on children and families that we desire? How do we relate to the other people and organizations working in our communities? Why haven’t we made more progress against our goals than we would like? How can those with power and influence help improve the outcomes for all?

What We Model

Let’s first assume a hypothetical program for our mathematical representation. Let’s say that we administer this program with the goal of creating a desirable outcome for individuals in our system. Let’s also assume that we can measure whether each individual achieves the outcome or not (pass/fail).

When modeling a problem like this where the possible values of the outcome are binary (pass/fail), researchers often use a methodology called logistic regression. This mathematical tool seeks to represent the odds of a successful outcome. When the model predicts high odds (greater than one), we know we are more likely than not to get our desired outcome whereas low odds (less than one) predict the opposite. Odds are always greater than zero and “near-infinite” odds indicate we are nearly certain of an outcome.

The math used within this tool is fairly complex, so I will attempt to simplify things quite a bit by formulating the model in a way that we can draw conclusions from its structure. I will not do anything radical such as eliminating terms or assuming negligent contributions. I only wish to reframe for understanding. Bear with me on this, and if you need a more thorough understanding, a Google search will turn up far more explanation.

How We Model

For our hypothetical program, the odds of success for an individual are calculated at the population level and are defined as the number of individuals that succeed divided by the number of individuals who do not succeed in achieving the desirable outcome.

Our hypothetical program is designed to achieve this outcome by modifying (hopefully strengthening) one or more influencers or determinants of the odds of individual success. In the real world, there are many determinants (x) that influence the outcome for an individual. Some are within the impact of our program and others are outside our influence. The challenge for researchers is 1) finding those relevant determinants and 2) expressing their relative contribution to the odds of success. If done effectively, a researcher can tease out the impact of a program or situation.

The expression typically solved in logistic regression analysis looks like this:

Where:

xi are the values of each determinant for an individual in the system.

βi represent constant coefficients that define the magnitude of influence for each of the determinants.

While xi are unique to each individual, βi are discoverable truths within our population and program set up. At least that is what the model assumes.

Simplifying for Understanding

That’s about as complex as the math needs to get because we are not, at the moment, trying to discern relevant determinants (x) or solve for the coefficients. Instead, we would like to take this valid modeling approach and see what its structure might mean for us in the real world. Using the Product Rule of Exponents, the equation can be rewritten:

Then using some simple substitutions, it can be written as

Where:

Ϲ = eβo is a constant.

fi = eβixi are functions representing the influence of each of our determinants on the odds of success.

There you have it, complex math made simple. It is this form of the mathematical model that I believe holds the key to much basic understanding of our work. In layman’s terms: The chances of success are governed by the influence of many factors multiplied together. The significant phrase here is “multiplied together.” The phrase is not “added together” or a more-complex mathematical relationship. For binary-outcome modeling problems, the relevant relationship is multiplication.

If you can accept that, then we can move forward with our questions. What implications does this model’s structure create? What can we glean about how best to improve outcomes for children and families? What policy implications can be drawn?

Over the course of the next articles in this series, I will show how the model implies three important pre-conditions for change in complex social and political systems:

Those with power and influence must recognize that they cannot solve everything and that there is a problem with the current structure.
Those with power and influence must yield some of that power and influence to make room for others.
Change must be safe for all.

While these conditions may apply to any situation dominated by status and privilege, I will draw them from the specific example of health and well-being for children and families, where the healthcare sector has a dominant, perhaps outsized, leadership role. Let’s start making some observations.

In the next article in this series, I will explore the first of these pre-conditions: recognizing the problem with the current structure.

Author’s Note: My deepest gratitude to Amy Hunter, PhD, MPH, who reviewed this series, challenged my thinking and offered feedback and insight.