Today’s post ventures into unfamiliar territory for many of us, the pedagogy of equations and statistics. It is part of a series summarizing an economic article in HHTM’s Journal section entitled Supply and Demand for Audiologists.  

If you think we’re down in the weeds again, where Econ 202 posts usually end up, you aren’t wrong. But there is method in the madness so please read on.

Besides, it’s the 4th of July so we don’t expect many readers of posts this week. Tune in next week when we pick up the summary of Supply and Demand of Audiologists again.


Central Tendencies and Parametric Stats — It’s a Start


Most audiologists take a basic course in statistics or research methods at some point in college, or as part of their graduate training. How else to read research in our own journals and understand how to interpret results described in terms of means, medians, variance, confidence levels, significance levels, risk levels, etc? How else to implement best practices in clinical settings, with a working understanding that what is best for 95 patients through the door may not be what is best for another 5?   

But clinicians know that a single variable and measures of central tendency do not a patient, or a successful outcome, make. Anyone who has worked for even a short time with those who report hearing difficulties knows that the audiogram is a starting point at best and a red herring at worst when it comes to hearing aid fittings. Clinicians sense that some, maybe many, variables in the equation for successful treatment outcomes have yet to be nailed down, much less measured.


The Story Isn’t Complete and It’s Not Simple


A number of studies throughout the years have taken statistical stabs at identifying the formulary for successful hearing aid fitting outcomes, using survey responses (e.g., MarkeTrak), retrospective and occasionally prospective studies. Most recently, our field has taken up large data sets made available from ongoing epidemiological studies (e.g., NHANES) to grab more variables and see if they contribute to the formula for fitting success.  

It is hard to keep so many variables in one’s head, much less give each its due, even when the equations signal their relative importance or lack thereof. Perhaps this is why there is a strong tendency to single out a heuristic1 variable (e.g., price) as the main mover and shaker even though it not justified by the analyses.

This natural, but erroneous, tendency to pare down and explain all in “simple” terms is manifest even among those who navigate comfortably in the statistical maze and perform some of the epidemiological work. How else to explain why the present social policy discussions identify  cost and access as the clear and present villains which prevent all from wearing hearing aids, when even the most basic analyses make it equally clear that this thinking is over simplified at best and almost certainly wrong in most cases. 

Could it be that many people do not prefer hearing aids at any cost point when they assess other choices for how to spend their time and money?


Using Models to Predict Outcomes


All of the above is a thought exercise bringing readers to the point of today’s post. In order to keep many potentially relevant variables in our heads, we need to step up our game and familiarize ourselves with a tool or two from basic econometrics, which Investopedia straight-forwardly defines as “…the application of statistical and mathematical theories … for the purpose of testing hypotheses and forecasting future trends.”


Count the Squares


Here’s a brief primer on one tool called OLS regression, where OLS = Ordinary Least Squares. The goal of OLS, like other regression methods, is to go beyond correlation and find causation by identifying determinants of outcomes.

Figure 1. Examples of relative success of a linear model at “fitting” observed data.  The smaller the variance of observed responses (the squares of standard deviations) from the line, the closer the fit. Hence the name, “least squares.”

Start with a single regressor linear model:  

 Y = ß0 + ß1X1 +  ε    

and consider how Y changes with every 1-unit change in X. 

Algebraically, we know that this plots as a line which intersects the y axis at ß0 and has a slope of ß1. What OLS does is compare actual values of Y (e.g., our hearing aid outcomes for N subjects) to predicted values of Y, based on the line estimated by the equation. 

The success of the OLS model hinges on how small the average squared differences (the variances) are between actual and predicted values – the smaller the variance, the better the predicted fit. That is measured by “R2”, the coefficient of determination (c.f. Figure 1).

The model breaks down as follows: 

Dependent Variable Y: This is the effect, the outcome, we’re measuring. For instance, Y could be a scale of patient-reported hearing aid satisfaction from 1 to 10.

Constant/Intercept β0: Y depends on some constant which we don’t care much about that is β0  (it is what it is). For example, all those observations in the dataset may come from subjects who at least 60 years old, so the regression “starts” at 60.

Independent (Regressor) Variable X: What we do care about is what measureable variable (e.g., patient age) could “cause” our outcome to change. That potential influencer is X1, where the subscript denotes each subject in the dataset, from 1 to n.  

Regression Coefficient ß1: Maybe age does predict hearing aid satisfaction, but if it does, then by how much and in what direction (i.e., does satisfaction grow or diminish with age)? ß1 multiples the regressor variable and therefore determines its magnitude and sign. When it’s 0, we know that X (age) has no effect on Y (hearing aid satisfaction) . The bigger ß1 is, the more our model “expects” that X affects Y.  When ß1 is negative,  the model predicts an inverse relationship between X and Y; when positive, we expect that Y grows as X grows. 

Going back to the model and plugging in some made-up numbers, if  

 Y = 60 + (-0.5)X1 +  ε

then we expect hearing aid satisfaction to decline a half-point on the 1 to 10 scale with every 1-year increase in age, all other things held constant.  

Error term ε: It’s simplistic to expect age or any other single variable to explain 100% of hearing aid satisfaction. Which is why the model has  ε . That “noise” variable is everything else that independently influences hearing aid satisfaction, essentially everything that our single-regressor variable model doesn’t explain.  

With the simple model above, you can be sure that ε is going to be big and a lot of “other things” have to be held constant in order for age to “predict” hearing aid satisfaction.


More Independent Influencers, Less Noise, Better Prediction


Which is why it’s better to use a model with multiple regressors, so long as the regressors aren’t themselves related (e.g., age and blood pressure tend to vary together so they can’t serve as independent regressors in the same model). That would violate the assumption of collinearity, which is one of several assumptions required of the model (and not discussed here). 

Eventually, the linear model looks like this:

Y = ß0 + ß1X1 + β2X2 + β3X3 … βjXj + ε

Where there are 1 through j “vector” variables and the expected contribution of each variable on, for instance,  hearing aid satisfaction (when the others are held constant) can be seen by the size of the respective correlation coefficients, β1…j.

Finally, within the model, we can apply statistical tests to test the research and null hypotheses (H1 and H0) by calculating the probability and stating a level of confidence that a given variable does or does not significantly affect hearing aid outcome. As for the overall model, R2 tells us what percent of the variance in hearing aid satisfaction is explained by the model.


A  Tool Set We Can Use


Most audiologists are not going to run out and start modeling data, even if their math and stat skills allow for it.  But, all audiologists will feel good about being able to look at a regression equation and understand what variables are influencing whatever outcome the model is trying to predict.  

The take-home is that methods exist which allow us to test assumptions about what factors influence consumer decisions to purchase and use of hearing aids, choose a provider, etc. Price and access are likely determinants, but so are many other variables we can and should plug into our models.




1Heuristics in psychology refer to “mental shortcuts that usually involve focusing on one aspect of a complex problem and ignoring others.” Wikipedia


images from wikihow &  minsitab

by Kelli Marquardt, BS (applied mathematics and economics)

Kelli Marquardt, BS

 The supply of audiologists in the US is not on track to cover the rising demand in the next 30 years (Windmill & Freeman, 2013, 2017).  This post summarizes on-going economic analyses in a pilot study of the US market for audiologists. The study goals are to establish determinants of supply and demand and estimate effects of those determinants on the market. Click here to access the full study.  

Preliminary results outline three main findings:

  1. Demand for audiologists is price elastic while supply is relatively price inelastic.
  2. Costco Hearing Centers have become a substitute for hearing aid testing and fitting services and their entrants into the market are decreasing the demand for audiologists.
  3. The most significant determinant of supply are academic variables, specifically the number of audiology programs available to students.


 Limitations of Equilibrium Analyses of Demand and Supply


As the average age in the US rises, so too does the demand for health care services geared to an aging population.1  Paired with this increase in demand is a static or even decreasing audiology work force (Windmill & Freeman, 2013, 2017;  Hosford-Dunn,2017a,b,c). These statements are examples of conclusions based on observed data.

Observed data is reflective of market equilibrium. Demand and supply represent relationships between wage and employment. As the wage increases, more people are willing to work, and thus we expect to see an increase in quantity supplied. On the other hand, employers must pay their workers more, and thus we expect to see a decrease in quantity demanded. In equilibrium, the equilibrium wage equates quantity demanded and quantity supplied.

Predictions as to the reasons underlying observed inadequate supply of audiologists have been made based on observational data. Likewise, observational data have been used to make predictions for future shortfalls of audiologists in the labor market (Windmill & Freeman, 2013).

But, without a complete analysis of what factors have affected supply and demand in the past, we cannot distinguish the different determinants of quantity supplied and quantity demanded in the audiology workforce.  


An Econometric Approach, Summarized


The present pilot study is the first step in identifying determinants of supply and demand in the audiology labor market. Instrumental variables and a two stage least squares regression are used to estimate both the demand and the supply equations for the Audiology labor market. All equations together with in depth description of  econometric rationale can be found in the complete journal article (see introductory paragraph of this post).


Developing a Model


We first developed a general model and identification strategy to estimate quantity demanded and quantity supplied. Both were assumed to be linear functions of wage and exogenous variables. 2 A 2 stage least squares regression with instrumental variables was used to ensure estimates were consistent and unbiased (Angrist et al. 2000).

Variables were defined as follows:

  • Controls:  unemployment rate, per capita income, population, year and state fixed effects. These variables are exogenous to the model and affect both the supply and the demand for audiologists
  • Instrumental variables (shift Supply or Demand, but not both):  
    •  Number of audiology training programs, considered a supply shifter but not a demand shifter, as discussed below.
    • Number of Costco shopping centers, considered a demand shifter but not a supply shifter.
    • Percent of population 85 years and older, a demand shifter but not a supply shifter.


Identifying a Supply-shifting Variable


Supply-shifting variables affect the supply of audiologists but do not directly determine demand. We claim that the number of audiology programs is an appropriate supply-shifting variable because:

  • The only way to practice as an audiologist is to obtain a degree from an accredited audiology training program.
  • If the number of training programs increases, at any given wage, more students have the opportunity to attend a program and pursue an audiology career.

Therefore, the supply of audiologists will increase along with the number of audiology programs.

On the other hand, we claim that the number of audiology programs does not directly affect demand.  While it is intuitive that as demand for labor rises, so too will the demand for educational programs, it is also the case that schools are slow to respond to these changes in demand due to long lags or inertia in opening new AuD programs. The effect of demand on the number of audiology programs would not be realized until several years later, if at all.

Thus, we claim that in a given state and given year, the number of programs available is not correlated with the demand for audiologists, which makes number of programs a valid instrument to use in estimating the demand equation.


Identifying Demand-shifting Variables


Demand-shifting variables shift demand for audiologists but have no direct effect on supply.  We claim that both the percent of the population over the age of 85 and the number of Costcos are demand shifting variables.

Half of the population over 85 years of age experience significant hearing loss and are likely to demand hearing health services, which makes this variable a demand shifter.

Treating Costco as a demand shifter is based on the following assumptions and logic:

  • As time progresses, hearing aid technology advances.
  • Demand for hearing aids increases as the aging population increases.
  • Costco’s corporate policy since at least 1998 has been to add Hearing Centers to existing and future stores3, such that almost all US Costco stores provide hearing aid tests and sell hearing aids to adults. By report, there are currently 482 Hearing Centers (Swearinger, 2017) in 506 US warehouses (2016 Costco Annual Report to shareholders). 
  • Costco hires state-licensed hearing aid dispensers and/or trains existing employees in-house to be licensed hearing aid dispensers,in addition to seeking to hire audiologists.
  • Therefore, many people who traditionally sought out audiologists to obtain hearing aids can now chose to see a hearing aid specialist at Costco, a close substitute in demand for audiologists.

With this emerging substitute for hearing aids and associated services, we expect to see a decrease in demand for audiologists, making Costco a demand-shifting determinant.

Neither Age or Costco variables directly affect supply. The quantity of audiologists supplied is essentially the number of licensed audiologists who want to work as audiologists. This is an individual choice that entails extensive schooling and high cost to obtain the required degree and licensure.  At least in the short run, it is unlikely that the percentage of people 85 years and older or the number Costco shopping centers in a state will influence an individual’s complex academic/professional decision to work as an audiologist.

Therefore, the decision to work as an audiologist is independent of the percentage of older individuals and the number of Costcos per state, making Age and Costco valid instrumental variables that shift demand but not supply.   


Testing the Model


Part 2 of the study summary describes data collection methods and explains results of first-round modeling of supply and demand in the audiology workforce.




1 According to the recent study by Goman & Lin (2016), approximately 81% of people who are 80 years or over experience hearing loss, 55% of individuals 70-79 years of age experience hearing loss, and 27% of individuals 60-69 years of age experience hearing loss

In econometrics, exogenous variables are independent factors which are not causally related “within” the model (i.e., non-correlation with other independent variables in the regression equation).  Click here for an intuitive video explaining exogeneity.

See Costco Annual Reports to shareholders for 1998, 2002, 2016.; also Swearinger, 2017.

4 Costco’s corporate strategy of in-house talent cultivation is manifest on its Career’s  website: “Exciting opportunities, Personal and career growth, Friendly and supportive work environment, Stability, A workplace focused on ethics and obeying the law, and Great benefits”(Yang et al. undated).   




Angrist, J and Krueger, AB, 2001. “Instrumental variables and the search for identification: From supply and demand to natural experiments” (No. w8456). National Bureau of Economic Research.

Goman, AM and Lin, FR, 2016. “Prevalence of Hearing Loss by Severity in the United States.” American journal of public health, 106(10), pp.1820-1822.

Hosford-Dunn, H, 2017a. New Year’s Resolution: Demand an Audiologist. Jan 3, Hearing Economics,

Hosford-Dunn, H, 2017b. How much and how low, Audiology workforce part 2. Jan 10, Hearing Economics,

Hosford-Dunn, H, 2017c. Supply and demand in the audiology labor market, part 4. Feb 21, Hearing Economics,

Swearinger, G. April 18, 2017. FTC Workshop Transcript Now Hear This: Competition, Innovation, and Consumer Protection Issues in Hearing Health Care. Federal Trade Commission.

Windmill, IM and Freeman, BA, 2013. “Demand for audiology services: 30-yr projections and      impact on academic programs.” Journal of the American Academy of Audiology, 24(5), pp.407-416.

Windmill, IM and Freeman, BA. 2017. Demand an audiologist, but will there by one available? Hearing Economics,

Yang, J, Whitfield, M, McKee, C. et al. (undated). Competition with both quality and quantity – a case study. 


Kelli Marquardt is an Economics PhD Candidate at the University of Arizona. After growing up in Colorado, she attended the University of Dayton and received a Bachelors of Science in Applied Mathematics and Economics in 2016. She has completed her first year in the University of Arizona Economics PhD program and plans for future research in the fields of labor and health economics. Between classes and research, Kelli also works as an teaching assistant and will teach her first college course this summer.


feature image from Bureau of Labor Statistics