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##### How to Interpret Your Results

Public libraries are only one node in a web of assets governed by institutions: laws, regulations, and policies which influence how, when, and whether people can access resources. And each service we provide is only impactful on some community level outcomes when our community members use the service. For instance, we offer motivating incentives to read during summer months because we know that children’s literacy doesn’t improve unless the books move from shelves to little hands.

The percent change displayed here tells a specific story of service impact: **given the slice of influence public library service has on this outcome, changing the service mix locally would improve the effectiveness of the library’s contribution by the resulting percentage.**

This percent change result is designed to help you decide between and within your host of useful services. Absent from the services you can select is staff hours. Staffing, like facilities, influences the level of service a library can provide across these specific offerings. There is also another, difficult to see in national data, element to library worker influence: relationship building. Across research from this project, its antecedents, and many others, we hear that it is the availability and willingness of library workers to interact with visitors and community members that creates feelings of safety and belonging. And that these two feelings are the context required for robust use, and therefore robust outcomes.

In a similar vein, the number of hours open influences across these dimensions as well. In order to capture the influence statistically, there has to be variation and throughout most of the US, there simply isn’t enough variation in hours open that we can tie to varied community level outcomes. That said, we hope that even when the influence in the calculator is 0, you will consider how increasing your hours will aid in your increasing access to your collections or attendance at your programs.

##### Technical Notes: What's Going on Here?

To read a pretty long .pdf document which says basically the same thing as this section, click here.

### What, precisely, is the calculator doing?

This calculator formula is a static optimization which uses a Cobb-Douglass production function format. This general production function is a measure of productive outputs a firm can make given a set of inputs where a total productive factor is multiplied by factors of production, each raised to a power defined by the level of output responsiveness to inputs (formally: elasticity). We can express this with a formula as below, where ^ means raised to the power and * mean multiply:

Output = A * (Library Programs ^ Program Elasticity with Output) * (Library Items for Circulation ^ Collection Elasticity with Output) * … and so on.

Let’s go over each part of this function and how it is estimated so you can decide whether or not it is a sensible approach for thinking about your work in public libraries.

#### Defining Parts of the Library Co-Production Function

Wait! The Cobb-Douglass production function is referred to above. What is co-production and why is the calculator displaying co-productive results instead of more traditional input-output measures?

Co-production is a way of thinking about how outcomes aren’t created by services or interventions on their own. People aren’t empty vessals which then do exactly as directed to. Each individual lives within their own context and Libraries in Community Systems is interested in understanding and incorporating that complexity into the way we think of public libraries and service.

###### ELASTICITY

In our model, a measure of co-production enters in with our estimates of elasticity. Before thinking in math about elasticity, think of a stretchy band. The more elastic the band, the more it will stretch when you pull it apart, and the less elastic, the more difficult it will be to pull and it won’t get quite as big. When talking about inputs and outputs, elasticity refers to how stretchy (responsive) the output is to a pull (change) in the input.

Each elasticity in our function is a number estimated using library and community data from 2010 – 2019 or 2021 (depending on the index). To incorporate the importance and magnitude of individual use of a service into the measure of service impact (level of mediation), we employ structural equation modeling. Theoretically, we can think of our estimation approach as a mediator model, which is described in more detail below. In practice, the estimation using these data is a set of simultaneously calculated regression equations which are then combined into a single measure of mediated elasticity.

Deriving the magnitude of responsiveness of an outcome to a library service (as mediated by use) is conducted for every combination of single or multiple outlet system status, level of rurality or urbanity, within each region of the US. Sometimes there isn’t enough information available for a certain combination to get a useful result. In such cases, that combination takes the result of the next level up. For example, I don’t derive interpretable Economic Wellbeing results for single outlet libraries serving small towns in the “Mid East” region stretching from NY to DC along the coast. In order to use the information I do have, these libraries take the elasticities from single outlet libraries serving small towns nationwide.

###### MEDIATOR MODEL

One way people have measured the relationship between libraries and some outcome is in a regression with control variables. In that setting, the service of interest is in the model and use of that service is either absent or included as a distinct independant variable. The typical (imprecise) form is Outcome = Service + Use + Community Characteristics + Library Characteristics + error. With this set up interpretation of the results would read, “holding usage (and all else) constant, service has XX impact on outcome”. Hold on! Levels of use and levels of service are linked! Why would we want a measure about a service that somehow strips usage?

Hypothesis: Public Library Service Provision Contributes to Community Level Outcomes both Directly and Through its Use

Now, imagine a triangle, and each point is given a letter, A for the left point, B for the top point, C for the right point. Transform the lines between the points into arrows. You should now have A ->C, A ->B, and B ->C. A is our library service. C is our outcome of interest. B is service use. Each of these arrows is an equation we estimate in our analysis. A -> C: Service (like collections) influence on Outcome (like economic wellbeing); A -> B: Service (like collections) on one paired Use (like circulation); B -> C: Use on Outcome.

Once all three equations are estimated, called path analysis, results are combinesd like this: A -> C + (A ->B * B ->C). This process gives us the elasticty used in the service choice calculator, and tells us the relative influence of use on outcome. Some outcomes, given our data, are more strongly influenced by the library providing the service than on its use. Others are driven almost entirely by use. The results and possible library policy interpretations are given on each Index’s in-depth page in the section “Users Produce Outcomes”.

#### Outcomes & Indexes

An index gives a single value to a bundle of measurements. Most people are familiar with the Consumer Price Index, which takes the prices of a bundle of consumer goods and estimates a single index value. The purpose is to make something complex and multi-dimensional into something we can compare over time or over space (or both!).

The index values are based on density in region standardization (rural southeast places are compared to other rural southeast places). Each Index has its own detailed documentation including each thinking and doing step in the process of getting to the Outcome measures used in the calculator. As a starting point, indexes are replicated from the IMLS study Understanding the Social Wellbeing Impacts of the Nation’s Libraries & Museums (Norton, et al, 2021) following their detailed technical appendix.