Home > Articles

This chapter is from the book

This chapter is from the book

A Full Example

Let’s put some meat on these bones. Suppose that you’re interested in the relationship between body weight and LDL cholesterol levels. You have a simple, straightforward hypothesis that, other things being equal, there is a direct relationship between a person’s body weight and his or her LDL level. To take advantage of the tools that quap gives you, you’ll need an alist, one that looks something like the following code:

adult.weight <- read.csv("Sample Weight Data.csv")

The read.csv statement attempts to open the file named Sample Weight Data.csv in the working directory. You can include the file’s path in the argument to read.csv; if you handle it that way, keep in mind that R uses forward slashes, not back slashes, to delimit folder names in file addresses. Or, you could save the data file in what you know to be R’s current working directory.

sample.mean.wt <- mean(adult.weight$Weight)
  ldl.model <- quap(
        LDL ~ dnorm( mu , st.dev.wt ) ,
        mu <- alpha + beta *( adult.weight$Weight - sample.mean.wt ) ,

I have given these two variables in mu’s definition the names of alpha and beta, because that’s how they are normally referred to in the literature on simple (i.e., not multiple: only one predictor) regression: alpha is the intercept and beta is the regression coefficient.

Now we need to establish the central tendency and the spread of the alpha and beta priors. We can tell quap that alpha, the equation’s intercept, has a mean of 20 and a standard deviation of 20:

alpha ~ dnorm( 20 , 20 ) ,

and that beta, the regression coefficient, has a mean of 0 and a standard deviation of 1:

beta ~ dnorm( 0 , 1 ) ,

and that the standard deviation of body weight follows a uniform distribution with a mean of 0 and its own standard deviation of 50:

        st.dev.wt ~ dunif( 0 , 50 )
    ) , data = adult.weight)

And you’ll need a set of observations that are stored in the csv file Sample Weight Data.csv. They are temporarily stored by R in the structure named adult.weight. Here are the first few observations in adult.weight. Note that the first row of the csv file contains field names. The read.csv’s header argument is set to True when the table’s first row contains one fewer field name than the table’s number of columns. (This is often true when the first column contains row numbers but is not the case here.)













And here are the results of running the code, shown in precis form:

> precis(ldl.model)




















You can compare R’s results to Excel’s by running LINEST. The LINEST function (entered normally in current Excel versions, by selecting a single cell and using Enter rather than Ctrl+Shift+Enter) is


Here are the results of running the Excel LINEST function:











The LINEST results require some mapping:

  • The value in LINEST’s first row and rightmost column (here, 58.52) is always the equation’s intercept. Notice that the quap model returns a value of 58.27 (second row, second column of the precis summary). The two values are quite close, and the difference is easily attributable to sampling error in the quap model.

  • The value in LINEST’s first row and leftmost column (here, 0.064) is always the final regression coefficient. In this case, because we have called for one coefficient only, it is also the equation’s first and only coefficient.

  • The value in LINEST’s second row and rightmost column (here, 1.655) is the standard error of the intercept. It is always in that cell, of LINEST’s results, directly below the intercept. Its value is quite close to that returned by precis in its third column, second row, 1.63.

  • The value in LINEST’s second row and leftmost column (here, 0.071) is the standard error of the regression coefficient. Values in the second row of LINEST results are always the standard error of the statistic in the same column, first row.

  • The value in LINEST’s third row and rightmost column (here, 11.7) is the standard error of estimate, and it is quite close to the precis estimate of 11.53. Suppose that you took all the observations at a given value of the predictor and found the standard deviation of the difference between their actual and the predicted values on the predicted variable. That’s the standard error of estimate, and it helps you decide whether the prediction equation is more accurate at some levels of the predictor than at others.

Compare the results of the regression analysis as returned by Excel with those returned by quap via precis. It’s clear that where the two approaches return the same analyses (e.g., intercepts, coefficients, standard errors), the Bayesian approach and the frequentist approach are either identical or very nearly so.

And you can get those results without risking the slippery slope of multicollinearity. Which makes this a good point to go further into multiple regression.

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.


Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.


If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information

Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.


This site is not directed to children under the age of 13.


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure

Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020