# Regression Using Bayesian Methods

This chapter is from the book

## Arranging a Bayesian Multiple Regression

Earlier in this chapter I described how to provide arguments to a quap function that support a single-predictor regression. I’ll review it briefly here. You supply these arguments:

• A variable that represents the outcome for each case, such as a car’s MPG, usually the name of the outcome variable. For example:

`MPG <- dnorm ( mu, sigma)`

specifies that MPG’s density is normally distributed (dnorm) with a mean of mu and a standard deviation of sigma. This outcome variable is usually input in a data frame along with the predictor (see below).

• A parameter, often but not necessarily termed mu, that represents the result of the regression equation. For example:

`mu <- alpha + beta ( predictor )`
• Parameters, usually but not necessarily named alpha and beta, that represent the constant (or the intercept) and the coefficient (or the slope) in the regression equation.

• A parameter, often but not necessarily termed sigma, which represents the standard deviation of the outcome variable. This determines the spread of the outcome variable’s distribution across its x-axis.

• A data frame that contains, at a minimum, the values for the outcome variable (in this example, MPG) and for a predictor variable such as Speed. The data frame might be named CarData.

Here’s how the quap function might appear for an analysis of MPG given a single predictor variable, Speed:

```CarQuap <- quap(
alist(
MPG ~ dnorm ( mu, sigma )
mu <- alpha + beta ( Speed )
alpha ~ dnorm ( 0, 1 )
beta ~ dnorm ( 0, 1 )
sigma ~ dexp (1)
), data = CarData )```

• As I mentioned earlier in this chapter, it’s usually a good idea to standardize the values that you supply for the outcome variable and the predictor variable(s) before passing them along to quap. Doing so minimizes the effects that numeric overflows can have on the results of the analysis. You can use an R function, standardize, to handle this for you, or you can subtract the mean value of a variable from each actual value and divide each result by the variable’s standard deviation. (The results are often termed z-scores.)

• One result of this standardization is that the z-scores will have a mean of 0 and a standard deviation of 1. It often works out well, especially if you have standardized the predictors and the outcome variable, to use 0 and 1 as the mean and sigma of the dnorm arguments that describe the distributions of alpha and beta.

• Notice the use of the tilde instead of an assignment operator in several lines of the quap code. This simply indicates that a parameter is to be distributed as the density of, in this case, a normal curve.

• In this example, sigma is specified as sigma ~ dexp(1). The dexp function returns the density of the exponential distribution, which is the parent for a variety of other continuous distributions such as the Gaussian-normal, the Gamma, the Poisson, and the Binomial.

The exponential distribution has one parameter, rate (or lambda); by contrast, the Gaussian distribution has two: the mean and the standard deviation. In R syntax, the exponential distribution’s rate parameter is 1 by default, and the dexp function returns the density probability for the associated quantile, x (or 1 as here). Among other reasons, the exponential distribution is handy for specifying sigma, because the exponential is constrained to positive returns, and the standard deviation is, by definition, a positive quantity.

That’s all you need for a simple regression of one outcome variable on one predictor. To add a predictor and analyze the simultaneous effect of two on one outcome variable, you need four items omitted from the single-predictor analysis:

1. The additional predictor named Weight should be added to the input data frame named CarData above.

2. The additional regression coefficient, for Weight, must be specified by the addition of this line of code:

`Weight_beta ~ dnorm ( 0, 1 )`
3. In addition, for clarity it makes sense to edit the existing specification for the Speed coefficient to this:

`Speed_beta ~ dnorm ( 0, 1 )`
4. The Weight predictor and its coefficient should be added to the regression equation. In the single-variable example, that equation looks like this:

`mu <- alpha + beta ( Speed )`

In the two-variable example the equation looks like this:

`mu <- alpha + Speed_beta ( Speed ) + Weight_beta (Weight)`

The full code example might look like this:

```library(rethinking)
setwd("C:/Users/conra/Documents/Pearson Bayes/Drafts/Ch 6")
#You may need to adjust the path to the .csv file on your computer
#The three variables are named Spd, Wt and Mileage
#in the csv file. They are saved as newly standardized data
#with new names (Speed, Weight, and MPG) in the
#same steps that standardize them.
CarDataFrame\$Speed <- standardize( CarDataFrame\$Spd )
CarDataFrame\$Weight <- standardize( CarDataFrame\$Wt )
CarDataFrame\$MPG <- standardize( CarDataFrame\$Mileage )
regmodel <- quap(
alist(
MPG ~ dnorm( mu , sigma ) ,
mu <- a + ( Speed_beta * Speed ) + ( Weight_beta * Weight ) ,
a ~ dnorm( 0 , 1 ) ,
Weight_beta ~ dnorm ( 0, 1 ) ,
Speed_beta ~ dnorm ( 0, 1 ) ,
sigma ~ dexp( 1 )
) , data = CarDataFrame )```

You can get a smattering of summary information using the rethinking library’s precis function. Simply supply it with the name of the quap model you just created, and specify the number of significant figures if you wish:

`precis(regmodel, digits=6)`

Here’s what precis returns:

 mean sd 5.50% 94.50% a -1.1E-05 0.131111 -0.20955 0.20953 Weight_beta -0.30059 0.137421 -0.52021 -0.08096 Speed_beta -0.01806 0.137417 -0.23768 0.201556 sigma 0.93517 0.092242 0.78775 1.08259

(The 5.50% and 94.50% limits are how the developer of the rethinking package chooses to protest the conventional and arbitrary criteria of, for example, 5% and 95% confidence intervals.)

To check your work, consider running a true regression package on the data that this section has analyzed. One convenient way, using continuous predictors and an outcome as here, is to use the lm package. If you do so after running your Bayesian analysis you can take advantage of the data frame you just created. For example, you can get quite a bit of summary information from these two statements, which return the results shown in Figure 6.4:

`Car_lm <- lm (CarDataFrame\$MPG ~ CarDataFrame\$Speed + CarDataFrame\$Weight) summary(Car_lm)`

Notice first that the intercept and coefficients returned by lm are close to the a (alpha) and Speed and Weight (betas) returned by quap and precis, but do not duplicate them precisely. This is largely due to traditional regression’s use of the maximum R2 as its criterion that a solution has been reached.

Furthermore, lm by default returns only three significant figures, but you can choose the number of digits with quap’s digits argument. You might want to compare as many as, say, eight digits in the regression coefficients. One way to do so is via the options function. For example, these functions:

```options(digits=4)
coef(Car_lm)```

return these results:

```(Intercept)  CarDataFrame\$Speed CarDataFrame\$Weight
-3.036e-16          -2.005e-02          -3.066e-01```

but these functions:

```options(digits=6)
coef(Car_lm)```

return these results:

```(Intercept)  CarDataFrame\$Speed CarDataFrame\$Weight
-3.03642e-16        -2.00472e-02        -3.06564e-01       ```

(In the latter two examples I’ve used the coef function instead of the summary function to save space by showing only the coefficients.)

There are lots of ways to specify numeric formats in R. The options statement, just discussed, belongs to R’s base functions, whereas the digits specification belongs, among many others, to the quap function. This situation tends to make matters more confused rather than less.

### 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.

## Overview

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.

### Surveys

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.

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.

## Security

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

## Children

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

## Marketing

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.

## Choice/Opt-out

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.

## 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