11月 212018
 

A data analyst asked how to compute parameter estimates in a linear regression model when the underlying data matrix is rank deficient. This situation can occur if one of the variables in the regression is a linear combination of other variables. It also occurs when you use the GLM parameterization of a classification variable. In the GLM parameterization, the columns of the design matrix are linearly dependent. As a result, the matrix of crossproducts (the X`X matrix) is singular. In either case, you can understand the computation of the parameter estimates learning about generalized inverses in linear systems. This article presents an overview of generalized inverses. A subsequent article will specifically apply generalized inverses to the problem of estimating parameters for regression problems with collinearities.

What is a generalized inverse?

Recall that the inverse matrix of a square matrix A is a matrix G such as A*G = G*A = I, where I is the identity matrix. When such a matrix exists, it is unique and A is said to be nonsingular (or invertible). If there are linear dependencies in the columns of A, then an inverse does not exist. However, you can define a series of weaker conditions that are known as the Penrose conditions:

  1. A*G*A = A
  2. G*A*G = G
  3. (A*G)` = A*G
  4. (G*A)` = G*A

Any matrix, G, that satisfies the first condition is called a generalized inverse (or sometimes a "G1" inverse) for A. A matrix that satisfies the first and second condition is called a "G2" inverse for A. The G2 inverse is used in statistics to compute parameter estimates for regression problems (see Goodnight (1979), p. 155). A matrix that satisfies all four conditions is called the Moore-Penrose inverse or the pseudoinverse. When A is square but singular, there are infinitely many matrices that satisfy the first two conditions, but the Moore-Penrose inverse is unique.

Computations with generalized inverses

In regression problems, the parameter estimates are obtained by solving the "normal equations." The normal equations are the linear system (X`*X)*b = (X`*Y), where X is the design matrix, Y is the vector of observed responses, and b is the parameter estimates to be solved. The matrix A = X`*X is symmetric. If the columns of the design matrix are linearly dependent, then A is singular. The following SAS/IML program defines a symmetric singular matrix A and a right-hand-side vector c, which you can think of as X`*Y in the regression context. The call to the DET function computes the determinant of the matrix. A zero determinant indicates that A is singular and that there are infinitely many vectors b that solve the linear system:

proc iml;
A = {100  50 20 10,
      50 106 46 23,
      20  46 56 28,
      10  23 28 14 };
c = {130, 776, 486, 243};
 
det = det(A);         /* demonstrate that A is singular */
print det;

For nonsingular matrices, you can use either the INV or the SOLVE functions in SAS/IML to solve for the unique solution of the linear system. However, both functions give errors when called with a singular matrix. SAS/IML provides several ways to compute a generalized inverse, including the SWEEP function and the GINV function. The SWEEP function is an efficient way to use Gaussian elimination to solve the symmetric linear systems that arise in regression. The GINV function is a function that computes the Moore-Penrose pseudoinverse. The following SAS/IML statements compute two different solutions for the singular system A*b = c:

b1 = ginv(A)*c;       /* solution even if A is not full rank */
b2 = sweep(A)*c;
print b1 b2;

The SAS/IML language also provides a way to obtain any of the other infinitely many solutions to the singular system A*b = c. Because A is a rank-1 matrix, it has a one-dimensional kernel (null space). The HOMOGEN function in SAS/IML computes a basis for the null space. That is, it computes a vector that is mapped to the zero vector by A. The following statements compute the unit basis vector for the kernel. The output shows that the vector is mapped to the zero vector:

xNull = homogen(A);   /* basis for nullspace of A */
print xNull (A*xNull)[L="A*xNull"];

All solutions to A*b = c are of the form b + α*xNull, where b is any particular solution.

Properties of the Moore-Penrose solution

You can verify that the Moore-Penrose matrix GINV(A) satisfies the four Penrose conditions, whereas the G2 inverse (SWEEP(A)) only satisfies the first two conditions. I mentioned that the singular system has infinitely many solutions, but the Moore-Penrose solution (b1) is unique. It turns out that the Moore-Penrose solution is the solution that has the smallest Euclidean norm. Here is a computation of the norm for three solutions to the system A*b = c:

/* GINV gives the estimate that has the smallest L2 norm: */
GINVnorm  = norm(b1);
sweepNorm = norm(b2);
/* you can add alpha*xNull to any solution to get another solution */
b3 = b1 + 2*xNull;  /* here's another solution (alpha=2) */
otherNorm = norm(b3);
print ginvNorm sweepNorm otherNorm;

Because all solutions are of the form b1 + α*xNull, where xNull is the basis for the nullspace of A, you can graph the norm of the solutions as a function of α. The graph is shown below and indicates that the Moore-Penrose solution is the minimal-norm solution:

alpha = do(-2.5, 2.5, 0.05);
norm = j(1, ncol(alpha), .);
do i = 1 to ncol(alpha);
   norm[i] = norm(b1 + alpha[i] * xNull);
end;
title "Euclidean Norm of Solutions b + alpha*xNull";
title2 "b = Solution from Moore-Penrose Inverse";
title3 "xNull = Basis for Nullspace of A";
call series(alpha, norm) 
     other="refline 0 / axis=x label='b=GINV';refline 1.78885 / axis=x label='SWEEP';";
Graph of norm of solutions to the singular system A*b=c. The norm is plotted for vectors b + alpha*x_Null where b is the Moore-Penrose solution and x_Null is a basis for the nullspace of A.

In summary, a singular linear system has infinitely many solutions. You can obtain a particular solution by using the sweep operator or by finding the Moore-Penrose solution. You can use the HOMOGEN function to obtain the full family of solutions. The Moore-Penrose solution is expensive to compute but has an interesting property: it is the solution that has the smallest Euclidean norm. The sweep solution is more efficient to compute and is used in SAS regression procedures such as PROC GLM to estimate parameters in models that include classification variables and use a GLM parameterization. The next blog post explores regression estimates in more detail.

The post Generalized inverses for matrices appeared first on The DO Loop.

11月 212018
 

I recently read an article that said a school in Asheville, North Carolina had the worst chickenpox outbreak in the state in 2 decades. The article was interesting, and it also let me know I had a hole in my knowledge ... "What?!? - There's a chickenpox vaccine?!?" When I [...]

The post Immunization rates in North Carolina schools appeared first on SAS Learning Post.

11月 192018
 

You might know that you can use the ODS SELECT statement to display only some of the tables and graphs that are created by a SAS procedure. But did you know that you can use a WHERE clause on the ODS SELECT statement to display tables that match a pattern? This article shows how to use wildcards, regular expressions, and pattern matching to select ODS tables in SAS.

ODS SELECT: Filter the output from SAS procedures

A SAS procedure might produce a dozen or more tables. You might be interested in displaying a subset of those tables. Recall that you can use the ODS TRACE ON statement to obtain a list of all the tables and graphs that a procedure creates. You can then use the ODS SELECT or the ODS EXCLUDE statement to control which tables and graphs are displayed.

Here's an example from the SAS/STAT documentation. The following PROC LOGISTIC call creates 27 tables and graphs, most of which are related to ROC curves. The ODS TRACE ON statement displays the names of each output object in the SAS log:

data roc;
   input alb tp totscore popind @@;
   totscore = 10 - totscore;
   datalines;
3.0 5.8 10 0   3.2 6.3  5 1   3.9 6.8  3 1   2.8 4.8  6 0
3.2 5.8  3 1   0.9 4.0  5 0   2.5 5.7  8 0   1.6 5.6  5 1
3.8 5.7  5 1   3.7 6.7  6 1   3.2 5.4  4 1   3.8 6.6  6 1
4.1 6.6  5 1   3.6 5.7  5 1   4.3 7.0  4 1   3.6 6.7  4 0
2.3 4.4  6 1   4.2 7.6  4 0   4.0 6.6  6 0   3.5 5.8  6 1
3.8 6.8  7 1   3.0 4.7  8 0   4.5 7.4  5 1   3.7 7.4  5 1
3.1 6.6  6 1   4.1 8.2  6 1   4.3 7.0  5 1   4.3 6.5  4 1
3.2 5.1  5 1   2.6 4.7  6 1   3.3 6.8  6 0   1.7 4.0  7 0
3.7 6.1  5 1   3.3 6.3  7 1   4.2 7.7  6 1   3.5 6.2  5 1
2.9 5.7  9 0   2.1 4.8  7 1   2.8 6.2  8 0   4.0 7.0  7 1
3.3 5.7  6 1   3.7 6.9  5 1   3.6 6.6  5 1
;
 
ods graphics on;
ods trace on;
proc logistic data=roc;
   model popind(event='0') = alb tp totscore / nofit;
   roc 'Albumin' alb;
   roc 'K-G Score' totscore;
   roc 'Total Protein' tp;
   roccontrast reference('K-G Score') / estimate e;
run;

The SAS log displays the names of the tables and graphs. A portion of log is shown below:

Output Added:
-------------
Name:       OddsRatios
Label:      Odds Ratios
Template:   Stat.Logistic.OddsRatios
Path:       Logistic.ROC3.OddsRatios
-------------

Output Added:
-------------
Name:       ROCCurve
Label:      ROC Curve
Template:   Stat.Logistic.Graphics.ROC
Path:       Logistic.ROC3.ROCCurve
-------------

Output Added:
-------------
Name:       ROCOverlay
Label:      ROC Curves
Template:   Stat.Logistic.Graphics.ROCOverlay
Path:       Logistic.ROCComparisons.ROCOverlay
-------------

Output Added:
-------------
Name:       ROCAssociation
Label:      ROC Association Statistics
Template:   Stat.Logistic.ROCAssociation
Path:       Logistic.ROCComparisons.ROCAssociation
-------------

Output Added:
-------------
Name:       ROCContrastCoeff
Label:      ROC Contrast Coefficients
Template:   Stat.Logistic.ROCContrastCoeff
Path:       Logistic.ROCComparisons.ROCContrastCoeff
-------------

Only a few of the 27 ODS objects are shown here. Notice that each ODS object has four properties: a name, a label, a template, and a path. Most of the time, the name is used on the ODS SELECT statement to filter the output. For example, if you want to display only the ROC curves and the overlay of the ROC curves, you can put the following statement prior to the RUN statement in the procedure:

   ods select ROCCurve ROCOverlay;     /* specify the names literally */

Use a WHERE clause in the ODS SELECT statement

Often the ODS objects that you want to display are related to each other. In the LOGISTIC example, you might want to display all the information about ROC curves. Fortunately, the SAS developers often use a common prefix or suffix, such as 'ROC', in the names of the ODS objects. That means that you can display all ROC-related tables and graphs be selecting the ODS objects whose name (or path) contains 'ROC' as a substring.

You can use the WHERE clause to select ODS objects whose name (or label or path) matches a particular pattern. The object's name is available in a special variable named _NAME_. Similarly, the object's label and path are available in variables named _LABEL_ and _PATH_, respectively. You cannot match patterns in the template string; there is no _TEMPLATE_ variable.

In SAS, the following operators and functions are useful for matching strings:

  • The CONTAINS keyword matches strings that contains a specified substring. The question mark (?) is an equivalent way to specify the CONTAINS operator.
  • The LIKE keyword matches strings to a pattern. The underscore (_) is a wildcard that matches any character. The percent sign (%) is a wildcard that matches one or more characters.
  • The "begins with" operators (=: and in:) match strings that begin with a certain pattern or set of patterns, respectively.
  • SAS functions such as FIND, INDEX, and SUBSTR can be used to match patterns.
  • The SAS PRXMATCH function, which enables you to use Perl regular expressions to match patterns.

For example, the following statements select ODS tables and graphs from the previous PROC LOGISTIC call. You can put one of these statements before the RUN statement in the procedure:

   /* use any one of the following statements inside the PROC LOGISTIC call */
   ods select where=(_name_ =: 'ROC');                 /* name starts with 'ROC' */
   ods select where=(_name_ like 'ROC%');              /* name starts with 'ROC' */
   ods select where=(_path_ ? 'ROC');                  /* path contains 'ROC' */
   ods select where=(_label_ ? 'ROC');                 /* label contains 'ROC' */
   ods select where=(_name_ in: ('Odds', 'ROC'));      /* name starts with 'Odds' or 'ROC' */
   ods select where=(substr(_name_,4,8)='Contrast');   /* name has subtring 'Contrast' at position 4 */

For additional examples of using pattern matching to select ODS objects, see Warren Kuhfeld's graphics-focused blog post and the section of the SAS/STAT User's Guide that discusses selecting ODS graphics.

Use PRXMATCH to match regular expressions

Although the CONTAIN and LIKE operators are often sufficient for selecting a table, SAS provides the powerful PRXMATCH function for more complex pattern-matching tasks. The PRXMATCH function uses Perl regular expressions to match strings. SAS provides a Perl Regular Expression "cheat sheet" that summarizes the syntax and commons search queries for the PRXMATCH function.

You can put any of the following statements inside the PROC LOGISTIC call:

   /* use any one of the following PRXMATCH expressions inside the PROC LOGISTIC call */
   ods select where=(prxmatch('/ROC/',       _name_)); /* name contains 'ROC' anywhere */
   ods select where=(prxmatch('/^ROC/',      _name_)); /* name starts with 'ROC' */
   ods select where=(prxmatch('/Odds|^ROC/', _name_)); /* name contains 'Odds' anywhere or 'ROC' at the beginning */
   ods select where=(prxmatch('/ROC/',      _name_)=0);     /* name does NOT contain 'ROC' anywhere */
   ods select where=(prxmatch('/Logistic\.ROC2/', _path_)); /* escape special wildcard character '.' */

In summary, the WHERE= option on the ODS SELECT (and ODS EXCLUDE) statement is quite powerful. Many SAS programmers know how to list the names of tables and graphs on the ODS SELECT statement to display only a subset of the output. However, the WHERE= option enables you to use wildcards and regular expressions to select objects whose names or paths match a certain pattern. This can be a quick and efficient way to select tables that are related to each other and share a common prefix or suffix in their name.

The post Select ODS tables by using wildcards and regular expressions in SAS appeared first on The DO Loop.

11月 172018
 

Disclaimer: this article does not cover or promote any political views. It’s all about data and REST APIs.

I am relieved, thankful, elated, glad, thrilled, joyful (I could go on with more synonyms from my thesaurus.com search for 'happy') November 6, 2018 has come and gone. Election day is over. This means no more political ads on TV, and those signs lining the streets will be coming down! It is a joy to now watch commercials about things that matter. Things like injury lawyers who are on your side or discovering a copper colored pan is going to cook my food better than a black one.

The data in this article pertains to advertising expenditures in the 2018 elections. This is the second of three articles in a series outlining the use of REST APIs and SAS. The first article, Using SAS Viya REST APIs to access images from SAS Visual Analytics, I used SAS Viya REST APIs to download an image from a flight data SAS report. In this article I use Cloud Analytics Service (CAS) REST APIs to run statistical methods on political ad spending data. The third article will bring both APIs together in an application.

The data

In the closing days of the election season, while being inundated with political advertising, I thought about how much money is spent during each cycle. The exact numbers vary depending on the resource, but the range for this year’s mid-term elections is between four and five billion dollars.

A little research reveals that outside the candidates themselves, the biggest spenders on political ads are political action committees, aka PACs. The Center for Responsive Politics compiled the data set used in this article, and derives from a larger data set released by the Federal Election Commission. The data set lists a breakdown of PAC contributions to campaign finances.

CAS REST APIs

As I explained in the previous article, SAS publishes two sets of APIs. Which APIs to use depends on the service, the data organization, or the intended use of the data. Please refer to the SAS Viya REST API article for more information on each set of APIs.

CAS REST APIs use CAS actions to perform statistical methods across a variety of SAS products. You can also use the CAS REST APIs to configure and maintain the SAS Viya environment. Here, I focus on the CAS actions. Calling the CAS actions via the REST API allow users to access SAS data and procedures and integrate them into their applications.

The process

How to construct the API call

I start with the API documentation for information on how to construct and use the CAS REST APIs. The REST API can submit actions and return the results. Parameters and result data are in JSON format. To specify your parameters, encapsulate the attributes in a JSON object, then submit a POST method on the action. The URL for your action will include the UUID of your session in the format: /cas/sessions/{uuid}/actions/{action}. Replace {uuid} and action with the appropriate values.

Create a session

The first requirement is to create a session. I use the following cURL command to create the session.

curl -X POST http://sasserver.demo.sas.com:8777/cas/sessions \
    -H 'Authorization: Bearer <access-token-goes-here>'

The response is a JSON object with a session ID:

{
    "session": "16dd9ee7-3189-1e40-8ba7-934a4a257fd7"
}

I’ll use the UUID for the session to build the URLs for the remainder of the REST calls.

Build the CAS REST API call body

Now we know the general structure of the CAS REST API call. We can browse the CAS actions by name to determine how to build the body text.

Using the simple.summary action definition, I build a JSON body to access the PAC spending from a CASTable, create a new table grouped by political views, and calculate total spending. The resulting code is below:

{
	"table":{"caslib":"CASUSER(sasdemo)","name":"politicalspending2018","groupBy":{"name":"view"}},
	"casout":{"caslib":"CASUSER(sasdemo)","name":"spendingbyaffiliation","promote":true},
	"inputs":"total",
	"subset":["SUM","N"],
}

Each line of code above contributes to running the CAS action:

  1. Define the table to use and how to group the data
  2. The output of the API call will create a new CASTable
  3. Dictate the column to summarize.
  4. The statistical method(s) to include in the result table; in this case I want to sum the Total column and count the number of PACs by group.

Send the CAS REST API

Next, I send the body of the text with the curl call below. Notice the session ID obtained earlier is now part of the URL:

curl -X POST http://sasserver.demo.sas.com:8777/cas/sessions/16dd9ee7-3189-1e40-8ba7-934a4a257fd7/actions/simple.summary \
  -H 'Authorization: Bearer <access-token-goes-here>' \
  -H 'Accept = application/json' \
  -H 'Content-Type = application/json'

The REST call creates a new CASTable, SPENDINGBYAFFILIATION. Refer to the screen shot below.

New table

SAS CASTable created by the simple.summary action

I also have the option of returning the data to create the SPENDINGBYAFFILIATION table in JSON format. To accomplish this, remove the casout{} line from the preceding call. Below is a snippet of the JSON response.

JSON response

JSON response to the simple.summary REST call

After parsing the JSON response code, it is now ready for utilization by a web application, software program, or script.

Moving on

The Thanksgiving Day holiday is fast approaching here in the United States. I plan to eat a lot of turkey and sweet potato pie, welcome the out-of-town family, and watch football. It will be refreshing to not hear the back-and-forth banter and bickering between candidates during commercial breaks. Oh, but wait, Thanksgiving is the start of the holiday season. This means one thing: promotions on Black Friday deals for items I may not need will start airing and last through year's-end. I guess if it is not one thing filling the advertising air waves, it is another. I'll just keep the remote handy and hope I can find another ball game on.

What’s next?

I understand and appreciate political candidates’ needs to communicate their stance on issues and promote their agendas. This takes money. I don't see the spending trend changing direction in the coming years. I can only hope the use of the funds will promote candidates' qualifications, beliefs, and ideas, and not to bash or belittle their opponents.

My next article will demonstrate how to use both the SAS Viya and the CAS REST APIs under the umbrella of one web application. And I promise, no politics.

Using SAS Cloud Analytics Service REST APIs to run CAS Actions was published on SAS Users.

11月 172018
 

Disclaimer: this article does not cover or promote any political views. It’s all about data and REST APIs.

I am relieved, thankful, elated, glad, thrilled, joyful (I could go on with more synonyms from my thesaurus.com search for 'happy') November 6, 2018 has come and gone. Election day is over. This means no more political ads on TV, and those signs lining the streets will be coming down! It is a joy to now watch commercials about things that matter. Things like injury lawyers who are on your side or discovering a copper colored pan is going to cook my food better than a black one.

The data in this article pertains to advertising expenditures in the 2018 elections. This is the second of three articles in a series outlining the use of REST APIs and SAS. The first article, Using SAS Viya REST APIs to access images from SAS Visual Analytics, I used SAS Viya REST APIs to download an image from a flight data SAS report. In this article I use Cloud Analytics Service (CAS) REST APIs to run statistical methods on political ad spending data. The third article will bring both APIs together in an application.

The data

In the closing days of the election season, while being inundated with political advertising, I thought about how much money is spent during each cycle. The exact numbers vary depending on the resource, but the range for this year’s mid-term elections is between four and five billion dollars.

A little research reveals that outside the candidates themselves, the biggest spenders on political ads are political action committees, aka PACs. The Center for Responsive Politics compiled the data set used in this article, and derives from a larger data set released by the Federal Election Commission. The data set lists a breakdown of PAC contributions to campaign finances.

CAS REST APIs

As I explained in the previous article, SAS publishes two sets of APIs. Which APIs to use depends on the service, the data organization, or the intended use of the data. Please refer to the SAS Viya REST API article for more information on each set of APIs.

CAS REST APIs use CAS actions to perform statistical methods across a variety of SAS products. You can also use the CAS REST APIs to configure and maintain the SAS Viya environment. Here, I focus on the CAS actions. Calling the CAS actions via the REST API allow users to access SAS data and procedures and integrate them into their applications.

The process

How to construct the API call

I start with the API documentation for information on how to construct and use the CAS REST APIs. The REST API can submit actions and return the results. Parameters and result data are in JSON format. To specify your parameters, encapsulate the attributes in a JSON object, then submit a POST method on the action. The URL for your action will include the UUID of your session in the format: /cas/sessions/{uuid}/actions/{action}. Replace {uuid} and action with the appropriate values.

Create a session

The first requirement is to create a session. I use the following cURL command to create the session.

curl -X POST http://sasserver.demo.sas.com:8777/cas/sessions \
    -H 'Authorization: Bearer <access-token-goes-here>'

The response is a JSON object with a session ID:

{
    "session": "16dd9ee7-3189-1e40-8ba7-934a4a257fd7"
}

I’ll use the UUID for the session to build the URLs for the remainder of the REST calls.

Build the CAS REST API call body

Now we know the general structure of the CAS REST API call. We can browse the CAS actions by name to determine how to build the body text.

Using the simple.summary action definition, I build a JSON body to access the PAC spending from a CASTable, create a new table grouped by political views, and calculate total spending. The resulting code is below:

{
	"table":{"caslib":"CASUSER(sasdemo)","name":"politicalspending2018","groupBy":{"name":"view"}},
	"casout":{"caslib":"CASUSER(sasdemo)","name":"spendingbyaffiliation","promote":true},
	"inputs":"total",
	"subset":["SUM","N"],
}

Each line of code above contributes to running the CAS action:

  1. Define the table to use and how to group the data
  2. The output of the API call will create a new CASTable
  3. Dictate the column to summarize.
  4. The statistical method(s) to include in the result table; in this case I want to sum the Total column and count the number of PACs by group.

Send the CAS REST API

Next, I send the body of the text with the curl call below. Notice the session ID obtained earlier is now part of the URL:

curl -X POST http://sasserver.demo.sas.com:8777/cas/sessions/16dd9ee7-3189-1e40-8ba7-934a4a257fd7/actions/simple.summary \
  -H 'Authorization: Bearer <access-token-goes-here>' \
  -H 'Accept = application/json' \
  -H 'Content-Type = application/json'

The REST call creates a new CASTable, SPENDINGBYAFFILIATION. Refer to the screen shot below.

New table

SAS CASTable created by the simple.summary action

I also have the option of returning the data to create the SPENDINGBYAFFILIATION table in JSON format. To accomplish this, remove the casout{} line from the preceding call. Below is a snippet of the JSON response.

JSON response

JSON response to the simple.summary REST call

After parsing the JSON response code, it is now ready for utilization by a web application, software program, or script.

Moving on

The Thanksgiving Day holiday is fast approaching here in the United States. I plan to eat a lot of turkey and sweet potato pie, welcome the out-of-town family, and watch football. It will be refreshing to not hear the back-and-forth banter and bickering between candidates during commercial breaks. Oh, but wait, Thanksgiving is the start of the holiday season. This means one thing: promotions on Black Friday deals for items I may not need will start airing and last through year's-end. I guess if it is not one thing filling the advertising air waves, it is another. I'll just keep the remote handy and hope I can find another ball game on.

What’s next?

I understand and appreciate political candidates’ needs to communicate their stance on issues and promote their agendas. This takes money. I don't see the spending trend changing direction in the coming years. I can only hope the use of the funds will promote candidates' qualifications, beliefs, and ideas, and not to bash or belittle their opponents.

My next article will demonstrate how to use both the SAS Viya and the CAS REST APIs under the umbrella of one web application. And I promise, no politics.

Using SAS Cloud Analytics Service REST APIs to run CAS Actions was published on SAS Users.

11月 162018
 

In my first two posts of this blog series, we heard why two students chose to pursue a STEM field and what appealed to them about data science. We also heard how they put their knowledge to work on a real-world data science project. Today, we'll hear their advice to future [...]

Two students’ advice on data science, SAS and more was published on SAS Voices by Georgia Mariani

11月 162018
 

There has been a lot of controversy surrounding this year's midterm election, when it comes to counting the ballots ... and I kept hearing the term provisional ballots in the news. But I'm embarrassed to say that I didn't really know much about provisional ballots. I decided to do a [...]

The post Where do provisional ballots come from? appeared first on SAS Learning Post.