6月 182019
 

What is Item Response Theory?

Item Response Theory (IRT) is a way to analyze responses to tests or questionnaires with the goal of improving measurement accuracy and reliability.

A common application is in testing a student’s ability or knowledge. Today, all major psychological and educational tests are built using IRT. The methodology can significantly improve measurement accuracy and reliability while providing potential significant reductions in assessment time and effort, especially via computerized adaptive testing. For example, the SAT and GRE both use Item Response Theory for their tests. IRT takes into account the number of questions answered correctly and the difficulty of the question.

In recent years, IRT models have also become increasingly popular in health behavior, quality of life, and clinical research. There are many different models for IRT. Three of the most popular are:

The Rasch model

Two-parameter model

Graded Response model

Early IRT models (such as the Rasch model and two-parameter model) concentrate mainly on dichotomous responses. These models were later extended to incorporate other formats, such as ordinal responses, rating scales, partial credit scoring, and multiple category scoring.

Item Response Theory Models Using SAS

Ron Cody and Jeffrey K. Smith’s book, Test Scoring and Analysis Using SAS, uses SAS PROC IRT to show how to develop your own multiple-choice tests, score students, produce student rosters (in print form or Excel), and explore item response theory (IRT).

Aimed at non-statisticians working in education or training, the book describes item analysis and test reliability in easy-to-understand terms and teaches SAS programming to score tests, perform item analysis, and estimate reliability.

For those with a more statistical background, Bayesian Analysis of Item Response Theory Models Using SAS describes how to estimate and check IRT models using the SAS MCMC procedure. Written especially for psychometricians, scale developers, and practitioners, numerous programs are provided and annotated so that you can easily modify them for your applications.

Assessment has played, and continues to play, an integral part in our work and educational settings. IRT models continue to be increasingly popular in many other fields, such as medical research, health sciences, quality-of-life research, and even marketing research. With the use of IRT models, you can not only improve scoring accuracy but also economize test administration by adaptively using only the discriminative items.

Interested in learning more? Check out our chapter previews available for free. Want to learn more about SAS Press? Explore our online bookstore and subscribe to our newsletter to get all the latest discounts, news, and more.

Further resources

SAS Blogs:
New at SAS: Psychometric Testing by Charu Shankar
SAS author’s tip: Bayesian analysis of item response theory models

SAS Communities:
SAS Communities: Custom Task Tuesday: SAS Global Forum/PROC IRT Edition!

SAS Global Forum Paper:
Item Response Theory: What It Is and How You Can Use the IRTProcedure to Apply It by Xinming An and Yiu-Fai Yung

SAS Documentation:
The IRT Procedure
SAS/STAT 14.1 User Guide: The IRT Procedure
SAS/STAT 14.2 User Guide: Help Center

Understanding Item Response Theory with SAS was published on SAS Users.

6月 172019
 

My article about deletion diagnostics investigated how influential an observation is to a least squares regression model. In other words, if you delete the i_th observation and refit the model, what happens to the statistics for the model? SAS regression procedures provide many tables and graphs that enable you to examine the influence of deleting an observation. For example:

  • The DFBETAS are statistics that indicate the effect that deleting each observation has on the estimates for the regression coefficients.
  • The DFFITS and Cook's D statistics indicate the effect that deleting each observation has on the predicted values of the model.
  • The COVRATIO statistics indicate the effect that deleting each observation has on the variance-covariance matrix of the estimates.

These observation-wise statistics are typically used for smaller data sets (n ≤ 1000) because the influence of any single observation diminishes as the sample size increases. You can get a table of these (and other) deletion diagnostics by using the INFLUENCE option on the MODEL statement of PROC REG in SAS. However, because there is one statistic per observation, these statistics are usually graphed. PROC REG can automatically generate needle plots of these statistics (with heuristic cutoff values) by using the PLOTS= option on the PROC REG statement.

This article describes the DFBETAS statistic and shows how to create graphs of the DFBETAS in PROC REG in SAS. The next article discusses the DFFITS and Cook's D statistics. The COVRATIO statistic is not as popular, so I won't say more about that statistic.

DFBETAS: How the coefficient estimates change if an observation is excluded

The documentation for PROC REG has a section that describes the influence statistics, which is based on the book Regression Diagnostics by Belsley, Kuh, and Welsch (1980, p. 13-14). Among these, the DFBETAS statistics are perhaps the easiest to understand. If you exclude an observation from the data and refit the model, you will get new parameter estimates. How much do the estimates change? Notice that you get one statistic for each observation and also one for each regressor (including the intercept). Thus if you have n observations and k regressors, you get nk statistics.

Typically, these statistics are shown in a panel of k plots, with the DFBETAS for each regressor plotted against the observation number. Because "observation number" is an arbitrary number, I like to sort the data by the response variable. Then I know that the small observation numbers correspond to low values of the response variable and large observation numbers correspond to high values of the response variable. The following DATA step extracts a subset of n = 84 vehicles from the Sashelp.Cars data, creates a short ID variable for labeling observations, and sorts the data by the response variable, MPG_City:

data cars;
set sashelp.cars;
where Type in ('SUV', 'Truck');
/* make short ID label from Make and Model values */
length IDMakeMod $20;
IDMakeMod = cats(substr(Make,1,4), ":", substr(Model,1,5));
run;
 
proc sort data=cars;
   by MPG_City;
run;
 
proc print data=cars(obs=5) noobs;
   var Make Model IDMakeMod MPG_City;
run;

The first few observations are shown. Notice that the first observations correspond to small values of the MPG_City variable. Notice also a short label (IDMakeMod) identifies each vehicle.

There are two ways to generate the DFBETAS statistics: You can use the INFLUENCE option on the MODEL statement to generate a table of statistics, or you can use the PLOTS=DFBETAS option in the PROC REG statement to generate a panel of graphs. The following call to PROC REG generates a panel of graphs. The IMAGEMAP=ON option on the ODS GRAPHICS statement enables you to hover the mouse pointer over an observation and obtain a brief description of the observation:

ods graphics on / imagemap=on;              /* enable data tips (tooltips) */
proc reg data=Cars plots(only) = DFBetas; 
   model MPG_City = EngineSize HorsePower Weight;
   id IDMakeMod;
run; quit;
ods graphics / imagemap=off;

The panel shows the influence of each observation on the estimates of the four regression coefficients. The statistics are standardized so that all graphs can use the same vertical scale. Horizontal lines are drawn at ±2/sqrt(n) ≈ 0.22. Observations are called influential if they have a DFBETA statistic that exceeds that value. The graph shows a tool tip for one of the observations in the EngineSize graph, which shows that the influential point is observation 4, the Land Rover Discovery.

Each graph reveals a few influential observations:

  • For the intercept estimate, the most influential observations are numbers 1, 35, 83, and 84.
  • For the EngineSize estimates, the most influential observations are numbers 4, 35, and 38.
  • For the Horsepower estimates, the most influential observations are numbers 1, 4, and 38.
  • For the Weight estimates, the most influential observations are numbers 1, 24, 35, and 38.

Notice that several observations (such as 1, 35, and 38) are influential for more than one estimate. Excluding those observations causes several parameter estimates to change substantially.

Labeing the influential observations

For me, the panel of graphs is too small. I found it difficult to hover the mouse pointer exactly over the tip of a needle in an attempt to discover the observation number and name of the vehicle. Fortunately, if you want details like that, PROC REG supplies options that make the process easier. If you don't have too many observations, you can add labels to the DFBETAS plots by using the LABEL suboption. To plot each graph individually (instead of in a panel), use the UNPACK suboption, as follows:

proc reg data=Cars plots(only) = DFBetas(label unpack); 
   model MPG_City = EngineSize HorsePower Weight;
   id IDMakeMod;
quit;

The REG procedure creates four plots, but only the graph for the Weight variable is shown here. In this graph, the influential observations are labeled by the IDMakeMod variable, which enables you to identify vehicles rather than observation numbers. For example, some of the influential observations for the Weight variable are the Ford Excursion (1), the Toyota Tundra (24), the Mazda B400 (35), and the Volvo XC90 (38).

A table of influential observations

If you want a table that displays the most influential observations, you can use the INFLUENCE option to generate the OutputStatistics table, which contains the DFBETAS for all regressors. You can write that table to a SAS data set and exclude any that do not have a large DFBETAS statistic, where "large" means the magnitude of the statistic exceeds 2/sqrt(n), where n is the sample size. The following DATA step filters the observations and prints only the influential ones.

ods exclude all;
proc reg data=Cars plots=NONE; 
   model MPG_City = EngineSize HorsePower Weight / influence;
   id IDMakeMod;
   ods output OutputStatistics=OutputStats;      /* save influence statistics */
run; quit;
ods exclude none;
 
data Influential;
set OutputStats nobs=n;
array DFB[*] DFB_:;
cutoff = 2 / sqrt(n);
ObsNum = _N_;
influential = 0;
DFBInd = '0000';                   /* binary string indicator */
do i = 1 to dim(DFB);
   if abs(DFB[i])>cutoff then do;  /* this obs is influential for i_th regressor */
      substr(DFBInd,i,1) = '1';
      influential = 1;
   end;
end;
if influential;                    /* output only influential obs */
run;
 
proc print data=Influential noobs;
   var ObsNum IDMakeMod DFBInd cutoff DFB_:;
run;

The DFBInd variable is a four-character binary string that indicates which parameter estimates are influenced by each observation. Some observations are influential only for one coefficient; others (1, 3, 35, and 38) are influential for many variables. Creating a binary string for each observation is a useful trick.

By the way, did you notice that the name of the statistic ("DFBETAS") has a large S at the end? Until I researched this article, I assumed it was to make the word plural since there is more than one "DFBETA" statistic. But, no, it turns out that the S stands for "scaled." You can define the DFBETA statistic (without the S) to be the change in parameter estimates bb(i), but that statistic depends on the scale of the variables. To standardize the statistic, divide by the standard error of the parameter estimates. That scaling is the reason for the S as the end of DFBETAS. The same is true for the DFFITS statistic: S stands for "scaled."

The next article describes how to create similar graphs for the DFFITS and Cook's D statistics.

---------------

DFFITS: How the predicted values change if an observation is excluded

The DFFITS statistic measures, for each observation, how the predicted value at that observation changes if you exclude the observation and refit the model.

Cook's D: How the sum of the predicted values change if an observation is excluded

Cook's distance (D) statistic measures, for each observation, the sum of the differences in the predicted values (summed over all observations) if you exclude the observation and refit the model.

The post Influential observations in a linear regression model: The DFBETAS statistics appeared first on The DO Loop.

6月 122019
 

For linear regression models, there is a class of statistics that I call deletion diagnostics or leave-one-out statistics. These observation-wise statistics address the question, "If I delete the i_th observation and refit the model, what happens to the statistics for the model?" For example:

  • The PRESS statistic is similar to the residual sum of squares statistic but is based on fitting n different models, where n is the sample size and the i_th model excludes the i_th observation.
  • Cook's D statistic measures the influence of the i_th observation on the fit.
  • The DFBETAS statistics measure how the regression estimates change if you delete the i_th observation.

Although most references define these statistics in terms of deleting an observation and refitting the model, you can use a mathematical trick to compute the statistics without ever refitting the model! For example, the Wikipedia page on the PRESS statistic states, "each observation in turn is removed and the model is refitted using the remaining observations. The out-of-sample predicted value is calculated for the omitted observation in each case, and the PRESS statistic is calculated as the sum of the squares of all the resulting prediction errors." Although this paragraph is conceptually correct, theSAS/STAT documentation for PROC GLMSELECT states that the PRESS statistic "can be efficiently obtained without refitting the model n times."

A rank-1 update to the inverse of a matrix

Recall that you can use the "normal equations" to obtain the least squares estimate for the regression problem with design matrix X and observed responses Y. The normal equations are b = (X`X)-1(X`Y), where X`X is known as the sum of squares and crossproducts (SSCP) matrix and b is the least squares estimate of the regression coefficients. For data sets with many observations (very large n), the process of reading the data and forming the SSCP is a relatively expensive part of fitting a regression model. Therefore, if you want the PRESS statistic, it is better to avoid rebuilding the SSCP matrix and computing its inverse n times. Fortunately, there is a beautiful result in linear algebra that relates the inverse of the full SSCP matrix to the inverse when a row of X is deleted. The result is known as the Sherman-Morrison formula for rank-1 updates.

The key insight is that one way to compute the SSCP matrix is as a sum of outer products of the rows of X. Therefore if xi is the i_th row of X, the SCCP matrix for data where xi is excluded is equal to X`X - xi`xi. You have to invert this matrix to find the least squares estimates after excluding xi.

The Sherman-Morrison formula enables you to compute the inverse of X`X - xi`xi when you already know the inverse of X`X. For brevity, set A = X`X. The Sherman-Morrison formula for deleting a row vector xi` is
(A – xi`xi)-1 = A-1 + A-1 xi`xi A-1 / (1 – xiA-1xi`)

Implement the Sherman-Morrison formula in SAS

The formula shows how to compute the inverse of the updated SSCP by using a matrix-vector multiplication and an outer product. Let's use a matrix language to demonstrate the update method. The following SAS/IML program reads in a small data set, forms the SSCP matrix (X`X), and computes its inverse:

proc iml;
use Sashelp.Class;   /* read data into design matrix X */
read all var _NUM_ into X[c=varNames];  
close;
XpX = X`*X;          /* form SSCP */
XpXinv = inv(XpX);   /* compute the inverse */

Suppose you want to compute a leave-one-out statistic such as PRESS. For each observation, you need to estimate the parameters that result if you delete that observation. For simplicity, let's just look at deleting the first row of the X matrix. The following program creates a new design matrix (Z) that excludes the row, forms the new SSCP matrix, and finds its inverse:

/* Inefficient: Manually delete the row from the X matrix 
   and recompute the inverse */
n = nrow(X);
Z = X[2:n, ];       /* delete first row */
ZpZ = Z`*Z;         /* reform the SSCP matrix */
ZpZinv = inv(ZpZ);  /* recompute the inverse */
print ZpZinv[c=varNames r=varNames L="Inverse of SSCP After Deleting First Row"];

The previous statements essentially repeat the entire least squares computation. To compute a leave-one-out statistic, you would perform a total of n similar computations.

In contrast, it is much cheaper to apply the Sherman-Morrison formula to update the inverse of the original SSCP. The following statements apply the Sherman-Morrison formula as it is written:

/* Alternative: Do not change X or recompute the inverse. 
   Use the Sherman-Morrison rank-1 update formula.
   https://en.wikipedia.org/wiki/Sherman–Morrison_formula */
r = X[1, ];          /* first row */
rpr = r`*r;          /* outer product */
/* apply Sherman-Morrison formula */
NewInv = XpXinv + XPXinv*rpr*XPXinv / (1 - r*XpXinv*r`);
print NewInv[c=varNames r=varNames L="Inverse from Sherman-Morrison Formula"];

These statements compute the new inverse by using the old inverse, an outer product, and a few matrix multiplications. Notice that the denominator of the Sherman-Morrison formula includes the expression r*(X`X)-1*r`, which is the leverage statistic for the i_th row.

The INVUPDT function in SAS/IML

Because it is important to be able to update an inverse matrix quickly when an observation is deleted (or added!), the SAS/IML language supports the IMVUPDT function, which implements the Sherman-Morrison formula. You merely specify the inverse matrix to update, the vector (as a column vector) to use for the rank-one update, and an optional scalar value, which is usually +1 if you are adding a new observation and -1 if you are deleting an observation. For example, the following statements are the easiest way to implement the Sherman-Morrison formula in SAS for a leave-one-out statistic:

NewInv2 = invupdt(XpXinv, r`, -1);
print NewInv2[c=varNames r=varNames L="Inverse from INVUPDT Function"];

The output is not displayed because the matrix NewInv2 is the same as the matrix NewInv in the previous section. The documentation includes additional examples.

The general Sherman-Morrison-Woodbury formula

The Sherman-Morrison formula shows how to perform a rank-1 update of an inverse matrix. There is a more general formula, called the Sherman-Morrison-Woodbury formula, which enables you to update an inverse for any rank-k modification of the original matrix. The general formula (Golub and van Loan, p. 51 of 2nd ed. or p. 65 of 4th ed.) shows how to find the matrix of a rank-k modification to a nonsingular matrix, A, in terms of the inverse of A. The general formula is
(A + U VT)-1 = A-1 – A-1 U (I + VT A-1 U) VT A-1
where U and V are p x k and all inverses are assumed to exist. When k = 1, the matrices U and V become vectors and the k x k identify matrix becomes the scalar value 1. In the previous section, U equals -xiT and V equals xiT.

The Sherman-Morrison-Woodbury formula is one of my favorite results in linear algebra. It shows that a rank-k modification of a matrix results in a rank-k modification of its inverse. It is not only a beautiful theoretical result, but it has practical applications to leave-one-out statistics because you can use the formula to quickly compute the linear regression model that results by dropping an observation from the data. In this way, you can study the influence of each observation on the model fit (Cook's D, DFBETAS,...) and perform leave-one-out cross-validation techniques, such as the PRESS statistic.

The post Leave-one-out statistics and a formula to update a matrix inverse appeared first on The DO Loop.