data analysis

1月 132020

Did you add "learn something new" to your list of New Year's resolutions? Last week, I wrote about the most popular articles from The DO Loop in 2019. The most popular articles are about elementary topics in SAS programming or univariate statistics because those topics have broad appeal.

Advanced topics and multivariate statistics are less popular but no less important. If you want to learn something new, check out this "Editor's Choice" list of articles that will broaden your statistical knowledge and enhance your SAS programming skills. I've grouped the articles into three broad categories.

Regression statistics

Schematic diagram of outliers in bivariate normal data. The point 'A' has large univariate z scores but a small Mahalanobis distance. The point 'B' has a large Mahalanobis distance. Only 'b' is a multivariate outlier.

High-dimensional analyses and visualization:

Low-dimensional data visualization

The tips and techniques in these articles are useful, so read a few articles today and teach yourself something new in this New Year!

Do you have a favorite article from 2019 that did not make either list? Share it in a comment!

The post 10 posts from 2019 that deserve a second look appeared first on The DO Loop.

1月 062020

Last year, I wrote more than 100 posts for The DO Loop blog. The most popular articles were about SAS programming tips for data analysis, statistical analysis, and data visualization. Here are the most popular articles from 2019 in each category.

SAS programming tips

The Essential Guide to Binning

  • Create training, testing, and validation data sets:: This post shows how to create training, validation, and test data sets in SAS. This technique is popular in data science and machine learning because you typically want to fit the model on one set of data and then evaluate the goodness of fit by using a different set of data.
  • 5 reasons to use PROC FORMAT to recode variables in SAS: Often SAS programmers use PROC SQL or the SAS DATA step to create new data variables to recode raw data. This is not always necessary. It is more efficient to use PROC FORMAT to recode the raw data. Learn five reasons why you should use PROC FORMAT to recode variables.
  • Conditionally append observations to a data set: In preparing data for graphing. you might want to add additional data to the end of a data set. (For example, to plot reference lines or text.) You can do this by using one DATA step to create the new observations and another DATA step to merge the new observations to the end of the original data. However, you can also use the END= option and end-of-file processing to append new observations to data. Read about how to use the END= option to append data. The comments at the end of the article show how to perform similar computations by using a hash table, PROC SQL, and a DOW loop.
  • Use PROC HPBIN to bin numerical variables: In machine learning, it is common to bin numerical variables into a set of discrete values. You can use PROC HPBIN to bin multiple variables in a single pass through the data. PROC HPBIN can bin data into equal-length bins (called bucket binning) or by using quantiles of the data. This article and the PROC FORMAT article are both referenced in my Essential Guide to Binning in SAS.

Statistical analyses

Geometric Meaning of Kolmogorov's D

Data visualization

Visualize an Interaction Effect

I always enjoy learning new programming methods, new statistical ideas, and new data visualization techniques. If you like to learn new things, too, read (or re-read!) these popular articles from 2019. Then share this page with a friend. I hope we both have many opportunities to learn and share together in the new year.

The post Top posts from <em>The DO Loop</em> in 2019 appeared first on The DO Loop.

12月 182019
Conditional bin plot in SAS

A 2-D "bin plot" counts the number of observations in each cell in a regular 2-D grid. The 2-D bin plot is essentially a 2-D version of a histogram: it provides an estimate for the density of a 2-D distribution. As I discuss in the article, "The essential guide to binning in SAS," sometimes you are more interested in displaying bins that contain (approximately) an equal number of observations. This leads to a quantile bin plot.

A SAS programmer on the SAS Support Communities recently requested a variation of the quantile bin plot. In the variation, the bins for the Y variable are computed as usual, but the bins for the X variable are conditional on the Y quantiles. This leads to a situation where the bins do not form a regular grid, but rather vary in both weight and width, as shown to the right. I call this a conditional quantile bin plot. In the graph, there are 12 rectangles. Each contains approximately the same number of observations. The three rectangles across each row contain a quartile of Y values.

This article shows how to construct a conditional quantile bin plot in SAS by first finding the quantiles for the Y variable and then finding the quantiles for the X variable (conditioned on the Y) for each Y quantile. You can solve this problem by using the SAS/IML language (as shown in the Support Community post) or by using PROC RANK and the DATA step, as shown in this article.

Although it is straightforward to compute conditional quantiles, it is less clear how to display the rectangles that represent the cells in the graph. On the Support Community, one solution uses the VECTOR statement in PROC SGPLOT, whereas another solution uses the POLYLINE command in an annotate data set. In this article, I present a third alternative, which is to use a POLYGON statement in PROC SGPLOT. I think that the POLYGON statement is the most powerful option because you can easily display the rectangles as filled, unfilled, in the foreground, in the background, and you can color the polygons in many informative ways.

Compute the unconditional and conditional quantiles

To compute quantiles, you can use the QNTL function in PROC IML or you can call PROC RANK and use the GROUPS= option. The following statements use PROC RANK. ("KSharp" used this method for his answer on the Support Community.) One noteworthy idea in the following program is to use the CATX or CATT function to form a grouping variable that combines the group names for the X and Y variables.

%let XVar = MomWtGain; /* name of X variable */
%let YVar = Weight;    /* name of Y variable */
%let NumXDiv = 3;      /* number of subdivisions (quantiles) for the X variable */
%let NumYDiv = 4;      /* number of subdivisions (quantiles) for the Y variable */
/* create the example data */
data Have;
set sashelp.bweight(obs=5000 keep=&XVar &YVar);
if ^cmiss(&XVar, &YVar);   /* only use complete cases */
/* group the Y values into quantiles */
proc rank data=Have out=RankY groups=&NumYDiv ties=high;
   var &YVar;
   ranks rank_&YVar;
proc sort data=RankY;  by rank_&YVar;  run;
/* Conditioned on each Y quantile, group the X values into quantiles */
proc rank data=RankY out=RankBoth groups=&NumXDiv ties=high;
   by rank_&YVar;
   var &XVar;
   ranks rank_&XVar;
proc sort data=RankBoth;  by rank_&YVar rank_&XVar;  run;
/* Trick: Use the CATT fuction to create a group for each unique (RankY, RankX) combination */
data Points;
set RankBoth;
Group = catt(rank_&YVar, rank_&XVar);
proc sgplot data=Points;
   scatter x=&XVar y=&YVar / group=Group markerattrs=(symbol=CircleFilled) transparency=0.75;
   /* These REFLINESs are hard-coded for these data. They are the unconditional Y quantiles. */
   refline 3061 3402 3716 / axis=Y;

The SCATTER statement uses the GROUP= option to color each observation according to the quantile bin to which it belongs. The horizontal reference lines are the 25th, 50th, and 75th percentiles of the Y variable. These lines separate the Y variable into four groups. Within each group, the X variable is divided into three groups. The remainder of the program shows how to construct and plot the 12 rectangles that bound the groups.

Create polygons for the quantile bins

You can use PROC MEANS and a BY statement to find the minimum and maximum values of the X and Y variables for each cell. As a bonus, PROC MEANS also tells you how many observations are in each cell. If the data are unique values, each cell will have approximately N / 12 observations, where N=5000 is the total number of observations in the data set. If the data have duplicate values, remember that some quantile groups might have more observations than others.

/* Create polygons from ranks. Each polygon is a rectangle of the form
   [min(X), max(X)] x [min(Y), max(Y)] 
/* generate min/max for each group. Results are in wide form. */
proc means data=Points noprint;
   by Group;
   var &XVar &YVar;
   output out=Limits(drop=_Type_) min= max= / autoname;  
proc print data=Limits noobs;
Coordinate of rectangular bins in a conditional quantile bin plot

For these data, there are between 362 and 500 observations in each cell. If there were no tied values, you would expect 5000/12 ≈ 417 observations in each cell.

The Limits data set contains all the information that you need to construct the cells of the conditional quantile bin plot. PROC MEANS writes the ranges of the variables in a wide format. In order to use the POLYGON statement in PROC SGPLOT, you need to convert the data to long form and use the GROUP variable as an ID variable for the polygon plot:

/* generate polygons from min/max of groups. Write data in long form. */
data Rectangles;
set Limits;
xx = &XVar._min; yy = &YVar._min; output;
xx = &XVar._max; yy = &YVar._min; output;
xx = &XVar._max; yy = &YVar._max; output;
xx = &XVar._min; yy = &YVar._max; output;
xx = &XVar._min; yy = &YVar._min; output;   /* repeat the first obs to close the polygon */
drop &XVar._min &XVar._max &YVar._min &YVar._max;
/* combine the data and the polygons. Plot the results */
data All;
set Points Rectangles;
label _FREQ_ = "Frequency";

Create polygons for the quantile bins

The advantage of creating polygons is that you can experiment with many ways to visualize the conditional quantile plot. For example, the simplest quantile plot merely overlays rectangles on a scatter plot of the bivariate data, as follows:

title "Conditional Quantile Bin Plot";
proc sgplot data=All noautolegend;
   scatter x=&XVar y=&YVar / markerattrs=(symbol=CircleFilled) transparency=0.9;
   polygon x=xx y=yy ID=Group;  /* plot in the foreground, no fill */

On the other hand, you can also display the rectangles in the background and use the GROUP= option to display the rectangles in different colors.

proc sgplot data=All noautolegend;
   polygon x=xx y=yy ID=Group / fill outline group=Group; /* background, filled */
   scatter x=&XVar y=&YVar / markerattrs=(symbol=CircleFilled) transparency=0.9;

The graph that results from these statements is shown at the top of this article.

To give a third example, you can plot the rectangles in the background and use the COLORRESPONSE= option to color the rectangles according to the number of observations inside each bin. This is shown in the following graph, which uses the default three-color color ramp to assign colors to rectangles.

proc sgplot data=All;
   polygon x=xx y=yy ID=Group / fill outline colorresponse=_FREQ_; /* background, filled */
   scatter x=&XVar y=&YVar / markerattrs=(symbol=CircleFilled) transparency=0.9;


In summary, this article shows how to construct a 2-D quantile bin plot where the quantiles in the horizontal direction are conditional on the quantiles in the vertical direction. In such a plot, the cells are irregularly spaced but have approximately the same number of observations in each cell. You can use polygons to create the rectangles. The polygons give you a lot of flexibility about how to display the conditional quantile plot.

You can download the complete SAS program that creates the conditional quantile bin plot.

The post Create a conditional quantile bin plot in SAS appeared first on The DO Loop.

12月 112019

Binary matrices are used for many purposes. I have previously written about how to use binary matrices to visualize missing values in a data matrix. They are also used to indicate the co-occurrence of two events. In ecology, binary matrices are used to indicate which species of an animal are present in which ecological site. For example, if you remember your study of Darwin's finches in high school biology class, you might recall that certain finches (species) are present or absent on various Galapagos islands (sites). You can use a binary matrix to indicate which finches are present on which islands.

Recently I was involved in a project that required performing a permutation test on rows of a binary matrix. As part of the project, I had to solve three smaller problems involving rows of a binary matrix:

  1. Given two rows, find the locations at which the rows differ.
  2. Given two binary matrices, determine how similar the matrices are by computing the proportion of elements that are the same.
  3. Given two rows, swap some of the elements that differ.

This article shows how to solve each problem by using the SAS/IML matrix language. A future article will discuss permutation tests for binary matrices. For clarity, I introduce the following macro that uses a temporary variable to swap two sets of values:

/* swap values of A and B. You can use this macro in the DATA step or in the SAS/IML language */
%macro SWAP(A, B, TMP=_tmp);
   &TMP = &A; &A = &B; &B = &TMP;

Where do binary matrices differ?

The SAS/IML matrix language enables you to treat matrices as high-level objects. You often can answer questions about matrices without writing loops that iterate over the elements of the matrices. For example, if you have two matrices of the same dimensions, you can determine the cells at which the matrices are unequal by using the "not equal" (^=) operator. The following SAS/IML statements define two 2 x 10 matrices and use the "not equal" operator to find the cells that are different:

proc iml;
A = {0 0 1 0 0 1 0 1 0 0 ,
     1 0 0 0 0 0 1 1 0 1 };
B = {0 0 1 0 0 0 0 1 0 1 ,
     1 0 0 0 0 1 1 1 0 0 };
colLabel = "Sp1":("Sp"+strip(char(ncol(A))));
rowLabel = "A1":("A"+strip(char(nrow(A))));
/* 1. Find locations where binary matrices differ */
Diff = (A^=B);
print Diff[c=colLabel r=rowLabel];

The matrices A and B are similar. They both have three 1s in the first row and fours 1s in the second row. However, the output shows that the matrices are different for the four elements in the sixth and tenth columns. Although I used entire matrices for this example, the same operations work on row vectors.

The proportion of elements in common

You can use various statistics to measure the similarity between the A and B matrices. A simple statistic is the proportion of elements that are in common. These matrices have 20 elements, and 16/20 = 0.8 are common to both matrices. You can compute the proportion in common by using the express (A=B)[:], or you can use the following statements if you have previously computed the Diff matrix:

/* 2. the proportion of elements in B that are the same as in A */
propDiff = 1 - Diff[:];
print propDiff;

As a reminder, the mean subscript reduction operator ([:]) computes the mean value of the elements of the matrix. For a binary matrix, the mean value is the proportion of ones.

Swap elements of rows

The first two tasks were easy. A more complicated task is swapping values that differ between rows. The swapping operation is not difficult, but it requires finding the k locations where the rows differ and then swapping all or some of those values. In a permutation test, the number of elements that you swap is a random integer between 1 and k, but for simplicity, this example uses the SWAP macro to swap two cells that differ. For clarity, the following example uses temporary variables such as x1, x2, d1, and d2 to swap elements in the matrix A:

/* specify the rows whose value you want to swap */
i1 = 1;                         /* index of first row to compare and swap */
i2 = 2;                         /* index of second row to compare and swap */
/* For clarity, use temp vars x1 & x2 instead of A[i1, ] and A[i2, ] */
x1 = A[ i1, ];                  /* get one row */
x2 = A[ i2, ];                  /* get another row */
idx = loc( x1^=x2 );            /* find the locations where rows differ */
if ncol(idx) > 0 then do;       /* do the rows differ? */
   d1 = x1[ ,idx];              /* values at the locations that differ */
   d2 = x2[ ,idx];
   print (d1//d2)[r={'r1' 'r2'} L='Original'];
   /* For a permutation test, choose a random number of locations to swap */
   /* numSwaps = randfun(1, "Integer", 1, n);
      idxSwap = sample(1:ncol(idx), numSwaps, "NoReplace");  */
   idxSwap = {2 4};              /* for now, hard-code locations to swap */
   %SWAP(d1[idxSwap], d2[idxSwap]);
   print (d1//d2)[r={'d1' 'd2'} L='New'];
   /* update matrix */
   A[ i1, idx] = d1;
   A[ i2, idx] = d2;
print A[c=colLabel r=rowLabel];

The vectors x1 and x2 are the rows of A to compare. The vectors d1 and d2 are the subvectors that contain only the elements of x1 and x2 that differ. The example swaps the second and fourth columns of d1 and d2. The new values are then inserted back into the matrix A. You can compare the final matrix to the original to see that the process swapped two elements in each of two rows.

Although the examples are for binary matrices, these techniques work for any numerical matrices.

The post Swap elements in binary matrices appeared first on The DO Loop.

12月 052019

This is a second article about analyzing longitudinal data, which features measurements that are repeatedly taken on subjects at several points in time. The previous article discusses a response-profile analysis, which uses an ANOVA method to determine differences between the means of an experimental group and a placebo group. The response-profile analysis has limitations, including the fact that longitudinal data are autocorrelated and so do not satisfy the independence assumption of ANOVA. Furthermore, the method does not enable you to model the response profile of individual subjects; it produces only a mean response-by-time profile.

This article shows how a mixed model can produce a response profile for each subject. A mixed model also addresses other limitations of the response-profile analysis. This blog post is based on the introductory article, "A Primer in Longitudinal Data Analysis", by G. Fitzmaurice and C. Ravichandran (2008), Circulation, 118(19), p. 2005-2010.

The data (from Fitzmaurice and C. Ravichandran, 2008) are the blood lead levels for 100 inner-city children who were exposed to lead in their homes. Half were in an experimental group and were given a compound called succimer as treatment. The other half were in a placebo group. Blood-lead levels were measured for each child at baseline (week 0), week 1, week 4, and week 6. The data are visualized in the previous article. You can download the SAS program that creates the data and graphs in this article.

Advantages of the mixed model for longitudinal data

The main advantage of a mixed-effect model is that each subject is assumed to have his or her own mean response curve that explains how the response changes over time. The individual curves are a combination of two parts: "fixed effects," which are common to the population and shared by all subjects, and "random effects," which are specific to each subject. The term "mixed" implies that the model incorporates both fixed and random effects.

You can use a mixed model to do the following:

  1. Model the individual response-by-time curves.
  2. Model autocorrelation in the data. This is not discussed further in this blog post.
  3. Model time as a continuous variable, which is useful for data that includes mistimed observations and parametric models of time, such as a quadratic model or a piecewise linear model.

The book Applied Longitudinal Analysis (G. Fitzmaurice, N. Laird, and J. Ware, 2011, 2nd Ed.) discusses almost a dozen ways to model the data for blood-lead level in children. This blog post briefly shows how to implement three models in SAS that incorporate random intercepts. The models are the response-profile model, a quadratic model, and a piecewise linear model.

Visualize mixed models

I've previously written about how to visualize mixed models in SAS. One of the techniques is to create a spaghetti plot that shows the predicted response profiles for each subject in the study. Because we will examine three different models, the following statements define a macro that will sort the predicted values and plot them in a spaghetti plot:

%macro SortAndPlot(DSName);
proc sort data=&DSName;
   by descending Treatment ID Time;
proc sgplot data=&DSName dattrmap=Order;
   series x=Time y=Pred / group=ID groupLC=Treatment break lineattrs=(pattern=solid)
   legenditem type=line name="P" / label="Placebo (P)" lineattrs=GraphData1; 
   legenditem type=line name="A" / label="Succimer (A)" lineattrs=GraphData2; 
   keylegend "A" "P";
   xaxis values=(0 1 4 6) grid;
   yaxis label="Predicted Blood Lead Level (mcg/dL)";

A response-profile model with a random intercept

In the response-profile analysis, the data were analyzed by using PROC GLM, although these data do not satisfy the assumptions of PROC GLM. This article uses PROC MIXED in SAS/STAT software for the analyses. You can use the REPEATED statement in PROC MIXED to specify that the measurements for individuals are autocorrelated. We will use an unstructured covariance matrix for the model (TYPE=UN), but Fitzmaurice, Laird, and Ware (2011) discuss other options.

In the response-profile analysis, the model predicts the mean response for each treatment group. However, the baseline measurements for each subject are all different. For example, some start the trial with a blood-lead level that is higher than the mean, others start lower than the mean. To adjust for this variation among subjects, you can use the RANDOM INTERCEPT statement in PROC MIXED. Use the OUTPRED= option on the MODEL statement to write the predicted values for each subject to a data set, then plot the results. The following statements repeat the response-profile model of the previous blog post but include an intercept for each subject.

/* Repeat the response-profile analysis, but 
   use the RANDOM statement to add random intercept for each subject */
proc mixed data=TLCView;
   class id Time(ref='0') Treatment(ref='P');
   model y = Treatment Time Treatment*Time / s chisq outpred=MixedOut;
   repeated Time / type=un subject=id r;   /* measurements are repeated for subjects */
   random intercept / subject=id;          /* each subject gets its own intercept */
title "Predicted Individual Growth Curves";
title2 "Random Intercept Model";
Random intercept model or response profiles

In this model, the shape of the response-profile curve is the same for all subjects in each treatment group. However, the curves are shifted up or down to better match the subject's individual profile.

A mixed model with a quadratic response curve

From the shape of the predicted response curve in the previous section, you might conjecture that a quadratic model might fit the data. You can fit a quadratic model in PROC MIXED by treating Time as a continuous variable. However, Time is also used in the REPEATED statement, and that statement requires a discrete CLASS variable. A resolution to this problem is to create a copy of the Time variable (call it T). You can include T in the CLASS and REPEATED statements and use Time in the MODEL statement as a continuous effect. The third analysis will require a linear spline variable, so the DATA VIEW also creates that variable, called T1.

/* Make "discrete time" (t) to use in REPEATED statement.
   Make spline effect with knot at t=1. */
data TLCView / view=TLCView;
set tlc;
t = Time;                       /* discrete copy of time */
T1 = ifn(Time<1, 0, Time - 1);  /* knot at Time=1 for PWL analysis */

You can now use Time as a continuous effect and T to specify that measurements are repeated for the subjects.

/* Model time as continuous and use a quadratic model in Time. 
   For more about quadratic growth models, see */
proc mixed data=TLCView;
   class id t(ref='0') Treatment(ref='P');
   model y = Treatment Time Time*Time Treatment*Time / s outpred=MixedOutQuad;
   repeated t / type=un subject=id r;      /* measurements are repeated for subjects */
   random intercept / subject=id;          /* each subject gets its own intercept */
title2 "Random Intercept; Quadratic in Time";
Quadratic model for response profiles with random intercepts

The graph shows the individual predicted responses for each subject, but the quadratic model does not seem to capture the dramatic drop in the blood-lead level at T=1.

A mixed model with a piecewise linear response curve

The next model uses a piecewise linear model instead of a quadratic model. I've previously written about how to use spline effects in SAS to model data by using piecewise polynomials.

For the four time points, the mean response profile seems to go down for the experimental (succimer) group until T=1, and then increase at approximately a constant rate. Is this behavior caused by an underlying biochemical process? I don't know, but if you believe (based on domain knowledge) that this reflects an underlying process, you can incorporate that belief in a piecewise linear model. The first linear segment models the blood-lead level for T ≤ 1; the other segment models the blood-lead level for T > 1.

/* Piecewise linear (PWL) model with knot at Time=1.
   For more about PWL models, see Hwang (2015) 
   "Hands-on Tutorial for Piecewise Linear Mixed-effects Models Using SAS PROC MIXED"    */
proc mixed data=TLCView;
   class id t(ref='0') Treatment(ref='P');
   model y = Treatment Time T1 Treatment*Time Treatment*T1 / s outpred=MixedOutPWL;
   repeated t / type=un subject=id r;      /* measurements are repeated for subjects */
   random intercept / subject=id;          /* each subject gets its own intercept */
title2 "Random Intercept; Piecewise Linear Model";
Piecewise linear model for response profiles with random intercepts

This graph shows a model that is piecewise linear. It assumes that the blood-lead level falls constantly during the first week of the treatment, then either falls or rises constantly during the remainder of the study. You could use the slopes of the lines to report the average rate of change during each time period.

Further reading

There are many papers and many books written about mixed models in SAS. This article presents data and ideas that are discussed in detail in the book Applied Longitudinal Analysis (2012, 2nd Ed) by G. Fitzmaurice, N. Laird, and J. Ware. For an informative article about piecewise-linear mixed models, see Hwang (2015) "Hands-on Tutorial for Piecewise Linear Mixed-effects Models Using SAS PROC MIXED" For a comprehensive discussion of mixed models and repeated-measures analysis, I recommend SAS for Mixed Models, either the 2nd edition or the new edition.

Many people have questions about how to model longitudinal data in SAS. Post your questions to the SAS Support Communities, which has a dedicated community for statistical analysis.

The post Longitudinal data: The mixed model appeared first on The DO Loop.

12月 032019

Longitudinal data are used in many health-related studies in which individuals are measured at multiple points in time to monitor changes in a response variable, such as weight, cholesterol, or blood pressure. There are many excellent articles and books that describe the advantages of a mixed model for analyzing longitudinal data. Recently, I encountered an introductory article that summarizes the essential issues in a little more than five pages! You can download the article for free: "A Primer in Longitudinal Data Analysis", by G. Fitzmaurice and C. Ravichandran (2008), Circulation, 118(19), p. 2005-2010.

The article analyzes a set of longitudinal data in two ways. First, the authors use a traditional linear model to perform an "analysis of response profiles." Then, the authors discuss how a mixed model can correct some of the deficiencies of the analysis. This blog post analyzes the same data by using PROC GLM in SAS. A subsequent blog post analyzes the same data by using PROC MIXED in SAS.

Longitudinal Data: Treatment of lead-exposed children

Fitzmaurice and C. Ravichandran analyze data for a randomized trial involving toddlers who were exposed to high levels of lead. The article analyzes a subset of 100 children. Half the children were given a treatment (called succimer) and the other half were given a placebo. The blood lead levels were measured for each child at baseline (week 0), week 1, week 4, and week 6. The main scientific question is "whether the two treatment groups differ in their patterns of change from baseline in mean blood lead levels" (p. 2006).

The children in the subset were measured at all four time points. There are no missing values or mistimed measurements. (This situation is fairly unusual in longitudinal data, which is often plagued by missed appointments or individuals who leave the study.) The following "spaghetti plot" shows the individual measurements for the 100 children at each time point. Each line represents a child. Blue lines indicate that the child was in the placebo group; red lines indicate the experimental group that was given succimer.

You can download the SAS program that contains the data and the programs that create the graphs and tables in this article.

Analysis of response profiles

In an analysis of response profiles, you compare the mean response-by-time profile for the treatment and placebo groups. You can visualize the mean response over time for each group by using the VBAR statement in PROC SGPLOT. The following graph shows the mean value and standard error for each time point for each treatment group:

If the treatment is ineffective, the line segments for the two treatment groups will be approximately parallel. The graph shows that this is not that case for these data. The visualization indicates that the mean blood-lead value for the treatment group (Treatment='A') is lower than for the placebo group at 1 and 4 weeks.

You can use PROC GLM to confirm that these differences are statistically significant and to estimate the effect that taking succimer had on the mean blood-lead level:

proc glm data=tlc;
   class Time(ref='0') Treatment(ref='P');
   model y = Treatment Time Treatment*Time / solution;
   output out=GLMOut Predicted=Pred;

According to the Type 3 statistics, all three effects in the model are significant. The parameter estimates (outlined in red) indicate that the mean blood-lead level for children in the Treatment='A' group is 11.4 mcg/dL lower than the children in the placebo group after 1 week. Similarly, after four weeks the mean of the experimental group is 8.8 mcg/dL lower than the placebo group. These are both significant differences, so the response-profile analysis provides a positive answer to the research question: the profile for the treatment group is lower than the placebo group.

Advantages and disadvantages of the analysis of response profiles

As discussed in Fitzmaurice and Ravichandran (2008), the analysis of the response profile has several advantages:

  • It is familiar to researchers who have experience with ANOVA.
  • It does not require any advanced statistical modeling , such as modeling the covariance of the repeated measurements.

However, this simple method suffers from several statistical problems:

  • Longitudinal data do not satisfy the assumptions of linear regression and ANOVA. Because the data contains repeated measures from the same individuals, the residual errors are neither independent nor do they have constant variance (homoscedastic).
  • Some participants in a study might miss an appointment or drop out of the study. Others might be measured at time points that were not part of the design (for example, at 2 or 3 weeks). These two problems are known as incomplete data and mistimed measurements, respectively. Although the first can be handled by using an unbalanced ANOVA, the second is a problem that does not have a simple solution within an ANOVA model that uses discrete time points.
  • The response-profile analysis does not enable you to model each individual's response as a function of time.

Prediction of individual response profiles

The inability to model individual trajectories is often the reason that researchers abandon the response-profile analysis in favor of a more complicated mixed model. To be clear, the GLM model can make predictions, but the predicted values for every child in the placebo group are the same. Similarly, the predicted values for every child in the experimental group are the same. This is shown in the following panel of graphs, which shows the predicted response curves for six children in the study, three from each treatment group.

/* Look at predictions for each individual. They are identical! */
proc sort data=GLMOut(where=(ID in (1,2,4,5,6,7))) out=GLMSubset;
   by Treatment ID;
title2 "Fixed Effect Model";
proc sgpanel data=GLMSubset dattrmap=Order;
   panelby Treatment ID;
   scatter x=Time y=y / group=Treatment attrid=Treat;
   series x=Time y=Pred / group=Treatment attrid=Treat;

Notice that the predicted responses are the same across the top row (ID=2, 5, and 6). These children were all in the experimental group. Although the predicted values seem to fit the actual observed response for ID=2, the predicted responses for ID=5 and ID=6 are much higher than the observed responses. Although the predicted response is the best prediction for an "average patient," it does not account for individual differences in the study participants.

The same is true for the patients in the placebo group, three of which are plotted in the second row. The predicted values are "too low" for ID=1 and are "too high" for ID=4.

If modeling the individual profiles is important, then clearly this method is not sufficient. If you want to model the individual profiles, you can use a linear mixed model. The mixed model also addresses other deficiencies of the response-profile analysis. The mixed model is described in the next blog post.

You can download the SAS code and data for the response-profile analysis.

Further reading

This blog post is a brief summary of the article "A Primer in Longitudinal Data Analysis" (Fitzmaurice and Ravichandran, 2008). See the article for more details. Also, these data and these ideas are also discussed in the book Applied Longitudinal Analysis (2012, 2nd Ed) by G. Fitzmaurice, N. Laird, and J. Ware. You can download data (and SAS programs) from the book at the book's web site.

The post Longitudinal data: The response-profile model appeared first on The DO Loop.

11月 202019

In a linear regression model, the predicted values are on the same scale as the response variable. You can plot the observed and predicted responses to visualize how well the model agrees with the data, However, for generalized linear models, there is a potential source of confusion. Recall that a generalized linear model (GLIM) has two components: a linear predictor, which is a linear combination of the explanatory variables, and a transformation (called the inverse link function) that maps the linear predictor onto the scale of the data. Consequently, SAS regression procedures support two types of predicted values and prediction limits. In the SAS documentation, the first type is called "predictions on the linear scale" whereas the second type is called "predictions on the data scale."

For many SAS procedures, the default is to compute predicted values on the linear scale. However, for GLIMs that model nonnormal response variables, it is more intuitive to predict on the data scale. The ILINK option, which is shorthand for "apply the inverse link transformation," converts the predicted values to the data scale. This article shows how the ILINK option works by providing an example for a logistic regression model, which is the most familiar generalized linear models.

Review of generalized linear models

The SAS documentation provides an overview of GLIMs and link functions. The documentation for PROC GENMOD provides a list of link functions for common regression models, including logistic regression, Poisson regression, and negative binomial regression.

Briefly, the linear predictor is
η = X*β
where X is the design matrix and β is the vector of regression coefficients. The link function (g) is a monotonic function that relates the linear predictor to the conditional mean of the response. Sometimes the symbol μ is used to denote the conditional mean of the response (μ = E[Y|x]), which leads to the formula
g(μ) = X*β

In SAS, you will often see options and variables names (in output data sets) that contains the substring 'XBETA'. When you see 'XBETA', it indicates that the statistic or variable is related to the LINEAR predictor. Because the link function, g, is monotonic, it has an inverse, g-1. For generalized linear models, the inverse link function maps the linear-scale predictions to data-scale predictions: if η = x β is a predicted value on the linear scale, then g-1(η) is the predicted value for x on the data scale.

When the response variable is binary, the GLIM is the logistic model. If you use the convention that Y=1 indicates an event and Y=0 indicates the absence of an event, then the "data scale" is [0, 1] and the GLIM predicts the probability that the event occurs. For the logistic GLIM, the link function is the logit function:
g(μ) = logit(μ) = log( μ / (1 - μ) )
The inverse of the logit function is called the logistic function:
g-1(η) = logistic(η) = 1 / (1 + exp(-η))

To demonstrate the ILINK option, the next sections perform the following tasks:

  1. Use PROC LOGISTIC to fit a logistic model to data. You can use the STORE statement to store the model to an item store.
  2. Use the SCORE statement in PROC PLM to score new data. This example scores data by using the ILINK option.
  3. Score the data again, but this time do not use the ILINK option. Apply the logistic transformation to the linear estimates to demonstrate that relationship between the linear scale and the data scale.

The transformation between the linear scale and the data scale is illustrated by the following graph:

Fit a logistic model

The following data are from the documentation for PROC LOGISTIC. The model predicts the probability of Pain="Yes" (the event) for patients in a study, based on the patients' sex, age, and treatment method ('A', 'B', or 'P'). The STORE statement in PROC LOGISTIC creates an item store that can be used for many purposes, including scoring new observations.

data Neuralgia;
   input Treatment $ Sex $ Age Duration Pain $ @@;
   DROP Duration;
P F 68  1 No  B M 74 16 No  P F 67 30 No  P M 66 26 Yes B F 67 28 No  B F 77 16 No
A F 71 12 No  B F 72 50 No  B F 76  9 Yes A M 71 17 Yes A F 63 27 No  A F 69 18 Yes
B F 66 12 No  A M 62 42 No  P F 64  1 Yes A F 64 17 No  P M 74  4 No  A F 72 25 No
P M 70  1 Yes B M 66 19 No  B M 59 29 No  A F 64 30 No  A M 70 28 No  A M 69  1 No
B F 78  1 No  P M 83  1 Yes B F 69 42 No  B M 75 30 Yes P M 77 29 Yes P F 79 20 Yes
A M 70 12 No  A F 69 12 No  B F 65 14 No  B M 70  1 No  B M 67 23 No  A M 76 25 Yes
P M 78 12 Yes B M 77  1 Yes B F 69 24 No  P M 66  4 Yes P F 65 29 No  P M 60 26 Yes
A M 78 15 Yes B M 75 21 Yes A F 67 11 No  P F 72 27 No  P F 70 13 Yes A M 75  6 Yes
B F 65  7 No  P F 68 27 Yes P M 68 11 Yes P M 67 17 Yes B M 70 22 No  A M 65 15 No
P F 67  1 Yes A M 67 10 No  P F 72 11 Yes A F 74  1 No  B M 80 21 Yes A F 69  3 No
title 'Logistic Model on Neuralgia';
proc logistic data=Neuralgia;
   class Sex Treatment;
   model Pain(Event='Yes')= Sex Age Treatment;
   store PainModel / label='Neuralgia Study';  /* store model for post-fit analysis */

Score new data by using the ILINK option

There are many reasons to use PROC PLM, but an important purpose of PROC PLM is to score new observations. Given information about new patients, you can use PROC PLM to predict the probability of pain if these patients are given a specific treatment. The following DATA step defines the characteristics of four patients who will receive Treatment B. The call to PROC PLM scores and uses the ILINK option to predict the probability of pain:

/* Use PLM to score new patients */
data NewPatients;
   input Treatment $ Sex $ Age Duration;
   DROP Duration;
B M 67 15 
B F 73  5 
B M 74 12 
B F 79 16 
/* predictions on the DATA scale */
proc plm restore=PainModel noprint;
   score data=NewPatients out=ScoreILink
         predicted lclm uclm / ilink; /* ILINK gives probabilities */
proc print data=ScoreILink; run;

The Predicted column contains probabilities in the interval [0, 1]. The 95% prediction limits for the predictions are given by the LCLM and UCLM columns. For example, the prediction interval for the 67-year-old man is approximately [0.03, 0.48].

These values and intervals are transformations of analogous quantities on the linear scale. The logit transformation maps the predicted probabilities to the linear estimates. The inverse logit (logistic) transformation maps the linear estimates to the predicted probabilities.

Linear estimates and the logistic transformation

The linear scale is important because effects are additive on this scale. If you are testing the difference of means between groups, the tests are performed on the linear scale. For example, the ESTIMATE, LSMEANS, and LSMESTIMATE statements in SAS perform hypothesis testing on the linear estimates. Each of these statements supports the ILINK option, which enables you to display predicted values on the data scale.

To demonstrate the connection between the predicted values on the linear and data scale, the following call to PROC PLM scores the same data according to the same model. However, this time the ILINK option is omitted, so the predictions are on the linear scale.

/* predictions on the LINEAR scale */
proc plm restore=PainModel noprint;
   score data=NewPatients out=ScoreXBeta(
         rename=(Predicted=XBeta LCLM=LowerXB UCLM=UpperXB))
         predicted lclm uclm;         /* ILINK not used, so linear predictor */
proc print data=ScoreXBeta; run;

I have renamed the variables that PROC PLM creates for the estimates on the linear scale. The XBeta column shows the predicted values. The LowerXB and UpperXB columns show the prediction interval for each patient. The XBeta column shows the values you would obtain if you use the parameter estimates table from PROC LOGISTIC and apply those estimates to the observations in the NewPatients data.

To demonstrate that the linear estimates are related to the estimates in the previous section, the following SAS DATA step uses the logistic (inverse logit) transformation to convert the linear estimates onto the predicted probabilities:

/* Use the logistic (inverse logit) to transform the linear estimates
     (XBeta) into probability estimates in [0,1], which is the data scale.
   You can use the logit transformation to go the other way. */
data LinearToData;
   set ScoreXBeta;   /* predictions on linear scale */
   PredProb   = logistic(XBeta);
   LCLProb    = logistic(LowerXB);
   UCLProb    = logistic(UpperXB); 
proc print data=LinearToData;
   var Treatment Sex Age PredProb LCLProb UCLProb;

The transformation of the linear estimates gives the same values as the estimates that were obtained by using the ILINK option in the previous section.


In summary, there are two different scales for predicting values for a generalized linear model. When you report predicted values, it is important to specify the scale you are using. The data scale makes intuitive sense because it is the same scale as the response variable. You can use the ILINK option in SAS to get predicted values and prediction intervals on the data scale.

The post Predicted values in generalized linear models: The ILINK option in SAS appeared first on The DO Loop.

11月 132019

Biplots are two-dimensional plots that help to visualize relationships in high dimensional data. A previous article discusses how to interpret biplots for continuous variables. The biplot projects observations and variables onto the span of the first two principal components. The observations are plotted as markers; the variables are plotted as vectors. The observations and/or vectors are not usually on the same scale, so they need to be rescaled so that they fit on the same plot. There are four common scalings (GH, COV, JK, and SYM), which are discussed in the previous article.

This article shows how to create biplots in SAS. In particular, the goal is to create the biplots by using modern ODS statistical graphics. You can obtain biplots that use the traditional SAS/GRAPH system by using the %BIPLOT macro by Michael Friendly. The %BIPLOT macro is very powerful and flexible; it is discussed later in this article.

There are four ways to create biplots in SAS by using ODS statistical graphics:

  • You can use PROC PRINQUAL in SAS/STAT software to create the COV biplot.
  • If you have a license for SAS/GRAPH software (and SAS/IML software), you can use Friendly's %BIPLOT macro and use the OUT= option in the macro to save the coordinates of the markers and vectors. You can then use PROC SGPLOT to create a modern version of Friendly's biplot.
  • You can use the matrix computations in SAS/IML to "manually" compute the coordinates of the markers and vectors. (These same computations are performed internally by the %BIPLOT macro.) You can use the Biplot module to create a biplot, or you can use the WriteBiplot module to create a SAS data set that contains the biplot coordinates. You can then use PROC SGPLOT to create the biplot.

For consistency with the previous article, all methods in this article standardize the input variables to have mean zero and unit variance (use the SCALE=STD option in the %BIPLOT macro). All biplots show projections of the same four-dimensional Fisher's iris data. The following DATA step assigns a blank label. If you do not supply an ID variable, some biplots display observations numbers.

data iris;
   set Sashelp.iris;
   id = " ";        /* create an empty label variable */

Use PROC PRINQUAL to compute the COV biplot

The PRINQUAL procedure can perform a multidimensional preference analysis, which is visualized by using a MDPREF plot. The MDPREF plot is closely related to biplot (Jackson (1991), A User’s Guide to Principal Components, p. 204). You can get PROC PRINQUAL to produce a COV biplot by doing the following:

  • Use the N=2 option to specify you want to compute two principal components.
  • Use the MDPREF=1 option to specify that the procedure should not rescale the vectors in the biplot. By default, MDPREF=2.5 and the vectors appear 2.5 larger than they should be. (More on scaling vectors later.)
  • Use the IDENTITY transformation so that the variables are not transformed in a nonlinear manner.

The following PROC PRINQUAL statements produce a COV biplot (click to enlarge):

proc prinqual data=iris plots=(MDPref) 
              n=2       /* project onto Prin1 and Prin2 */
              mdpref=1; /* use COV scaling */
   transform identity(SepalLength SepalWidth PetalLength PetalWidth);  /* identity transform */
   id ID;
   ods select MDPrefPlot;
COV biplot, created in SAS by using PROC PRINQUAL

Use Friendly's %BIPLOT macro

Friendly's books [SAS System for Statistical Graphics (1991) and Visualizing Categorical Data (2000)] introduced many SAS data analysts to the power of using visualization to accompany statistical analysis, and especially the analysis of multivariate data. His macros use traditional SAS/GRAPH graphics from the 1990s. In the mid-2000s, SAS introduced ODS statistical graphics, which were released with SAS 9.2. Although the %BIPLOT macro does not use ODS statistical graphics directly, the macro supports the OUT= option, which enables you to create an output data set that contains all the coordinates for creating a biplot.

The following example assumes that you have downloaded the %BIPLOT macro and the %EQUATE macro from Michael Friendly's web site.

/* A. Use %BIPLOT macro, which uses SAS/IML to compute the biplot coordinates. 
      Use the OUT= option to get the coordinates for the markers and vectors.
   B. Transpose the data from long to wide form.
   C. Use PROC SGPLOT to create the biplot
%let FACTYPE = SYM;   /* options are GH, COV, JK, SYM */
title "Biplot: &FACTYPE, STD";
        var=SepalLength SepalWidth PetalLength PetalWidth, 
        factype=&FACTYPE,  /* GH, COV, JK, SYM */
        std=std,           /* NONE, MEAN, STD  */
        scale=1,           /* if you do not specify SCALE=1, vectors are auto-scaled */
        out=biplotFriendly,/* write SAS data set with results */ 
        symbols=circle dot, inc=1);
/* transpose from long to wide */
data Biplot;
set biplotFriendly(where=(_TYPE_='OBS') rename=(dim1=Prin1 dim2=Prin2 _Name_=_ID_))
    biplotFriendly(where=(_TYPE_='VAR') rename=(dim1=vx dim2=vy _Name_=_Variable_));
proc sgplot data=Biplot aspect=1 noautolegend;
   refline 0 / axis=x; refline 0 / axis=y;
   scatter x=Prin1 y=Prin2 / datalabel=_ID_;
   vector  x=vx    y=vy    / datalabel=_Variable_
                             lineattrs=GraphData2 datalabelattrs=GraphData2;
   xaxis grid offsetmin=0.1 offsetmax=0.2;
   yaxis grid;
SYM biplot, created in SAS by using ODS statistical graphics

Because you are using PROC SGPLOT to display the biplot, you can easily configure the graph. For example, I added grid lines, which are not part of the output from the %BIPLOT macro. You could easily change attributes such as the size of the fonts or add additional features such as an inset. With a little more work, you can merge the original data and the biplot data and color-code the markers by a grouping variable (such as Species) or by a continuous response variable.

Notice that the %BUPLOT macro supports a SCALE= option. The SCALE= option applies an additional linear scaling to the vectors. You can use this option to increase or decrease the lengths of the vectors in the biplot. For example, in the SYM biplot, shown above, the vectors are long relative to the range of the data. If you want to display vectors that are only 25% as long, you can specify SCALE=0.25. You can specify numbers greater than 1 to increase the vector lengths. For example, SCALE=2 will double the lengths of the vectors. If you omit the SCALE= option or set SCALE=0, then the %BIPLOT macro automatically scales the vectors to the range of the data. If you use the SCALE= option, you should tell the reader that you did so.

SAS/IML modules that compute biplots

The %BIPLOT macro uses SAS/IML software to compute the locations of the markers and vectors for each type of biplot. I wrote three SAS/IML modules that perform the three steps of creating a biplot:

  • The CalcBiplot module computes the projections of the observations and scores onto the first few principal components. This module (formerly named CalcPrinCompBiplot) was written in the mid-2000s and has been distributed as part of the SAS/IML Studio application. It returns the scores and vectors as SAS/IML matrices.
  • The WriteBiplot module calls the CalcBiplot module and then writes the scores to a SAS data set called _SCORES and the vectors (loadings) to a SAS data set called _VECTORS. It also creates two macro variables, MinAxis and MaxAxis, which you can use if you want to equate the horizontal and vertical scales of the biplot axes.
  • The Biplot function calls the WriteBiplot module and then calls PROC SGPLOT to create a biplot. It is the "raw SAS/IML" version of the %BIPLOT macro.

You can use the CalcBiplot module to compute the scores and vectors and return them in IML matrices. You can use the WriteBiplot module if you want that information in SAS data sets so that you can create your own custom biplot. You can use the Biplot module to create standard biplots. The Biplot and WriteBiplot modules are demonstrated in the next sections.

Use the Biplot module in SAS/IML

The syntax of the Biplot module is similar to the %BIPLOT macro for most arguments. The input arguments are as follows:

  • X: The numerical data matrix
  • ID: A character vector of values used to label rows of X. If you pass in an empty matrix, observation numbers are used to label the markers. This argument is ignored if labelPoints=0.
  • varNames: A character vector that contains the names of the columns of X.
  • FacType: The type of biplot: 'GH', 'COV', 'JK', or 'SYM'.
  • StdMethod: How the original variables are scaled: 'None', 'Mean', or 'Std'.
  • Scale: A numerical scalar that specifies additional scaling applied to vectors. By default, SCALE=1, which means the vectors are not scaled. To shrink the vectors, specify a value less than 1. To lengthen the vectors, specify a value greater than 1. (Note: The %BIPLOT macro uses SCALE=0 as its default.)
  • labelPoints: A binary 0/1 value. If 0 (the default) points are not labeled. If 1, points are labeled by the ID values. (Note: The %BIPLOT macro always labels points.)

The last two arguments are optional. You can specify them as keyword-value pairs outside of the parentheses. The following examples show how you can call the Biplot module in a SAS/IML program to create a biplot:

ods graphics / width=480px height=480px;
proc iml;
/* assumes the modules have been previously stored */
load module=(CalcBiplot WriteBiplot Biplot);
use sashelp.iris;
read all var _NUM_ into X[rowname=Species colname=varNames];
title "COV Biplot with Scaled Vectors and Labels";
run Biplot(X, Species, varNames, "COV", "Std") labelPoints=1;   /* label obs */
title "JK Biplot: Relationships between Observations";
run Biplot(X, NULL, varNames, "JK", "Std");
title "JK Biplot: Automatic Scaling of Vectors";
run Biplot(X, NULL, varNames, "JK", "Std") scale=0;            /* auto scale; empty ID var */
title "SYM Biplot: Vectors Scaled by 0.25";
run Biplot(X, NULL, varNames, "SYM", "Std") scale=0.25;        /* scale vectors by 0.25 */
SYM biplot with auto-scaled vectors. Biplot created by using ODS statistical graphics

The program creates four biplots, but only the last one is shown. The last plot uses the SCALE=0.25 option to rescale the vectors of the SYM biplot. You can compare this biplot to the SYM biplot in the previous section, which did not rescale the length of the vectors.

Use the WriteBiplot module in SAS/IML

If you prefer to write an output data set and then create the biplot yourself, use the WriteBiplot module. After loading the modules and the data (see the previous section), you can write the biplot coordinates to the _Scores and _Vectors data sets, as follows. A simple DATA step appends the two data sets into a form that is easy to graph:

run WriteBiplot(X, NULL, varNames, "JK", "Std") scale=0;   /* auto scale vectors */
data Biplot;
   set _Scores _Vectors;    /* append the two data sets created by the WriteBiplot module */
title "JK Biplot: Automatic Scaling of Vectors";
title2 "FacType=JK; Std=Std";
proc sgplot data=Biplot aspect=1 noautolegend;
   refline 0 / axis=x; refline 0 / axis=y;
   scatter x=Prin1 y=Prin2 / ; 
   vector  x=vx    y=vy    / datalabel=_Variable_
                             lineattrs=GraphData2 datalabelattrs=GraphData2;
   xaxis grid offsetmin=0.1 offsetmax=0.1 min=&minAxis max=&maxAxis;
   yaxis grid min=&minAxis max=&maxAxis;
JK biplot, created in SAS by using ODS statistical graphics

In the program that accompanies this article, there is an additional example in which the biplot data is merged with the original data so that you can color-code the observations by using the Species variable.


This article shows four ways to use modern ODS statistical graphics to create a biplot in SAS. You can create a COV biplot by using the PRINQUAL procedure. If you have a license for SAS/IML and SAS/GRAPH, you can use Friendly's %BIPLOT macro to write the biplot coordinates to a SAS data set, then use PROC SGPLOT to create the biplot. This article also presents SAS/IML modules that compute the same biplots as the %BIPLOT macro. The WriteBiplot module writes the data to two SAS data sets (_Score and _Vector), which can be appended and used to plot a biplot. This gives you complete control over the attributes of the biplot. Or, if you prefer, you can use the Biplot module in SAS/IML to automatically create biplots that are similar to Friendly's but are displayed by using ODS statistical graphics.

You can download the complete SAS program that is used in this article. For convenience, I have also created a separate file that defines the SAS/IML modules that create biplots.

The post Create biplots in SAS appeared first on The DO Loop.

11月 112019

In grade school, students learn how to round numbers to the nearest integer. In later years, students learn variations, such as rounding up and rounding down by using the greatest integer function and least integer function, respectively. My sister, who is an engineer, learned a rounding method that rounds half-integers to the nearest even number. This method is called the round-to-even method. (Other names include the round-half-to-even method, the round-ties-to-even method, and "bankers' rounding.") When people first encounter the round-to-even method, they are often confused. Why would anyone use such a strange rounding scheme? This article describes the round-to-even method, explains why it is useful, and shows how to use SAS software to apply the round-to-even method.

What is the round-to-even method of rounding?

The round-to-even method is used in engineering, finance, and computer science to reduce bias when you use rounded numbers to estimate sums and averages. The round-to-even method works like this:

  • If the difference between the number and the nearest integer is less than 0.5, round to the nearest integer. This familiar rule is used by many rounding methods.
  • If the difference between the number and the nearest integer is exactly 0.5, look at the integer part of the number. If the integer part is EVEN, round towards zero. If the integer part of the number is ODD, round away from zero. In either case, the rounded number is an even integer.

All rounding functions are discontinuous step functions that map the real numbers onto the integers. The graph of the round-to-even function is shown to the right.

I intentionally use the phrase "round away from zero" instead of "round up" because you can apply the rounding to positive or negative numbers. If you round the number -2.5 away from zero, you get -3. If you round the number -2.5 towards zero, you get -2. However, for simplicity, the remainder of the article uses positive numbers.

Examples of the round-to-even method of rounding

For the number 2.5, which integer is closest to it? Well, it's a tie: 2.5 is a "half-integer" that is just as close to 2 as it is to 3. So which integer should you choose as THE rounded value? The traditional rounding method rounds the midpoint between two integers away from zero, so 2.5 is traditionally rounded to 3. This produces a systematic bias: all half-integers are rounded away from zero. This fact leads to biased estimates when you use the rounded data in an analysis.

To reduce this systematic bias, you can use the round-to-even method, which rounds some half-integers away from zero and others towards zero. For the round-to-even method, the number 2.5 rounds down to 2, whereas the number 3.5 rounds up to 4.

The table at the right shows some decimal values and the results of rounding the values under the standard method and the round-to-even method. The second column (Round(x)) shows the result of the traditional rounding method where all half-integers are rounded away from 0. The third column (RoundE(x)) is the round-to-even method that rounds half-integers to the nearest even integer. The red boxes indicate numbers for which the Round and RoundE functions produce different answers. Notice that for the round-to-even method, 50% of the half-integers round towards 0 and 50% round away from 0. In the table, 0.5 and 2.5 are rounded down by the round-to-even method, whereas 1.5 and 3.5 are rounded up.

Why use the round-to-even method of rounding?

The main reason to use the round-to-even method is to avoid systematic bias when calculating with rounded numbers. One application involves mental arithmetic. If you want to estimate the sum (or mean) of a list of numbers, you can mentally round the numbers to the nearest integer and add up the numbers in your head. The round-to-even method helps to avoid bias in the estimate, especially if many of the values are half-integers.

Most computers use the round-to-even method for numerical computations. The round-to-even method has been a part of the IEEE standards for rounding since 1985.

How to use the round-to-even method in SAS?

SAS software supports the ROUND function for standard rounding of numbers and the ROUNDE function ('E' for 'even') for round-to-even rounding. For example, the following DATA step produces the table that is shown earlier in this article:

data Seq;
keep x Round RoundEven;
label Round = "Round(x)" RoundEven="RoundE(x)";
do x = 0 to 3.5 by 0.25;
   Round     = round(x);    /* traditional: half-integers rounded away from 0 */
   RoundEven = rounde(x);   /* round half-integers to the nearest even integer */
proc print data=Seq noobs label;

An application: Estimate the average length of lengths with SAS

Although the previous sections discuss rounding values like 0.5 to the nearest integer, the same ideas apply when you round to the nearest tenth, hundredth, thousandth, etc. The next example rounds values to the nearest tenth. Values like 0.95, 1.05, 1.15, etc., are equidistant from the nearest tenth and can be rounded up or down, depending on the rounding method you choose. In SAS, you can use an optional argument to the ROUND and ROUNDE functions to specify the unit to which you want to round. For example, the expression ROUND(x, 0.1) rounds x to the nearest tenth.

An example in the SAS documentation for PROC UNIVARIATE contains the effective channel length (in microns) for 425 transistors from "Lot 1" of a production facility. In the data set, the lengths are recorded to two decimal places. What would be the impact on statistical measurements if the engineer had been lazy and decided to round the measurements to one decimal place, rather than typing all those extra digits?

The following SAS DATA step rounds the data to one decimal place (0.1 microns) by using the ROUND and ROUNDE functions. The call to PROC MEANS computes the mean and sum of the unrounded and rounded values. For the full-precision data, the estimate of the mean length is 1.011 microns. If you round the data by using the standard rounding method, the estimate shoots up to 1.018, which overestimates the average. In contrast, if you round the data by using the round-to-even method, the estimate is 1.014, which is closer to the average of the unrounded numbers (less biased). Similarly, the Sum column shows that the sum of the round-to-even values is much closer to the sum of the unrounded values.

/* round real data to the nearest 0.1 unit */
data rounding;
set Channel1;
   Round     = round(Length, 0.1);    /* traditional: round to nearest tenth */
   RoundEven = rounde(Length, 0.1);   /* use round-to-even method to round to nearest tenth */
   /* create a binary indicator variable: Was x rounded up or down? */
   RoundUp     = (Round > Length);    /* 1 if rounded up; 0 if rounded down */
   RoundEvenUp = (RoundEven > Length);
proc means data=rounding sum mean ndec=3;
   label Length="True values" Round ="Rounded values" RoundEven="Round-to-even values";
   var Length Round RoundEven;

As mentioned earlier, when you use the traditional rounding method, you introduce a bias every time you encounter a "half-unit" datum such as 0.95, 1.05, or 1.15. For this real data, you can count how many data were rounded up versus rounded down by each method. To get an unbiased result, you should round up the half-unit data about as often as you round down. The following call to PROC MEANS shows the proportion of data that are rounded up and rounded down by each method. The output shows that about 55% of the data are rounded up by the traditional rounding method, whereas a more equitable 50.1% of the values are rounded up by the round-to-even method.

proc means data=rounding mean ndec=3;
   label RoundUp    = "Proportion rounded up for ROUND"
         RoundEvenUp= "Proportion rounded up for ROUNDE";
   var RoundUp RoundEvenUp;

This example illustrates a general truism: The round-to-even method is a less biased way to round data.


This article explains the round-to-even method of rounding. This method is not universally taught, but it is taught to college students in certain disciplines. The method rounds most numbers to the nearest integer. However, half-integers, which are equally close to two integers, are rounded to the nearest EVEN integer, thus giving the method its name. This method reduces the bias when you use rounded values to estimate quantities such as sums, means, standard deviations, and so forth.

Whereas SAS provides separate ROUND and ROUNDE functions, other languages might default to the round-to-even method. For example, the round() function in python and the round function in R both implement the round-to-even method. Because some users are unfamiliar with this method of rounding, the R documentation provides an example and then explicitly states, "this is *good* behaviour -- do *NOT* report it as bug!"

You can download the SAS programs and data that are used in this article.

The post Round to even appeared first on The DO Loop.

11月 062019
COV biplot of Fisher's iris data. Commputed in SAS.

Principal component analysis (PCA) is an important tool for understanding relationships in multivariate data. When the first two principal components (PCs) explain a significant portion of the variance in the data, you can visualize the data by projecting the observations onto the span of the first two PCs. In a PCA, this plot is known as a score plot. You can also project the variable vectors onto the span of the PCs, which is known as a loadings plot. See the article "How to interpret graphs in a principal component analysis" for a discussion of the score plot and the loadings plot.

A biplot overlays a score plot and a loadings plot in a single graph. An example is shown at the right. Points are the projected observations; vectors are the projected variables. If the data are well-approximated by the first two principal components, a biplot enables you to visualize high-dimensional data by using a two-dimensional graph.

In general, the score plot and the loadings plot will have different scales. Consequently, you need to rescale the vectors or observations (or both) when you overlay the score and loadings plots. There are four common choices of scaling. Each scaling emphasizes certain geometric relationships between pairs of observations (such as distances), between pairs of variables (such as angles), or between observations and variables. This article discusses the geometry behind two-dimensional biplots and shows how biplots enable you to understand relationships in multivariate data.

Some material in this blog post is based on documentation that I wrote in 2004 when I was working on the SAS/IML Studio product and writing the SAS/IML Studio User's Guide. The documentation is available online and includes references to the literature.

The Fisher iris data

A previous article shows the score plot and loadings plot for a PCA of Fisher's iris data. For these data, the first two principal components explain 96% of the variance in the four-dimensional data. Therefore, these data are well-approximated by a two-dimensional set of principal components. For convenience, the score plot (scatter plot) and the loadings plot (vector plot) are shown below for the iris data. Notice that the loadings plot has a much smaller scale than the score plot. If you overlay these plots, the vectors would appear relatively small unless you rescale one or both plots.

Score plot for Fisher's iris data; first two principal components. Loadings plot for Fisher's iris data; first two principal components.

The mathematics of the biplot

You can perform a PCA by using a singular value decomposition of a data matrix that has N rows (observations) and p columns (variables). The first step in constructing a biplot is to center and (optionally) scale the data matrix. When variables are measured in different units and have different scales, it is usually helpful to standardize the data so that each column has zero mean and unit variance. The examples in this article use standardized data.

The heart of the biplot is the singular value decomposition (SVD). If X is the centered and scaled data matrix, then the SVD of X is
X = U L V`
where U is an N x N orthogonal matrix, L is a diagonal N x p matrix, and V is an orthogonal p x p matrix. It turns out that the principal components (PCs) of X`X are the columns of V and the PC scores are the columns of U. If the first two principal components explain most of the variance, you can choose to keep only the first two columns of U and V and the first 2 x 2 submatrix of L. This is the closest rank-two approximation to X. In a slight abuse of notation,
X ≈ U L V`
where now U, L, and V all have only two columns.

Since L is a diagonal matrix, you can write L = Lc L1-c for any number c in the interval [0, 1]. You can then write
X ≈ (U Lc)(L1-c V`)
   = A B

This the factorization that is used to create a biplot. The most common choices for c are 0, 1, and 1/2.

The four types of biplots

The choice of the scaling parameter, c, will linearly scale the observations and vectors separately. In addition, you can write X ≈ (β A) (B / β) for any constant β. Each choice for c corresponds to a type of biplot:

  • When c=0, the vectors are represented faithfully. This corresponds to the GH biplot. If you also choose β = sqrt(N-1), you get the COV biplot.
  • When c=1, the observations are represented faithfully. This corresponds to the JK biplot.
  • When c=1/2, the observations and vectors are treated symmetrically. This corresponds to the SYM biplot.

The GH biplot for variables

GHbiplot of Fisher's iris data. Commputed in SAS.

If you choose c = 0, then A = U and B = L V`. The literature calls this biplot the GH biplot. I call it the "variable preserving" biplot because it provides the most faithful two-dimensional representation of the relationship between vectors. In particular:

  • The length of each vector (a row of B) is proportional to the variance of the corresponding variable.
  • The Euclidean distance between the i_th and j_th rows of A is proportional to the Mahalanobis distance between the i_th and j_th observations in the data.

In preserving the lengths of the vectors, this biplot distorts the Euclidean distance between points. However, the distortion is not arbitrary: it represents the Mahalanobis distance between points.

The GH biplot is shown to the right, but it is not very useful for these data. In choosing to preserve the variable relationships, the observations are projected onto a tiny region near the origin. The next section discusses an alternative scaling that is more useful for the iris data.

The COV biplot

If you choose c = 0 and β = sqrt(N-1), then A = sqrt(N-1) U and B = L V` / sqrt(N-1). The literature calls this biplot the COV biplot. This biplot is shown at the top of this article. It has two useful properties:

  • The length of each vector is equal to the variance of the corresponding variable.
  • The Euclidean distance between the i_th and j_th rows of A is equal to the Mahalanobis distance between the i_th and j_th observations in the data.

In my opinion, the COV biplot is usually superior to the GH biplot.

The JK biplot

JK biplot of Fisher's iris data. Commputed in SAS.

If you choose c = 1, you get the JK biplot, which preserves the Euclidean distance between observations. Specifically, the Euclidean distance between the i_th and j_th rows of A is equal to the Euclidean distance between the i_th and j_th observations in the data.

In faithfully representing the observations, the angles between vectors are distorted by the scaling.

The SYM biplot

If you choose c = 1/2, you get the SYM biplot (also called the SQ biplot), which attempts to treat observations and variables in a symmetric manner. Although neither the observations nor the vectors are faithfully represented, often neither representation is very distorted. Consequently, some people prefer the SYM biplot as a compromise between the COV and JK biplots. The SYM biplot is shown in the next section.

How to interpret a biplot

SYM biplot of Fisher's iris data. Commputed in SAS.

As discussed in the SAS/IML Studio User's Guide, you can interpret a biplot in the following ways:

  • The cosine of the angle between a vector and an axis indicates the importance of the contribution of the corresponding variable to the principal component.
  • The cosine of the angle between pairs of vectors indicates correlation between the corresponding variables. Highly correlated variables point in similar directions; uncorrelated variables are nearly perpendicular to each other.
  • Points that are close to each other in the biplot represent observations with similar values.
  • You can approximate the relative coordinates of an observation by projecting the point onto the variable vectors within the biplot. However, you cannot use these biplots to estimate the exact coordinates because the vectors have been centered and scaled. You could extend the vectors to become lines and add tick marks, but that becomes messy if you have more than a few variables.

If you want to faithfully interpret the angles between vectors, you should equate the horizontal and vertical axes of the biplot, as I have done with the plots on this page.

If you apply these facts to the standardized iris data, you can make the following interpretations:

  • The PetalLength and PetalWidth variables are the most important contributors to the first PC. The SepalWidth variable is the most important contributor to the second PC.
  • The PetalLength and PetalWidth variables are highly correlated. The SepalWidth variable is almost uncorrelated with the other variables.
  • Although I have suppressed labels on the points, you could label the points by an ID variable or by the observation number and use the relative locations to determine which flowers had measurements that were most similar to each other.


This article presents an overview of biplots. A biplot is an overlay of a score plot and a loadings plot, which are two common plots in a principal component analysis. These two plots are on different scales, but you can rescale the two plots and overlay them on a single plot. Depending upon the choice of scaling, the biplot can provide faithful information about the relationship between variables (lengths and angles) or between observations (distances). It can also provide approximates relationships between variables and observations.

In a subsequent post, I will show how to use SAS to create the biplots in this article.

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