I would also like to make some comments on Rick's blog you have mentioned. Rick mentioned that he didn't know where the misconception that the variables (including the independent variables and the dependent variables) has to be normally distributed came from. Neither do I. Still, the following two statements are essentially the same: (1) the residual follows a normal distribution, (2) the dependent variable follows a normal distribution. As I had stated earlier, the independent variables do not have to be random variables, as you can assign the values of the independent variables, which means that the values of the independent variables are not always results of random experiments. Therefore, the independent variables can be deemed as constants. Now that the residual follows a normal distribution, the dependent variable follows a normal distribution as well. So, in theory, when expressed as a binary outcome (follow or not follow), the normality testing results of the dependent variable and the residual should be identical. However, it is plausible that in practice, the results may be different, as errors may have an impact. I use the word "plausible" because I never test the dependent variable in my data analysis, so I do not know that if such a circumstance could really happen. Whatever my speculation is correct, one thing is certain: the residual follows a normal distribution with a mean of 0 is the hypothesis of multivariate linear regression, whereas the dependent variable follows a normal distribution is a theorem. It should be noted that statistics is a discipline dealing with data, with variation being the everlasting topic. Therefore, definiteness that something holds true only on few occasions in statistics. This is a major difference between statistics and mathematics. That is why I speculate that in practice, the normality results of the dependent variable and the residual could be different. If that is really the case, normality testing on the residual should be trusted, as the residual, not the dependent variable follows a normal distribution with a mean of 0 is the hypothesis of multivariate linear regression is the hypothesis of multivariate linear regression. Finally, I would like to point out that normality testing results (like Shapiro-Wilk and Kolmogorov-Smirnov tests) may not be reliable in large samples, as these tests tend to produce small P-values. Therefore, judgement of normality should not be solely based on the results of the normality tests. Statistical plots like histogram, P-P plot and Q-Q plot provide valuable information when the sample size is large. The information they provide deserve a larger weight in the judgement of normality when sample size is large. ... View more

Thank you for your information you provided. But I am afraid that you still had not get what I mean. I use t-test or nonparametric tests like Wilcoxon to conduct intergroup comparison of the variable I am interested in, namely variable k. Normality only helps me decide which type of test I should choose, not whether or not I should include the candidate variable as the independent variable in the regression model. ... View more

Hello, I've already tried your code featured on CALL EXECUTE routine. It is absolutely fantastic! It is concise (only a few lines longer than you perform one UNIVARIATE procedure), efficient and easier to understand than SAS macro, at least for people like me, who are green hands on SAS macros. By the way, I tried with my own code once again and used "options mprint;" argument to see what is wrong. SAS log stated in CLASS statement of PROC UNIVARIATE that the variable (e.g. I1, I2) does not exist. So I guess that the values I typed in as observations of "VAR" in dataset a were still not transformed into macro variables, despite the presence of CALL SYMPUT. I still don't know why they are not transformed, but never mind, your code works well. Still, please allow me to attach notes to your code, so that other reader of our conversation can understand the code thoroughly. I found out simply reading the part of SAS Help regarding CALL EXECUTE is not enough for understanding that code. What is more, I will add arguments on outputting my results to a PDF file. Here is the entire code of how to achieve parts of the goal I mentioned in my original post. I will explain why I add "parts of" in the preceding sentence later. /*Type in all of the grouping variables in dataset a*/ data a; input var$ @@; cards; aa ab ac ad ae af ag ah ai aj ; run; data b; merge c a;/*c is the dataset in which all of the data are stored*/ run; libname d "";/*Assign a pseudo- (or false) SAS library to store all of the dataset generated by outtable= option in PROC UNIVARIATE*/ libname n "";/*Assign another pseudo-SAS library*/ /*Run PROC UNIVARIATE repetitively using CALL EXECUTE*/ ods pdf file="";/*Type in the entire name of the pdf file between the quotation marks*/ data _null_; set b; /*I think that using b (the combined dataset) instead of c (the original dataset) is necessary, as you can see in the following steps, CALL EXECUTE runs PROC UNIVARIATE repetitively, with each observation of VAR being the grouping variable (variable following the CLASS statement) every time. Therefore, you should provide a list of grouping variables for CALL EXECUTE to read from*/ call execute ('proc univariate data=C outtable=');/*The user can use the original dataset (dataset c) here, as c contained all of the information except the variable list, which is not needed here. In addition, I ran the code successfully with dataset c*/ call execute ('d'||var);/*CALL EXECUTE stores all of the datasets generated from outtable=option into the temporary SAS library (i.e. Work), with a prefix of "d" (e.g. daa, dab); yet the CALL EXECUTE argument itself treats what is does as putting the datasets in a SAS library (e.g. d.aa), which is not the case. That's why I call the SAS library ("d") a "pseudo-SAS library" in advance*/ call execute ('freq normal;'); call execute ('class '||var||';');/*Don't forget to type in the space after 'class'*/ call execute ('var k; ppplot; cdfplot; qqplot/normal(mu=est sigma=est); output out='); call execute ('n'||var); call execute ('normaltest=n probn=p; run;');/*There is a subtle difference here compared to the situation in the outtable= option above. Despite the pseudo-SAS library nature of SAS library "n", CALL EXECUTE does create a SAS dataset entitiled "normaltest" in the file path designated in "libname n "";" statement after the entire process, yet it is an empty one. All of the noramlity test results are still in Work (temporary SAS dataset), with the prefix being "n" (e.g. naa, nbb)*/ run; ods pdf close; So here is my question: I wish to output all of the dataset generated by outtable= option and output out= option into an Excel file, with each dataset being a sheet of the Excel. Any suggestion? Thank you very much! ... View more

Yes, you are correct when it comes to building multivariate linear regression models. It is the residual (or the independent variable, as some books on statistics have stated; but they are essentially the same, since only the independent variable and the residual are random variables, while all of the independent variables do not have to be random variables). But I am working on the process of selecting potential variables prior to model building, which is also called as "univariate analysis" in epidemiology. I need to test for statistical significance of the differences of each and every variable one by one. It is true that whether or not univariate analysis is a must before regression model building (also called as "multivariate analysis") remains controversial. When I was a student, different teachers gave different opinions on that topic: some say that this process is unnecessary, since the variables that have been proven to be statistically significant in univariate analysis may become insignificant in multivariate analysis, and vice versa; while some say that this is a must, without citing any reason. For me, I choose to perform the so-called "univariate analysis" for the following reasons: (1) the teachers who suggested doing that process outnumber those who were against doing that process; (2) this process is essentially a model selection process beforehand. My data contains some 400 variables. I cannot simply put all of the variables into my regression model and let the P-values of the model generated after the model selection process help me pick out the variables I put in the regression model, since I do not have that many sample required to make each and every P-value "believable" (in a statistical language, I cannot guarantee the power of the test of significance of each independent variable; not to mention that collinearity may also have an impact on P-values). Variable selection based totally on professional knowledge is also not pragmatic, since there may be surprises (I have found several statistically significant variables that I deemed as insignificant based on my professional knowledge before I start the entire data analysis process). Therefore, performing univariate analysis by methods including t-tests and ANOVA is necessary for me. So, in one word, I need to conduct t-tests and ANOVA, so testing for normality is necessary. ... View more

OK, thank you for your reminder. I will try CALL EXECUTE and see if there is still anything wrong. @PaigeMiller wrote: options mprint; By the way, where should I put these arguments? Should I place them here? %macro st options mprint; ... View more

In a nutshell, to conduct t-tests or analysis of variance (ANOVA). Variable k is a continuous variable, while all of the grouping variables (e.g. aa, ab) are discrete (categorical) variables. I am building a multivariate linear regression model and is selecting potential independent variables. Therefore, I want to know that if there is a statistically significant difference between (among) the different groups. I need t-tests and ANOVA to help me figure out the problems. But normality of k in each and every group of each and every grouping variable is a prerequisite for conducting t-tests and ANOVA. So I am doing the work I described in my original post. ... View more

Thank you for the tip you offer regarding the usage of _null_ in the data step! ... View more

Thank you for your detailed reply and suggestion! I admit that merging datasets b and c could be unnecessary, and that I did not look up SAS Help to figure out the correct usage of MERGE statement before I posted my question. I did this because as I have stated in my original post, my method failed. I didn't know if that was caused by generating macro variables from a dataset in which only variable names and nothing else was included (i.e. dataset a). So, I merged the datasets to exclude the possibility of that circumstance. The second reason why I did this was that: by merging the two datasets, I get a dataset (b) with the variable names that I wish to use as grouping factors when it comes to analyzing variable k appended in the last column of the combined dataset. In this way, I can use the CALL SYMPUT routine to generate macro variables. I read the part of CALL SYMPUT of SAS Help carefully before I posted my question online and found that this argument is suitable for producing multiple macro variables at one time, with the variables aligned in a column. Now that each and every of the grouping variables themselves were a column in the original dataset (dataset c), I typed the variable names on my own and appended them to the original dataset so that I can make sure that CALL SYMPUT works successfully (please refer to the example of generating macro variables POS1, POS2 and POS3 in CALL SYMPUT routine, if you need). Currently, I have one question regarding the code you composed: the dataset name of your code in the first sentence is "_null_", I previously learnt from my teacher that this name is not suitable for being a dataset name in SAS, for it is specially designated as being a dataset with nothing inside. Therefore, is there any special consideration regarding the nomenclature of the dataset name in your first sentence of the code? I will take a closer look at CALL EXECUTE, since I haven't heard about that argument before. Many thanks for your explicit explanation! ... View more

Since you have proposed the "matrix version of the indicators of collinearity", I think it necessary for you to delve a bit deeper into the essence of tolerance and variance inflation factor (VIF). The essence I wish to talk about is: How are the two statistics computed? I think you will find it unnecessary to generate a "matrix version of the indicators of collinearity" once you have mastered the way that the two statistics are computed (at least I think so). Before we talk about the two statistics, let's talk about the definition of collinearity first. In short, for a given set of variables, namely , if a series of constants, is existent in making this equation true: , then we say that collinearity exists among . Let's talk about tolerance now. How is it computed? In short, if we want to calculate the tolerance of   (p=1, 2, 3, ..., n), then we use   as the dependent variable and all of the other variables (i.e.  ) as the independent variable to form a multivariate linear regression. We can calculate a regression coefficient for this linear regression. Tolerance is the result of 1 subtracts that regression coefficient. As for variance inflation index, understanding that statistic would be easy once you've come to know the calculation formula of tolerance. VIF is the inverse of tolerance (i.e. ). So now you can see, for each independent variable, we can generate a pair of statistics that measure its collinearity. Since the tolerance as well as VIF of each and every independent variable take both the information of that variable and other independent variables in the model into account simultaneously, there is no need for you to compute a matrix that specifically tells you the collinearity between one independent variable and another one. In addition to tolerance and VIF, conditional index is also another indicator of collinearity. The computation of condition index is more complex than tolerance and VIF, since eigenvectors are key players in the computation of conditional index. You need mathematical knowledge on matrix algebra before you can understand the computation of condition index. From a practical perspective, a condition index larger than 10 suggests collinearity, while a condition index larger than 30 suggests severe collinearity. You can let SAS compute condition index by using the method demonstrated by @StatDave. In practice, SAS generates a table (two tables if you specify the collinoint option) and I usually look from the bottom of the table(s) to the top, since the largest condition index is at the bottom of the table(s). If the largest condition index is smaller than 10, then the result suggest that no collinearity exist. Collinearity is an aspect of statistical diagnostics (or regression diagnostics). You can refer to a book on multivariate statistics or a monograph on statistical diagnostics if you want more information. But please be aware of the fact that many statistical methods (like the computation of condition index) are based on matrix algebra, so it may be necessary in some circumstances that you master (at least know something about) matrix algebra before you study statistics. ... View more

Thank you very much for your information and step-by-step guide! I have tried your method on my data. It worked very well in the diagnostics of collinearity. But there is another question that follows: should we standardized all of the independent variables (including the interaction terms) or only the original independent variables (leaving the interaction terms unstandardized)? I guess that standardizing both the original independent variables and their interaction terms included in the model is the correct answer, but I am not sure about it, so I come to seek for your help. Many thanks! ... View more

OK, thank you very, very much for your precious information! I will investigate PROC GLMMOD in details. ... View more

Thank you for your help again!😀 @PaigeMiller wrote: No different than if you have no interaction terms. The interaction term is simply another variable in the model. But there are several details I would like to consult on. I found out that several interaction terms were statistically (and of course professionally) significant in my logistic regression model. The interaction terms were both statistically significant when I used the unstandardized or the standardized variables as independent variables in the model. I tried with the Hessian weight building process in PROC GENMOD demonstrated in the note you provided and found out that the Hessian weights generated when I included the interaction terms were different from the Hessian weights generated when all of the interaction terms were not included. However, both results showed that collinearities exists among several independent variables and the intercept, while no collinearity was observed among the independent variables themselves. In addition, I tried to use PROC REG to generate tolerance, VIF and condition index and found out that PROC REG does not support adding interaction terms, no matter in the format of "a|b" or "a b a*b". Therefore, despite the fact that the Hessian weights were computed with the interaction terms in the model, there was no way to take them into account when it comes to the computation of collinearity indicators (i.e. tolerance, VIF and condition index). To change my lengthy words into a simpler question, please help me pick the correct code from the following two codes I compose. Suppose (1) we have a, b and c as the independent variables and y as the independent variable; (2) among the variables, a*c is statistically significant; (3) collinearity exists only between one or more independent variable(s) and the intercept. Here are the following two codes of dealing with collinearity.   /*Code 1*/ ods select none; proc genmod data=log; model y(event='1')=a|c b/dist=binomial corrb itprint scoring=50;/*Computation of Hessian weight, with interaction term included*/ output out=col hesswgt=w; run; ods select collindiag collindiagnoint ParameterEstimates; proc reg data=col; weight w; model y=a b c/collin collinoint vif tol;/*PROC REG does not support putting interaction terms in the model*/ run; quit; ods select collindiag collindiagnoint ParameterEstimates; proc reg data=col; model gh=a b c/collin collinoint vif tol; run; quit; /*Correction of collinearity (assuming that only collinearities among independent variables and the intercept exist)*/ proc standard data=col s=0.5 out=std; var a b c;/*PROC STANDARD does not support putting interaction terms in the var argument*/ run; /*Check if collinearity were successfully eliminated via the variable standardization process*/ ods select ParameterEstimates ; proc genmod data=std; model y(event='1')=a|c b/dist=binomial corrb itprint scoring=50; output out=col1 hesswgt=w; run; ods select collindiag collindiagnoint ParameterEstimates; proc reg data=col1; weight w; model gh=a b c/collin collinoint vif tol; run; quit; ods select collindiag collindiagnoint ParameterEstimates; proc reg data=col1; model gh=a b c/collin collinoint vif tol; run; quit; /*Logistic regression model building*/ proc logistic data=std; model y(event='1')=a|c b/parmlabel lackfit aggregate scale=pearson covb pcorr corrb ctable stb rsq; roc; run; /*Code 2*/ ods select none; proc genmod data=log; model y(event='1')=a c b/dist=binomial corrb itprint scoring=50;/*Computation of Hessian weight, without including interaction term, resulting a result different from the corresponding process in Code 1*/ output out=col hesswgt=w; run; ods select collindiag collindiagnoint ParameterEstimates; proc reg data=col; weight w; model y=a b c/collin collinoint vif tol;/*PROC REG does not support putting interaction terms in the model*/ run; quit; ods select collindiag collindiagnoint ParameterEstimates; proc reg data=col; model gh=a b c/collin collinoint vif tol; run; quit; /*Correction of collinearity (assuming that only collinearities among independent variables and the intercept exist)*/ proc standard data=col s=0.5 out=std; var a b c;/*PROC STANDARD does not support putting interaction terms in the var argument*/ run; /*Check if collinearity were successfully eliminated via the variable standardization process*/ ods select ParameterEstimates ; proc genmod data=std; model y(event='1')=a c b/dist=binomial corrb itprint scoring=50;/*Whether or not including the interaction term here may have no impact on the Hessian weight, as results generated from my own data show*/ output out=col1 hesswgt=w; run; ods select collindiag collindiagnoint ParameterEstimates; proc reg data=col1; weight w; model gh=a b c/collin collinoint vif tol; run; quit; ods select collindiag collindiagnoint ParameterEstimates; proc reg data=col1; model gh=a b c/collin collinoint vif tol; run; quit; /*Logistic regression model building*/ proc logistic data=std; model y(event='1')=a|c b/parmlabel lackfit aggregate scale=pearson covb pcorr corrb ctable stb rsq; roc; run; Which code is the correct one?   Many thanks! ... 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Thank you, Paige, for giving me your suggestion as well as a detailed explanation once again. Your information has benefited me a lot. But I am afraid that you were not fully aware of the specific question I raised. I talked about correlation among variables when I was translating the definition on collinearity in a reply to @StatDave. The reply did not contain any question at all, but was instead a record of the procedure I solved the problem I raised on my own in the past few days. I wrote the passage to remind other readers of the thread on the details of tackling collinearity in logistic regression models. The question I consulted was written in other reply in which I @ you. In brief, I found some of the interaction terms of my logistic regression model statistically and professionally significant. So I am wondering the exact way of dealing with collinearity when interaction terms are included. Is there anything different from the way mentioned in the note provided by @StatDave? Please refer to the thread in which I @ you for details. Finally, I would like to ask on the sample size requirement of Logistic Partial Least Squares Regression. I have come to known that LASSO requires a larger sample size, yet there was no conclusion on the specific calculation formula of sample size, at least in the case in which a logistic regression model is used as a clinical prediction model. What about Logistic Partial Least Squares Regression? Is there a specific calculation formula? Many thanks! ... View more

But now I am confronted with another problem. I found out that several interaction terms were statistically (and of course professionally) significant in my logistic regression model. The interaction terms were both statistically significant when I used the unstandardized or the standardized variables as independent variables in the model. I tried with the Hessian weight building process in PROC GENMOD demonstrated in the note you provided and found out that the Hessian weights generated when I included the interaction terms were different from the Hessian weights generated when all of the interaction terms were not included. However, both results showed that collinearities exists among several independent variables and the intercept, while no collinearity was observed among the independent variables themselves. In addition, I tried to use PROC REG to generate tolerance, VIF and condition index and found out that PROC REG does not support adding interaction terms, no matter in the format of "a|b" or "a b a*b". Therefore, despite the fact that the Hessian weights were computed with the interaction terms in the model, there was no way to take them into account when it comes to the computation of collinearity indicators (i.e. tolerance, VIF and condition index). So, what is the guide to diagnostics and management of collinearity in logistic regression when interaction terms are included? I also wonder if @PaigeMiller could provide some suggestions on this issue. Thank you for your help! ... View more