## Introduction

This page shows how to perform a number of statistical tests using SAS. Each section gives a brief description of the aim of the statistical test, when it is used, an example showing the SAS commands and SAS output (often excerpted to save space) with a brief interpretation of the output. You can see the page Choosing the Correct Statistical Test for a table that shows an overview of when each test is appropriate to use. In deciding which test is appropriate to use, it is important to consider the type of variables that you have (i.e., whether your variables are categorical, ordinal or interval and whether they are normally distributed), see What is the difference between categorical, ordinal and interval variables? for more information on this.

Please note that the information on this page is intended only as a very brief introduction to each analysis. This page may be a useful guide to suggest which statistical techniques you should further investigate as part of the analysis of your data. This page does not include necessary and important information on many topics, such as the assumptions of the statistical techniques, under what conditions the results may be questionable, etc. Such information may be obtained from a statistics text or journal article. Also, the interpretation of the results given on this page is very minimal and should not be used as a guide for writing about the results. Rather, the intent is to orient you to a few key points. For many analyses, the output has been abbreviated to save space, and potentially important information is not presented here.

## About the hsb data file

Most of the examples in this page will use a data file called

hsb2. This data file contains 200 observations from a sample of high school students with demographic information about the students, such as their gender (female), socio-economic status (ses) and ethnic background (race). It also contains a number of scores on standardized tests, including tests of reading (read), writing (write), mathematics (math) and social studies (socst). You can get thehsb2file as a SAS version 8 data file by clicking here . You can store this file anywhere on your computer, but in the examples we show, we will assume the file is stored in a folder namedc:/mydata/sas/notes/hsb2.sas7bdat. If you store the file in a different location, changec:/mydatato the location where you stored the file on your computer.

## One sample t-test

A one sample t-test allows us to test whether a sample mean (from a normally distributed interval variable) significantly differs from a hypothesized value. For example, using the hsb2 data file, say we wish to test whether the average writing score (

write) differs significantly from 50. We can do this as shown below.proc ttest data = "c:/mydata/hsb2" h0 = 50; var write; run;The TTEST Procedure Statistics Lower CL Upper CL Lower CL Upper CL Variable N Mean Mean Mean Std Dev Std Dev Std Dev Std Err write 200 51.453 52.775 54.097 8.6318 9.4786 10.511 0.6702 T-Tests Variable DF t Value Pr > |t| write 199 4.14 <.0001The mean of the variable

writefor this particular sample of students is 52.775, which is statistically significantly different from the test value of 50. We would conclude that this group of students has a significantly higher mean on the writing test than 50.

## One sample median test

A one sample median test allows us to test whether a sample median differs significantly from a hypothesized value. We will use the same variable,

write, as we did in the one sample t-test example above, but we do not need to assume that it is interval and normally distributed (we only need to assume thatwriteis an ordinal variable). We will test whether the median writing score (write) differs significantly from 50. Theloccountoption on theproc univariatestatement provides the location counts of the data shown at the bottom of the output.proc univariate data = "c:/mydata/hsb2" loccount mu0 = 50; var write; run;Basic Statistical Measures Location Variability Mean 52.77500 Std Deviation 9.47859 Median 54.00000 Variance 89.84359 Mode 59.00000 Range 36.00000 Interquartile Range 14.50000 Tests for Location: Mu0=50 Test -Statistic- -----p Value------ Student's t t 4.140325 Pr > |t| <.0001 Sign M 27 Pr >= |M| 0.0002 Signed Rank S 3326.5 Pr >= |S| <.0001 Location Counts: Mu0=50.00 Count Value Num Obs > Mu0 12 Num Obs ^= Mu0 198 Num Obs < Mu0 72You can use either the sign test or the signed rank test. The difference between these two tests is that the signed rank requires that the variable be from a symmetric distribution. The results indicate that the median of the variable

writefor this group is statistically significantly different from 50.## See also

## Binomial test

A one sample binomial test allows us to test whether the proportion of successes on a two-level categorical dependent variable significantly differs from a hypothesized value. For example, using the hsb2 data file, say we wish to test whether the proportion of females (

female) differs significantly from 50%, i.e., from .5. We will use theexactstatement to produce the exact p-values.proc freq data = "c:/mydata/hsb2"; tables female / binomial(p=.5); exact binomial; run;The FREQ Procedure Cumulative Cumulative female Frequency Percent Frequency Percent ----------------------------------------------------------- 0 91 45.50 91 45.50 1 109 54.50 200 100.00 Binomial Proportion for female = 0 ----------------------------------- Proportion (P) 0.4550 ASE 0.0352 95% Lower Conf Limit 0.3860 95% Upper Conf Limit 0.5240 Exact Conf Limits 95% Lower Conf Limit 0.3846 95% Upper Conf Limit 0.5267 Test of H0: Proportion = 0.5 ASE under H0 0.0354 Z -1.2728 One-sided Pr < Z 0.1015 Two-sided Pr > |Z| 0.2031 Exact Test One-sided Pr <= P 0.1146 Two-sided = 2 * One-sided 0.2292 Sample Size = 200The results indicate that there is no statistically significant difference (p = .2292). In other words, the proportion of females in this sample does not significantly differ from the hypothesized value of 50%.

## See also

## Chi-square goodness of fit

A chi-square goodness of fit test allows us to test whether the observed proportions for a categorical variable differ from hypothesized proportions. For example, let’s suppose that we believe that the general population consists of 10% Hispanic, 10% Asian, 10% African American and 70% White folks. We want to test whether the observed proportions from our sample differ significantly from these hypothesized proportions. The hypothesized proportions are placed in parentheses after the

testp=option on thetablesstatement.proc freq data = "c:/mydata/hsb2"; tables race / chisq testp=(10 10 10 70); run;The FREQ Procedure Test Cumulative Cumulative race Frequency Percent Percent Frequency Percent --------------------------------------------------------------------- 1 24 12.00 10.00 24 12.00 2 11 5.50 10.00 35 17.50 3 20 10.00 10.00 55 27.50 4 145 72.50 70.00 200 100.00 Chi-Square Test for Specified Proportions ------------------------- Chi-Square 5.0286 DF 3 Pr > ChiSq 0.1697 Sample Size = 200These results show that racial composition in our sample does not differ significantly from the hypothesized values that we supplied (chi-square with three degrees of freedom = 5.0286, p = .1697).

## Two independent samples t-test

An independent samples t-test is used when you want to compare the means of a normally distributed interval dependent variable for two independent groups. For example, using the hsb2 data file, say we wish to test whether the mean for

writeis the same for males and females.proc ttest data = "c:/mydata/hsb2"; class female; var write; run;The TTEST Procedure Statistics Lower CL Upper CL Lower CL Upper CL Variable female N Mean Mean Mean Std Dev Std Dev Std Dev Std Err write 0 91 47.975 50.121 52.267 8.9947 10.305 12.066 1.0803 write 1 109 53.447 54.991 56.535 7.1786 8.1337 9.3843 0.7791 write Diff (1-2) -7.442 -4.87 -2.298 8.3622 9.1846 10.188 1.3042 T-Tests Variable Method Variances DF t Value Pr > |t| write Pooled Equal 198 -3.73 0.0002 write Satterthwaite Unequal 170 -3.66 0.0003 Equality of Variances Variable Method Num DF Den DF F Value Pr > F write Folded F 90 108 1.61 0.0187The results indicate that there is a statistically significant difference between the mean writing score for males and females (t = -3.73, p = .0002). In other words, females have a statistically significantly higher mean score on writing (54.991) than males (50.121).

## See also

## Wilcoxon-Mann-Whitney test

The Wilcoxon-Mann-Whitney test is a non-parametric analog to the independent samples t-test and can be used when you do not assume that the dependent variable is a normally distributed interval variable (you need only assume that the variable is at least ordinal). We will use the same data file (the hsb2 data file) and the same variables in this example as we did in the independent t-test example above and will not assume that

write, our dependent variable, is normally distributed.

proc npar1way data = "c:/mydata/hsb2" wilcoxon; class female; var write; run;The NPAR1WAY Procedure Wilcoxon Scores (Rank Sums) for Variable write Classified by Variable female Sum of Expected Std Dev Mean female N Scores Under H0 Under H0 Score ---------------------------------------------------------------------- 0 91 7792.0 9145.50 406.559086 85.626374 1 109 12308.0 10954.50 406.559086 112.917431 Average scores were used for ties. Wilcoxon Two-Sample Test Statistic 7792.0000 Normal Approximation Z -3.3279 One-Sided Pr < Z 0.0004 Two-Sided Pr > |Z| 0.0009 t Approximation One-Sided Pr < Z 0.0005 Two-Sided Pr > |Z| 0.0010 Z includes a continuity correction of 0.5.The results suggest that there is a statistically significant difference between the underlying distributions of the

writescores of males and thewritescores of females (z = -3.329, p = 0.0009).## See also

- FAQ: Why is the Mann-Whitney significant when the medians are equal?
- Exact Methods in the NPAR1WAY Procedure
- Sample Size Computations and Power Analysis with the SAS System

## Chi-square test

A chi-square test is used when you want to see if there is a relationship between two categorical variables. In SAS, the

chisqoption is used on thetablesstatement to obtain the test statistic and its associated p-value. Using the hsb2 data file, let’s see if there is a relationship between the type of school attended (schtyp) and students’ gender (female). Remember that the chi-square test assumes that the expected value for each cell is five or higher. This assumption is easily met in the examples below. However, if this assumption is not met in your data, please see the section on Fisher’s exact test below.proc freq data = "c:/mydata/hsb2"; tables schtyp*female / chisq; run;The FREQ Procedure Table of schtyp by female schtyp(type of school) female Frequency| Percent | Row Pct | Col Pct | 0| 1| Total ---------+--------+--------+ 1 | 77 | 91 | 168 | 38.50 | 45.50 | 84.00 | 45.83 | 54.17 | | 84.62 | 83.49 | ---------+--------+--------+ 2 | 14 | 18 | 32 | 7.00 | 9.00 | 16.00 | 43.75 | 56.25 | | 15.38 | 16.51 | ---------+--------+--------+ Total 91 109 200 45.50 54.50 100.00 Statistics for Table of schtyp by female Statistic DF Value Prob ------------------------------------------------------ Chi-Square 1 0.0470 0.8283 Likelihood Ratio Chi-Square 1 0.0471 0.8281 Continuity Adj. Chi-Square 1 0.0005 0.9815 Mantel-Haenszel Chi-Square 1 0.0468 0.8287 Phi Coefficient 0.0153 Contingency Coefficient 0.0153 Cramer's V 0.0153 Sample Size = 200These results indicate that there is no statistically significant relationship between the type of school attended and gender (chi-square with one degree of freedom = 0.0470, p = 0.8283).

Let’s look at another example, this time looking at the relationship between gender (

female) and socio-economic status (ses). The point of this example is that one (or both) variables may have more than two levels, and that the variables do not have to have the same number of levels. In this example,femalehas two levels (male and female) andseshas three levels (low, medium and high).proc freq data = "c:/mydata/hsb2"; tables female*ses / chisq; run;The FREQ Procedure Table of female by ses female ses Frequency| Percent | Row Pct | Col Pct | 1| 2| 3| Total ---------+--------+--------+--------+ 0 | 15 | 47 | 29 | 91 | 7.50 | 23.50 | 14.50 | 45.50 | 16.48 | 51.65 | 31.87 | | 31.91 | 49.47 | 50.00 | ---------+--------+--------+--------+ 1 | 32 | 48 | 29 | 109 | 16.00 | 24.00 | 14.50 | 54.50 | 29.36 | 44.04 | 26.61 | | 68.09 | 50.53 | 50.00 | ---------+--------+--------+--------+ Total 47 95 58 200 23.50 47.50 29.00 100.00 Statistics for Table of female by ses Statistic DF Value Prob ------------------------------------------------------ Chi-Square 2 4.5765 0.1014 Likelihood Ratio Chi-Square 2 4.6789 0.0964 Mantel-Haenszel Chi-Square 1 3.1098 0.0778 Phi Coefficient 0.1513 Contingency Coefficient 0.1496 Cramer's V 0.1513 Sample Size = 200Again we find that there is no statistically significant relationship between the variables (chi-square with two degrees of freedom = 4.5765, p = 0.1014).

## See also

## Fisher’s exact test

The Fisher’s exact test is used when you want to conduct a chi-square test, but one or more of your cells has an expected frequency of less than five. Remember that the chi-square test assumes that each cell has an expected frequency of five or more, but the Fisher’s exact test has no such assumption and can be used regardless of how small the expected frequency is. In the example below, we have cells with observed frequencies of two and one, which may indicate expected frequencies that could be below five, so we will use Fisher’s exact test with the

fisheroption on thetablesstatement.proc freq data = "c:/mydata/hsb2"; tables schtyp*race / fisher; run;The FREQ Procedure Table of schtyp by race schtyp(type of school) race Frequency| Percent | Row Pct | Col Pct | 1| 2| 3| 4| Total ---------+--------+--------+--------+--------+ 1 | 22 | 10 | 18 | 118 | 168 | 11.00 | 5.00 | 9.00 | 59.00 | 84.00 | 13.10 | 5.95 | 10.71 | 70.24 | | 91.67 | 90.91 | 90.00 | 81.38 | ---------+--------+--------+--------+--------+ 2 | 2 | 1 | 2 | 27 | 32 | 1.00 | 0.50 | 1.00 | 13.50 | 16.00 | 6.25 | 3.13 | 6.25 | 84.38 | | 8.33 | 9.09 | 10.00 | 18.62 | ---------+--------+--------+--------+--------+ Total 24 11 20 145 200 12.00 5.50 10.00 72.50 100.00 Statistics for Table of schtyp by race Statistic DF Value Prob ------------------------------------------------------ Chi-Square 3 2.7170 0.4373 Likelihood Ratio Chi-Square 3 2.9985 0.3919 Mantel-Haenszel Chi-Square 1 2.3378 0.1263 Phi Coefficient 0.1166 Contingency Coefficient 0.1158 Cramer's V 0.1166 WARNING: 38% of the cells have expected counts less than 5. Chi-Square may not be a valid test. Fisher's Exact Test ---------------------------------- Table Probability (P) 0.0077 Pr <= P 0.5975 Sample Size = 200These results suggest that there is not a statistically significant relationship between race and type of school (p = 0.5975). Note that the Fisher’s exact test does not have a “test statistic”, but computes the p-value directly.

## See also

## One-way ANOVA

A one-way analysis of variance (ANOVA) is used when you have a categorical independent variable (with two or more categories) and a normally distributed interval dependent variable and you wish to test for differences in the means of the dependent variable broken down by the levels of the independent variable. For example, using the hsb2 data file, say we wish to test whether the mean of

writediffers between the three program types (prog). We will also use the means statement to output the mean of write for each level of program type. Note that this will not tell you if there is a statistically significant difference between any two sets of means.proc glm data = "c:/mydata/hsb2"; class prog; model write = prog;means prog; run; quit;The GLM Procedure Class Level Information Class Levels Values prog 3 1 2 3 Number of observations 200 Dependent Variable: write writing score Sum of Source DF Squares Mean Square F Value Pr > F Model 2 3175.69786 1587.84893 21.27 <.0001 Error 197 14703.17714 74.63542 Corrected Total 199 17878.87500 R-Square Coeff Var Root MSE write Mean 0.177623 16.36983 8.639179 52.77500 Source DF Type I SS Mean Square F Value Pr > F prog 2 3175.697857 1587.848929 21.27 <.0001 Source DF Type III SS Mean Square F Value Pr > F prog 2 3175.697857 1587.848929 21.27 <.0001 Level of ------------write------------ prog N Mean Std Dev 1 45 51.3333333 9.39777537 2 105 56.2571429 7.94334333 3 50 46.7600000 9.31875441The mean of the dependent variable differs significantly among the levels of program type. However, we do not know if the difference is between only two of the levels or all three of the levels. (The F test for the model is the same as the F test for

progbecauseprogwas the only variable entered into the model. If other variables had also been entered, the F test for theModelwould have been different fromprog.) We can also see that the students in the academic program have the highest mean writing score, while students in the vocational program have the lowest.## See also

## Kruskal Wallis test

The Kruskal Wallis test is used when you have one independent variable with two or more levels and an ordinal dependent variable. In other words, it is the non-parametric version of ANOVA. It is also a generalized form of the Mann-Whitney test method, as it permits two or more groups. We will use the same data file as the one way ANOVA example above (the hsb2 data file) and the same variables as in the example above, but we will not assume that

writeis a normally distributed interval variable.proc npar1way data = "c:/mydata/hsb2"; class prog; var write; run;The NPAR1WAY Procedure Wilcoxon Scores (Rank Sums) for Variable write Classified by Variable prog Sum of Expected Std Dev Mean prog N Scores Under H0 Under H0 Score -------------------------------------------------------------------- 1 45 4079.0 4522.50 340.927342 90.644444 3 50 3257.0 5025.00 353.525185 65.140000 2 105 12764.0 10552.50 407.705133 121.561905 Average scores were used for ties. Kruskal-Wallis Test Chi-Square 34.0452 DF 2 Pr > Chi-Square <.0001The results indicate that there is a statistically significant difference among the three type of programs (chi-square with two degrees of freedom = 34.0452, p = 0.0001).

## See also

## Paired t-test

A paired (samples) t-test is used when you have two related observations (i.e., two observations per subject) and you want to see if the means on these two normally distributed interval variables differ from one another. For example, using the hsb2 data file we will test whether the mean of

readis equal to the mean ofwrite.proc ttest data = "c:/mydata/hsb2"; paired write*read; run;The TTEST Procedure Statistics Lower CL Upper CL Lower CL Upper CL Difference N Mean Mean Mean Std Dev Std Dev Std Dev Std Err write - read 200 -0.694 0.545 1.7841 8.0928 8.8867 9.8546 0.6284 T-Tests Difference DF t Value Pr > |t| write - read 199 0.87 0.3868These results indicate that the mean of

readis not statistically significantly different from the mean ofwrite(t = 0.87, p = 0.3868).## See also

## Wilcoxon signed rank sum test

The Wilcoxon signed rank sum test is the non-parametric version of a paired samples t-test. You use the Wilcoxon signed rank sum test when you do not wish to assume that the difference between the two variables is interval and normally distributed (but you do assume the difference is ordinal). We will use the same example as above, but we will not assume that the difference between

readandwriteis interval and normally distributed. We will first do a data step to create the difference of the two scores for each subject. This is necessary because SAS will not calculate the difference for you inproc univariate.data hsb2a; set 'c:/mydata/hsb2'; diff = read - write; run; proc univariate data = hsb2a; var diff; run;The UNIVARIATE Procedure Variable: diff Basic Statistical Measures Location Variability Mean -0.54500 Std Deviation 8.88667 Median 0.00000 Variance 78.97284 Mode 6.00000 Range 45.00000 Interquartile Range 13.00000 Tests for Location: Mu0=0 Test -Statistic- -----p Value------ Student's t t -0.86731 Pr > |t| 0.3868 Sign M -4.5 Pr >= |M| 0.5565 Signed Rank S -658.5 Pr >= |S| 0.3677The results suggest that there is not a statistically significant difference between

readandwrite.If you believe the differences between

readandwritewere not ordinal but could merely be classified as positive and negative, then you may want to consider a sign test in lieu of sign rank test. Note that the SAS output gives you the results for both the Wilcoxon signed rank test and the sign test without having to use any options. Using the sign test, we again conclude that there is no statistically significant difference betweenreadandwrite(p=.5565).

## McNemar test

You would perform McNemar’s test if you were interested in the marginal frequencies of two binary outcomes. These binary outcomes may be the same outcome variable on matched pairs (like a case-control study) or two outcome variables from a single group. Let us consider two questions, Q1 and Q2, from a test taken by 200 students. Suppose 172 students answered both questions correctly, 15 students answered both questions incorrectly, 7 answered Q1 correctly and Q2 incorrectly, and 6 answered Q2 correctly and Q1 incorrectly. These counts can be considered in a two-way contingency table. The null hypothesis is that the two questions are answered correctly or incorrectly at the same rate (or that the contingency table is symmetric).

data set1; input Q1correct Q2correct students; datalines; 1 1 172 0 1 6 1 0 7 0 0 15 run; proc freq data=set1; table Q1correct*Q2correct; exact mcnem; weight students; run;The FREQ Procedure Table of Q1correct by Q2correct Q1correct Q2correct Frequency| Percent | Row Pct | Col Pct | 0| 1| Total ---------+--------+--------+ 0 | 15 | 6 | 21 | 7.50 | 3.00 | 10.50 | 71.43 | 28.57 | | 68.18 | 3.37 | ---------+--------+--------+ 1 | 7 | 172 | 179 | 3.50 | 86.00 | 89.50 | 3.91 | 96.09 | | 31.82 | 96.63 | ---------+--------+--------+ Total 22 178 200 11.00 89.00 100.00 Statistics for Table of Q1correct by Q2correct McNemar's Test ---------------------------- Statistic (S) 0.0769 DF 1 Asymptotic Pr > S 0.7815 Exact Pr >= S 1.0000 Simple Kappa Coefficient -------------------------------- Kappa 0.6613 ASE 0.0873 95% Lower Conf Limit 0.4901 95% Upper Conf Limit 0.8324 Sample Size = 200McNemar’s test statistic suggests that there is not a statistically significant difference in the proportions of correct/incorrect answers to these two questions.

## See also

## One-way repeated measures ANOVA

You would perform a one-way repeated measures analysis of variance if you had one categorical independent variable and a normally distributed interval dependent variable that was repeated at least twice for each subject. This is the equivalent of the paired samples t-test, but allows for two or more levels of the categorical variable. The one-way repeated measures ANOVA tests whether the mean of the dependent variable differs by the categorical variable. We have an example data set called rb4wide, which is used in Kirk’s book Experimental Design. In this data set,

y1 y2 y3andy4represent the dependent variable measured at the 4 levels ofa, the repeated measures independent variable.proc glm data = 'c:/mydata/rb4wide'; model y1 y2 y3 y4 = ; repeated a ; run; quit;The GLM Procedure Repeated Measures Analysis of Variance Repeated Measures Level Information Dependent Variable Y1 Y2 Y3 Y4 Level of a 1 2 3 4 Manova Test Criteria and Exact F Statistics for the Hypothesis of no a Effect H = Type III SSCP Matrix for a E = Error SSCP Matrix S=1 M=0.5 N=1.5 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.24580793 5.11 3 5 0.0554 Pillai's Trace 0.75419207 5.11 3 5 0.0554 Hotelling-Lawley Trace 3.06821705 5.11 3 5 0.0554 Roy's Greatest Root 3.06821705 5.11 3 5 0.0554 Repeated Measures Analysis of Variance Univariate Tests of Hypotheses for Within Subject Effects Adj Pr > F Source DF Type III SS Mean Square F Value Pr > F G - G H - F a 3 49.00000000 16.33333333 11.63 0.0001 0.0015 0.0003 Error(a) 21 29.50000000 1.40476190 Greenhouse-Geisser Epsilon 0.6195 Huynh-Feldt Epsilon 0.8343The results indicate that the model as well as both factors (

aands) are statistically significant. The p-value given in this output fora(0.0001) is the “regular” p-value and is the p-value that you would get if you assumed compound symmetry in the variance-covariance matrix.## See also

## Repeated measures logistic regression

If you have a binary outcome measured repeatedly for each subject and you wish to run a logistic regression that accounts for the effect of multiple measures from single subjects, you can perform a repeated measures logistic regression. In SAS, this can be done by using the

genmodprocedure and indicating binomial as the probability distribution and logit as the link function to be used in the model. The exercise data file contains three pulse measurements from each of 30 people assigned to two different diet regiments and three different exercise regiments. If we define a “high” pulse as being over 100, we can then predict the probability of a high pulse using diet regiment.proc genmod data='c:/mydata/exercise' descending; class id diet / descending; model highpulse = diet / dist = bin link = logit; repeated subject = id / type = exch; run;Response Profile Ordered Total Value highpulse Frequency 1 1 27 2 0 63 PROC GENMOD is modeling the probability that highpulse='1'. Parameter Information Parameter Effect diet Prm1 Intercept Prm2 diet 2 Prm3 diet 1 Algorithm converged. GEE Model Information Correlation Structure Exchangeable Subject Effect id (30 levels) Number of Clusters 30 The GENMOD Procedure GEE Model Information Correlation Matrix Dimension 3 Maximum Cluster Size 3 Minimum Cluster Size 3 Algorithm converged. Exchangeable Working Correlation Correlation 0.3306722695 GEE Fit Criteria QIC 113.9859 QICu 111.3405 Analysis Of GEE Parameter Estimates Empirical Standard Error Estimates Standard 95% Confidence Parameter Estimate Error Limits Z Pr > |Z| Intercept -1.2528 0.4328 -2.1011 -0.4044 -2.89 0.0038 diet 2 0.7538 0.6031 -0.4283 1.9358 1.25 0.2114 diet 1 0.0000 0.0000 0.0000 0.0000 . .These results indicate that diet is not statistically significant (Z = -1.25, p = 0.2114).

## Factorial ANOVA

A factorial ANOVA has two or more categorical independent variables (either with or without the interactions) and a single normally distributed interval dependent variable. For example, using the hsb2 data file we will look at writing scores (

write) as the dependent variable and gender (female) and socio-economic status (ses) as independent variables, and we will include an interaction offemalebyses. Note that in SAS, you do not need to have the interaction term(s) in your data set. Rather, you can have SAS create it/them temporarily by placing an asterisk between the variables that will make up the interaction term(s).proc glm data = "c:/mydata/hsb2"; class female ses; model write = female ses female*ses; run; quit;The GLM Procedure Dependent Variable: write writing score Sum of Source DF Squares Mean Square F Value Pr > F Model 5 2278.24419 455.64884 5.67 <.0001 Error 194 15600.63081 80.41562 Corrected Total 199 17878.87500 R-Square Coeff Var Root MSE write Mean 0.127427 16.99190 8.967476 52.77500 Source DF Type I SS Mean Square F Value Pr > F female 1 1176.213845 1176.213845 14.63 0.0002 ses 2 1080.599437 540.299718 6.72 0.0015 female*ses 2 21.430904 10.715452 0.13 0.8753 Source DF Type III SS Mean Square F Value Pr > F female 1 1334.493311 1334.493311 16.59 <.0001 ses 2 1063.252697 531.626349 6.61 0.0017 female*ses 2 21.430904 10.715452 0.13 0.8753These results indicate that the overall model is statistically significant (F = 5.67, p = 0.001). The variables

femaleandsesare also statistically significant (F = 16.59, p = 0.0001 and F = 6.61, p = 0.0017, respectively). However, that interaction betweenfemaleandsesis not statistically significant (F = 0.13, p = 0.8753).## See also

- Annotated Output of Proc Glm
- SAS FAQ: How can I do test of simple main effects?
- SAS Textbook Examples: Applied Linear Statistical Models
- SAS Textbook Examples: Fox, Applied Regression Analysis, Chapter 8
- SAS Learning Module: Comparing SAS and Stata Side by Side
- SAS Textbook Examples from Design and Analysis: Chapter 10

## Friedman test

You perform a Friedman test when you have one within-subjects independent variable with two or more levels and a dependent variable that is not interval and normally distributed (but at least ordinal). We will use this test to determine if there is a difference in the reading, writing and math scores. The null hypothesis in this test is that the distribution of the ranks of each type of score (i.e., reading, writing and math) are the same. To conduct a Friedman test, the data need to be in a long format; we will use

proc transposeto change our data from the wide format that they are currently in to a long format. We create a variable to code for the type of score, which we will callrwm(forread,write,math), andcol1that contains the score on the dependent variable, that is the reading, writing or math score. To obtain the Friedman test, you need to use thecmh2option on thetablesstatement inproc freq.proc sort data = "c:/mydata/hsb2" out=hsbsort; by id; run; proc transpose data=hsbsort out=hsblong name=rwm; by id; var read write math; run; proc freq data=hsblong; tables id*rwm*col1 / cmh2 scores=rank noprint; run;The FREQ Procedure Summary Statistics for rwm by COL1 Controlling for id Cochran-Mantel-Haenszel Statistics (Based on Rank Scores) Statistic Alternative Hypothesis DF Value Prob --------------------------------------------------------------- 1 Nonzero Correlation 1 0.0790 0.7787 2 Row Mean Scores Differ 2 0.6449 0.7244 Total Sample Size = 600The Row Mean Scores Differ is the same as the Friedman’s chi-square, and we see that with a value of 0.6449 and a p-value of 0.7244, it is not statistically significant. Hence, there is no evidence that the distributions of the three types of scores are different.

## Ordered logistic regression

Ordered logistic regression is used when the dependent variable is ordered, but not continuous. For example, using the hsb2 data file we will create an ordered variable called

write3. This variable will have the values 1, 2 and 3, indicating a low, medium or high writing score. We do not generally recommend categorizing a continuous variable in this way; we are simply creating a variable to use for this example. We will use gender (female), reading score (read) and social studies score (socst) as predictor variables in this model. Thedescoption on theproc logisticstatement is used so that SAS models the odds of being in the lower category. The Response Profile table in the output shows the value that SAS used when conducting the analysis (given in the Ordered Value column), the value of the original variable, and the number of cases in each level of the outcome variable. (If you want SAS to use the values that you have assigned the outcome variable, then you would want to use theorder = dataoption on theproc logisticstatement.) The note below this table reminds us that the “Probabilities modeled are cumulated over the lower Ordered Values.” It is helpful to remember this when interpreting the output. Theexpboption on the model statement tells SAS to show the exponentiated coefficients (i.e., the proportional odds ratios).data hsb2_ordered; set "c:/mydata/hsb2"; if 30 <= write <=48 then write3 = 1; if 49 <= write <=57 then write3 = 2; if 58 <= write <=70 then write3 = 3; run; proc logistic data = hsb2_ordered desc; model write3 = female read socst / expb; run;The LOGISTIC Procedure Model Information Data Set WORK.HSB2_ORDERED Response Variable write3 Number of Response Levels 3 Model cumulative logit Optimization Technique Fisher's scoring Number of Observations Read 200 Number of Observations Used 200 Response Profile Ordered Total Value write3 Frequency 1 3 78 2 2 61 3 1 61 Probabilities modeled are cumulated over the lower Ordered Values. Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Score Test for the Proportional Odds Assumption Chi-Square DF Pr > ChiSq 2.1211 3 0.5477 Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 440.627 322.553 SC 447.224 339.044 -2 Log L 436.627 312.553 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 124.0745 3 <.0001 Score 93.1890 3 <.0001 Wald 76.6752 3 <.0001 Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Exp(Est) Intercept 3 1 -11.8007 1.3122 80.8702 <.0001 0.000 Intercept 2 1 -9.7042 1.2026 65.1114 <.0001 0.000 FEMALE 1 1.2856 0.3225 15.8901 <.0001 3.617 READ 1 0.1177 0.0215 29.8689 <.0001 1.125 SOCST 1 0.0802 0.0190 17.7817 <.0001 1.083 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits FEMALE 3.617 1.922 6.805 READ 1.125 1.078 1.173 SOCST 1.083 1.044 1.125 Association of Predicted Probabilities and Observed Responses Percent Concordant 83.8 Somers' D 0.681 Percent Discordant 15.7 Gamma 0.685 Percent Tied 0.6 Tau-a 0.453 Pairs 13237 c 0.840The results indicate that the overall model is statistically significant (p < .0001), as are each of the predictor variables (p < .0001). There are two intercepts for this model because there are three levels of the outcome variable. We also see that the test of the proportional odds assumption is non-significant (p = .5477). One of the assumptions underlying ordinal logistic (and ordinal probit) regression is that the relationship between each pair of outcome groups is the same. In other words, ordinal logistic regression assumes that the coefficients that describe the relationship between, say, the lowest versus all higher categories of the response variable are the same as those that describe the relationship between the next lowest category and all higher categories, etc. This is called the proportional odds assumption or the parallel regression assumption. Because the relationship between all pairs of groups is the same, there is only one set of coefficients (only one model). If this was not the case, we would need different models (such as a generalized ordered logit model) to describe the relationship between each pair of outcome groups.

## See also

## Factorial logistic regression

A factorial logistic regression is used when you have two or more categorical independent variables but a dichotomous dependent variable. For example, using the hsb2 data file we will use

femaleas our dependent variable, because it is the only dichotomous variable in our data set; certainly not because it common practice to use gender as an outcome variable. We will use type of program (prog) and school type (schtyp) as our predictor variables. Because neitherprognorschtypare continuous variables, we need to include them on theclassstatement. Thedescoption on theproc logisticstatement is necessary so that SAS models the odds of being female (i.e., female = 1). Theexpboption on the model statement tells SAS to show the exponentiated coefficients (i.e., the odds ratios).proc logistic data = "c:/mydata/hsb2" desc; class prog schtyp; model female = prog schtyp prog*schtyp / expb; run;The LOGISTIC Procedure Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 277.637 284.490 SC 280.935 304.280 -2 Log L 275.637 272.490 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 3.1467 5 0.6774 Score 2.9231 5 0.7118 Wald 2.6036 5 0.7608 Type III Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq prog 2 1.1232 0.5703 schtyp 1 0.4132 0.5203 prog*schtyp 2 2.4740 0.2903 Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Exp(Est) Intercept 1 0.3331 0.3164 1.1082 0.2925 1.395 prog 1 1 0.4459 0.4568 0.9532 0.3289 1.562 prog 2 1 -0.1964 0.3438 0.3264 0.5678 0.822 schtyp 1 1 -0.2034 0.3164 0.4132 0.5203 0.816 prog*schtyp 1 1 1 -0.6269 0.4568 1.8838 0.1699 0.534 prog*schtyp 2 1 1 0.3400 0.3438 0.9783 0.3226 1.405The results indicate that the overall model is not statistically significant (LR chi2 = 3.1467, p = 0.6774). Furthermore, none of the coefficients are statistically significant either. In addition, there is no statistically significant effect of program (p = 0.5703), school type (p = 0.5203) or of the interaction (p = 0.2903).

## Correlation

A correlation is useful when you want to see the linear relationship between two (or more) normally distributed interval variables. For example, using the hsb2 data file we can run a correlation between two continuous variables,

readandwrite.proc corr data = "c:/mydata/hsb2"; var read write; run;The CORR Procedure 2 Variables: read write Pearson Correlation Coefficients, N = 200 Prob > |r| under H0: Rho=0 read write read 1.00000 0.59678 reading score <.0001 write 0.59678 1.00000 writing score <.0001In the second example below, we will run a correlation between a dichotomous variable,

female, and a continuous variable,write. Although it is assumed that the variables are interval and normally distributed, we can include dummy variables when performing correlations.proc corr data = "c:/mydata/hsb2"; var female write; run;The CORR Procedure 2 Variables: female write Pearson Correlation Coefficients, N = 200 Prob > |r| under H0: Rho=0 female write female 1.00000 0.25649 0.0002 write 0.25649 1.00000 writing score 0.0002In the first example above, we see that the correlation between

readandwriteis 0.59678. By squaring the correlation and then multiplying by 100, you can determine what percentage of the variability is shared. Let’s round 0.59678 to be 0.6, which when squared would be .36, multiplied by 100 would be 36%. Hencereadshares about 36% of its variability withwrite. In the output for the second example, we can see the correlation betweenwriteandfemaleis 0.25649. Squaring this number yields .0657871201, meaning thatfemaleshares approximately 6.5% of its variability withwrite.## See also

## Simple linear regression

Simple linear regression allows us to look at the linear relationship between one normally distributed interval predictor and one normally distributed interval outcome variable. For example, using the hsb2 data file, say we wish to look at the relationship between writing scores (

write) and reading scores (read); in other words, predictingwritefromread.proc reg data = "c:/mydata/hsb2"; model write = read / stb; run; quit;The REG Procedure Model: MODEL1 Dependent Variable: write writing score Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 6367.42127 6367.42127 109.52 <.0001 Error 198 11511 58.13866 Corrected Total 199 17879 Root MSE 7.62487 R-Square 0.3561 Dependent Mean 52.77500 Adj R-Sq 0.3529 Coeff Var 14.44788 Parameter Estimates Parameter Standard Standardized Variable Label DF Estimate Error t Value Pr > |t| Estimate Intercept Intercept 1 23.95944 2.80574 8.54 <.0001 0 read reading score 1 0.55171 0.05272 10.47 <.0001 0.59678We see that the relationship between

writeandreadis positive (.55171) and based on the t-value (10.47) and p-value (0.000), we conclude this relationship is statistically significant. Hence, there is a statistically significant positive linear relationship between reading and writing.## See also

## Non-parametric correlation

A Spearman correlation is used when one or both of the variables are not assumed to be normally distributed and interval (but are assumed to be ordinal). The values of the variables are converted in ranks and then correlated. In our example, we will look for a relationship between

readandwrite. We will not assume that both of these variables are normal and interval. Thespearmanoption on theproc corrstatement is used to tell SAS to perform a Spearman rank correlation instead of a Pearson correlation.proc corr data = "c:/mydata/hsb2" spearman; var read write; run;The CORR Procedure 2 Variables: read write Spearman Correlation Coefficients, N = 200 Prob > |r| under H0: Rho=0 read write read 1.00000 0.61675 reading score <.0001 write 0.61675 1.00000 writing score <.0001The results suggest that the relationship between

readandwrite(rho = 0.61675, p = 0.000) is statistically significant.

## Simple logistic regression

Logistic regression assumes that the outcome variable is binary (i.e., coded as 0 and 1). We have only one variable in the hsb2 data file that is coded 0 and 1, and that is

female. We understand thatfemaleis a silly outcome variable (it would make more sense to use it as a predictor variable), but we can usefemaleas the outcome variable to illustrate how the code for this command is structured and how to interpret the output. The first variable listed on the model statement is the outcome (or dependent) variable, and all of the rest of the variables are listed after the equals sign and are predictor (or independent) variables. You can use theexpboption on themodelstatement if you want to see the odds ratios. In our example,femalewill be the outcome variable, andreadwill be the predictor variable. As with OLS regression, the predictor variables must be either dichotomous or continuous; they cannot be categorical.proc logistic data = "c:/mydata/hsb2" desc; model female = read / expb; run;The LOGISTIC Procedure Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Exp(Est) Intercept 1 0.7261 0.7420 0.9577 0.3278 2.067 read 1 -0.0104 0.0139 0.5623 0.4533 0.990 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits read 0.990 0.963 1.017 Association of Predicted Probabilities and Observed Responses Percent Concordant 50.3 Somers' D 0.069 Percent Discordant 43.4 Gamma 0.073 Percent Tied 6.3 Tau-a 0.034 Pairs 9919 c 0.534The results indicate that reading score (

read) is not a statistically significant predictor of gender (i.e., being female), Wald chi-square = 0.5623, p = 0.4533.See also

## Multiple regression

Multiple regression is very similar to simple regression, except that in multiple regression you have more than one predictor variable in the equation. For example, using the hsb2 data file we will predict writing score from gender (

female), reading, math, science and social studies (socst) scores. Thestboption on themodelstatement tells SAS to display the standardized regression coefficients (seen on the far right of the output).proc reg data = "c:/mydata/hsb2"; model write = female read math science socst / stb; run; quit;The REG Procedure Model: MODEL1 Dependent Variable: write writing score Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 5 10757 2151.38488 58.60 <.0001 Error 194 7121.95060 36.71109 Corrected Total 199 17879 Root MSE 6.05897 R-Square 0.6017 Dependent Mean 52.77500 Adj R-Sq 0.5914 Coeff Var 11.48075 Parameter Estimates Parameter Standard Standardized Variable Label DF Estimate Error t Value Pr > |t| Estimate Intercept Intercept 1 6.13876 2.80842 2.19 0.0300 0 female 1 5.49250 0.87542 6.27 <.0001 0.28928 read reading score 1 0.12541 0.06496 1.93 0.0550 0.13566 math math score 1 0.23807 0.06713 3.55 0.0005 0.23531 science science score 1 0.24194 0.06070 3.99 <.0001 0.25272 socst social studies score 1 0.22926 0.05284 4.34 <.0001 0.25967The results indicate that the overall model is statistically significant (F = 58.60, p = 0.0001). Furthermore, all of the predictor variables are statistically significant except for

read.## See also

## Analysis of covariance

Analysis of covariance is like ANOVA, except in addition to the categorical predictors you have continuous predictors as well. For example, the one way ANOVA example used

writeas the dependent variable andprogas the independent variable. Let’s addreadas a continuous variable to this model.proc glm data = "c:/mydata/hsb2"; class prog; model write = prog read; run; quit;The GLM Procedure Dependent Variable: write writing score Sum of Source DF Squares Mean Square F Value Pr > F Model 3 7017.68123 2339.22708 42.21 <.0001 Error 196 10861.19377 55.41425 Corrected Total 199 17878.87500 R-Square Coeff Var Root MSE write Mean 0.392512 14.10531 7.444075 52.77500 Source DF Type I SS Mean Square F Value Pr > F prog 2 3175.697857 1587.848929 28.65 <.0001 read 1 3841.983376 3841.983376 69.33 <.0001 Source DF Type III SS Mean Square F Value Pr > F prog 2 650.259965 325.129983 5.87 0.0034 read 1 3841.983376 3841.983376 69.33 <.0001The results indicate that even after adjusting for reading score (

read), writing scores still significantly differ by program type (prog) F = 5.87, p = 0.0034.See also

## Multiple logistic regression

Multiple logistic regression is like simple logistic regression, except that there are two or more predictors. The predictors can be interval variables or dummy variables, but cannot be categorical variables. If you have categorical predictors, they should be coded into one or more dummy variables. We have only one variable in our data set that is coded 0 and 1, and that is

female. We understand thatfemaleis a silly outcome variable (it would make more sense to use it as a predictor variable), but we can usefemaleas the outcome variable to illustrate how the code for this command is structured and how to interpret the output. In our example,femalewill be the outcome variable, andreadandwritewill be the predictor variables. Thedescoption on theproc logisticstatement is necessary so that SAS models the probability of being female (i.e., female = 1). Theexpboption on themodelstatement tells SAS to display the exponentiated coefficients (i.e., the odds ratios).proc logistic data = "c:/mydata/hsb2" desc; model female = read write / expb; run;The LOGISTIC Procedure Model Information Data Set WORK.HSB2 Response Variable female Number of Response Levels 2 Number of Observations 200 Model binary logit Optimization Technique Fisher's scoring Response Profile Ordered Total Value female Frequency 1 1 109 2 0 91 Probability modeled is female=1. Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 277.637 253.818 SC 280.935 263.713 -2 Log L 275.637 247.818 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 27.8186 2 <.0001 Score 26.3588 2 <.0001 Wald 23.4135 2 <.0001Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Exp(Est) Intercept 1 -1.7061 0.9234 3.4137 0.0647 0.182 read 1 -0.0710 0.0196 13.1251 0.0003 0.931 write 1 0.1064 0.0221 23.0748 <.0001 1.112 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits read 0.931 0.896 0.968 write 1.112 1.065 1.162 Association of Predicted Probabilities and Observed Responses Percent Concordant 69.3 Somers' D 0.396 Percent Discordant 29.7 Gamma 0.400 Percent Tied 1.0 Tau-a 0.198 Pairs 9919 c 0.698These results show that both

read(Wald chi-square = 13.1251, p = 0.0003) andwrite(Wald chi-square = 23.0748, p = 0.0001) are significant predictors offemale.## See also

## Discriminant analysis

Discriminant analysis is used when you have one or more normally distributed interval independent variables and a categorical dependent variable. It is a multivariate technique that considers the latent dimensions in the independent variables for predicting group membership in the categorical dependent variable. For example, using the hsb2 data file, say we wish to use

read,writeandmathscores to predict the type of program (prog) to which a student belongs.proc discrim data = "c:/mydata/hsb2" can; class prog; var read write math; run;The SAS System The DISCRIM Procedure Observations 200 DF Total 199 Variables 3 DF Within Classes 197 Classes 3 DF Between Classes 2 Class Level Information Variable Prior prog Name Frequency Weight Proportion Probability 1 _1 45 45.0000 0.225000 0.333333 2 _2 105 105.0000 0.525000 0.333333 3 _3 50 50.0000 0.250000 0.333333 Pooled Covariance Matrix Information Natural Log of the Covariance Determinant of the Matrix Rank Covariance Matrix 3 12.18440 Pairwise Generalized Squared Distances Between Groups 2 _ _ -1 _ _ D (i|j) = (X - X )' COV (X - X ) i j i j Generalized Squared Distance to prog From prog 1 2 3 1 0 0.73810 0.31771 2 0.73810 0 1.90746 3 0.31771 1.90746 0 Canonical Discriminant Analysis Adjusted Approximate Squared Canonical Canonical Standard Canonical Correlation Correlation Error Correlation 1 0.512534 0.502546 0.052266 0.262691 2 0.067247 0.031181 0.070568 0.004522 Test of H0: The canonical correlations in the current row and all Eigenvalues of Inv(E)*H that follow are zero = CanRsq/(1-CanRsq) Likelihood Approximate Eigenvalue Difference Proportion Cumulative Ratio F Value Num DF Den DF Pr > F 1 0.3563 0.3517 0.9874 0.9874 0.73397507 10.87 6 390 <.0001 2 0.0045 0.0126 1.0000 0.99547788 0.45 2 196 0.6414 Total Canonical Structure Variable Label Can1 Can2 read reading score 0.822122 -0.167318 write writing score 0.818851 0.572893 math math score 0.933429 -0.239761 Between Canonical Structure Variable Label Can1 Can2 read reading score 0.999644 -0.026693 write writing score 0.995813 0.091410 math math score 0.999433 -0.033682 Pooled Within Canonical Structure Variable Label Can1 Can2 read reading score 0.778465 -0.184093 write writing score 0.775344 0.630310 math math score 0.912889 -0.272463 Total-Sample Standardized Canonical Coefficients Variable Label Can1 Can2 read reading score 0.299373057 -0.449624188 write writing score 0.363246854 1.298397979 math math score 0.659035164 -0.743012325 Pooled Within-Class Standardized Canonical Coefficients Variable Label Can1 Can2 read reading score 0.272852441 -0.409793246 write writing score 0.331078354 1.183414147 math math score 0.581553807 -0.655657953 Raw Canonical Coefficients Variable Label Can1 Can2 read reading score 0.0291987615 -.0438532096 write writing score 0.0383228947 0.1369822435 math math score 0.0703462492 -.0793100780 Class Means on Canonical Variables prog Can1 Can2 1 -.3120021323 0.1190423066 2 0.5358514591 -.0196809384 3 -.8444861449 -.0658081053 Linear Discriminant Function _ -1 _ -1 _ Constant = -.5 X' COV X Coefficient Vector = COV X j j j Linear Discriminant Function for prog Variable Label 1 2 3 Constant -24.47383 -30.60364 -20.77468 read reading score 0.18195 0.21279 0.17451 write writing score 0.38572 0.39921 0.33999 math math score 0.40171 0.47236 0.37891 Generalized Squared Distance Function 2 _ -1 _ D (X) = (X-X )' COV (X-X ) j j j Posterior Probability of Membership in Each prog 2 2 Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X)) j k k Number of Observations and Percent Classified into prog From prog 1 2 3 Total 1 11 17 17 45 24.44 37.78 37.78 100.00 2 18 68 19 105 17.14 64.76 18.10 100.00 3 14 7 29 50 28.00 14.00 58.00 100.00 Total 43 92 65 200 21.50 46.00 32.50 100.00 Priors 0.33333 0.33333 0.33333 Error Count Estimates for prog 1 2 3 Total Rate 0.7556 0.3524 0.4200 0.5093 Priors 0.3333 0.3333 0.3333Clearly, the SAS output for this procedure is quite lengthy, and it is beyond the scope of this page to explain all of it. However, the main point is that two canonical variables are identified by the analysis, the first of which seems to be more related to program type than the second.

## See also

## One-way MANOVA

MANOVA (multivariate analysis of variance) is like ANOVA, except that there are two or more dependent variables. In a one-way MANOVA, there is one categorical independent variable and two or more dependent variables. For example, using the hsb2 data file, say we wish to examine the differences in

read,writeandmathbroken down by program type (prog). Themanovastatement is necessary in theproc glmto tell SAS to conduct a MANOVA. Theh=on themanovastatement is used to specify the hypothesized effect.proc glm data = "c:/mydata/hsb2"; class prog; model read write math = prog; manova h=prog; run; quit;The GLM Procedure Dependent Variable: read reading score Sum of Source DF Squares Mean Square F Value Pr > F Model 2 3716.86127 1858.43063 21.28 <.0001 Error 197 17202.55873 87.32263 Corrected Total 199 20919.42000 R-Square Coeff Var Root MSE read Mean 0.177675 17.89136 9.344658 52.23000 Source DF Type I SS Mean Square F Value Pr > F prog 2 3716.861270 1858.430635 21.28 <.0001 Source DF Type III SS Mean Square F Value Pr > F prog 2 3716.861270 1858.430635 21.28 <.0001Dependent Variable: write writing score Sum of Source DF Squares Mean Square F Value Pr > F Model 2 3175.69786 1587.84893 21.27 <.0001 Error 197 14703.17714 74.63542 Corrected Total 199 17878.87500 R-Square Coeff Var Root MSE write Mean 0.177623 16.36983 8.639179 52.77500 Source DF Type I SS Mean Square F Value Pr > F prog 2 3175.697857 1587.848929 21.27 <.0001 Source DF Type III SS Mean Square F Value Pr > F prog 2 3175.697857 1587.848929 21.27 <.0001Dependent Variable: math math score Sum of Source DF Squares Mean Square F Value Pr > F Model 2 4002.10389 2001.05194 29.28 <.0001 Error 197 13463.69111 68.34361 Corrected Total 199 17465.79500 R-Square Coeff Var Root MSE math Mean 0.229140 15.70333 8.267019 52.64500 Source DF Type I SS Mean Square F Value Pr > F prog 2 4002.103889 2001.051944 29.28 <.0001 Source DF Type III SS Mean Square F Value Pr > F prog 2 4002.103889 2001.051944 29.28 <.0001Multivariate Analysis of Variance Characteristic Roots and Vectors of: E Inverse * H, where H = Type III SSCP Matrix for prog E = Error SSCP Matrix Characteristic Characteristic Vector V'EV=1 Root Percent read write math 0.35628297 98.74 0.00208033 0.00273039 0.00501196 0.00454266 1.26 -0.00312441 0.00975958 -0.00565061 0.00000000 0.00 -0.00904826 0.00054800 0.00823531 MANOVA Test Criteria and F Approximations for the Hypothesis of No Overall prog Effect H = Type III SSCP Matrix for prog E = Error SSCP Matrix S=2 M=0 N=96.5 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.73397507 10.87 6 390 <.0001 Pillai's Trace 0.26721285 10.08 6 392 <.0001 Hotelling-Lawley Trace 0.36082563 11.70 6 258.23 <.0001 Roy's Greatest Root 0.35628297 23.28 3 196 <.0001 NOTE: F Statistic for Roy's Greatest Root is an upper bound. NOTE: F Statistic for Wilks' Lambda is exact.This command produces four different test statistics that are used to evaluate the statistical significance of the relationship between the independent variable and the outcome variables. According to all four criteria, the students in the different programs differ in their joint distribution of

read,writeandmath.## See also

## Multivariate multiple regression

Multivariate multiple regression is used when you have two or more dependent variables that are to be predicted from two or more predictor variables. In our example, we will predict

writeandreadfromfemale,math,scienceand social studies (socst) scores. Themteststatement in theproc regis used to test hypotheses in multivariate regression models where there are several independent variables fit to the same dependent variables. If no equations or options are specified, themteststatement tests the hypothesis that all estimated parameters except the intercept are zero. In other words, the multivariate tests test whether the independent variable specified predicts the dependent variables together, holding all of the other independent variables constant. You can put a label in front of themteststatement to aid in the interpretation of the output (this is particularly useful when you have multiplemteststatements).proc reg data = "c:/mydata/hsb2"; model write read = female math science socst; female: mtest female; math: mtest math; science: mtest science; socst: mtest socst; run; quit;The REG Procedure Model: MODEL1 Dependent Variable: write writing score Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 4 10620 2655.02312 71.32 <.0001 Error 195 7258.78251 37.22453 Corrected Total 199 17879 Root MSE 6.10119 R-Square 0.5940 Dependent Mean 52.77500 Adj R-Sq 0.5857 Coeff Var 11.56076 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 6.56892 2.81908 2.33 0.0208 female 1 5.42822 0.88089 6.16 <.0001 math math score 1 0.28016 0.06393 4.38 <.0001 science science score 1 0.27865 0.05805 4.80 <.0001 socst social studies score 1 0.26811 0.04919 5.45 <.0001 Model: MODEL1 Dependent Variable: read reading score Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 4 12220 3054.91459 68.47 <.0001 Error 195 8699.76166 44.61416 Corrected Total 199 20919 Root MSE 6.67938 R-Square 0.5841 Dependent Mean 52.23000 Adj R-Sq 0.5756 Coeff Var 12.78840 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 3.43000 3.08624 1.11 0.2678 female 1 -0.51261 0.96436 -0.53 0.5956 math math score 1 0.33558 0.06999 4.79 <.0001 science science score 1 0.29276 0.06355 4.61 <.0001 socst social studies score 1 0.30976 0.05386 5.75 <.0001 Model: MODEL1 Multivariate Test: female Multivariate Statistics and Exact F Statistics S=1 M=0 N=96 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.83011470 19.85 2 194 <.0001 Pillai's Trace 0.16988530 19.85 2 194 <.0001 Hotelling-Lawley Trace 0.20465280 19.85 2 194 <.0001 Roy's Greatest Root 0.20465280 19.85 2 194 <.0001 Model: MODEL1 Multivariate Test: math Multivariate Statistics and Exact F Statistics S=1 M=0 N=96 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.84006791 18.47 2 194 <.0001 Pillai's Trace 0.15993209 18.47 2 194 <.0001 Hotelling-Lawley Trace 0.19037995 18.47 2 194 <.0001 Roy's Greatest Root 0.19037995 18.47 2 194 <.0001 Model: MODEL1 Multivariate Test: science Multivariate Statistics and Exact F Statistics S=1 M=0 N=96 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.83357462 19.37 2 194 <.0001 Pillai's Trace 0.16642538 19.37 2 194 <.0001 Hotelling-Lawley Trace 0.19965265 19.37 2 194 <.0001 Roy's Greatest Root 0.19965265 19.37 2 194 <.0001 Model: MODEL1 Multivariate Test: socst Multivariate Statistics and Exact F Statistics S=1 M=0 N=96 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.77932902 27.47 2 194 <.0001 Pillai's Trace 0.22067098 27.47 2 194 <.0001 Hotelling-Lawley Trace 0.28315509 27.47 2 194 <.0001 Roy's Greatest Root 0.28315509 27.47 2 194 <.0001With regard to the univariate tests, each of the independent variables is statistically significant predictor for writing. All of the independent variables are also statistically significant predictors for reading except

female(t = -0.53, p = 0.5956). All of the multivariate tests are also statistically significant.

## Canonical correlation

Canonical correlation is a multivariate technique used to examine the relationship between two groups of variables. For each set of variables, it creates latent variables and looks at the relationships among the latent variables. It assumes that all variables in the model are interval and normally distributed. In SAS, one group of variables is placed on the

varstatement and the other group on thewithstatement. There need not be an equal number of variables in the two groups. Thealloption on theproc cancorrstatement provides additional output that many researchers might find useful.proc cancorr data = "c:/mydata/hsb2" all; var read write; with math science; run;The CANCORR Procedure VAR Variables 2 WITH Variables 2 Observations 200 Means and Standard Deviations Standard Variable Mean Deviation Label read 52.230000 10.252937 reading score write 52.775000 9.478586 writing score math 52.645000 9.368448 math score science 51.850000 9.900891 science scoreCorrelations Among the Original Variables Correlations Among the VAR Variables read write read 1.0000 0.5968 write 0.5968 1.0000 Correlations Among the WITH Variables math science math 1.0000 0.6307 science 0.6307 1.0000 Correlations Between the VAR Variables and the WITH Variables math science read 0.6623 0.6302 write 0.6174 0.5704Canonical Correlation Analysis Adjusted Approximate Squared Canonical Canonical Standard Canonical Correlation Correlation Error Correlation 1 0.772841 0.771003 0.028548 0.597283 2 0.023478 . 0.070849 0.000551 Test of H0: The canonical correlations in the current row and all Eigenvalues of Inv(E)*H that follow are zero = CanRsq/(1-CanRsq) Likelihood Approximate Eigenvalue Difference Proportion Cumulative Ratio F Value Num DF Den DF Pr > F 1 1.4831 1.4826 0.9996 0.9996 0.40249498 56.47 4 392 <.0001 2 0.0006 0.0004 1.0000 0.99944876 0.11 1 197 0.7420 Multivariate Statistics and F Approximations S=2 M=-0.5 N=97 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.40249498 56.47 4 392 <.0001 Pillai's Trace 0.59783426 42.00 4 394 <.0001 Hotelling-Lawley Trace 1.48368501 72.58 4 234.16 <.0001 Roy's Greatest Root 1.48313347 146.09 2 197 <.0001 NOTE: F Statistic for Roy's Greatest Root is an upper bound. NOTE: F Statistic for Wilks' Lambda is exact. Raw Canonical Coefficients for the VAR Variables V1 V2 read reading score 0.063261313 0.1037907932 write writing score 0.0492491834 -0.12190836 Raw Canonical Coefficients for the WITH Variables W1 W2 math math score 0.0669826768 -0.120142451 science science score 0.0482406314 0.1208859811Standardized Canonical Coefficients for the VAR Variables V1 V2 read reading score 0.6486 1.0642 write writing score 0.4668 -1.1555 Standardized Canonical Coefficients for the WITH Variables W1 W2 math math score 0.6275 -1.1255 science science score 0.4776 1.1969Canonical Structure Correlations Between the VAR Variables and Their Canonical Variables V1 V2 read reading score 0.9272 0.3746 write writing score 0.8539 -0.5205 Correlations Between the WITH Variables and Their Canonical Variables W1 W2 math math score 0.9288 -0.3706 science science score 0.8734 0.4870 Correlations Between the VAR Variables and the Canonical Variables of the WITH Variables W1 W2 read reading score 0.7166 0.0088 write writing score 0.6599 -0.0122 Correlations Between the WITH Variables and the Canonical Variables of the VAR Variables V1 V2 math math score 0.7178 -0.0087 science science score 0.6750 0.0114Canonical Redundancy Analysis Raw Variance of the VAR Variables Explained by Their Own The Opposite Canonical Variables Canonical Variables Canonical Variable Cumulative Canonical Cumulative Number Proportion Proportion R-Square Proportion Proportion 1 0.7995 0.7995 0.5973 0.4775 0.4775 2 0.2005 1.0000 0.0006 0.0001 0.4777 Raw Variance of the WITH Variables Explained by Their Own The Opposite Canonical Variables Canonical Variables Canonical Variable Cumulative Canonical Cumulative Number Proportion Proportion R-Square Proportion Proportion 1 0.8100 0.8100 0.5973 0.4838 0.4838 2 0.1900 1.0000 0.0006 0.0001 0.4839 Standardized Variance of the VAR Variables Explained by Their Own The Opposite Canonical Variables Canonical Variables Canonical Variable Cumulative Canonical Cumulative Number Proportion Proportion R-Square Proportion Proportion 1 0.7944 0.7944 0.5973 0.4745 0.4745 2 0.2056 1.0000 0.0006 0.0001 0.4746 Standardized Variance of the WITH Variables Explained by Their Own The Opposite Canonical Variables Canonical Variables Canonical Variable Cumulative Canonical Cumulative Number Proportion Proportion R-Square Proportion Proportion 1 0.8127 0.8127 0.5973 0.4854 0.4854 2 0.1873 1.0000 0.0006 0.0001 0.4855 Squared Multiple Correlations Between the VAR Variables and the First M Canonical Variables of the WITH Variables M 1 2 read reading score 0.5135 0.5136 write writing score 0.4355 0.4356 Squared Multiple Correlations Between the WITH Variables and the First M Canonical Variables of the VAR Variables M 1 2 math math score 0.5152 0.5153 science science score 0.4557 0.4558The output above shows the linear combinations corresponding to the first canonical correlation. At the bottom of the output are the two canonical correlations. These results indicate that the first canonical correlation is .772841. The F-test in this output tests the hypothesis that the first canonical correlation is equal to zero. Clearly, F = 56.47 is statistically significant. However, the second canonical correlation of .0235 is not statistically significantly different from zero (F = 0.11, p = 0.7420).

## Factor analysis

Factor analysis is a form of exploratory multivariate analysis that is used to either reduce the number of variables in a model or to detect relationships among variables. All variables involved in the factor analysis need to be continuous and are assumed to be normally distributed. The goal of the analysis is to try to identify factors which underlie the variables. There may be fewer factors than variables, but there may not be more factors than variables. For our example, let’s suppose that we think that there are some common factors underlying the various test scores. We will use the principal components method of extraction, use a varimax rotation, extract two factors and obtain a scree plot of the eigenvalues. All of these options are listed on the

proc factorstatement.proc factor data = "c:/mydata/hsb2" method=principal rotate=varimax nfactors=2 scree; var read write math science socst; run;The FACTOR Procedure Initial Factor Method: Principal Components Prior Communality Estimates: ONE Eigenvalues of the Correlation Matrix: Total = 5 Average = 1 Eigenvalue Difference Proportion Cumulative 1 3.38081982 2.82344156 0.6762 0.6762 2 0.55737826 0.15058550 0.1115 0.7876 3 0.40679276 0.05062495 0.0814 0.8690 4 0.35616781 0.05732645 0.0712 0.9402 5 0.29884136 0.0598 1.0000 2 factors will be retained by the NFACTOR criterion. The FACTOR Procedure Initial Factor Method: Principal Components Factor Pattern Factor1 Factor2 READ reading score 0.85760 -0.02037 WRITE writing score 0.82445 0.15495 MATH math score 0.84355 -0.19478 SCIENCE science score 0.80091 -0.45608 SOCST social studies score 0.78268 0.53573 Variance Explained by Each Factor Factor1 Factor2 3.3808198 0.5573783 Final Communality Estimates: Total = 3.938198 READ WRITE MATH SCIENCE SOCST 0.73589906 0.70373337 0.74951854 0.84945810 0.89958900 The FACTOR Procedure Rotation Method: Varimax Orthogonal Transformation Matrix 1 2 1 0.74236 0.67000 2 -0.67000 0.74236 Rotated Factor Pattern Factor1 Factor2 READ reading score 0.65029 0.55948 WRITE writing score 0.50822 0.66742 MATH math score 0.75672 0.42058 SCIENCE science score 0.90013 0.19804 SOCST social studies score 0.22209 0.92210 Variance Explained by Each Factor Factor1 Factor2 2.1133589 1.8248392 Final Communality Estimates: Total = 3.938198 READ WRITE MATH SCIENCE SOCST 0.73589906 0.70373337 0.74951854 0.84945810 0.89958900Communality (which is the opposite of uniqueness) is the proportion of variance of the variable (i.e.,

read) that is accounted for by all of the factors taken together, and a very low communality can indicate that a variable may not belong with any of the factors. From the factor pattern table, we can see that all five of the test scores load onto the first factor, while all five tend to load not so heavily on the second factor. The purpose of rotating the factors is to get the variables to load either very high or very low on each factor. In this example, because all of the variables loaded onto factor 1 and not on factor 2, the rotation did not aid in the interpretation. Instead, it made the results even more difficult to interpret. The scree plot may be useful in determining how many factors to retain.

Screen Plot of Eigenvalues | | | | | | 3.5 + | | 1 | | | | | 3.0 + | | | | | | | 2.5 + | | | | | | E | i 2.0 + g | e | n | v | a | l | u | e 1.5 + s | | | | | | | 1.0 + | | | | | | | 2 0.5 + | 3 | 4 | 5 | | | | 0.0 + | | | | | | -------+-----------+-----------+-----------+-----------+-----------+------ 0 1 2 3 4 5 Number

## See also