What statistical analysis should I use?

Statistical analyses using SAS

1. One Sample t-test

2. One sample median test
3. Binomial Test
4. Chi-square goodness of fit
5. Two independent samples t-test
6. Wilcoxon-Mann-Whitney test
7. Chi-square test
8. Fisher's exact test
9. One-way ANOVA
10. Kruskal Wallis test
11. Paired t-test
12. Wilcoxon signed rank sum test
13. McNemar test
14. One-way repeated measures ANOVA
15. Repeated measures logistic regression
16. Factorial ANOVA
17. Friedman test
18. Factorial logistic regression
19. Correlation
20. Simple linear regression
21. Non-parametric correlation
22. Simple Logistic regression
23. Multiple regression
24. Analysis of Covariance
25. Multiple logistic regression
26. Discriminant analysis
27. One-way MANOVA
28. Multivariate multiple regression
29. Canonical correlation
30. Factor Analysis
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 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 for more information on this.

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 the hsb2
file as a SAS version 8 data file by clicking . 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 named c:\
mydata\hsb2.sas7bdat. If you store the file in a different location, change c:\mydata to 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 <.0001
The mean of the variable write for 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 that write is an ordinal variable). We will test whether the median writing
score (write) differs significantly from 50. The loccount option on the proc univariate
statement 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 72
You 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 write for 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 the exact statement 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 = 200

The 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 the tables statement.
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 = 200

These 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 write is 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 0.0187
The 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 write scores of males and the write scores of females (z = -3.329, p =
0.0009).
See also
· FAQ: Why is the Mann-Whitney significant when the medians are equal?
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 chisq option is used on the tables statement 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 = 200

These 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, female has two levels (male and female) and ses has 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 = 200

Again 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 five or less. 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 fisher option on
the tables statement.
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 = 200

 These 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 write differs 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 9.31875441
The 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 prog because prog was the only
variable entered into the model. If other variables had also been entered, the F test for the
Model would have been different from prog.) 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 write is 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 <.0001
The 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 read is equal to the mean of write.
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 0.3868

These results indicate that the mean of read is not statistically significantly different from the
mean of write (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 read and write is 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 in proc univariate.
data hsb2a;
set 'c:\mydatahsb2';
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| 0.3677
The results suggest that there is not a statistically significant difference between read and write.
If you believe the differences between read and write were 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 between read and write (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 = 200

McNemar'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 y3 and y4 represent the dependent variable measured at the 4
levels of a, 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 0.8343
The results indicate that the model as well as both factors (a and s) are statistically significant.
The p-value given in this output for a (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
genmod procedure and indicating binomial as the probability distribution and logit as the link
function to be used in the model. The exercise data file contains 3 pulse measurements from
each of 30 people assigned to 2 different diet regiments and 3 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;
run;
GEE Model Information

Correlation Structure Independent

Subject Effect id (30 levels)
Number of Clusters 30
Correlation Matrix Dimension 3
Maximum Cluster Size 3
Minimum Cluster Size 3

Algorithm converged.

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 of female by ses. 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
0.8753
These results indicate that the overall model is statistically significant (F = 5.67, p = 0.001). The
variables female and ses are also statistically significant (F = 16.59, p = 0.0001 and F = 6.61, p
= 0.0017, respectively). However, that interaction between female and ses is not statistically
significant (F = 0.13, p = 0.8753).
See also
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 transpose to 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 call rwm (for read, write, math), and col1 that contains the score on the dependent
variable, that is the reading, writing or math score. To obtain the Friedman test, you need to use
the cmh2 option on the tables statement in proc 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 = 600

The 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.
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 female as 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 both prog
and schtyp are not continuous variables, we need to include them on the class statement. The
desc option on the proc logistic statement is necessary so that SAS models the odds of being
female (i.e., female = 1). The expb option 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
1.405
The 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, read and write.
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 <.0001
In 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 0.0002
In the first example above, we see that the correlation between read and write is 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%. Hence read shares about 36% of its variability with write. In
the output for the second example, we can see the correlation between write and female is
0.25649. Squaring this number yields .0657871201, meaning that female shares approximately
6.5% of its variability with write.
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, predicting write from read.
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 0.59678
We see that the relationship between write and read is 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
read and write. We will not assume that both of these variables are normal and interval. The
spearman option on the proc corr statement 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 <.0001
The results suggest that the relationship between read and write (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 that female is a silly outcome variable (it would make more sense to use it as a
predictor variable), but we can use female as 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 the expb
option on the model statement if you want to see the odds ratios. In our example, female will be
the outcome variable, and read will 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 0.534
The 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. The stb option on the model statement 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 0.25967
The 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 write as the
dependent variable and prog as the independent variable. Let's add read as 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
<.0001
The 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 that female is a silly outcome variable (it would make more sense to use
it as a predictor variable), but we can use female as the outcome variable to illustrate how the
code for this command is structured and how to interpret the output. In our example, female
will be the outcome variable, and read and write will be the predictor variables. The desc
option on the proc logistic statement is necessary so that SAS models the probability of being
female (i.e., female = 1). The expb option on the model statement 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 <.0001
Analysis 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 0.698
These results show that both read (Wald chi-square = 13.1251, p = 0.0003) and write (Wald
chi-square = 23.0748, p = 0.0001) are significant predictors of female.
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, write and math scores 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 0.3333
Clearly, 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
· discriminant function analysis

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, write and math broken down by program type (prog). The manova
statement is necessary in the proc glm to tell SAS to conduct a MANOVA. The h= on the
manova statement 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
<.0001
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
Dependent 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
<.0001
Multivariate 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, write and math.
See also

Multivariate multiple regression

Multivariate multiple regression is used when you have two or more variables that are to be
predicted from two or more predictor variables. In our example, we will predict write and read
from female, math, science and social studies (socst) scores. The mtest statement in the proc
reg is used to test hypotheses in multivariate regression models where there are several
dependent variables fit to the same regressors (i.e., independent variables). If no equations or
options are specified, the mtest statement tests the hypothesis that all estimated parameters
except the intercept are zero. In other words, the multivariate tests test whether there the single
dependent variable specified predicts the independent variables together, holding all of the other
dependent variables constant. You can put a label in front of the mtest statement to aid in the
interpretation of the output (this is useful when you have multiple mtest statements).
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
<.0001
With regard to the univariate tests, each of the dependent variables is statistically significant
predictor for writing. All of the dependent 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 var statement and the
other group on the with statement. There need not be an equal number of variables in the two
groups. The all option on the proc cancorr statement 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 score
Correlations 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 0.5704
Canonical 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 0.1208859811
Standardized 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 1.1969
Canonical 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 0.0114
Canonical 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 0.4558
The 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 factor statement.
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 0.89958900
Communality (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
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See also