Table of Contents
The question “What statistical analyses should I use using SAS?” pertains to the appropriate statistical methods to be employed while utilizing the SAS software. SAS is a widely used statistical program that offers a wide range of statistical procedures and techniques. Therefore, the statistical analyses to be used may vary depending on the research question, data type, and study design. It is important to carefully consider the available options and select the most suitable analyses to ensure accurate and meaningful results. Some commonly used statistical techniques in SAS include descriptive statistics, regression analysis, ANOVA, and cluster analysis. Additionally, SAS also offers advanced features such as data manipulation and visualization, making it a versatile tool for statistical analysis.
What statistical analysis should I use?Statistical analyses using SAS
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 the hsb2 file as a SAS version 8 data file by clicking
here https://stats.idre.ucla.edu/wp-content/uploads/2016/02/hsb2.sas7bdat
. 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/sas/notes/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 <.0001The 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 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 write for this group is
statistically significantly different from 50.
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 = 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%.
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 = 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 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 0.0187The 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).
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).
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 = 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, 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 = 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).
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 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 = 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.
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 9.31875441The 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 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.
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 <.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).
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 0.3868These results indicate that the mean of read is not statistically significantly
different from the mean of write (t = 0.87, p = 0.3868).
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:/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 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 = 200McNemar’s test statistic suggests that there is not a statistically significant difference in the proportions of correct/incorrect answers to these two questions.
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 0.8343The 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.
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
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 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 0.8753These 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).
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 = 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. The desc option on the proc
logistic statement 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 the order
= data option on the proc logistic statement.) 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. The expb option 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.
SAS Annotated
Output: Ordered logistic regression
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
neither
prog nor schtyp are 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 1.405The 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, 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 <.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 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.
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 0.59678We 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.
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 <.0001The 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 0.534The 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. 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 0.25967The 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.
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 <.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 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 0.698These 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.
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 0.3333Clearly, 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.
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 <.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, write and math.
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 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 independent variables fit to the same dependent 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
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 the mtest statement to
aid in the interpretation of the output (this is particularly 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 <.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 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 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 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 0.89958900Communality (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
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NumberCite this article
stats writer (2024). What statistical analyses should I use using SAS?. PSYCHOLOGICAL SCALES. Retrieved from https://scales.arabpsychology.com/stats/what-statistical-analyses-should-i-use-using-sas/
stats writer. "What statistical analyses should I use using SAS?." PSYCHOLOGICAL SCALES, 28 Jun. 2024, https://scales.arabpsychology.com/stats/what-statistical-analyses-should-i-use-using-sas/.
stats writer. "What statistical analyses should I use using SAS?." PSYCHOLOGICAL SCALES, 2024. https://scales.arabpsychology.com/stats/what-statistical-analyses-should-i-use-using-sas/.
stats writer (2024) 'What statistical analyses should I use using SAS?', PSYCHOLOGICAL SCALES. Available at: https://scales.arabpsychology.com/stats/what-statistical-analyses-should-i-use-using-sas/.
[1] stats writer, "What statistical analyses should I use using SAS?," PSYCHOLOGICAL SCALES, vol. X, no. Y, ص Z-Z, June, 2024.
stats writer. What statistical analyses should I use using SAS?. PSYCHOLOGICAL SCALES. 2024;vol(issue):pages.