Introduction

This page shows how to perform a number of statistical tests using SPSS.  Each
section gives a brief description of the aim of the statistical test, when it is used, an
example showing the SPSS commands and SPSS (often abbreviated) output 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.

About the hsb data file

Most of the examples in this page will use a data file called hsb2, high school
and beyond.  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 hsb data file by clicking on hsb2.

One sample t-test

A one sample t-test allows us to test whether a sample mean (of 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.

t-test
 /testval = 50
 /variable = write.

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).

nptests
/onesample test (write) wilcoxon(testvalue = 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 can do this as shown below.

npar tests
 /binomial (.5) = female.

The results indicate that there is no statistically significant difference (p =
.229).  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.

npar test 
 /chisquare = race
 /expected = 10 10 10 70.

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 =
5.029, p = .170).

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.

t-test groups = female(0 1) 
 /variables = write.

Because the standard deviations for the two groups are similar (10.3 and
8.1), we will use the “equal variances assumed” test.  The results indicate that there is a statistically significant difference between the
mean writing score for males and females (t = -3.734, p = .000).  In other words,
females have a statistically significantly higher mean score on writing (54.99) than males
(50.12).

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 only assume that the variable is at least ordinal).  You
will notice that the SPSS syntax for the Wilcoxon-Mann-Whitney test is almost identical
to that of the independent samples t-test.  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.

npar test
 /m-w = write by female(0 1).

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 = 0.001).

Chi-square test

A chi-square test is used when you want to see if there is a relationship between two
categorical variables.  In SPSS, the chisq option is used on the
statistics subcommand of the crosstabs
command 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.

crosstabs
 /tables = schtyp by female
 /statistic = chisq.

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 =
0.047, p
= 0.828).

Let’s look at another example, this time looking at the linear 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).

crosstabs
 /tables = female by ses
 /statistic = chisq.

Again we find that there is no statistically significant relationship between the
variables (chi-square with two degrees of freedom = 4.577, p = 0.101).

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 SPSS unless you have the SPSS Exact Test Module, you
can only perform a Fisher’s exact test on a 2×2 table, and these results are
presented by default.  Please see the results from the chi squared
example above.

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).  The command for this test
would be:

oneway write by prog.

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.)  To see the mean of write for each level of
program type,

means tables = write by prog.

From this we can 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 and a generalized form of the Mann-Whitney test method since 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.

npar tests
 /k-w = write by prog (1,3).

If some of the scores receive tied ranks, then a correction factor is used, yielding a
slightly different value of chi-squared.  With or without ties, the results indicate
that there is a statistically significant difference among the three type of programs.

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.

t-test pairs = read with write (paired).

These results indicate that the mean of read is not statistically significantly
different from the mean of write (t = -0.867, p = 0.387).

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.

npar test
 /wilcoxon = write with read (paired).

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.  Again, we will use the same variables in this
example and assume that this difference is not ordinal.

npar test
 /sign = read with write (paired).

We
conclude that no statistically significant difference was found (p=.556).

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.  Continuing with the hsb2 dataset used
in several above examples, let us create two binary outcomes in our dataset:
himath and
hiread. These outcomes can be considered in a
two-way contingency table.  The null hypothesis is that the proportion
of students in the himath group is the same as the proportion of
students in hiread group (i.e., that the contingency table is
symmetric).

compute himath = (math>60).
compute hiread = (read>60).
execute.

crosstabs
  /tables=himath  BY hiread
  /statistic=mcnemar
  /cells=count.

McNemar’s chi-square statistic suggests that there is not a statistically
significant difference in the proportion of students in the
himath group
and the proportion of students in the
hiread group.

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. This
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, y is the
dependent variable, a is the repeated measure and s is the variable that
indicates the subject number.

glm y1 y2 y3 y4
 /wsfactor a(4).

You will notice that this output gives four different p-values.  The
output labeled “sphericity assumed”  is the p-value (0.000) that you would get if you assumed compound
symmetry in the variance-covariance matrix.  Because that assumption is often not
valid, the three other p-values offer various corrections (the Huynh-Feldt, H-F,
Greenhouse-Geisser, G-G and Lower-bound).  No matter which p-value you
use, our results indicate that we have a statistically significant effect of a at
the .05 level.

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
SPSS, this can be done using the
GENLIN command 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.

GET FILE='C:mydatahttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/exercise.sav'.

GENLIN highpulse (REFERENCE=LAST)
  BY diet (order = DESCENDING)
/MODEL diet 
  DISTRIBUTION=BINOMIAL
  LINK=LOGIT
/REPEATED SUBJECT=id CORRTYPE = EXCHANGEABLE.

These results indicate that diet is not statistically
significant (Wald Chi-Square = 1.562, p = 0.211).

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
SPSS,
you do not need to have the interaction term(s) in your data set.  Rather, you can
have SPSS create it/them temporarily by placing an asterisk between the variables that
will make up the interaction term(s).

glm write by female ses.

These results indicate that the overall model is statistically significant (F =
5.666, p
= 0.00).  The variables female and ses are also statistically
significant (F = 16.595, p = 0.000 and F = 6.611, p = 0.002, respectively).  However,
that interaction between female and ses is not statistically significant (F
= 0.133, p = 0.875).

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.  SPSS handles this for you, but in other
statistical packages you will have to reshape the data before you can conduct
this test.

npar tests
 /friedman = read write math.

Friedman’s chi-square has a value of 0.645 and a p-value of 0.724 and 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.  We will use a logit link and on the
print subcommand we have requested the parameter estimates, the (model)
summary statistics and the test of the parallel lines assumption.

if write ge 30 and write le 48  write3 = 1.
if write ge 49 and write le 57  write3 = 2.
if write ge 58 and write le 70  write3 = 3.
execute.

plum write3 with female read socst
/link = logit
/print = parameter summary tparallel.

The results indicate that the overall model is statistically significant
(p < .000), as are each of the predictor variables (p < .000).  There are
two thresholds 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 = .563).  One of the assumptions underlying ordinal
logistic (and ordinal probit) regression is that the relationship between
each pair of outcome groups is the same.  In other words, ordinal logistic
regression assumes that the coefficients that describe the relationship
between, say, the lowest versus all higher categories of the response
variable are the same as those that describe the relationship between the
next lowest category and all higher categories, etc.  This is called the
proportional odds assumption or the parallel regression assumption.  Because
the relationship between all pairs of groups is the same, there is only one
set of coefficients (only one model).  If this was not the case, we would
need different models (such as a generalized ordered logit model) to
describe the relationship between each pair of outcome groups.

See also

Factorial logistic regression

A factorial logistic regression is used when you have two or more categorical
independent variables but a dichotomous dependent variable.  For example, using the hsb2 data file we will use 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 prog is a
categorical variable (it has three levels), we need to create dummy codes for it.
SPSS will do this for you by making dummy codes for all variables listed after
the keyword with.  SPSS will also create the interaction term;
simply list the two variables that will make up the interaction separated by
the keyword by.

logistic regression female with prog schtyp prog by schtyp
 /contrast(prog) = indicator(1).

The results indicate that the overall model is not statistically significant (LR chi2 =
3.147, p = 0.677).  Furthermore, none of the coefficients are statistically
significant either.  This shows that the overall effect of prog
is not significant.

See also

Correlation

A correlation is useful when you want to see the 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.

correlations
 /variables = read write.

In the second example, 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.

correlations
 /variables =  female write.

In the first example above, we see that the correlation between read and write
is 0.597.  By squaring the correlation and then multiplying by 100, you can
determine what percentage of the variability is shared.  Let’s round
0.597 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
0.256.  Squaring this number yields .065536, 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.

regression variables = write read
 /dependent = write
 /method = enter.

We see that the relationship between write and read is positive
(.552)
and based on the t-value (10.47) and p-value (0.000), we would conclude this
relationship is statistically significant.  Hence, we would say 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.

nonpar corr
 /variables = read write
 /print = spearman.

The results suggest that the relationship between read and write
(rho = 0.617, 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 after the logistic
command is the outcome (or dependent) variable, and all of the rest of
the variables are predictor (or independent) variables.  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.

logistic regression female with read.

The results indicate that reading score (read) is not a statistically
significant predictor of gender (i.e., being female), Wald = .562, p = 0.453.
Likewise, the test of the overall model is not statistically significant, LR chi-squared –
0.56, p = 0.453.

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.

regression variable = write female read math science socst
 /dependent = write
 /method = enter.

The results indicate that the overall model is statistically significant (F = 58.60, p
= 0.000).  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 also 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,
as shown below.

glm write with read by prog.

The results indicate that even after adjusting for reading score (read), writing
scores still significantly differ by program type (prog), F = 5.867, p =
0.003.

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.  The first variable listed
after the logisticregression command is the outcome (or dependent)
variable, and all of the rest of the variables are predictor (or independent)
variables (listed after the keyword with).  In
our example, female will be the outcome variable, and read and write
will be the predictor variables.

logistic regression female with read write.

These results show that both read and write 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 a student belongs to (prog).

discriminate groups = prog(1, 3)
 /variables = read write math.

Clearly, the SPSS output for this procedure is quite lengthy, and it is
beyond the scope of this page to explain all of it.  However, the main
point is that two canonical variables are identified by the analysis, the
first of which seems to be more related to program type than the second.

See also

One-way MANOVA

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

glm read write math by prog.

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
dependent variables that are
to be predicted from two or more independent variables.  In our example using the hsb2 data file, we will
predict write and read from female, math, science and
social studies (socst) scores.

glm write read with female math science socst.

These results
show that all of  the variables in the model have a statistically significant relationship with the joint distribution of write
and read.

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.  SPSS requires that
each of the two groups of variables be separated by the keyword with.  There need not be an
equal number of variables in the two groups (before and after the with).

manova read write with math science
 /discrim.
* * * * * * A n a l y s i s   o f   V a r i a n c e -- design   1 * * * * * *

 EFFECT .. WITHIN CELLS Regression
 Multivariate Tests of Significance (S = 2, M = -1/2, N = 97 )

 Test Name         Value  Approx. F Hypoth. DF   Error DF  Sig. of F

 Pillais          .59783   41.99694       4.00     394.00       .000
 Hotellings      1.48369   72.32964       4.00     390.00       .000
 Wilks            .40249   56.47060       4.00     392.00       .000
 Roys             .59728
 Note.. F statistic for WILKS' Lambda is exact.

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 EFFECT .. WITHIN CELLS Regression (Cont.)
 Univariate F-tests with (2,197) D. F.

 Variable    Sq. Mul. R  Adj. R-sq.  Hypoth. MS    Error MS           F

 READ            .51356      .50862  5371.66966    51.65523   103.99081
 WRITE           .43565      .42992  3894.42594    51.21839    76.03569

 Variable     Sig. of F

 READ              .000
 WRITE             .000

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 Raw canonical coefficients for DEPENDENT variables
           Function No.

 Variable            1

 READ             .063
 WRITE            .049

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 Standardized canonical coefficients for DEPENDENT variables
           Function No.

 Variable            1

 READ             .649
 WRITE            .467

* * * * * * A n a l y s i s   o f   V a r i a n c e -- design   1 * * * * * *

 Correlations between DEPENDENT and canonical variables
           Function No.

 Variable            1

 READ             .927
 WRITE            .854

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 Variance in dependent variables explained by canonical variables

 CAN. VAR.  Pct Var DE Cum Pct DE Pct Var CO Cum Pct CO

        1       79.441     79.441     47.449     47.449

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 Raw canonical coefficients for COVARIATES
           Function No.

 COVARIATE           1

 MATH             .067
 SCIENCE          .048

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 Standardized canonical coefficients for COVARIATES
           CAN. VAR.

 COVARIATE           1

 MATH             .628
 SCIENCE          .478

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 Correlations between COVARIATES and canonical variables
           CAN. VAR.

 Covariate           1

 MATH             .929
 SCIENCE          .873

* * * * * * A n a l y s i s   o f   V a r i a n c e -- design   1 * * * * * *

 Variance in covariates explained by canonical variables

 CAN. VAR.  Pct Var DE Cum Pct DE Pct Var CO Cum Pct CO

        1       48.544     48.544     81.275     81.275

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 Regression analysis for WITHIN CELLS error term
 --- Individual Univariate .9500 confidence intervals
 Dependent variable .. READ              reading score

 COVARIATE            B        Beta   Std. Err.     t-Value   Sig. of t

 MATH            .48129      .43977        .070       6.868        .000
 SCIENCE         .36532      .35278        .066       5.509        .000

 COVARIATE   Lower -95%  CL- Upper

 MATH              .343        .619
 SCIENCE           .235        .496
 Dependent variable .. WRITE             writing score

 COVARIATE            B        Beta   Std. Err.     t-Value   Sig. of t

 MATH            .43290      .42787        .070       6.203        .000
 SCIENCE         .28775      .30057        .066       4.358        .000

 COVARIATE   Lower -95%  CL- Upper

 MATH              .295        .571
 SCIENCE           .158        .418

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

* * * * * * A n a l y s i s   o f   V a r i a n c e -- design   1 * * * * * *

 EFFECT .. CONSTANT
 Multivariate Tests of Significance (S = 1, M = 0, N = 97 )

 Test Name         Value    Exact F Hypoth. DF   Error DF  Sig. of F

 Pillais          .11544   12.78959       2.00     196.00       .000
 Hotellings       .13051   12.78959       2.00     196.00       .000
 Wilks            .88456   12.78959       2.00     196.00       .000
 Roys             .11544
 Note.. F statistics are exact.

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 EFFECT .. CONSTANT (Cont.)
 Univariate F-tests with (1,197) D. F.

 Variable   Hypoth. SS   Error SS Hypoth. MS   Error MS          F  Sig. of F

 READ        336.96220 10176.0807  336.96220   51.65523    6.52329       .011
 WRITE      1209.88188 10090.0231 1209.88188   51.21839   23.62202       .000

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 EFFECT .. CONSTANT (Cont.)
 Raw discriminant function coefficients
           Function No.

 Variable            1

 READ             .041
 WRITE            .124

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 Standardized discriminant function coefficients
           Function No.

 Variable            1

 READ             .293
 WRITE            .889

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 Estimates of effects for canonical variables
           Canonical Variable

  Parameter          1

        1        2.196

* * * * * * A n a l y s i s   o f   V a r i a n c e -- design   1 * * * * * *

 EFFECT .. CONSTANT (Cont.)
 Correlations between DEPENDENT and canonical variables
           Canonical Variable

 Variable            1

 READ             .504
 WRITE            .959

 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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 .7728.  The F-test in this output tests the hypothesis that the first canonical correlation is
equal to zero.  Clearly, F = 56.4706 is statistically significant.  However, the
second canonical correlation of .0235 is not statistically significantly different from
zero (F = 0.1087, 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
interval 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 using the hsb2 data file, let’s
suppose that we think that there are some common factors underlying the various test
scores.  We will include subcommands for varimax rotation and a plot of
the eigenvalues.  We will use a principal components extraction and will
retain two factors. (Using these options will make our results compatible with
those from SAS and Stata and are not necessarily the options that you will
want to use.)

factor
  /variables read write math science socst  
  /criteria factors(2) 
  /extraction pc
  /rotation varimax
  /plot eigen.

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.  The
scree plot may be useful in determining how many factors to retain.  From the component matrix 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.

See also