SPSS
for Windows: Descriptive and Inferential Statistics
Section
1: Summarizing Data
Descriptive
Statistics
Frequencies
Crosstabulation
Section
2: Inferential Statistics
Chi-Square
T
test
Correlation
Regression
General
Linear Model
Some Further Resources
This
course describes the use of SPSS to obtain descriptive and
inferential statistics. In the first section, you will be introduced to
procedures used to obtain several descriptive statistics, frequency tables, and
crosstabulations. In the second section, the chi-square test of independence,
independent and paired sample t tests, bivariate correlations,
regression, and the general linear model will be covered. If you are not
familiar with SPSS or need more information about how to get SPSS
to read your data, you may wish to attend our SPSS for
Windows: Getting Started course. This set of documents uses a sample
dataset, Employee data.sav, that SPSS provides. It can be found
in the root SPSS directory. If you installed SPSS
in the default location, then this file will be located in the following
location: C:\Program Files\SPSS\Employee Data.sav.
Section 1: Summarizing Data
A common
first step in data analysis is to summarize information about variables in your
dataset, such as the averages and variances of variables. Several summary or
descriptive statistics are available under the Descriptives option
available from the Analyze and Descriptive Statistics menus:
Analyze
Descriptive Statistics
Descriptives...
After
selecting the Descriptives option, the following dialog box will appear:
This
dialog box allows you to select the variables for which descriptive statistics
are desired. To select variables, first click on a variable name in the box on
the left side of the dialog box, then click on the arrow button that will move
those variables to the Variable(s) box. For example, the variables salbegin
and salary have been selected in this manner in the above example. To
view the available descriptive statistics, click on the button labeled Options.
This will produce the following dialog box:
Clicking on
the boxes next to the statistics' names will result in these statistics being
displayed in the output for this procedure. In the above example, only the
default statistics have been selected (mean, standard deviation, minimum, and
maximum); however, there are several others that could be selected. After
selecting all of the statistics you desire, output can be generated by first
clicking on the Continue button in the Options dialog box, then
clicking on the OK button in the Descriptives dialog box. The statistics
that you selected will be printed in the Output Viewer. For example, the
selections from the preceding example would produce the following output:
The
number of cases in the dataset is recorded under the column labeled N.
Information about the range of variables is contained in the Minimum and
Maximum columns. The average salary is contained in the Mean
column. Variability can be assessed by examining the values in the Std.
Deviation column. The more that individual data points differ from the
mean, the larger the standard deviation will be. Conversely, if there is a
great deal of similarity between data points, the standard deviation will be
quite small. Examining differences in variability could be useful for
anticipating further analyses: in the above example, it is clear that there is
much greater variability in the current salaries than beginning salaries.
Because equal variances is an assumption of many inferential statistics, this
information is important to a data analyst.
As a side
note, if your distribution is “normal,” almost all (96%) of your observations
should fall within +/- 2 standard deviations from the mean. A starting salary
of $32,757.37 is two standard deviations above the mean of $17,016.09, while a
starting salary of $1,274.81 is two standard deviations below the mean;
accordingly, 96% of salaries should
fall between these values, with a few people (2%) earning salaries below
$1,274.81 and a few (2%) earning salaries above $32,757.37. Given that the
minimum value is $9,000 and the maximum is $79,980, however, we can see that
these data may not follow the normal distribution. (For the purposes of this
tutorial, we will treat salary, educational level, and a number of other
variables as though they were normally-distributed continuous variables. In
your own research, however, if your outcome variables are not normally
distributed, you may need to pursue an alternate analysis. Feel free to direct
your questions on this topic to us at
While the
descriptive statistics procedure described above is useful for summarizing data
with an underlying continuous distribution, the Descriptives procedure
will not prove helpful for interpreting categorical data. Instead, it is more
useful to investigate the numbers of cases that fall into various categories.
The Frequencies option allows you to obtain the number of people within
each employment category in the dataset. The Frequencies procedure is
found under the Analyze menu:
Analyze
Descriptives Statistics
Frequencies...
Selecting
this menu item produces the following dialog box:
Select
variables by clicking on them in the left box, then clicking the arrow in
between the two boxes. Frequencies will be obtained for all of the variables in
the box labeled Variable(s). This is the only step necessary for
obtaining frequency tables; however, there are several other descriptive
statistics available, many of which are described in the preceding section. The
example in the above dialog box would produce the following output:
Going
back to the Frequencies dialog box, you may click on the Statistics
button to request additional descriptive statistics. Clicking on the Charts
button produces the following box which allows you to graphically examine their
data in several different formats:
Each of
the available options provides a visual display of the data. For example,
clicking on the Bar charts button produces the following output:
If you
have continuous data (such as salary) you can also use the Histograms option and its suboption, With normal curve, to
allow you to assess whether your data are normally distributed, which is an
assumption of several inferential statistics. (You can also use the Explore
procedure, available from the Descriptive Statistics menu, to obtain the
Kolmogorov-Smirnov test, which is a hypothesis test to determine if your
data are normally distributed.)
While
frequencies show the numbers of cases in each level of a categorical variable,
they do not give information about the relationship between categorical
variables. For example, frequencies can give you the number of men and women in
a company AND the number of people in each employment category, but not the
number of men and women IN each employment category. The Crosstabs
procedure is useful for investigating this type of information because it can
provide information about the intersection of two variables. The Crosstabs
procedure is found in the Analyze menu:
Analyze
Descriptive Statistics
Crosstabs…
After
selecting Crosstabs from the menu, the dialog box shown above will
appear on your monitor. The box on the left side of the dialog box contains a
list of all of the variables in the working dataset. If theory suggests that
one variable may cause the other, then the causal variable would typically be
placed in the Row, while the outcome variable would be placed in the Column.
For example, selecting the variable gender for the rows of the table and
jobcat for the columns would produce a crosstabulation of gender by job
category.
The
options available by selecting the Statistics and Cells buttons
provide you with several additional output features. Selecting the Cells
button will produce a menu that allows you to add additional values to your
table; it is often most informative to select the Row percentages.
The
combination of the two dialog boxes shown above will produce the following
output table:
This table shows that 95.4% of females are
clerical workers, while only 60.9% of males are clerical workers. It also seems
that men are much more likely to be custodians (5.7%) or managers (28.7%) than
are women. However, we do not yet know whether this apparent difference is
statistically significant.
Section 2:
Inferential Statistics
In the
section above, it appeared that there were some differences between men and
women in terms of their distribution among the three employment categories.
Conducting a Chi-square test of
independence would tell us if the observed pattern is statistically
different from the pattern expected due to chance.
The
Chi-square test of independence can be obtained through the Crosstabs
dialog boxes that were used above to get a crosstabulation of the data. After
opening the Crosstabs dialog box as described in the preceding section,
click the Statistics button to get the following dialog box:
By
clicking on the box labeled Chi-Square, you will obtain the Chi-square
test of independence for the variables you have crosstabulated. This will
produce the following table in the Output Viewer:
Inspecting
the table in the previous section, it appears that the two variables, gender
and employment category, are related to each other in some way. For example, if
gender and employment classification were unrelated, we would expect 17.7% of
women to be in the manager classification as opposed to the observed
percentage, 4.6%. The output above provides a statistical hypothesis test for
the hypothesis that gender and employment category are independent of each
other. The large Chi-Square statistic (79.28) and its small significance level
(p < .001) indicate that it is very unlikely that these variables are
independent of each other. Thus, you can conclude that there is a relationship
between a person's gender and their employment classification.
The t
test is a useful technique for comparing mean values of two sets of numbers.
The comparison will provide you with a statistic for evaluating whether the
difference between two means is statistically significant. T tests can
be used either to compare two independent groups (independent-samples t
test) or to compare observations from two measurement occasions for the same
group (paired-samples t test). To conduct a t test, your outcome
data should be a sample drawn from a continuous underlying distribution. If you
are using the t test to compare two groups, the groups should be
randomly drawn from normally distributed and independent populations. For
example, if you were comparing clerical and managerial salaries, the independent
populations are clerks and managers, which are two nonoverlapping groups.
If you have more than two groups or more than two variables in a single group
that you want to compare, you should use one of the General Linear Model
procedures in SPSS, which are described below.
There are
three types of t tests; the options are all located under the Analyze
menu item:
Analyze
Compare Means
One-Sample T test...
Independent-Samples
T test...
Paired-Samples T test...
While
each of these t tests compares mean values of two sets of numbers, they
are designed for distinctly different situations:
- The one-sample t test is
used compare a single sample with a population value. For example, a test
could be conducted to compare the average salary of managers within a
company with a value that was known to represent the national average for
managers.
- The independent-sample t test
is used to compare two groups' scores on the same variable. For example,
it could be used to compare the salaries of clerks and managers to
evaluate whether there is a difference in their salaries.
- The paired-sample t test
is used to compare the means of two variables within a single group. For
example, it could be used to see if there is a statistically significant
difference between starting salaries and current salaries among the
custodial staff in an organization.
To conduct an independent sample t test, first select the Independent-Samples
T test option to produce the following dialog box:
To select
variables for the analysis, first highlight them by clicking on them in the box
on the left. Then move them into the appropriate box on the right by clicking
on the arrow button in the center of the box. Your independent variable should
go in the Grouping Variable box, which is a variable that defines which
groups are being compared. For example, because employment categories are being
compared in this analysis, the jobcat variable is selected. However,
because jobcat has more than two levels, you will need to click on Define
Groups to specify the two levels of jobcat that you want to compare.
This will produce another dialog box as is shown below:
Here, the
groups to be compared are limited to the groups with the values 2 and 3, which
represent the clerical and managerial groups. After selecting the groups to be
compared, click the Continue button, and then click the OK button
in the main dialog box. The above choices will produce the following output:
The first output table, labeled Group Statistics,
displays descriptive statistics. The second output table, labeled Independent
Samples Test, contains the statistics that are critical to evaluating the
current research question. This table contains two sets of analyses: the first
assumes equal variances and the second does not. To assess whether you should
use the statistics for equal or unequal variances, use the significance level
associated with the value under the heading, Levene's Test for Equality of
Variances. It tests the hypothesis that the variances of the two groups are
equal. A small value (<.05) in the column labeled Sig. indicates that
this hypothesis is false and that the groups do indeed have unequal variances.
In the above case, the value <.05 in that column indicates that the variance
of the two groups, clerks and managers, is not equal. Thus, you should use the
t-test statistics in the row labeled Equal variances not assumed.
The SPSS output reports a t statistic and
degrees of freedom for all t test procedures. Every unique value
of the t statistic and its associated degrees of freedom have a
significance value. In the above example in which the hypothesis is that clerks
and managers do not differ in their salaries, the t statistic under the
assumption of unequal variances has a value of -16.26, and the degrees of
freedom has a value of 89.58 with an associated significance level of .000. The
significance level tells us that the probability that there is no difference between
clerical and managerial salaries is very small: specifically, less than one
time in a thousand would we obtain a mean difference of $33,038 or larger
between these groups if there were really no differences in their salaries.
To obtain
a paired-samples t test, select Paired-Samples T test and
the following dialog box will appear:
The above
example illustrates a t test between the variables salbegin and salary,
which represent employees' beginning salary and their current salary. To set up
a paired-samples t test, click on the two variables that you want to
compare. The variable names will appear in the section of the box labeled Current
Selections. When these variable names appear there, click the arrow in the
middle of the dialog box and they will appear in the Paired Variables box.
Clicking the OK button with the above variables selected will produce
output for the paired-samples t test. The following output is an example
of the statistics you would obtain from the above example.
As with the independent samples t
test, there is a t statistic and degrees of freedom that has a
significance level associated with it. The t test in this example tests
the hypothesis that there is no difference in clerks' beginning and current
salaries. The t statistic (35.04) and its associated significance level
(p < .001) indicate that this in not the case. In fact, the observed
mean difference of $17,403.48 between beginning and current salaries would
occur fewer than once in a thousand times if there really were no difference
between clerks' beginning and current salaries.
Correlation
is one of the most common forms of data analysis because it underlies many
other analyses. Correlations measure the linear relationship between two
variables. A correlation coefficient has a value ranging from -1 to 1. Values
that are closer to the absolute value of 1 indicate that there is a strong
relationship between the variables being correlated, whereas values closer to 0
indicate that there is little or no linear relationship. The sign of a
correlation coefficient describes the type of relationship between the
variables being correlated. A positive correlation coefficient indicates that
there is a positive linear relationship between the variables: as one variable increases
in value, so does the other. An example of two variables that are likely to be
positively correlated are the number of days a student attended class and test
grades, because as the number of classes attended increases in value, so do
test grades. A negative value indicates a negative linear relationship between
variables: as one variable increases in value, the other variable decreases in
value. The number of days students miss class and their test scores are likely
to be negatively correlated because as the number of days of missed classed
increases, test scores typically decrease.
Prior to conducting a
correlation analysis, it is advisable to plot the two variables to visually
inspect the relationship between them. To produce scatter
plots, select the following menu option:
Graphs
Scatter/Dot...
This will produce the following dialog box:
Although several plot options are listed, only
the Simple Scatter method is discussed here.
To create a scatter plot of current salary by education, select the Simple
Scatter option and then click the Define button to produce the
following dialog box:
If one variable can be theoretically
conceptualized as causing the other, then the causal variable would typically
be placed on the X axis, and the outcome variable on the Y axis. Click on the
current salary variable to select it from the list on the left side of the
dialog box and then click the right arrow next to the Y Axis box to move
it over to the right. Next, click on the
education variable and then click on the right arrow next to the X Axis
box to move it over to this box.
Finally, click the OK button to create the output shown below:
This plot indicates that there is a
positive, fairly linear relationship between current salary and education
level. In order to test whether this apparent relationship is statistically
significant, we could run a correlation.
To obtain
a correlation in SPSS, start at the Analyze menu. Select the Correlate
option from this menu. By selecting this menu item, you will see that there are
three options for correlating variables: (1) Bivariate, (2) Partial,
and (3) Distances. This document will cover only bivariate
correlations, which is used in situations
where you are interested only in the relationship between two variables. To obtain a bivariate correlation, choose the
following menu option:
Analyze
Correlate
Bivariate...
This will
produce the following dialog box:
To obtain
correlations, first click on the variable names in the variable list on the
left side of the dialog box. Next, click on the arrow between the two white
boxes, which will move the selected variables into the Variables box.
Each variable listed in the Variables box will be correlated with every
other variable in the box. For example, with the above selections, we would
obtain correlations between Education Level and Current Salary,
between Education Level and Previous Experience, and between Current
Salary and Previous Experience. We will maintain the default options
shown in the above dialog box in this example. The first option to consider is
the type of correlation coefficient. Pearson's is appropriate for continuous
data as noted in the above example, whereas the other two correlation
coefficients,
This output gives us a correlation matrix for the three
correlations requested in the above dialog box. Note that despite there being
nine cells in the above matrix, there are only three correlation coefficients
of interest: (1) the correlation between current salary and educational level,
the correlation between previous experience and educational level, and the
correlation between current salary and previous experience. The reason only
three of the nine correlations are of interest is because the diagonal consists
of correlations of each variable with itself, always resulting in a value of
1.00 and the values on each side of the diagonal replicate the values on the
opposite side of the diagonal. For example, the three unique correlation
coefficients show there is a positive correlation between employees' number of
years of education and their current salary. This positive correlation
coefficient (.661) indicates that there is a statistically significant (p
< .001) linear relationship between these two variables such that the more
education a person has, the larger that person's salary is. Also observe that
there is a statistically significant (p < .001) negative correlation
coefficient (-.252) for the
association between education level and previous experience, indicating that
the linear relationship between these two variables is one in which the values
of one variable decrease as the other increases. The third correlation coefficient
(-.097) also indicates a negative association between employee's current
salaries and their previous work experience, although this correlation is
fairly weak.
Regression
is a technique that can be used to investigate the effect of one or more
predictor variables on an outcome variable. Regression allows you to make
statements about how well one or more independent variables will predict the
value of a dependent variable. For example, if you were interested in
investigating which variables in the employee database were good predictors of
employees' current salaries, you could create a regression equation that would
use several of the variables in the dataset to predict employees' salaries. By
doing this you will be able to make statements about whether variables such as
employees' number of years of education, their starting salary, or their number
of months on the job are good predictors of their current salaries.
To
conduct a regression analysis, select the following from the Analyze
menu:
Analyze
Regression
Linear...
This will
produce the following dialog box:
This
dialog box illustrates an example regression equation. As with other analyses,
you select variables from the box on the left by clicking on them, then moving
them to the boxes on the right by clicking the arrow next to the box where you
want to enter a particular variable. Here, employees' current salary has been
entered as the dependent variable. In the Independent(s) box, four
predictor variables have been entered: educational level, previous experience,
beginning salary, and months since hire.
NOTE:
Before you run a regression model, you should consider the method that you use
for selecting or rejecting variables in that model. The box labeled Method
allows you to select from one of five methods: Enter, Remove, Forward,
Backward, and Stepwise. Unfortunately, we cannot offer a
comprehensive discussion of the characteristics of each of these methods here,
but you have several options regarding the method you use to remove and retain
predictor variables in your regression equation. In this example, we will use
the SPSS
default method, Enter, which is a standard approach in regression
models. If you have questions about which method is most appropriate for your
data analysis, consult a regression text book, the SPSS
help facilities, or contact a consultant.
The
following output assumes that only the default options have been requested. If
you have selected options from the Statistics, Plots, or Options
boxes, then you will have more output than is shown below and some of your
tables may contain additional statistics not shown here.
The first
table in the output, shown below, includes information about the quantity of
variance that is explained by your predictor variables. The first statistic, R,
is the multiple correlation coefficient between all of the predictor variables
and the dependent variable. In this model, the value is .90, which indicates
that there is a great deal of variance shared by the independent variables and
the dependent variables. The next value,
The
second table in the output is an ANOVA table that
describes the overall variance accounted for in the model. The F
statistic represents a test of the null hypothesis that the expected values of
the regression coefficients are equal to each other and that they equal zero.
Put another way, this F statistic tests whether the R square proportion
of variance in the dependent variable accounted for by the predictors is zero.
If the null hypothesis were true, then that would indicate that there is not a
regression relationship between the dependent variable and the predictor
variables. But, instead, it appears that the four predictor variables in the
present example are not all equal to each other and could be used to predict
the dependent variable, current salary, as is indicated by a large F
value and a small significance level.
The third table in standard regression output provides information
about the effects of individual predictor variables. Generally, there are two
types of information in the Coefficients table: coefficients and
significance tests. The unstandardized coefficients indicate the increase in
the value of the dependent variable for each unit increase in the predictor
variable. For example, the unstandardized coefficient for Educational Level
in the example is 669.91, which indicates to us that for each year of
education, a person's predicted salary will increase by $669.91. A well-known
problem with the interpretation of unstandardized coefficients is that their
values are dependent on the scale of the variable for which they were
calculated, which makes it difficult to assess the relative influence of
independent variables through a comparison of unstandardized coefficients. For
example, comparing the unstandardized coefficient of Education Level,
669.91, with the unstandardized coefficient of the variable Beginning Salary,
1.77, it could appear that Educational Level is a greater predictor of a
person's current salary than is Beginning Salary. We can see that this
is deceiving, however, if we examine the standardized coefficients, or Beta
coefficients. Beta coefficients are based on data expressed in
standardized, or z score form. Thus, all variables have a mean of zero
and a standard deviation of one and are thus expressed in the same units of
measurement. Examining the Beta coefficients for Education Level and Beginning
Salary, we can see that when these two variables are expressed in the same
scale, Beginning Salary is more obviously the better predictor of Current
Salary.
In
addition to the coefficients, the table also provides a significance test for
each of the independent variables in the model. The significance test evaluates
the null hypothesis that the unstandardized regression coefficient for the
predictor is zero when all other predictors' coefficients are fixed to zero.
This test is presented as a t statistic. For example, examining the t
statistic for the variable, Months Since Hire, you can see that it is
associated with a significance value of .000, indicating that the null
hypothesis, that states that this variable's regression coefficient is zero
when all other predictor coefficients are fixed to zero, can be rejected.
The
majority of procedures used for conducting analysis of variance (ANOVA) in SPSS
can be found under the General Linear Model (GLM) menu item in the Analyze
menu. Analysis of variance can be used in many situations to determine whether
there are differences between groups on the basis of one or more outcome
variables or if a continuous variable is a good predictor of one or more
dependent variables. There are three varieties of the general linear model
available in SPSS: univariate, multivariate, and repeated
measures. The univariate general linear model is used in
situations where you only have a single dependent variable, but may have
several independent variables that can be fixed between-subjects factors,
random between-subjects factors, or covariates. The multivariate general
linear model is used in situations where there is more than one dependent
variable and independent variables are either fixed between-subjects factors or
covariates. The repeated measures general linear model is used in
situations where you have more than one measurement occasion for a dependent
variable and have fixed between-subjects factors or covariates as independent
variables. Because it is beyond the scope of this document to cover all three
varieties of the general linear model in detail, we will focus on the
univariate version of the general linear model with some attention given to
topics that are unique to the repeated measures general linear model. Several
features of the univariate general linear model are useful for understanding
other varieties of the model that are provided in SPSS:
understanding the univariate model will prove useful for understanding other GLM
options.
The
univariate general linear model is used to compare differences between group
means and estimating the effect of covariates on a single dependent variable.
For example, you may want to see if there are differences between men and
women's salaries in a sample of employee data. To do this, you would want to
demonstrate that the average salary is significantly different between men and
women. However, in doing such an analysis, you are likely aware that there are
other factors that could affect a person's salary that need to be controlled
for in such an analysis. For example, educational background and starting
salary are some such variables. By including these variables in our analysis,
you will be able to evaluate the differences between men and women's salaries
while controlling for the influence of these other variables.
To
specify a univariate general linear model in SPSS, go to the
analyze menu and select univariate from the general linear model menu:
Analyze
General Linear Model
Univariate...
This will
produce the following dialog box:
The above
box demonstrates a model with multiple types of independent variables. The
variable gender has been designated as a fixed factor because it
contains all of the levels of interest.
In
contrast, random variables are variables that represent a random sample
of the possible levels that could be sampled. There are not any true random
variables in our dataset; therefore, this input box has been left blank here.
However, you could imagine a situation similar to the above example where you
sampled data from multiple corporations for our employee database. In that
case, you would have introduced a random variable into the model--the
corporation to which an employee belongs. Corporation is a random factor
because you would only be sampling a few of the many possible corporations to
which you would want to generalize your results. (It should be noted, however,
that the GLM method for handling random factors is inferior to the method used
in the Mixed Models procedure.)
The next
input box contains the covariates in your model. A covariate is a
quantitative independent variable. Covariates are often entered in models to
reduce error variance: by removing the effects of the relationship between the
covariate and the dependent variable, you can often get a better estimate of
the amount of variance that is being accounted for by the factors in the model.
Covariates can also be used to measure the linear association between the
covariate and a dependent variable, as is done in regression models. In this
situation, a linear relationship indicates that the dependent variable
increases or decreases in value as the covariate increases or decreases in
value.
The box
labeled WLS Weight can contain a variable that is used to weight other
variables in a weighted least-squares analysis. This procedure is infrequently
used however, and is not discussed in any detail here.
The
default model for the SPSS univariate GLM will include
main effects for all independent variables and will provide interaction terms
for all possible combinations of fixed and random factors. You may not want
this default model, or you may want to create interaction terms between your
covariates and some of the factors. In fact, if you intend to conduct an
analysis of covariance, you should test for interactions between covariates and
factors. Doing so will determine whether you have met the homogeneity of
regression slopes assumption, which states that the regression slopes for
all groups in your analysis are equal. This assumption is important because the
means for each group are adjusted by averaging the slopes for each group so
that group differences in the covariate are removed from the dependent
variable. Thus, it is assumed that the relationship between the covariate and
the dependent variable is the same at all levels of the independent variables.
To make changes in the default model, click on the Model button, which
will produce the following dialog box:
The first
step for modifying the default model is to click on the button labeled Custom,
to activate the grayed out areas of the dialog box. At this point, you can
begin to move variables in the Factors & Covariates box into the Model
box. First, move all of the main effects into the Model box. The
quickest way to do that is to double-click on their names in the Factors
& Covariates box. After entering all of the main effects, you can begin
building interaction terms. To build the interactions, click on the arrow
facing downwards in the Build Term(s) section and select interaction, as
shown in the figure above. After you have selected the interaction, you can
click on the names of the variables with which you would like to build an
interaction, then click on the arrow facing right under the Build Term(s)
heading. In the above example, the educ*gender term has already been
created. The salbegin*gender and salbegin*educ terms can be
created by highlighting two terms at a time as shown above, then clicking on
the right-facing arrow. Some of the other options in the Build Terms
list that you may find useful are the All n-way options. For example if
you highlighted all three variables in the Factors & Covariates box, you
could create all of the three possible 2-way interactions by selecting the All
2-way option from the Build Terms(s) drop-down menu, then clicking
the right-facing arrow.
If you
are testing the homogeneity of regression slopes assumption, you should examine
your group by covariate interactions, as well as any covariate by covariate
interactions. In order to meet the ANCOVA assumption, these interactions should
not be significant. Examining the output from the example above, we expect to
see nonsignificant effects for the gender*educ and the gender*salbegin
interaction effects:
Examining
the group by covariate effects, you can see that both were nonsignificant. The gender*salbegin
effect has a small F statistic (.660) and a large significance value
(.417), the educ*salbegin effect also has a small F statistic
(1.808) and large significance value (.369), and the salbegin*educ
effect also has a small F statistic (1.493) and large significance level
(.222). Because all of these significance levels are greater than .05, the
homogeneity of regression assumption has been met and you can proceed with the
ANCOVA.
Knowing
that the model does not violate the homogeneity of regression slopes
assumption, you can remove the interaction terms from the model by returning to
the GLM
Univariate dialog box, clicking the Model button, and selecting Full
Factorial. This will return the model to its default form in which there
are no interactions with covariates. After you have done this, click OK
in the GLM Univariate dialog box to produce the
following output:
The default output for the univariate general linear model contains
all main effects and interactions between fixed factors. The above output
contains no interactions because gender is the only fixed factor. Each factor,
covariate, or other source of variance is listed in the left column. For each
source of variance, there are several test statistics. To evaluate the
influence of each independent variable, look at the F statistic and its
associated significance level. Examining the first covariate, education level,
the F statistic (37.87) and its associated significance level (.000)
indicate that it has a significant linear relationship with the dependent
variable. The second covariate, salbegin, also has a significant F
statistic (696.12) as can be seen from its associated significance level
(.000). In both cases, this indicates that the values of the dependent
variable, salary, increase as the values of education level and
beginning salaries increase. The next source of variance, gender, provides us
with a test of the null hypothesis that there are no differences between gender
groups (i.e., that there are not differences between men and women's salaries)
when education and beginning salary are controlled. This test provides a small F
statistic (3.87) and a significance level that is just barely statistically significant
(p = .05). In the above model containing education level and beginning
salaries as covariates, we can say that there is a statistically significant
difference between men and women's salaries. To further interpret this effect,
go back to the Univariate dialog box
and click on the Options button.
Choose the variable gender and click it over to the Display Means For box, then choose Continue and OK. This
will present the estimated means for each group, taking into account the
covariates educational level and beginning salary, as shown below. Note that
after setting both groups equal to the mean educational and beginning salary
level, men in this company earn approximately $1,590 more than women in current
salary. It is possible that the inclusion of other covariates could explain
away the remaining difference between the two groups’ salaries.
The
repeated measures version of the general linear model has many similarities to
the univariate model described above. However, the key difference between the
models is that there are multiple measurement occasions of the dependent
variable in repeated measures models, whereas the univariate model only permits
a single dependent variable. You could conduct a similar model with repeated
measurements by using beginning salaries and current salaries as the repeated
measurement occasions.
To
conduct this analysis, you should select the Repeated Measures option
from the General Linear Model submenu of the Analyze menu:
Analyze
General Linear Model
Repeated
Measures...
Selecting
this option will produce the following dialog box:
This
dialog box is used for defining the repeated measures, or within-subjects,
dependent variables. You first give the within-subject factor a name in the box
labeled Within-Subject Factor Name. This name should be something that
describes the dependent variables you are grouping together. For example, in
this dialog box, the salaries that are being analyzed differ in terms of the
time in which the salary data were collected, so the within-subject factor was
given the name time. Next, specify the number of levels, or number
of measurement occasions, in the box labeled Number of Levels. This is
the number of times the dependent variable was measured. Thus, in the present
example, there are two measurement occasions for salary because you are
measuring beginning salaries and current salaries. After you have filled in the
Within-Subject Factor Name and the Number of Levels input boxes,
click the Add button to transfer the information in the input boxes into
the box below. Repeat this process until you have specified all of your
within-subject factors. Then, click on the Define button, and the
following dialog box will appear:
When this
box initially appears, you will see a slot for each level of the within-subject
factor variables that you specified in the previous dialog box. These slots are
labeled numerically for each level of the within-subject factor but do not
contain variable names. You still need to specify which variable fills each
slot of the within-subject factors. To do this, click the appropriate variable
name in the list on the left side of the
dialog box. Next, click on the arrow pointing towards the Within-Subject
Variables dialog box to move the variable name from the list to the top
slot in the within-subjects box. This process has been completed for salbegin,
the first level of the salaries within-subject factor. The same process
should be repeated for salary, the variable representing an employee's
current salary.
After you
have completed the specifications for the within-subjects factors, you can
define your independent variables. Between-subject factors, or fixed factors,
should be moved into the box labeled Between-Subjects Factors(s) by first
clicking on the variable name in the variable list, then clicking on the arrow
to the left of the Between-Subjects Factor(s) box. In this example,
gender has been selected as a between-subjects factor. Covariates, or
continuous predictor variables, can be moved into the Covariates box.
Above, educ, the variable representing employee's number of years of
education, has been specified as a covariate.
This will
produce several output tables, but we will focus here on the tables describing
between-subject and within-subject effects. However, these tables for
univariate analysis of variance may not always be the appropriate. The
univariate tests have an additional assumption: the assumption of sphericity.
If this assumption is violated, you should use the multivariate output or
adjust your results using one of the correction factors in the SPSS
output. For a more detailed discussion of this topic, see the usage note, Repeated
Measures ANOVA
Using SPSS
MANOVA
in the section, "Within-Subjects Tests: The Univariate versus the
Multivariate Approach." This usage note can be found at http://www.utexas.edu/cc/docs/stat38.html.
The
following output contains the statistics for the effects in the model specified
in the above dialog boxes:
This table contains information about the within-subject factor, time,
and its interactions with the independent variables. The main effect for
salaries is a test of the null hypothesis that all levels of within-subjects
factor are equal, or, more specifically, it is a test of the hypothesis that
beginning and current salaries are equal. The F statistic (18.915) and
its associated significance level (p < .001) indicate that you can
reject this hypothesis as false. In other words, it appears that there is a
statistically significant difference between beginning salaries and current
salaries. After you have tested this hypothesis, you can then investigate
whether the increase in salaries is the same across all values or levels of the
other independent variables that are included in the model. The first
interaction term in the table tests the hypothesis that the increase in
salaries is constant, regardless of educational background. The F
statistic (172.10) and its associated significance level (p < .001)
allow us to reject this hypothesis as well. The knowledge that this interaction
is significant indicates that it is worthwhile to examine characteristics of
the interaction, which we will do using plots in a later example. Finally, the
second interaction tests whether salary increase differs by gender. The F
statistic (25.07) and its significance level (p < .001) indicate that
the increase in salaries does vary by gender.
The output for the repeated measures general
linear model also provides statistics for between-subject effects. In this
example, the model contains two between-subjects factors: employees' education
level and their gender. Education level was entered as a covariate in the
model, and therefore the statistics associated with it are a measure of the
linear relationship between education level and salaries. In contrast, the
statistics for the between-subjects factor, gender, represents a comparison
between groups across all levels of the within-subjects factors. Specifically,
it is a comparison between males and females on differences between their
beginning and current salaries. In the above example, both education level and
gender are statistically significant. The F statistic (277.96) and
significance level (p < .001) associated with education level allows
us to reject the null hypothesis that there is not a linear relationship
between education and salaries. By rejecting the null hypothesis, you can
conclude that there is a positive linear relationship between the two
variables, indicating that as number of years of education increases, salaries
do as well. The F statistic (55.79) for gender and its associated
significance level (p < .001) represent a test of the null hypothesis
that there are no group differences in salaries. The significant F
statistic indicates that you can reject this null hypothesis and conclude that
there is a statistically significant difference between men and women's
salaries.
In this
analysis, we saw that there was a significant interaction for Time*Gender. It
is important to investigate the properties of your interactions through
graphical displays, mean comparisons, or statistical tests, because a
significant interaction can take on many forms. For this analysis, we will add
a plot for the Time*Gender interaction by returning to the Repeated Measures dialog box and clicking on the Plots button.
Typically,
within-subjects factors such as time are
placed in the Horizontal Axis box,
while between-subjects factors are placed in the Separate Lines box. After placing time and gender in the
appropriate box, click on Add to
place the interaction plot into the list in the bottom pane, then choose Continue and OK. This yields the plot below.
Here, the
interaction reflects the fact that male salaries tend to increase more sharply
over time than do female salaries. Note that the plot uses the estimated
marginal means for salary, which are the estimated means when the covariate (in
this case, education) is set at its average (13.49 years).
Some Further
Resources
For more
information on SPSS, try the following resources:
- To make a
free statistical appointment with the Statistical Support group, visit http://ssc.utexas.edu/consulting/free_consulting.html,
email us at stats@ssc.utexas.edu, or
schedule an appointment by phone through the ITS helpdesk: (512) 475-9400.
- Visit our
list of answers to frequently asked SPSS questions at: http://ssc.utexas.edu/consulting/answers/faqs.html
- Go to the
SPSS homepage, http://www.spss.com.
Their site includes a variety of helpful resources, including the SPSS
Answer Net for answers to frequently asked technical questions, and an FTP
site where you can find SPSS macros and an online version of the SPSS
Algorithms manual. The support website is available at http://support.spss.com.