# Discriminant Function Analysis in SPSS

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```					Discriminant Function Analysis in SPSS
To do DFA in SPSS, start from “Classify” in the “Analyze” menu (because we’re trying to
classify participants into different groups). In this case we’re looking at a dataset that describes
children with ADHD. Research tells us that the degree of symptom impairment in childhood
disorders is often in the eye of the beholder—some adults may think that a child has difficulty
behaving, and other adults may not see it as a problem. For instance mothers (who are still most
often the child’s primary caregiver) may think that a child has difficulty sitting still, while fathers
(who may see the child primarily in less-structured “play” settings) may believe the behavior is
“just kids being kids.” We’re going to look at whether the symptoms reported on four different
measures (q1, q2, q3, and q4) can tell us whether those symptom ratings were provided by the
child’s mother (“parent” = 1) or by the child’s father (“parent” = 2).
In the dialog box, put in “parent” as the “grouping variable” (in other words, the variable that
you think defines different groups—your criterion variable for the analysis). It will appear in the
box with two question marks after it—you have to tell SPSS what the codes are for the two
groups that you want to compare. Hit “Define Range” and type in “1” and “2” as the different
values of the “parent” variable that you want to compare. Then hit “Continue” to go on.
Now put the four “question” variables (q1-q4) in as the predictors (“independents”):

Hit the “Statistics” button to go on.
This sub-dialog box lets you see descriptive statistics on each predictor variable for the different
groups. Let’s check “Means” to see some basic descriptive stats.

Hit “Continue” to go back to the main dialog box.

Then, in the main dialog box, hit the “Classify” button to see the next sub-dialog.
Here’s the sub-dialog that you get when you hit the “Classify” button:

On this screen, check the box for the “summary table.” This will give you the classification table
(sensitivity, specificity, etc.) on your printout.

Hit “Continue” to go back to the main dialog box, and then hit “OK” to see the results of your
analysis.
Here are the results:

Analysis Case Processing Summary

Unweighted Cases                               N          Percent
Valid                                               89        97.8
Excluded           Missing or out-of-
range group codes                 0               .0
At least one
missing
2              2.2
discriminating
variable
Both missing or
out-of-range group
codes and at least
0               .0
one missing
discriminating
variable
Total                             2              2.2
Total                                               91        100.0

This table just tells you if there’s any missing data.

Group Statistics

Mother or                                                                    Valid N (listwise)
Father?                             Mean           Std. Deviation         Unweighted     Weighted
Father          Question 1              1.54                 .519                  13      13.000
Question 2              1.00                 .000                  13        13.000
Question 3              1.92               1.038                   13        13.000
Question 4              1.62                 .870                  13        13.000
Mother          Question 1              2.41                 .819                  76        76.000
Question 2              2.67                 .598                  76        76.000
Question 3              2.17                 .915                  76        76.000
Question 4              1.82                 .905                  76        76.000
Total           Question 1              2.28                 .839                  89        89.000
Question 2              2.43                 .810                  89        89.000
Question 3              2.13                 .932                  89        89.000
Question 4              1.79                 .898                  89        89.000

This table shows the means that we asked for—it gives means on each variable for people in
each sub-group, and also the overall means on each variable.
Summary of Canonical Discriminant Functions
Eigenvalues

Canonical
Function     Eigenvalue          % of Variance    Cumulative %        Correlation
1               1.387(a)                  100.0           100.0               .762
a First 1 canonical discriminant functions were used in the analysis.

This table tells you something about the “latent variable” that you’ve constructed (i.e., the
discriminant function), which helps you to differentiate between the groups. For more about
eigenvalues, see the website information on the topic of “Factor Analysis.”

Wilks' Lambda

Wilks'                                       Sig.
Test of Function(s)         Lambda       Chi-square        df
1                                .419         73.945              4          .000

Here’s the multivariate test—Wilks’ lambda, just like in MANOVA. Because p < .05, we can
say that the model is a good fit for the data. This multivariate test is a goodness of fit statistic,
just like the F-test is for regression.

Standardized Canonical Discriminant Function Coefficients

Function
1
Question 1             .149
Question 2            1.007
Question 3                .049
Question 4                .382

These “discriminant function coefficients” work just like the beta-weights in regression. Based
on these, we can write out the equation for the discriminant function:

DF = .149*q1 + 1.007*q2 + .049*q3 + .382*q4

Using this equation, given someone’s scores on q1, q2, q3, and q4, we can calculate their score
on the discriminant function. To figure out what that DF score means, look at the group
centroids, below.
Structure Matrix

Function
1
Question 2              .914
Question 1              .336
Question 3              .081
Question 4           .068
Pooled within-groups correlations between discriminating variables and standardized canonical discriminant functions
Variables ordered by absolute size of correlation within function.

Ignore this table—we don’t use it for anything. Make sure that you aren’t looking at the
“structure matrix” when you get the coefficients for the discriminant function. Those come from
the table called “standardized canonical discriminant function coefficients” (which looks a lot
like this one, but isn’t the same).

Functions at Group Centroids

Function
Mother or Father?              1
Father                         -2.815
Mother                           .482
Unstandardized canonical discriminant functions evaluated at group means

Here are the group centroids. If someone’s score on the discriminant function is closer to
–2.815, then those answers were probably the child’s father. If the person’s score on the DF is
closer to .482, then the data probably came from the child’s mother. In practical terms, we
usually figure out which group a person is in by calculating a cut score halfway between the two
centroids:

Cut Score = (-2.815 + .482) / 2 = -1.167

If an individual person’s score on the DF (calculated by plugging in their scores on q1, q2, q3,
and q4 to the DF equation we wrote out above) is above –1.167, then they were probably the
child’s mother. If their DF score is below –1.167, then they were probably the child’s father.
Classification Statistics
Classification Processing Summary

Processed                                                  91
Excluded                Missing or out-of-
range group codes                   0
At least one missing
discriminating                      2
variable
Used in Output                                             89

Prior Probabilities for Groups

Cases Used in
Mother or Father?        Prior          Analysis

Unweighted               Weighted
Father                      .500                      13               13.000
Mother                      .500                      76                     76.000
Total                      1.000                      89                     89.000

Classification Results(a)

Predicted Group
Membership
Mother or
Father?           Father           Mother           Total
Original      Count          Father                  13                  0               13
Mother                   6              70                  76
%              Father                100.0              .0              100.0
Mother                 7.9             92.1              100.0
a 93.3% of original grouped cases correctly classified.

Here’s the classification table that we got by selecting that option in the SPSS dialog box. Like
the table shown in this week’s class notes, it gives information about actual group membership
vs. predicted group membership.

--Overall % correctly classified = 93.3%
--Sensitivity = 13 / 13 = 100%
--Specificity = 70 / 76 = 92.1%

Looking at the columns in this table instead of the rows, you can also calculate PPV and NPV:
--PPV = 13 / (13+6) = 68.4%
--NPV = 70 / 70 = 100%
Now let’s go back to the main dialog box for DFA, and try a stepwise procedure instead.

The only change needed is to change this radio button to “use stepwise method.” Now SPSS will
select the best predictor or set of predictors from the four original possibilities.
If you click on the “Method” button (which was grayed-out before, but becomes active once you
select “Use Stepwise Method”), you will see the following dialog box. Take a look at it—it tells
SPSS how to do the steps, and when to stop adding steps. Just leave everything here on its
default settings—no need to make any changes.

Hit “Continue” here, and then “OK” in the main dialog box to see your results.
Discriminant
[information omitted]

Analysis 1

Stepwise Statistics

Variables Entered/Removed(a,b,c,d)

Step                                                              Wilks' Lambda
S                                                               Exact F
t
a
t
i
s
t
i
Entered     c     df1           df2          df3         Statistic    df1             df2      Sig.
1                       .
4
Question 2               1              1      87.000        100.720           1        87.000      .000
6
3
2         Question 4 .
4
2              1      87.000         58.050           2        86.000      .000
2
6
At each step, the variable that minimizes the overall Wilks' Lambda is entered.
a Maximum number of steps is 8.
b Minimum partial F to enter is 3.84.
c Maximum partial F to remove is 2.71.
d F level, tolerance, or VIN insufficient for further computation.

Here’s a new table that shows you the steps SPSS went through. Based on this table, “Question
2” is the best single predictor, and “Question 4” is the next-best one. If you were asked “how
would be “two of them: Q2 and Q4.”

Variables in the Analysis

Wilks'
Step                      Tolerance     F to Remove      Lambda
1           Question
1.000         100.720
2
2           Question
.902         114.829           .994
2
Question
.902            7.664          .463
4
Variables Not in the Analysis

Min.                         Wilks'
Step                      Tolerance      Tolerance      F to Enter      Lambda
0            Question
1.000            1.000      13.624               .865
1
Question
1.000            1.000     100.720               .463
2
Question
1.000            1.000           .785            .991
3
Question
1.000            1.000           .550            .994
4
1            Question
.981             .981          2.500            .450
1
Question
.986             .986           .039            .463
3
Question
.902             .902          7.664            .426
4
2            Question
.957             .873          1.226            .419
1
Question
.909             .831           .329            .424
3

These next two tables just show you which predictors were used in each step.

Wilks' Lambda

Exact F
Number of
Step       Variables      Lambda          df1           df2             df3            Statistic   df1             df2      Sig.
1                     1        .463               1             1               87      100.720           1        87.000      .000
2                     2        .426               2             1               87       58.050           2        86.000     .000

Here are the Wilks’ lambdas for each step. As you can see, the model is a good fit for the data
with just one predictor (Q2) or with two predictors (Q2 and Q4).

Summary of Canonical Discriminant Functions
Eigenvalues

Canonical
Function     Eigenvalue      % of Variance      Cumulative %         Correlation
1                1.350(a)             100.0             100.0                 .758
a First 1 canonical discriminant functions were used in the analysis.

Wilks' Lambda

Wilks'
Test of Function(s)       Lambda       Chi-square         df             Sig.
1                             .426           73.479              2            .000
Standardized Canonical Discriminant Function Coefficients

Function
1
Question 2          1.050
Question 4              .397

If you wanted to construct a predictive equation using just the two best predictors, it would be:

DF = 1.050*q2 + .397*q4

Structure Matrix

Function
1
Question 2              .926
Question
.186
1(a)
Question 4            .068
Question
.003
3(a)
Pooled within-groups correlations between discriminating variables and standardized canonical discriminant functions
Variables ordered by absolute size of correlation within function.
a This variable not used in the analysis.

Functions at Group Centroids

Function
Mother or Father?              1
Father                         -2.778
Mother                     .475
Unstandardized canonical discriminant functions evaluated at group means

If you wanted to know whether someone’s score on this new, simpler DF suggested that they
were the child’s mother or father, you’d compare their score on the DF to these centroids. If
their score were closer to –2.778, they were probably the child’s father; if their score were closer
to .475, they were probably the child’s mother. (Note that these centroids are different from the
ones we used when all four variables were in the model, just as the discriminant function
coefficients are different. You have to use the numbers that correspond to the model with the
specific variables that you want to include).
One last step—let’s get a graph of the different groups, based on the two best predictors (q2 and
q4). This gives us a visual representation that shows how the two groups separate out from one
another using these two predictors.

Graphs are found in the “Graphs” menu in SPSS. We’ll use a scatterplot to show the
differentiation between the two groups, so select “Scatter” from the “Graphs” menu.
This box appears. Leave the graph set on “Simple,” and hit the “Define” key to go on.
Here’s the dialog box where you enter variables for the graph. Put q2 and q4 on the two axes of
the graph (y and x-axes: it doesn’t really matter which one goes on which axis, but if you set
them up this way, your graph will look the same as mine. If you reverse them, it will look like
my graph turned on its side. Either way is correct).

The “Set Markers By…” command lets you show the two different groups on the DV (mothers
vs. fathers) in two different colors on the graph. This is how you will be able to see the
differentiation between the two groups.

Hit “OK” to see the output.
Here’s the graph:

3.5

3.0                                                              Members of Group 1

2.5

2.0

1.5
Question 2

Mother or Father?
1.0
Mother
Members of Group 2
.5                                              Father
.5   1.0     1.5   2.0   2.5   3.0   3.5

Question 4

You can see on this graph how the two groups are visually separated from one another, based on
people’s answers to q2 and q4. Not all discriminant functions will separate groups this perfectly.
Sometimes you can find predictors that statistically differentiate between the groups, while the
graphical representation still shows the groups as pretty jumbled together.

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