ASSOCIATION BETWEEN
VARIABLES:
SCATTERGRAMS
Topic #11
Like Father, Like Son
Though it is not especially relevant to political science,
suppose we want to research the following bivariate
hypothesis that involves two interval and continuous
variables.
FATHER’S HEIGHT
+
(actual height in inches) ======>
ADULT SON’S HEIGHT
[father-
(actual height in inches)
adult son
pairs]
Like Father, Like Son (cont.)
We select a random sample of n = 1078 pairs of fathers
and their adult sons and collect the relevant data, the
first five cases, as well as the last case, of which
appears below.
Note that the unit of analysis is father-adult son pairs and
observed values have been very precisely measured
and recorded, so probably each case has a unique
recorded value on each variable.
Pair ID
1
2
3
4
5
.
1078
Father’s Height (inches)
66.67
69.83
65.19
65.15
64.66
.
62.31
Son’s Height (inches)
68.42
70.32
69.76
73.85
70.17
.
62.09
Like Father, Like Son (cont.)
We cannot straightforwardly crosstabulate these variables, because the
variables are continuous, each having an infinite number of possible
values, so each case would be in a unique row and column.
So what should we do? One possibility is to create class intervals for
both variables (as discussed in Topic #5 on histograms) — in effect,
to turn them into discrete variables and then proceed as before.
For example, we might create class intervals for both variables with this
recoding scheme:
Short
Medium
Tall
Pair ID
1
2
3
4
5
less than 65 inches
65-70 inches
greater than 70 inches
66.67
69.83
65.19
65.15
64.66
FH
Med
Med
Med
Med
Short
68.42
70.32
69.76
73.85
70.17
SH
Med
Tall
Med
Tall
Tall
Like Father, Like Son (cont.)
• We then can set up a crosstabulation worksheet and
begin to tally cases as shown above.
• If hypothesis is true (and if on average fathers and sons
are about the same height), we would expect most cases
to fall in the main diagonal, i.e., in the S&S, M&M, and
T&T cells.
Like Father, Like Son (cont.)
But note that this approach is not very satisfactory.
Pair ID
1
2
3
4
5
F’s Height
66.67
69.83
65.19
65.15
64.66
S’s Height
68.42
70.32
69.76
73.85
70.17
Cell
M&M
M&T
M&M
M&T
S&T
Like Father, Like Son (cont.)
• Creating class intervals when you have gone to the
trouble of measuring continuous variable quite precisely
entails throwing away valuable information that bears on
the hypothesis of interest.
– This problem can be mitigated by creating more
refined class intervals.
– But what we really should do is have “infinitely”
refined class intervals that match the very precise
information we have collected.
• A very nice analytical device called a scattergram (or
scatterdiagram or scatterplot), similar in its basic logic to
a crosstabulation, allows us to do just this.
Like Father, Like Son (cont.)
• First, we need to set up a scattergram template or
worksheet, which is similar in logic to that for a crosstabulation but reflects the continuous character of both
variables.
• Figure 1 shows the general template for such a
scattergram.
• We draw a horizontal interval scale [just as in a
histogram] representing values of the independent
variable (corresponding to the column variable in a
crosstab).
– This scale should be appropriately labeled and calibrated to
encompass the full range of observed values found in the data
(but it needn’t, and probably shouldn’t, be much wider than this).
• We then erect a vertical interval scale that similarly
represents values of the dependent variable (corresponding to the row variable in a crosstab).
Like Father, Like Son (cont.)
• Just as cases are placed in cells of a crosstabulation
defined by the intersection of the row and column
corresponding to the particular combination of (discrete)
variable values that characterizes the case, each case in
a scattergram is plotted at the point defined by the
intersecting of horizontal and vertical lines corresponding
to the particular combination of (continuous) variable
values that characterizes the case.
– In a sense, each case falls (almost always) in its own unique and
tiny cell.
• Figure 2 shows the scattergram worksheet and the
plotted points for each of the five father-son pairs listed
in the previous slides.
– To facilitate locating each point, I have put a (1" x 1") grid over
the scattergram.
Some Guidelines for Creating
Scattergrams
• Graph paper is very useful for making hand-drawn
scattergrams, as you will do in Problem Set #11.
• Draw the interval measurement scales on the left and
bottom margins of the scattergram.
• The end points of each scale must accommodate the
maximum and minimum (or range of) values observed in
the data for each variable.
• But (as a general rule) the end points should not be much
more extreme than these maximums and minimums.
– In other words, when the data has been plotted, there should not
be a lot of unnecessary “white space” in the finished (presentation
grade) scattergram.
• Scale the axes so that the scattergram is either approximately square or somewhat wider than tall (SPSS does
the latter by default).
Like Father, Like Son (cont.)
• A statistician named Karl Pearson actually conducted
such a study of father-son pairs over 100 years ago in
England.
• Having collected height data on 1078 father-son pairs,
Pearson realized that a list of 1078 pairs of numbers
would be impossible to grasp as raw data and that a
crosstabulation using class intervals would have the
problems discussed previously.
• Pearson therefore developed the scattergram, an
alternate analytic device appropriate for analyzing
association in such data.
• Figure 3 shows the scattergram of Pearson’s data.
– This scattergram is taken from David Freedman et al., Statistics, p. 110.
Questions Pertaining to the
Pearson Scattergram
• What is the significance of 45° line in the scattergram?
• On average, are the two generations about the same
height? If not, which generation is taller on average?
• What is the approximate average height of the sons? Of
the fathers?
• What is the significance of the two vertical dotted lines?
• What is the average height of all sons whose fathers are
about 72 inches tall?
• How does the average height of sons vary with the
height of their fathers?
• Is the there an association between the variables?
• Is it positive or negative?
• How strong is the association between the two
variables?
Questions Pertaining to the
Pearson Scattergram (cont.)
• Points that lie (approximately) on the 45° are cases in
which it is (approximately) true that FH = SH.
• Also note that a clear majority (maybe 60%) of points lie
“northwest” of the 45° line, indicating that most sons are
taller than their fathers.
• Move a vertical line left and right until it appears that half
of the points lie on either side. This line corresponds to
the median height of father; their mean height is about
the same.
• There is a distinct “football-shaped” cloud of points
running from southwest (Short-Short) to “northeast” (TallTall), indicating a positive association.
Questions Pertaining to the
Pearson Scattergram (cont.)
• The points inside the two vertical dotted lines represent
all cases in which the father is about 72 inches tall.
– On average these 72” fathers have sons who are shorter than
their fathers, though at the same time taller than the average of
all sons.
• We can draw other vertical strips and find the average
height of the sons of these fathers.
• The line of averages indicates that the average height of
sons increases with fathers’ height (positive association).
• But there is still a lot of dispersion in sons’ height within
each vertical strip, so the association is far from perfect.
Association Visualized
• The following figures show other scattergrams for small
hypothetical data sets.
– These were produced by the Statistical “Applet” on
Correlation and Regression Demo available at the
course web site.
– These scattergrams correspond directly to Tables 1B1E near the beginning of the previous handout on
crosstabulations, in that they show differing degrees
of association running from high positive through zero
to moderate negative.
– The (Pearson) correlation coefficient is the standard
measure of association between two interval
variables.
Crosstabs vs. Scattergram
• It may at first blush seem
puzzling that
– in a standard
crosstabulation a
concentration of cases
running from
“northwest” to
“southeast” (the main
diagonal) reflects a
positive association,
while
– in a scattergram the
same pattern reflects a
negative association
(and vice versa).
Crosstabs vs. Scattergram (cont.)
• This reflects only a cosmetic
difference in setting things up:
– in a crosstabulation, the
values of the row (vertical)
variables are usually
placed in descending
order (lowest value on top,
highest values at the
bottom), while
– in a scattergram the
values of the vertical (row)
variable are invariably
(and more sensibly)
placed in ascending order
(lowest value at the
bottom, highest value on
top).
Crosstabs vs. Scattergram (cont.)
• But remember that scattergrams and crosstabulations are
essentially similar devices.
• We can show how directly they are logically connected, and also
– how the former is much more informative that the latter.
• Suppose that, having constructed Pearson’s scattergram, we want
also to construct a full crosstabulation of the data using the shortmedium-tall class intervals we previously worked with.
Crosstabs vs. Scattergram (cont.)
This can be
accomplished
simply by
• superimposing
the appropriate 3x3
grid (table)
on the
scattergram, and
then
• counting
the number
of plotted
points in
each
resulting
cell.
Test 1: Blue Book Score by MC Score
Here is
some
more
topical
(and
politically
relevant)
bivariate
data
Data as of 11/05/08
Obama Vote in 2008 Vote By Kerry
Vote in 2004
• A scattergram of more topical interest is shown on the
following slide.
• Since there are only 50 cases (states), the plotted points
can be individually labeled.
– What is the significance of the four quadrants of the
scattergram?
– What is the significance of the blue diagonal line?
– What is the significance of the green diagonal line?
Note: DC not included in scattergram
SPSS Scattergrams
• SPSS can readily create scattergrams.
– You can use SPSS (or Excel) it for PS #11 if you wish (see Note 5).
• For example, let’s open the PRESIDENTIAL ELECTION
data file, giving state by state vote totals for each
Presidential candidate in each election, and then
– find the variables dem2000, rep2000, dem2004, and rep2004;
– compute the DEMOCRATIC PERCENT OF THE TWO-PARTY
VOTE in each election,
• by clicking on Transform => Compute and entering this
expression in the Compute Variables dialog box:
– d2pc2004 = 100 * dem2000/(dem2000 + rep2004)
• and likewise for d2pc2004; and then
– produce the following scattergram by clicking on Graphs =>
Scatter... => Simple/Define and then in the Simple Scatterplot
dialog box put d2pc2004 on the Y Axis and d2pc2000 on the X
Axis.
Histogram of Scores on Test 2
Scattergram of P2 by MC2
Scattergram of Score2 by Score 1
Correlation Matrix