ASSOCIATION BETWEEN VARIABLES: SCATTERGRAMS

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ASSOCIATION BETWEEN VARIABLES SCATTERGRAMS

• Topic 11

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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.
• FATHERS HEIGHT ADULT SONS
  HEIGHT father-
• (actual height in inches) gt (actual
  height in inches) adult son
  pairs

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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 Fathers Height (inches) Sons Height
  (inches)
• 1 66.67 68.42
• 2 69.83 70.32
• 3 65.19 69.76
• 4 65.15 73.85
• 5 64.66 70.17
• . . .
• 1078 62.31 62.09

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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 less than 65 inches
• Medium 65-70 inches
• Tall greater than 70 inches
• Pair ID FH SH
• 1 66.67 Med 68.42 Med
• 2 69.83 Med 70.32 Tall
• 3 65.19 Med 69.76 Med
• 4 65.15 Med 73.85 Tall
• 5 64.66 Short 70.17 Tall

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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 SS, MM, and TT cells.

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Like Father, Like Son (cont.)

• But note that this approach is not very
  satisfactory.
• Pair ID Fs Height Ss Height Cell
• 1 66.67 68.42 MM
• 2 69.83 70.32 MT
• 3 65.19 69.76 MM
• 4 65.15 73.85 MT
• 5 64.66 70.17 ST

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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.

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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 cross-tabulation 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
  neednt, and probably shouldnt, be much wider
  than this).
• We then erect a vertical interval scale that
  similarly represents values of the dependent
  variable (corres-ponding to the row variable in a
  crosstab).

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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.

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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
  approx-imately square or somewhat wider than tall
  (SPSS does the latter by default).

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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 Pearsons data.
  
• This scattergram is taken from David Freedman et
  al., Statistics, p. 110.

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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?

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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 (Tall-Tall), indicating a positive
  association.

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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.

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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
  1B-1E 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.

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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).

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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).

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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 Pearsons
  scattergram, we want also to construct a full
  crosstabulation of the data using the
  short-medium-tall class intervals we previously
  worked with.

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Crosstabs vs. Scattergram (cont.)

• This can be accomplished simply by
• super-imposing the appro-priate 3x3 grid (table)
  on the scatter-gram, and then
• counting the number of plotted points in each
  resulting cell.

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Test 1 Blue Book Score by MC Score
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• Here is some more topical (and politically
  relevant) bivariate data
• Data as of 11/05/08

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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

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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, lets 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 gt 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 gt Scatter... gt Simple/Define and then in
  the Simple Scatterplot dialog box put d2pc2004 on
  the Y Axis and d2pc2000 on the X Axis.

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Histogram of Scores on Test 2
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Scattergram of P2 by MC2
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Scattergram of Score2 by Score 1
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Correlation Matrix