VARIABLES
Topic #3
Variables and the Unit of Analysis
•
Variables
are characteristics of the “things” that we are
studying.
–
These “things” are commonly called
cases
or
units
.
•
A “case study” focuses on a single “thing.”
–
The kind of “thing” that is being studied is called the
unit of analysis
.
•
Individuals
constitute the unit of analysis for much
empirical social science research (and almost all survey
research in political science).
•
A particular research project focuses on a particular set
or
population
of cases (individuals or other units),
–
often by studying a sample of cases drawn from the population.
American National Election Studies
•
ANES focuses on individuals as the units of analysis in
the American
voting age population
(VAP).
•
ANES variables pertain to these individuals
–
ANES variables include
•
gender, race, education, and other
demographic
variables;
•
party identification, voting intention, President approval,
ideology, abortion opinion, political trust, and other
attitudinal
variables;
•
whether registered/voted, candidate vote for, whether
contributed campaign $$, and other
behavioral
variables;
–
These are all variable properties of individuals,
•
not households, elections, nations, etc.
Other Populations of Individuals
•
Population = All Members of Congress
–
additional variables pertaining to this specialized
population of individuals include:
•
number of terms served, campaign expenditure in last
election, last re

election margin, party affiliation, committee
assignments, roll

call vote on specified bill, ADA (etc.) rating,
NOMINATE score, etc.
•
Annual Survey of Social Security and Medicare
Beneficiaries
•
British [etc.] Election Studies
Other Units of Analysis in Political Research
•
Presidential elections
: variables include
–
winning party, winner’s vote popular vote %, Dem. candidate’s popular vote %,
winner’s electoral vote margin, turnout %, whether the incumbent was running for
re

election, total campaign expenditures, etc.
•
States in a given Presidential election
: variables include
–
number of electoral votes, winning party/candidate, winner’s vote %; Rep.
candidate’s vote %, turnout %, etc.
•
States in all historical Presidential elections
: variables include
–
all of above for each election year
•
Nations
: variables include
–
population, GNP, per capita income, literacy rate, military spending as % of GNP,
size of army, type of party system, etc.
•
States, counties, other jurisdictions, precincts, legislatures, political
parties
, etc.
Households
•
Households are often the unit of analysis in
economic and sociological research
–
Variables include:
•
size (# of persons)
•
type (single

parent, no children, unrelated, etc.)
•
type of housing unit
•
household income
•
etc.
•
Current Population Survey (CPS)
•
Panel Study of Income Dynamics (PSID)
–
Rotating panel surveys of households
Variables vs. Values
•
Variables
that pertain to a given
unit of analysis
take on
different
values
from
case to case
[
cross

sectional
analysis].
–
Gender [
individuals
]
:
male, female
–
Education [
individuals
]
:
primary school only,
# years
completed
, etc.
–
Income [
individuals
or
households
]
:
dollar amount (or
dollar range), quintile,
etc.
–
Type of dwelling [
households
]
:
detached, townhouse,
apartment, etc.
–
Literacy rate [
nations
]
:
numerical %
–
Turnout [
elections
]
:
numerical %
•
Variables can also vary over time in the same case
[
longitudinal
analysis],
–
e.g., state democratic candidate vote % over time.
Variables are the building blocks of
empirical political science research
•
Researchers have to figure out how to
measure
the
variables they are interested in by designing
–
appropriate survey questions
–
or other kinds of measures
•
Researchers next need to actually collect the data, e.g.,
by carrying out
–
the survey they have designed
–
or other data collecting operations.
•
With the data at hand, researchers then ask such
questions as the following:
–
What is the
average
or
typical
value of a variable in a set of
cases?
•
For example, what is typical level of interest among voters, or the
average rate of turnout in recent elections?
Questions (cont.)
•
How are the values of a variable
distributed
in a set of data, i.e., do
most of the same cases have about the same value (
low dispersion
) or
do different cases have very different values (
high dispersion
). For
example:
–
Do all voters have about the same level of interest or are some very interested while
others not interested at all?
–
Do all elections have about the same level of turnout, or do some have very high
turnout while others have very low turnout?
–
Distribution
of income or wealth.
•
How are two variables
related
or
associated
in a set of data? For
example:
–
Is the level of interest among voters related to their level of education?
–
Does the level of turnout in elections depend on how close elections are expected to
be?
•
Does one variable have a (direct)
causal impact
on another variable?
For example:
–
Does higher education cause people to become more interested in politics?
–
Does the prospect of a close election cause more voters to turn out and vote?
•
Does one variable have an (indirect)
causal impact
on another
variable? For example:
–
Does the prospect of a close election cause greater activity by campaign
organizations that in turn causes more voters to turn out and vote?
Variables and Their Values
•
To repeat, variables
vary
—
they take on different values from case to case
[or from time to time]
•
Thus, associated with every
variable
is a
list or range
of possible
values
.
For example:
–
PARTY IDENTIFICATION
(pertaining to
individuals
) in the U.S
has
values: REPUBLICAN, DEMOCRAT, INDEPENDENT
(or
perhaps refinements like
STRONG REPUBLICAN, WEAK
DEMOCRAT
, etc., and/or other values like
MINOR PARTY
).
–
VOTED IN 2008 ELECTION?
is another variable pertaining to
individuals
, with just two possible values,
YES
and
NO
.
–
HEIGHT
is a physical variable pertaining to
individuals
with
values that
are real numbers
(expressed in units such as inches, centimeters, or
feet).
–
SIZE
(# of persons) is a variable pertaining to
households
with
values
that are whole numbers
>
1 (values are
counts
)
–
LEVEL OF TURNOUT
is a variable pertaining to
elections
(or to
different jurisdictions in a given election), with
values ranging potentially
from 0% to 100%
.
Naming Variables
•
As a reminder that any variable must have a range of
two or more possible values, it is useful to give variables
names like
–
LEVEL OF
EDUCATION
–
WHETHER OR NOT
VOTED IN 2000 ELECTION
–
SIZE OF
POPULATION
–
TYPE OF
POLITICAL REGIME
–
LEVEL OF
VOTING TURNOUT
–
DIRECTION OF
IDEOLOGY
–
ETC.
•
In quantitative research, variable names are often written
in capital letters (as above).
Observations/Observed Values
•
The actual value of a variable in a particular case is
called an
observation
(or
observed value
). For example,
–
we "observe“ [by asking the appropriate question(s) in
a survey] that
Joe Smith
(the case) has the
PARTY
IDENTIFICATION
(the variable)
WEAK DEMOCRAT
(the observed value), and likewise
–
we “observe” [by consulting the appropriate records]
that the
2008 Presidential election
(the case) has a
LEVEL OF TURNOUT
(the variable) of
61%
(the
observed value).
Identifying Variables (PS#3A)
Each of the following statements makes an empirical assertion (which
may or may not be true); each refers (at least implicitly) to
two
variables
(and asserts that there is some kind of relationship
between them). For each statement:
(a)
indicate to what
unit of analysis
(individuals, nations, elections, etc.) and,
as appropriate, what particular
population
the variables pertain;
(b)
identify the two
variables
, with appropriate names (probably TYPE OF
_____, LEVEL OF _____, DEGREE OF _____, AMOUNT OF _____”,
WHETHER OR NOT _____”); and
(c)
indicate a range of possible
values
for each variable (often, but certainly
not always, LOW and HIGH will do).
(
Note
: both variables in each sentence pertain to the same units.)
1.
Junior members of Congress are less pragmatic than their senior
colleagues.
2.
Education tends to undermine religious faith.
3.
Capital punishment deters murder.
8.
When times are bad, incumbent candidates are punished in elections. =>
11.
If you want to get ahead, stay in school.
CLASS LIST (Data Spreadsheet)
Case ID
Variable 1 Var2 Var3 Var4
Grad.
Name
SSN
Class
Major
GPA Cand?
Jones, R.
215

14

6609
Senior
POLI
3.12
No
Kim, S.
144

56

9231
Sophomore
PYSC
2.78
No
Smith. H.
502

45

2323
Junior
POLI
2.75
No
Williams, R.
212

16

7834
Senior
HIST
3.28
Yes
Etc.
What distinctions between different types of variables can
we make?
Types of Variables
•
Our concern here is with drawing distinctions among
variables with respect to their
logical properties
, not their
substantive nature (e.g., demographic, attitudinal, etc.)
•
Every variable has
at least two
possible values
(otherwise it could not vary).
•
A variable is
dichotomous
(also called a
dummy variable
)
if it has
exactly
two possible values (typically “yes” and
“no”), e.g.,
–
GRADUATION CANDIDATE?
[
Students
] (
Yes/No
)
–
WHETHER VOTED IN 2000 ELECTION
[
Inds
.] (
Yes/No
)
–
GENDER
[
Inds
.] (
M/F
)
•
However, most variables have three or more possible
values.
–
Some variables have an infinite number of possible values.
Qualitative Variables
•
A variable is
qualitative
if its values are given by
words
–
MAJOR
[
Students
]:
POLI, HIST, BIOL, etc.
–
TYPE OF REGIME
[
nations
]:
Free, Partly Free, Unfree
–
ABORTION OPINION
[
Inds.
]:
Never permit, etc.
•
In a data spreadsheet [e.g., SPSS], these verbal values
are typically recorded in terms of
numerical codes,
because this
–
saves space, and
–
facilitates machine processing.
–
Moreover, survey data from closed

form questions is often pre

coded (e.g., the Student Survey).
In a spreadsheet
Rows are
cases
Columns are
variables
Cell are values
(varying from
case to case)
Values (except V01
YEAR OF
SURVEY) in the
Student Survey and
SETUPS are
numerically
coded.
Quantitative Variables
•
A variable is
quantitative
if its (true, not coded) values
are given by
numbers
:
–
GPA
[
Students
]:
3.12, 2.78, etc.
–
LITERACY RATE
[
Nations
]:
98%, 55%, etc.
–
HEIGHT
[
Inds.
]:
72
"
, 62
",
etc.
–
SIZE
[
Households
]:
1 person, 2 persons, etc.
–
LEVEL OF TURNOUT
[
Elections
or
jurisdictions
]:
51%, etc.
•
The magnitude of these numbers may depend on the units of
measurement used (e.g., is HEIGHT given in inches, feet,
centimeters, etc.?).
•
In spreadsheet, such values are typically recorded in
terms of their actual numerical values.
•
The SETUPS data contains data pertaining to variables
that, while “truly” quantitative in nature, are
recoded
in
broad categories, e.g.,
–
AGE (V60)
[ 18

24, 25

34, etc.]
or
–
INCOME (V65A)
[0

16
th
percentile, 17

33
rd
percentile, etc.]
Truly Quantitative Data Need Not be Coded
Variables and the Unit of Analysis
•
Substantively related
variables
may be of different types
depending on the
unit of analysis
to which they pertain.
–
TURNOUT
pertaining to
individuals
is a dichotomous
variable with values
“yes
–
voted”
and
“no
–
did not
vote”.
–
[LEVEL OF] TURNOUT
pertaining to
elections
(or
jurisdictions, precincts, etc.
) is a quantitative variable
with possible values ranging from
0% to 100%
.
Types of Variables / Levels of
Measurement
•
It is useful to refine both qualitative and quantitative
variables further by distinguishing among four
–
different
types
of variables, or (equivalently)
–
different
levels of measurement
of pertaining to
variables.
•
Note
: these distinctions are relevant only as they pertain
to
non

dichotomous
variables.
–
Please take note of this with respect to PS #3B,
Question 2.
Nominal Variables
•
A
nominal
variable (or a variable
measured at the
nominal level
) has values that are
unordered
categories
.
•
Accordingly, nominal variable are qualitative in nature.
•
Given two cases and a nominal variable, we can observe
–
that they have the same value or they have different
values, but (if they have different values)
–
we cannot say that one has the “higher/bigger” value
and the other the “lower/smaller,” etc.
Nominal Variables (cont.)
•
A nominal variable typically has a name like
–
NAME OF ____
–
TYPE OF ____
–
NATURE OF ____
–
KIND OF ____
•
Examples:
–
(NAME OF) MAJOR
:
Political Science, Economics, History, etc.
–
(TYPE OF) RELIGIOUS AFFILIATION
:
Protestant, Catholic,
Jewish, etc.
–
PREFERENCE FOR REPUBLICAN NOMINATION
:
Giuliani,
McCain, Romney, etc.
•
In a data spreadsheet, numerical codes must be
assigned to values of nominal variables in an essentially
arbitrary manner,
–
so it is certainly illegitimate to do arithmetic on the numerical
code values.
–
Typically the numerical codes are consecutive whole numbers.
Ordinal Variables
•
An
ordinal
variable (or a variable
measured at the ordinal
level
) has values that fall into some kind of
natural
ordering
,
–
often (but not always) running from (in some sense) LOW to
HIGH.
–
Therefore, cases can be ranked or ordered with respect to their
values on an ordinal variable.
•
An ordinal variable is also qualitative in nature.
•
Given two cases and a ordinal variable, we can observe
–
that they have the same value or they have different values, and
also
(if they have different values)
–
that one has the “higher/bigger” value and the other
“lower/smaller,” etc., but
–
we
cannot
say
how much
higher/bigger or lower/smaller.
•
Given three cases with different values on an ordinal
variable,
•
we can identify the case with the observed value
between
the other two
•
but we cannot say which of the other value it is closer to.
Ordinal Variables (cont.)
•
An ordinal variable typically has a name like
–
DIRECTION OF ___
–
EXTENT OF ____
–
LEVEL OF ____
–
DEGREE of ____
•
Examples:
–
TYPE OF REGIME/DEGREE OF FREEDOM
[
nations
]:
Free, Partly Free,
Unfree
–
(LEVEL OF) INTEREST IN THE ELECTION CAMPAIGN
[
individuals
]
:
from “low”
to “high”
–
(DIRECTION OF) IDEOLOGY
[
individuals
]
:
from most liberal to most
conservative
–
(DEGREE OF) PRESIDENTIAL APPROVAL
[
individuals
]
:
from strongly approve
to strongly disapprove
–
DIRECTION OF ABORTION OPINION
[
individuals
]:
Never permit, . . . , Always
permit
–
(LEVEL OF) CLASS STANDING
[
students
]
:
freshman, sophomore, junior, senior
•
When data is recorded in coded form, numerical codes should be
assigned to values in a manner consistent with the natural ordering
of the values.
Ordinal Variables (cont.)
•
If the natural ordering is from LOW to HIGH, the codes
should likewise run from lower to higher numbers.
•
If the natural ordering is not from LOW to HIGH, e.g.,
DIRECTION OF IDEOLOGY,
–
the two extreme values (or “poles”), e.g., MOST LIBERAL and
MOST CONSERVATIVE, should be assigned the minimum and
maximum code values, but
–
which gets which is arbitrary ,
–
and intermediate values, e.g., MODERATE, should be assigned
intermediate codes).
•
In any event, values are typically assigned numerical
codes that are
consecutive integers
,
–
but this is not a logical necessity (because only their order
matters).
–
It remains illegitimate to do arithmetic on the
numerical code values
•
unless we are willing to attribute “interval” status to the code
values.
Ordinal Variables (cont.)
•
Note that DIRECTION OF IDEOLOGY could be
renamed
DEGREE OF LIBERALISM,
–
which does range from LOW (i.e., “least liberal” [or
“most conservative”]) to HIGH (“most liberal” or [“least
conservative”]).
•
We could also
reverse
the “polarity” of the renamed
variable and call it DEGREE OF CONSERVATISM,
–
ranging from LOW (i.e., “least conservative” [or “most
liberal”]) to HIGH (“most conservative” [or “least
liberal”]).
Ordinal Variables (cont.)
•
Opinion variables with closed

form values running from
(STRONGLY) AGREE (or APPROVE) to (STRONGLY) DISAGREE
(or DISAPPROVE) are ordinal in nature.
•
The value INDEPENDENT is usually deemed to fall “between”
DEMOCRAT and REPUBLICAN, so PARTY IDENTIFICATION is
usually deemed to be ordinal in nature.
–
But this works only if we treat cases with “minor party” or DK values as
missing data (since these values don’t fall in the natural ordering).
•
An SPSS spreadsheet normally displays a numerical code (rather
than a blank) for missing data (“unobserved” values), which must be
understood as not part of the natural ordering.
–
In the SETUPS and Student Survey data, missing data coded as (9).
–
SPSS must be told the “missing data” code(s) for each variable, so that
it can set cases so coded aside when it processes data.
Interval [Scale] Variables
•
An
interval
variable (or variable
measured at the interval level
) has
values that are
real numbers
that can appropriately be added
together, subtracted one from another, and averaged.
–
SPSS refers to
scale
variables
•
An interval variable is quantitative in nature.
•
Given two cases and an interval variable, we can say they have the
same value or they have different values, and
also
(if they have
different values)
–
that one has the higher value and the other lower, etc., and
also
–
how much
higher or lower one value is than the other, because
•
we can subtract one value from another,
•
i.e., we can determine the magnitude of the
interval
separating them and
thus say how “far apart” the cases are with respect to the variable.
–
Given three case with different values on an interval variable, we
can identify the case with the observed value between the other
two
and
we can
also
determine which of the to other cases it is
closer to.
•
But we
cannot
say how
many times greater
one value is than
another.
Interval Variables (cont.)
•
An interval variable typically has a name like
–
LEVEL OF ____
–
DEGREE OF
____
–
NUMBER OF ____
–
AMOUNT OF ____
•
In a spreadsheet, actual numerical values (rather than numerical
codes) are normally entered into a data array (e.g., Presidential
election data).
•
But sometimes (numerically coded)
class intervals
are used instead
(e.g., SETUPS V60 [AGE]), as will be discussed later. [See =>]
•
Variables like PARTY IDENTIFICATION,IDEOLOGY, and ISSUE
OPINIONS are often treated as interval variables (e.g., my Student
Survey/ANES longitudinal charts that showed changing
average
levels of Party ID, Ideology, etc., over time).
A Truly Interval Variable May Be Recoded
into An Ordinal One
Ordinal vs. Interval Variables
•
Example
–
Baseball Standings
–
Rank Standing
of a team (first place, second place, etc.) is
ordinal
information
–
Winning Percent
(or
Games Behind Leader
) is
interval
information
–
For the league playoffs:
•
the determination of division winners is based on
ordinal
information only; but
•
the determination of the “wild card” entry is based on
interval
information (best winning percent not otherwise in playoffs)
•
A team that fails to make the playoffs may have a higher
winning percent that a team that does make the playoffs
Ratio Variables
•
A
ratio
variable (or a variable
measured at the ratio level
)
is an interval variable (that has values that are
real
numbers
that can appropriately be added together,
subtracted one from another, and averaged) but in
addition
–
one can appropriately
divide
one value by another
(i.e., compute their
ratio
), and
–
say, for example, that one case has
twice
the
observed value of another.
•
This requires that the ratio variable have a
non

arbitrary
zero value
,
–
which usually represents in some sense the complete absence
of the characteristic or property to which the variable refers.
–
Even if negative values are possible, the zero value is non

arbitrary, e.g.,
•
level of profit
(of a
business
) may have a
negative value
, or
•
rate of economic growth
(over
years
) may have a
negative
value
.
Ratio Variables (cont.)
•
Examples of interval variables that are
not
ratio:
–
LEVEL OF SAT (or IQ) SCORE: there is no 0 score
–
DEGREE OF TEMPERATURE (Fahrenheit or
Celsius): while each has a 0
°
value,
•
0
°
F
and
0
°
C represent different temperatures, so
•
0
°
has no fundamental significance in either temperature scale
•
vs. Kelvin Temperature scale with
absolute
0
°
K.
–
IDEOLOGY, PARTY IDENTIFICATION and OPINION
variables
•
may perhaps be treated as interval rather than merely
ordinal,
•
but they certainly are not ratio.
Ratio Variables (cont.)
•
Examples of ratio variables include:
–
NUMBER OF CHILDREN
or
AGE
(
uncoded
) [
individuals
]
–
SIZE/NUMBER OF MEMBERS
[
households
or
legislatures
]
–
SIZE OF POPULATION
[
nations
]
–
LEVEL OF INCOME
[
individuals
or
households
]
–
PER CAPITA INCOME
[
nations
]
–
LEVEL OF PROFITS
[
firms
]
–
SIZE OF BUDGET SURPLUS
[
governments
or
fiscal years
]
–
NUMBER OF VOTES FOR DEM CAND
[
elections
,
states
]
–
PERCENT OF VOTES FOR DEM CAND
[
elections
,
states
]
•
Even though LEVEL OF PROFITS or SIZE OF BUDGET
SURPLUS can have negative values, their zero points
are not arbitrary.
–
However, ratio comparisons can only be made between
observed values with the same [positive or negative] sign.
Freeway Exits and Levels of Measurement
•
The identification of freeway exits has changed over the
years, progressing from lower to higher levels of
measurement.
•
Nominal
: exits were once only given
names
(e.g., name
of crossroad or town),
–
So you could tell only whether the upcoming exit is your exit or
not.
•
Ordinal
: Exits then were ordered (e.g., from east to west)
and consecutively numbered, so you could tell
–
whether you have passed your exit or not, and
–
how many exits there are between your exit and where you are
now.
–
(
Otherwise exit numbers are uninformative =>
)
•
Interval/Ratio
: Exits are now usually numbered in terms
of their
distance
in miles from the state line,
–
so can tell how far you have to go to get to your exit
–
(and also that your exit is X times as far from the state line as
where you are now).
Ordinal Information May Not Be Informative
But Ordinal Is Better Than Nominal
Discrete vs. Continuous Variables
•
Quantitative
[interval and ratio] variables may be either
discrete
or
continuous
.
–
[Qualitative variables are pretty much necessarily discrete.]
•
A
discrete
variable has a finite (and typically small)
number of possible values that usually (if the variable is
quantitative) correspond to
whole numbers
(or
integers
)
only.
–
NUMBER OF CHILDREN
[
households
]
–
NUMBER OF MEMBERS
[
councils
or
legislatures
]
–
NUMBER OF ELECTORAL VOTES WON BY DEM
CANDIDATE
[
Presidential elections
] vs.
–
PERCENT OF POPULAR VOTE WON BY DEM CANDIDATE
[
Presidential elections
]
Continuous Variables
•
A
continuous
variable can have
any real number
(at least
within some range) as a value (i.e., including fractional
values between the integers).
–
So a continuous variable has (at least in principle) an
infinite number
of possible values,
•
so that given two cases with distinct values of the continuous
variable, it is in principle always possible that there is another
case with an intermediate value of the variable.
–
“Discrete” vs. “Continuous” temperature controls on a kitchen
range.
–
Digital vs. “old fashioned” thermometer
Continuous Variables (cont.)
•
Examples:
–
LEVEL OF DAILY HIGH TEMPERATURE
[
places
(cross

sectional),
days
(longitudinal)]
–
HEIGHT
,
WEIGHT
, and
AGE
[
individuals
]
•
Because we typically round off the value of such
variables to the nearest degree, inch, pound, year, etc.,
such variables may “look” discrete.
–
IDEOLOGY might be thought of as a “truly” continuous variable.
•
Some interval variables are in principle discrete but are
“virtually” continuous because they have so many
possible (numerical) values, e.g.,
•
RATE OF TURNOUT
[elections]
•
PERCENT OF VOTE FOR DEMOCRATIC CANDIDATE
[elections]
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