Introduction: A Common Language for Researchers
Research in the social sciences is a diverse topic. In part, this is because the social
sciences represent a wide variety of disciplines, including (but not limited to) psychology,
sociology, political science, anthropology, communication, education, management, and
economics. Further, within each discipline, researchers can use a number of different
methods to conduct research. These methods can include unobtrusive observation,
participant observation, case studies, interviews, focus groups, surveys, ex post facto
studies, laboratory experiments, and field experiments.
Despite this diversity in methods used and topics investigated, most social science
research still shares a number of common characteristics. Regardless of field, most
research involves an investigator gathering data and performing analyses to determine
what the data mean. In addition, most social scientists use a common language in
conducting and reporting their research: researchers in psychology and management
speak of “testing null hypotheses” and “obtaining significant p values.”
The purpose of this chapter is to review some of the fundamental concepts and terms that
are shared across the social sciences. You should familiarize (or refamiliarize) yourself
Chapter 1: Basic Concepts in Research and Data Analysis 3
with this material before proceeding to the subsequent chapters, as most of the terms
introduced here will be referred to again and again throughout the text. If you are
currently taking your first course in statistics, this chapter provides an elementary
introduction. If you have already completed a course in statistics, it provides a quick
review.
Steps to Follow When Conducting Research
The specific steps to follow when conducting research depend, in part, on the topic of
investigation, where the researchers are in their overall program of research, and other
factors. Nonetheless, it is accurate to say that much research in the social sciences follows
a systematic course of action that begins with the statement of a research question and
ends with the researcher drawing conclusions about a null hypothesis. This section
describes the research process as a planned sequence that consists of the following six
steps:
- Developing a statement of the research question
- Developing a statement of the research hypothesis
- Defining the instrument (questionnaire, unobtrusive measures)
- Gathering the data
- Analyzing the data
- Drawing conclusions regarding the hypothesis.
The preceding steps reference a fictitious research problem. Imagine that you have been
hired by a large insurance company to find ways of improving the productivity of its
insurance agents. Specifically, the company would like you to find ways to increase the
dollar amount of insurance policies sold by the average agent. You begin a program of
research to identify the determinants of agent productivity.
The Research Question
The process of research often begins with an attempt to arrive at a clear statement of the
research question (or questions). The research question is a statement of what you hope to
have learned by the time you complete the program of research. It is good practice to
revise and refine the research question several times to ensure that you are very clear
about what it is you really want to know.
4 JMP for Basic Univariate and Multivariate Statistics: A Step-by-Step Guide
For example, in the present case, you might begin with the question
“What is the difference between agents who sell more insurance and agents who sell
less insurance?”
An alternative question might be
“What variables have a causal effect on the amount of insurance sold by agents?”
Upon reflection, you realize that the insurance company really only wants to know what
things management can do to cause the agents to sell more insurance. This realization
eliminates from consideration certain personality traits or demographic variables that are
not under management’s control, and substantially narrows the focus of the research
program. This narrowing, in turn, leads to a more specific statement of the research
question, such as
“What variables under the control of management have a causal effect on the amount
of insurance sold by agents?”
Once you have defined the research question more clearly, you are in a better position to
develop a good hypothesis that provides an answer to the question.
The Hypothesis
A hypothesis is a statement about the predicted relationships among events or variables.
A good hypothesis in the present case might identify which specific variable has a causal
effect on the amount of insurance sold by agents. For example, the hypothesis might
predict that the agents’ level of training has a positive effect on the amount of insurance
sold. Or, it might predict that the agents’ level of motivation positively affects sales.
In developing the hypothesis, you can be influenced by any of a number of sources, such
as an existing theory, related research, or even personal experience. Let’s assume that you
are influenced by goal-setting theory. This theory states, among other things, that higher
levels of work performance are achieved when difficult work-related goals are set for
employees. Drawing on goal-setting theory, you now state the following hypothesis:
“The difficulty of the goals that agents set for themselves is positively related to the
amount of insurance they sell.”
Chapter 1: Basic Concepts in Research and Data Analysis 5
Notice how this statement satisfies the definition for a hypothesis: it is a statement about
the relationship between two variables. The first variable could be labeled Goal
Difficulty, and the second, Amount of Insurance Sold. Figure 1.1 illustrates this
relationship.
Figure 1.1 Hypothesized Relationship between Goal Difficulty and Amount
of Insurance Sold
The same hypothesis can also be stated in a number of other ways. For example, the
following hypothesis makes the same basic prediction:
“Agents who set difficult goals for themselves sell greater amounts of insurance than
agents who do not set difficult goals.”
Notice that these hypotheses have been stated in the present tense. It is also acceptable to
state hypotheses in the past tense. For example, the preceding could have been stated,
“Agents who set difficult goals for themselves sold greater amounts of insurance than
agents who did not set difficult goals.”
You should also note that these two hypotheses are quite broad in nature. In many
research situations, it is helpful to state hypotheses that are more specific in the
predictions they make. A more specific hypothesis for the present study might be,
“Agents who score above 60 on the Smith Goal Difficulty Scale sell greater amounts
of insurance than agents who score below 40 on the Smith Goal Difficulty Scale.”
Defining the Instrument, Gathering Data, Analyzing Data, and
Drawing Conclusions
With the hypothesis stated, you can now test it by conducting a study in which you gather
and analyze some relevant data. Data can be defined as a collection of scores obtained
when a subject’s characteristics and/or performance are assessed. For example, you could
choose to test your hypothesis by conducting a simple correlational study.
6 JMP for Basic Univariate and Multivariate Statistics: A Step-by-Step Guide
Suppose you identify a group of 100 agents and determine
• the difficulty of the goals set for each agent
• the amount of insurance sold by each agent.
Different types of instruments result in different types of data. For example, a
questionnaire can assess goal difficulty, but company records measure amount of
insurance sold. Once the data are gathered, each agent has one score that indicates
difficulty of the goals, and a second score that indicates the amount of insurance the agent
sold.
With the data gathered, an analysis helps tell if the agents with the more difficult goals
did, in fact, sell more insurance. If yes, the study lends some support to your hypothesis;
if no, it fails to provide support. In either case, you can draw conclusions regarding the
tenability of the hypotheses, and you have made some progress toward answering your
research question. The information learned in the current study might then stimulate new
questions or new hypotheses for subsequent studies, and the cycle repeats. For example,
if you obtained support for your hypothesis with the current correlational study, you
could follow it up with a study using a different method, perhaps an experimental study.
The difference between correlational and experimental studies is described later. Over
time, a body of research evidence accumulates, and researchers can review this body to
draw general conclusions about the determinants of insurance sales.
Variables, Values, and Observations
When discussing data, you often hear the terms variables, values, and observations. It is
important to have these terms clearly defined.
Variables
For the type of research discussed here, a variable refers to some specific characteristic
of a subject that assumes one or more different values. For the subjects in the study just
described, amount of insurance sold is an example of a variable—some subjects sold a lot
of insurance and others sold less. A different variable was goal difficulty—some subjects
had more difficult goals, while others had less difficult goals. Age was a third variable,
and gender (male or female) was yet another.
Chapter 1: Basic Concepts in Research and Data Analysis 7
Values
A value refers to either a subject’s relative standing on a quantitative variable, or a
subject’s classification within a classification variable. For example, Amount of
Insurance Sold is a quantitative variable that can assume many values. One agent might
sell $2,000,000 worth of insurance in one year, another sell $100,000 worth of policies,
and another sell nothing ($0). Age is another quantitative variable that assumes a wide
variety of values. In the sample shown in Table 1.1, these values ranged from a low of 22
years to a high of 56 years.
Quantitative Variables versus Classification Variables
You can see that, in both amount of insurance sold and age, a given value is a type of
score that indicates where the subject stands on the variable of interest. The word “score”
is an appropriate substitute for the word “value” in these cases because both are
quantitative variables. They are variables in which numbers serve as values.
A different type of variable is a classification variable, also called a qualitative variable
or categorical variable. With classification variables, different values represent different
groups to which the subject belongs. Gender is a good example of a classification
variable, as it assumes only one of two values—a subject is classified as either male or
female. Race is another example of a classification variable, but it can assume a larger
number of values—a subject can be classified as Caucasian American, African American,
or Asian American, or as belonging to another group. These variables are classification
variables and not quantitative variables because values only represent group membership;
they do not represent a characteristic that some subjects possess in greater quantity than
others.
Observations
In discussing data, researchers often make references to observational units (or
observations), which can be defined as the individual subjects (or other objects) that
serve as the source of the data. Within the social sciences, a person is usually the
observational unit under study (although it is also possible to use some other entity, such
as an individual school or organization, as the observational unit). In this text, the person
is the observational unit in all examples. Researchers often refer to the number of
observations (or cases) included in their data, which simply refers to the number of
subjects who were studied. For a more concrete illustration of the concepts discussed so
far, consider the data in Table 1.1.
8 JMP for Basic Univariate and Multivariate Statistics: A Step-by-Step Guide
Table 1.1 Insurance Sales Data
Observation Name Gender Age
Goal
Difficulty
Score Rank Sales
1 Bob M 34 97 2 $598,243
2 Walt M 56 80 1 $367,342
3 Jane F 36 67 4 $254,998
4 Susan F 24 40 3 $80,344
5 Jim M 22 37 5 $40,172
6 Mack M 44 24 6 $0
This table reports information about six research subjects: Bob, Walt, Jane, Susan, Jim,
and Mack—the data table includes six observations. Information about a given
observation (subject) appears as a row running from left to right across the table. The first
column of the data set (running vertically) indicates the observation number, and the
second column reports the name of the subject who constitutes or identifies that
observation. The remaining five columns report information on the five research
variables under study.
• The Gender column reports subject gender, which assumes either “M” for male or “F”
for female.
• The Age column reports the subject’s age in years.
• The Goal Difficulty Score column reports the subject’s score on a fictitious goal
difficulty scale. Assume that each participant completed a 20-item questionnaire that
assessed the difficulty of the work goals. Depending on how they respond to the
questionnaire, subjects receive a score that can range from a low of 0 (meaning that
the subject’s work goals are quite easy to achieve) to a high of 100 (meaning that they
are quite difficult to achieve).
• The Rank column shows how the supervisor ranked the subjects according to their
overall effectiveness as agents. A rank of 1 represents the most effective agent, and a
rank of 6 represents the least effective.
• The Sales column lists the amount of insurance sold by each agent (in dollars) during
the most recent year.
The preceding example illustrates a very small data table with six observations and five
research variables (Gender, Age, Goal Difficulty, Rank, and Sales). Gender is a
classification variable and the others are quantitative variables. The numbers or letters
that appear within a column represent some of the values that these variables can have.
Chapter 1: Basic Concepts in Research and Data Analysis 9
Scales of Measurement and JMP Modeling Types
One of the most important schemes for classifying a variable involves its scale of
measurement. Researchers generally discuss four scales of measurement: nominal,
ordinal, interval, and ratio. In JMP, scales of measurement are designated using three
modeling types. Modeling types are discussed later, in the section “Modeling Types in
JMP.”
Before analyzing a data set, it is important to determine each variable’s scale of
measurement (modeling type) because certain types of statistical procedures require
certain scales of measurement. For example, one-way analysis of variance generally
requires that the independent variable be a nominal-level variable and the dependent
variable be an interval or ratio (continuous) variable. In this text, each chapter that deals
with a specific statistical procedure indicates what scale of measurement is required by
the variables under study. Then, you must decide whether your variables meet these
requirements.
Nominal Scales
A nominal scale is a classification system that places people, objects, or other entities into
mutually exclusive categories. A variable measured using a nominal scale is a
classification variable that indicates the group to which each subject belongs. The
examples of classification variables provided earlier (Gender and Race) also serve as
examples of nominal variables. They tell us to which group a subject belongs, but they do
not provide any quantitative information about the subjects. That is, the Gender variable
might tell us that some subjects are males and other are females, but it does not tell us
that some subjects possess more of a specific characteristic relative to others. However,
the remaining three scales of measurement provide some quantitative information.
Ordinal Scales
Values on an ordinal scale represent the rank order of the subjects with respect to the
variable being assessed. For example, the preceding table includes one variable called
Rank that represents the rank ordering of subjects according to their overall effectiveness
as agents. The values on this ordinal scale represent a hierarchy of levels with respect to
the construct of effectiveness. That is, we know that the agent ranked “1” was perceived
as being more effective than the agent ranked “2,” that the agent ranked “2” was more
effective than the one ranked “3,” and so forth.
10 JMP for Basic Univariate and Multivariate Statistics: A Step-by-Step Guide
Caution: An ordinal scale has a limitation, in that equal differences in scale
values do not necessarily have equal quantitative meaning. For example,
look at the following rankings:
Rank Name
1 Walt
2 Bob
3 Susan
4 Jane
5 Jim
6 Mack
Notice that Walt is ranked “1” while Bob is ranked “2.” The rank difference between
these two rankings is 1 (2 – 1 = 1), so there is one unit of rank difference between Walt
and Bob. Now notice that Jim is ranked “5” while Mack is ranked “6.” The rank
difference between them is also 1 (6 – 5 = 1), so there is also 1 unit of difference between
Jim and Mack. Putting the two together, the rank difference between Walt and Bob is
equal to the rank difference between Jim and Mack. However, that does not necessarily
mean that the difference in overall effectiveness between Walt and Bob is equal to the
difference in overall effectiveness between Jim and Mack. It is possible that Walt is just
barely superior to Bob in effectiveness, while Jim is substantially superior to Mack.
These rankings reveal very little about the quantitative differences between the subjects
with regard to the underlying construct (effectiveness, in this case). An ordinal scale
simply provides a rank order of the subjects.
Interval Scales
With an interval scale, equal differences between scale values do have equal quantitative
meaning. For this reason, an interval scale provides more quantitative information than
the ordinal scale. A good example of an interval scale is the Fahrenheit degree scale used
to measure temperature. With the Fahrenheit scale, the difference between 70 degrees and
75 degrees is equal to the difference between 80 degrees and 85 degrees: The units of
measurement are equal throughout the full range of the scale.
However, the interval scale also has an important limitation: it does not have a true zero
point. A true zero point means that a value of zero on the scale represents zero quantity of
the construct being assessed. The Fahrenheit scale does not have a true zero point. When
a Fahrenheit thermometer reads 0 degrees, that does not mean there is absolutely no heat
present in the environment.
Chapter 1: Basic Concepts in Research and Data Analysis 11
Researchers in the social sciences often assume that many of their man-made variables
are measured on an interval scale. For example, in the preceding study involving
insurance agents, you probably assume that scores from the goal difficulty questionnaire
constitute an interval-level scale. That is, you assume that the difference between a score
of 50 and 60 is approximately equal to the difference between a score of 70 and 80. Many
researchers also assume that scores from an instrument such as an intelligence test are
measured at the interval level of measurement.
On the other hand, some researchers are skeptical that instruments such as these have true
equal-interval properties, and prefer to call them quasi-interval scales. Disagreement
about the level of measurement achieved with such instruments continues to be a
controversial topic within the social sciences.
However, it is clear that neither of the preceding instruments has a true zero. A score of 0
on the goal difficulty scale does not indicate the complete absence of goal difficulty, and
a score of 0 on an intelligence test does not indicate the complete absence of intelligence.
A true zero point is found only with variables measured on a ratio scale.
Ratio Scales
Ratio scales are similar to interval scales in that equal differences between scale values
have equal quantitative meaning. However, ratio scales also have a true zero point, which
gives them an additional property. With ratio scales, it is possible to make meaningful
statements about the ratios between scale values. For example, the system of inches used
with a common ruler is an example of a ratio scale. There is a true zero point because
zero inches does in fact indicate a complete absence of length. With this scale, it is
possible to make meaningful statements about ratios. It is appropriate to say that an
object four inches long is twice as long as an object two inches long. Age, as measured in
years, is also on a ratio scale—a 10-year-old house is twice as old as a 5-year-old house.
Notice that it is not possible to make these statements about ratios with the interval-level
variables discussed above. You would not say that a person with an IQ of 160 is twice as
intelligent as a person with an IQ of 80 because there is no true zero point on the IQ
scale.
Although ratio-level scales might be easiest to find in the physical properties of objects,
such as height and weight, they are also common in the type of research discussed in this
manual. For example, the study discussed previously included the variables for age and
amount of insurance sold (in dollars). Both of these have true zero points, and are
measured as ratio scales.
12 JMP for Basic Univariate and Multivariate Statistics: A Step-by-Step Guide
Modeling Types in JMP
In JMP, each variable has a modeling type that designates its scale of measurement. In a
JMP analysis, the modeling types of the variables convey their scale of measurement.
The JMP modeling types are called nominal, ordinal, and continuous. Nominal and
ordinal modeling types have the same characteristics as those described above for the
same scales of measurement. The continuous modeling type in JMP encompasses the
characteristics of both the ratio and interval scales of measurement. Modeling types are
used by JMP analysis platforms to help determine the correct analysis that needs to be
done.
The discussions that follow refer to JMP modeling types, which are discussed in detail in
Chapter 3, “Working with JMP Data.”
Basic Approaches to Research
Nonexperimental Research
Much research can be described as being either nonexperimental or experimental in
nature. In nonexperimental research (also called nonmanipulative, correlational, or
observational research), the researcher studies the naturally occurring relationship
between two or more naturally occurring variables. A naturally occurring variable is a
variable that is not manipulated or controlled by the researcher. It is measured as it
normally exists.
The insurance study described previously is a good example of nonexperimental research
in that you measured two naturally occurring variables (goal difficulty and amount of
insurance sold) to determine whether they were related. Another example of
nonexperimental research would be an investigation of the relationship between IQ and
college grade point average (GPA).
With nonexperimental designs, researchers sometimes refer to response variables and
predictor variables.
• A response variable is an outcome variable or criterion variable, whose values you
want to predict from one or more predictor variables. The response variable is often
the main focus of a study because it is mentioned in the statement of the research
problem. In the previous example, the response variable is Amount of Insurance Sold.
In some experimental research, the response variable is also called the dependent
variable.
Chapter 1: Basic Concepts in Research and Data Analysis 13
• A predictor variable is the variable used to predict values of the response. In some
studies, you might even believe that the predictor variable has a causal effect on the
response. In the insurance study, for example, the predictor variable was Goal
Difficulty. Because you believed that Goal Difficulty could positively affect insurance
sales, you conducted a study in which Goal Difficulty was the predictor and Sales was
the response. You do not necessarily have to believe that there is a causal relationship
between two variables to conduct a study such as this—you might only be interested
in determining whether it is possible to predict one variable from the other. In
experimental research, the predictor variable is also known as the independent
variable.
Notice that nonexperimental research, which investigates the relationship between just
two variables, does not provide evidence concerning cause-and-effect relationships. The
reason for this can be seen by reviewing the insurance sales study. If a psychologist
conducts this study and finds that the agents with the more difficult goals also tend to sell
more insurance, it is not necessarily true that having difficult goals causes them to sell
more insurance. Perhaps selling a lot of insurance increases the agents’ self-confidence,
and this causes them to set higher work goals for themselves. Under this second scenario,
it is the insurance sales that had a causal effect on goal difficulty.
As this example shows, with nonexperimental research it is often possible to obtain a
single result that is consistent with a number of contradictory causal explanations. Hence,
a strong inference that variable A had a causal effect on variable B is rarely if ever valid
when you conduct simple correlational research with just two variables. To obtain
stronger evidence of cause and effect, researchers either analyze the relationships
between a larger number of variables using sophisticated statistical procedures that are
beyond the scope of this text, or drop the nonexperimental approach entirely and use
experimental research methods instead. The nature of experimental research is discussed
in the following section.
Experimental Research
Most experimental research can be identified by three important characteristics:
• Subjects are randomly assigned to experimental conditions.
• The researcher manipulates an independent predictor variable.
• Subjects in different experimental conditions are treated similarly with regard to all
variables except the independent variable.
14 JMP for Basic Univariate and Multivariate Statistics: A Step-by-Step Guide
To illustrate these concepts, assume that you conduct an experiment to test the hypothesis
that goal difficulty positively affects insurance sales. Assume that you identify a group of
100 agents to serve as subjects. You randomly assign 50 agents to a “difficult goal”
condition. Subjects in this group are told by their superiors to make at least 25 cold calls
(unexpected sales calls) to potential policyholders per week. The other 50 agents have
been randomly assigned to the “easy goal” condition. They have been told to make just 5
cold calls to potential policyholders per week. The design and results of this experiment
are illustrated in Table 1.2.
After one year, you determine how much new insurance each agent sold that year.
Assume that the average agent in the difficult goal condition sold $156,000 of new
policies, while the average agent in the easy goal condition sold just $121,000 worth.
Table 1.2 Design of the Experiment Used to Assess the Effects of Goal Difficulty
Group
Treatment Conditions under
the Independent Variable
(Goal Difficulty)
Results Obtained
with the Dependent Variable
(Amount of Insurance Sold)
Group 1
(N = 50)
Difficult Goal Condition $156,000 in Sales
Group 2
(N = 50)
Easy Goal Condition $121,000 in Sales
It is possible to use some of the terminology associated with nonexperimental research
when discussing this experiment. For example, it is appropriate to continue to refer to
Amount of Insurance Sold as being a response variable, because this is the outcome
variable of interest. You can also refer to Goal Difficulty as the predictor variable
because you believe that this variable, to some extent, predicts the amount of insurance
sold.
Notice, however, that Goal Difficulty is now a different kind of variable. In the
nonexperimental study, Goal Difficulty was a naturally occurring variable that could take
on a wide variety of values (whatever score the subject received on the goal difficulty
questionnaire). In the present experiment Goal Difficulty is a manipulated variable,
which means that you (as the researcher) determine what value of the variable is to be
assigned to each subject. In this experiment, Goal Difficulty assumes only one of two
values—subjects are in either the difficult goal group or the easy goal group. Therefore,
Goal Difficulty is now a classification variable, with a nominal modeling type.
Chapter 1: Basic Concepts in Research and Data Analysis 15
Although it is acceptable to speak of predictor and response variables within the context
of experimental research, it is more common to speak in terms of independent variables
and dependent variables.
• An independent variable is that variable whose values (or levels) the experimenter
selects to determine what effect this independent variable has on the dependent
variable. The independent variable is the experimental counterpart to a predictor
variable.
• A dependent variable is some aspect of the subject’s behavior assessed to reflect the
effects of the independent variable. The dependent variable is the experimental
counterpart to a response variable.
In the example shown in Table 1.2, Goal Difficulty is the independent variable and Sales
is the dependent variable.
Remember that the terms predictor variable and response variable can be used with
almost any type of research, but that the terms independent and dependent variable
should be used only with experimental research.
Researchers often refer to the different levels of the independent variable. These levels
are also referred to as experimental conditions or treatment conditions and correspond to
the different groups to which a subject can be assigned. The present example includes
two experimental conditions, a “difficult goal condition” and an “easy goal condition.”
With respect to the independent variable, you can speak in terms of the experimental
group versus the control group. Generally speaking, the experimental group receives the
experimental treatment of interest, while the control group is an equivalent group of
subjects that does not receive this treatment. The simplest type of experiment consists of
one experimental group and one control group. For example, the present study could have
been redesigned so that it consisted of an experimental group that was assigned the goal
of making 25 cold calls (the difficult goal condition) and a control group in which no
goals were assigned (the no-goal condition).
You can expand the study by creating more than one experimental group. You could do
this in the present case by assigning one experimental group the difficult goal of 25 cold
calls and the second experimental group the easy goal of just 5 cold calls, and include a
third group as the control group assigned no goals.
16 JMP for Basic Univariate and Multivariate Statistics: A Step-by-Step Guide
Descriptive versus Inferential Statistical Analysis
To understand the difference between descriptive and inferential statistics, you must first
understand the difference between populations and samples.
• A population is the entire collection of a carefully defined set of people, objects, or
events. For example, if the insurance company in question employed 10,000 insurance
agents in the U.S., then those 10,000 agents would constitute the population of agents
hired by that company.
• A sample is a subset of the people, objects, or events selected from that population.
For example, the 100 agents used in the experiment described earlier constitute a
sample.
Descriptive Analyses: What Is a Parameter?
A parameter is a descriptive characteristic of a population. For example, if you found the
average amount of insurance sold by all 10,000 agents in this company (the population of
agents in this company), the resulting average (also called the mean) would be a
population parameter. To obtain this average, you first need to tabulate the amount of
insurance sold by each and every agent. When calculating this mean, you are engaging in
descriptive statistical analysis. Descriptive statistical analysis focuses on the exhaustive
measurement of population characteristics. You define a population, assess each member
of that population, and compute a summary value (such as a mean or standard deviation)
based on those values.
Most people think of populations as being very large groups, such as all of the people in
the U.S. However, a group does not have to be large to be a population; it only has to be
the entire collection of the people or things being studied. For example, a teacher might
define as a population all 23 students taking an English course, and then calculate the
average score of these students on a measure of class satisfaction. The resulting average
is a parameter.
Inferential Analyses: What Is a Statistic?
A statistic is a numerical value that is computed from a sample, describes some
characteristic of that sample such as the mean, and can be used to make inferences about
the population from which the sample is drawn. For example, if you were to compute the
average amount of insurance sold by your sample of 100 agents, that average would be a
statistic because it summarizes a specific characteristic of the sample. Remember that the
Chapter 1: Basic Concepts in Research and Data Analysis 17
word “statistic” is generally associated with samples, while “parameter” is generally
associated with populations.
In contrast to descriptive statistics, inferential statistical analysis involves using
information from a sample to make inferences, or estimates, about the population. For
example, assume that you need to know how much insurance is sold by the average agent
in the company. Suppose it is impossible (or very difficult) to obtain the necessary
information from all 10,000 agents and then calculate a mean. An alternative is to draw a
random (and ideally representative) sample of 100 agents and determine the average
amount sold by this subset. If this group of 100 sold an average of $179,322 worth of
policies last year, then your best guess of the amount of insurance sold by all 10,000
agents would be $179,322. You have used characteristics of the sample to make
inferences about characteristics of the population. Using some simple statistical
procedures, you can even compute confidence intervals around the estimate, which
allows you to make statements such as
“There is a 95% chance that the actual population mean lies somewhere between
$172,994 and $185,650.”
This is the real value of inferential statistical procedures—they allow you to review
information obtained from a relatively small sample and then to make inferences about a
population.
Hypothesis Testing
Most of the procedures described in this manual are inferential procedures that let you
test specific hypotheses about the characteristics of populations. As an illustration,
consider the simple experiment, described earlier, in which 50 agents are assigned to a
difficult goal condition and 50 other agents to an easy goal condition. Assume that, after
one year, the difficult-goal agents sold an average of $156,000 worth of insurance, while
the easy-goal agents sold only $121,000 worth. On the surface, this seems to support your
hypothesis that agents sell more insurance when they have difficult goals. But, even if
goal setting had no effect at all, you don’t really expect the two groups of 50 agents to
sell exactly the same amount of insurance. You expect one group to sell somewhat more
than the other due to chance alone. The difficult-goal group did sell more insurance, but
did it sell enough more to make you confident that the difference was due to placing the
agents in different goal groups?
What’s more, you can argue that you don’t even care about the amount of insurance sold
by these two small samples. What really matters is the amount of insurance sold by the
larger populations they represent. Define the first population as “the population of agents
18 JMP for Basic Univariate and Multivariate Statistics: A Step-by-Step Guide
assigned difficult goals” and the second as “the population of agents assigned easy
goals.” Your real research question is whether the first population sells more than the
second. To address this question, you need hypothesis testing.
Types of Inferential Tests
Generally speaking, there are two types of tests conducted when using inferential
procedures:
• tests of group differences
• tests of association.
With a test of group differences, you want to know whether two populations differ with
respect to their mean scores on some response variable. The present experiment leads to a
test of group differences because you want to know whether the average amount of
insurance sold in the population of difficult-goal agents is different from the average
amount sold in the population of easy-goal agents. A different example of testing group
differences might involve a study in which the researcher wants to know whether
Caucasian Americans, African Americans, and Asian Americans differ with respect to
their mean scores on a locus of control scale. (Locus of control refers to the extent to
which people believe that their own actions determine the rewards they obtain.) Notice
that in both cases, two or more distinct populations are being compared with respect to
their mean scores on a single response variable.
With a test of association, there is a single population of individuals and you want to
know whether there is a relationship between two or more variables within this
population. Perhaps the best-known test of association involves testing the significance of
a correlation coefficient. Assume that you conduct a simple correlational study in which
you ask 100 agents to complete the 20-item goal difficulty questionnaire. Remember that,
with this questionnaire, subjects can receive a score that ranges from a low of 0 to a high
of 100. You can then correlate these goal difficulty scores with the amount of insurance
sold by the agents that year. Here, the goal difficulty scores constitute the predictor
variable and the amount of insurance sold serves as the response. Obtaining a strong
positive correlation between these two variables means that the more difficult the agents’
goals, the more insurance they tend to sell. This is called a test of association because you
determine whether there is an association, or relationship, between the predictor and
response variables. Notice also that there is only one population being studied—there is
no experimental manipulation that creates a difficult-goal population versus an easy-goal
population.
Chapter 1: Basic Concepts in Research and Data Analysis 19
For the sake of completeness, it is worth mentioning that there are some relatively
sophisticated procedures that also let you test whether the association between two
variables is the same across two or more populations. Analysis of covariance (ANCOVA)
is one procedure that allows such a test.
For example, you could form a hypothesis that the association between self-reported goal
difficulty and insurance sales is stronger in the population of agents assigned difficult
goals than it is in the population assigned easy goals. To test this hypothesis, you
randomly assign a group of insurance agents to either an easy-goal condition or a
difficult-goal condition (as described earlier). Each agent completes the 20-item self report goal difficulty scale and is then given the group assignment (treatment) to make
more or fewer cold calls. Subsequently, you could record each agent’s sales. Analysis of
covariance allows you to determine whether the relationship between questionnaire
scores and sales is stronger in the difficult-goal population than it is in the easy-goal
population.
ANCOVA also allows you to test a number of additional hypotheses and is beyond the
scope of this text. For more information about ANCOVA in JMP, see the JMP Statistics
and Graphics Guide (2003).
Types of Hypotheses
Two different types of hypotheses are relevant to most statistical tests. The first is called
the null hypothesis, which is often abbreviated as H0. The null hypothesis is a statement
that, in the population(s) being studied, there are either (a) no differences between the
group means, or (b) no relationships between the measured variables. For a given
statistical test, either (a) or (b) applies, depending on whether the test is to detect group
differences or is a test of association.
With a test of group differences, the null hypothesis states that, in the population, there
are no differences between any of the groups being studied with respect to their mean
scores on the response variable. In the experiment in which a difficult-goal condition is
being compared to an easy-goal condition, the following null hypothesis might be used:
H0: In the population, the amount of insurance sold by individuals
assigned difficult goals does not differ from the amount of
insurance sold by individuals assigned easy goals.
This null hypothesis can also be expressed with symbols in the following way:
H0: M1 = M2
20 JMP for Basic Univariate and Multivariate Statistics: A Step-by-Step Guide
where
H0 represents the null hypothesis
M1 represents mean sales for the difficult-goal population
M2 represents mean sales for the easy-goal population.
In contrast to the null hypothesis, there is also an alternative hypothesis (H1) that states
the opposite of the null. The alternative hypothesis is a statement that there is a difference
between the means, or that there is a relationship between the variables, in the
population(s) being studied.
Perhaps the most common alternative hypothesis is a nondirectional alternative
hypothesis. With a test of group differences, a nondirectional alternative hypothesis
predicts that the means for the various populations differ, but makes no specific
prediction as to which mean will be relatively high and which will be relatively low. In
the preceding experiment, the following nondirectional null hypothesis might be used:
H1: In the population, individuals assigned difficult goals differ from
individuals assigned easy goals with respect to the mean amount
of insurance sold.
This alternative hypothesis can also be expressed with symbols in the following way:
H1: M1 ≠ M2
In contrast, a directional alternative hypothesis makes a more specific prediction
regarding the expected outcome of the analysis. With a test of group differences, a
directional alternative hypothesis not only predicts that the population means differ, but
also predicts which population means will be relatively high and which will be relatively
low.
Here is a directional alternative hypothesis for the preceding experiment.
H1: The average amount of insurance sold is higher in the population of individuals
assigned difficult goals than in the population of individuals assigned easy goals.
This hypothesis can be symbolically represented as follows:
H1: M1 > M2
Chapter 1: Basic Concepts in Research and Data Analysis 21
If you believe that the easy-goal population sells more insurance, you replace the “greater
than” symbol (>) with the “less than” symbol (<) in the alternative hypothesis, as follows:
H1: M1 < M2
Null and alternative hypotheses are also used with tests of association. For the study in
which you correlated goal-difficulty questionnaire scores with the amount of insurance
sold, you might use the following null hypothesis:
H0: In the population, the correlation between goal difficulty scores and the amount
of insurance sold is zero.
You could state a nondirectional alternative hypothesis that corresponds to this null
hypothesis as follows:
H1: In the population, the correlation between goal difficulty scores and the amount
of insurance sold is not equal to zero.
Notice that the preceding is an example of a nondirectional alternative hypothesis
because it does not specifically predict whether the correlation is positive or negative,
only that it is not zero. A directional alternative hypothesis, on the other hand, might
predict a positive correlation between the two variables. You could state such a prediction
as follows:
H1: In the population, the correlation between goal difficulty scores and the amount
of insurance sold is greater than zero.
There is an important advantage associated with the use of directional alternative
hypotheses compared to nondirectional hypotheses. Directional hypotheses allow
researchers to perform one-sided statistical tests (also called one-tailed tests), which are
relatively powerful. Here, “powerful” means the ability of a test to detect significant
differences between group means when differences really do exist. In contrast, non directional hypotheses allow only two-sided statistical tests (also called two-tailed tests),
which are less powerful.
Because they lead to more powerful tests, directional hypotheses are generally preferred
over nondirectional hypotheses. However, directional hypotheses should be stated only
when they can be justified on the basis of theory, prior research, or some other grounds.
For example, you should state the directional hypothesis that
“The average amount of insurance sold is higher in the population of individuals
assigned difficult goals than in the population of individuals assigned easy goals,”
22 JMP for Basic Univariate and Multivariate Statistics: A Step-by-Step Guide
only if there are theoretical or empirical reasons to believe that the difficult-goal group
will indeed score higher on insurance sales. The same should be true when you
specifically predict a positive correlation rather than a negative correlation (or vice
versa).
The p Value
Hypothesis testing is a process to determine whether you can reject a null hypothesis with
an acceptable level of confidence. When analyzing data with JMP, you look at the results
for two pieces of information that are critical for this purpose: - the obtained (calculated) statistic
- the probability (p) value associated with that statistic.
Consider the experiment in which you compared the difficult-goal group to the easy-goal
group. One way to test the null hypothesis associated with this study is to perform an
independent samples t test. When the data analysis for this study is complete, you
compute a t statistic and its corresponding p value. The p value indicates the probability
that you would obtain the present results if the null hypothesis were true. If the p value is
very small, you reject the null hypothesis. Recall that the null hypothesis states that there
is no difference between groups.
For example, assume that you obtain a t statistic of 0.14 and a corresponding p value of
0.90. This p value means that there are 90 chances in 100 that you would obtain a t
statistic of 0.14 (or larger) if the null hypothesis were true. Because this probability is so
high, you report that there is very little evidence to refute the null hypothesis. In other
words, you fail to reject the null hypothesis, and, instead, conclude that there is not
sufficient evidence to support a statistically significant difference between the two
groups.
On the other hand, assume that the research project instead produces a t value of 8.45 and
a corresponding p value of 0.001. The p value of 0.001 means that there is only one
chance in 1000 that you would obtain a t value of 8.45 (or larger) if the null hypothesis
were true. This is so unlikely that you are fairly confident that the null hypothesis is not
true. You therefore reject the null hypothesis and conclude that there is a difference in
mean sales between the two populations. In rejecting the null hypothesis, you have
tentatively accepted the alternative hypothesis.
Technically, the p value does not really provide the probability that the null hypothesis is
true. Instead, it provides the probability that you would obtain the present results (the
Chapter 1: Basic Concepts in Research and Data Analysis 23
present t statistic, in this case) if the null hypothesis were true. This might seem like a
trivial difference, but it is important to know the meaning of the p value.
Notice that you are able to reject the null hypothesis only when the p value is a small
number (0.001, in the above example). But how small must a p value be before you can
reject the null hypothesis? A p value of 0.05 is one of the most commonly accepted cutoff
values. Typically, when researchers obtain a p value larger than 0.05 (such as 0.13 or
0.37), they fail to reject the null hypothesis, and instead conclude that the differences or
relationships being studied are not statistically significant (there is no significant
difference between groups). When researchers obtain a p value smaller than 0.05, they
reject the null and conclude that differences or relationships being studied are statistically
significant (there is a significant difference between groups). The 0.05 level of
significance is not an absolute rule that must be followed in all cases, but it is serviceable
for most types of investigations likely to be conducted in the social sciences and other
areas.
Fixed Effects versus Random Effects
Experimental designs can be represented by mathematical models described as fixed effects models, random-effects models, or mixed-effects models. The use of these terms
refers to the way that the levels of the independent variable (or predictor variable) were
selected.
When the researcher arbitrarily selects the levels of the independent variable, the
independent variable is called a fixed-effects factor, and the resulting model is a fixed effects model. For example, assume that in the current study you arbitrarily decided that
the subjects in your easy-goal condition would be told to make just 5 cold calls per week,
and that the subjects in the difficult-goal condition would be told to make 25 cold calls
per week. In this case, you have fixed (arbitrarily selected) the levels of the independent
variable. Your experiment therefore represents a fixed-effects model.
In contrast, when the researcher randomly selects levels of the independent variable from
a population of possible levels, the independent variable is called a random-effects factor,
and the model is a random-effects model. For example, assume you know that the number
of cold calls an insurance agent could possibly place in one week ranges from 0 to 45.
This range represents the population of cold calls that you could possibly research.
Assume you use some random procedure to select two values from this population of
possible calls, and that those two randomly selected values are 12 and 32. In conducting
your study, one group of subjects is assigned to make at least 12 cold calls per week,
while the second is assigned to make 32 calls. In this case, your study represents a
24 JMP for Basic Univariate and Multivariate Statistics: A Step-by-Step Guide
random-effects model because the levels of the independent variable were randomly
selected from all possible levels.
As an illustration of a fixed-effects model, assume that you want to conduct research on
the effectiveness of hypnosis in reducing anxiety among subjects who suffer from
phobias. Specifically, you want to perform an experiment that compares the effectiveness
of 10 sessions of relaxation training versus 10 sessions of relaxation training plus
hypnosis. In this study, the independent variable might be labeled “Type of Therapy.”
Notice that you did not randomly select these two treatment conditions from the
population of all possible treatment conditions because you know which treatments you
want to compare, and design the study accordingly. This is experimental research and
your study represents a fixed-effects model.
To provide a nonexperimental example, assume that you want to conduct a study to
determine whether Caucasian Americans score significantly higher than African
Americans on internal locus of control. The predictor variable in your study is race, and
the response variable is scores on some index of locus of control. Most likely, you chose
“Caucasian American” versus “African American” as predictor variable groups because
you are particularly interested in these two races. You did not randomly select these
groups from all possible races. Therefore, the study is another example of a fixed-effects
model.
Random-effects factors do sometimes appear in research. For example, in a repeated measures investigation more than one measure of the response variable is taken from
each subject. Subjects are viewed as a random-effects factor, assuming they were
randomly selected. Some studies include both fixed-effects factors and random-effects
factors. Those models are called mixed-effects models.
This distinction between fixed and random effects has important implications for the
types of inferences that can be drawn from statistical tests. When analyzing a fixed effects model, you can generalize the results of the analysis only to the specific levels of
the independent variable manipulated in that study. This means that if you arbitrarily
selected 5 cold calls versus 25 cold calls for your two treatment conditions, once the data
are analyzed you can draw conclusions only about the population of agents assigned 5
cold calls versus the population assigned 25 cold calls.
On the other hand, if you randomly selected two values for your treatment conditions
(say, 12 versus 32 cold calls) from the population of possible numbers of calls, the model
is a random-effects model. This means that you can draw conclusions about the entire
population of possible values that the independent variable can assume. Inferences are
not restricted to just the two treatment conditions investigated in the study. In other
Chapter 1: Basic Concepts in Research and Data Analysis 25
words, you can draw inferences about the relationship between the population of the
possible number of cold calls that agents could be assigned and the response variable
(insurance sales).
Summary
Regardless of discipline, researchers need a common language to use when discussing
their work with others. This chapter has reviewed the basic concepts and terminology of
research that will be referred to throughout this text. Now that you can speak the
language, you are ready to move on to Chapter 2, “Getting Started with JMP,” where you
learn how to do simple JMP analyses.
References
SAS Institute Inc. 2003. JMP Statistics and Graphics Guide. Cary, NC: SAS Institute Inc.
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