Association and Causation
Associations Between Variables
2
Many interesting examples of the use of statistics involve relationships
between pairs of variables.
Two variables measured on the same cases are associated
if knowing the value of one of the variables tells you
something that you would not otherwise know about the
value of the other variable.
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Height and weight of same individual
Smoking habits and life expectancy
Age and bone-density of individuals
Gender and political affiliation
Gender and Smoking
2
Does Association = Causation?
3
 Caution: Often there are spurious, other
variables lurking in the background
 Shorter women have lower risk of heart attack
 Countries with more TV sets have better life expectancy
rates
 More deaths by drowning occur when ice cream sales
peak
Is your purpose simply to explore the nature of the relationship, or do you
wish to show that one of the variables can explain variation in the other?
3
Key Characteristics of a Data Set
Certain characteristics of a data set are key to exploring the relationship
between two variables. These should include the following:

Cases: Identify the cases and how many there are in the data set.

Label: Identify what is used as a label variable if one is present.
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Categorical or quantitative: Classify each variable as categorical or quantitative.
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Values: Identify the possible values for each variable.

Explanatory or response: If appropriate, classify each variable as explanatory or
response.
A response variable measures an outcome of a study. An
explanatory variable explains or causes changes in the
response variable.
4

Different amount of alcohol given to mice, body
temperature noted (belief: drop in body
temperature with increasing amount of alcohol)
◦ Response variable?
◦ Explanatory variable?

SAT scores used to predict college GPA
◦ Response variable?
◦ Explanatory variable?
Explaining Association: Causation
Association, however strong, does NOT imply causation.
Some possible explanations for an observed association
The dashed lines show an association. The solid arrows show a causeand-effect link. x is explanatory, y is response, and z is a lurking variable.
6
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Association does not imply causation!

Causation: Variable X directly causes a change
in Variable Y

Example:
◦ An association between x and y, even if it is very
strong, is not itself good evidence that changes in x
actually cause changes in y.
◦
◦
X = plant food
Y = plant’s growth

Other variables may affect the relationship
between X and Y
Beware of lurking variables

Example: for children,

◦ X = height
◦ Y = Math Score
◦ Z=

Other variables may affect the relationship
between X and Y
Can’t separate effects of X and Z on Y

Example:

◦ X = number years of education
◦ Y = income
◦ Z=
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Anecdotal data
Available data
Sample surveys and experiments
Observation vs. experiment
10

Exploratory Data Analysis
◦ reveals interesting features of data but may not be
sufficient for definite conclusions – more graphical
 What is the credibility of what I see?
 How trustworthy are my findings?

Statistical Inference
◦ gives answers with a controlled degree of confidence –
more numerical
 Relies heavily on properly collected data

Available data are data that were produced in the past for some
other purpose but that may help answer a present question
inexpensively. The library and the Internet are sources of available
data.

Some questions require data produced specifically to answer them.
This leads to designing observational or experimental studies.

Trustworthy data  Trustworthy Inference
Beware of drawing conclusions from our own experience or hearsay.
Anecdotal data represent individual cases that often come to our
attention because they are striking in some way. We tend to remember
these cases because they are unusual. The fact that they are
unusual means that they may not be representative of any larger
group of cases.
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
“Smoking isn't harmful. My grandmother smoked a pack a day and
lived to age 92.
“Last week an airplane crashed and 113 people were killed. So, I’m
driving home for break instead of flying.”
Sample Surveys
Sample surveys are a special type of designed experiment that
usually aim to discover the opinions of people on certain topics.

In a sample survey, a sample of individuals is selected from a
larger population of individuals.
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One can study a small part of the population in order to gain
information about the population as a whole.
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Example: Gallup poll – originally sampled 3000 adult residents of
the U.S. to represent the entire U.S. population
Conclusions drawn from a sample are valid only when the
sample is drawn in a well-defined way, to be discussed in
Section 3.3.
14
When our goal is to understand cause and effect, experiments are
the only source of fully convincing data.
The distinction between observational study and experiment is one of
the most important in statistics.
An observational study observes individuals and measures
variables of interest but does not attempt to influence the
responses. The purpose is to describe some group or situation.
An experiment deliberately imposes some treatment on
individuals to measure their responses. The purpose is to study
whether the treatment causes a change in the response.
15
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Observational studies cannot show causation.
◦ Example: smoking and lung cancer
◦ Could there be another lurking variable such as genetics,
which causes a person to both want to smoke and develop
lung cancer?

However, experiments can give good evidence for
causation. Why?
◦ Designed to answer specific questions
◦ Can control factors and levels

Then why have observational studies?
◦ Can we assign people to smoke?
Observational studies of the effect of one variable on another often fail
because of confounding between the explanatory variable and one or
more lurking variables.
A lurking variable is a variable that is not among the explanatory
or response variables in a study but that may influence the
response variable.
Confounding occurs when two variables are associated in such a
way that their effects on a response variable cannot be
distinguished from each other.
Well-designed experiments take steps to avoid confounding.
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Experimental units, subjects, treatments
Comparative experiments
Bias
Principles of experimental design
Statistical significance
Matched pairs design
Block design
18
An experiment is a study in which we actually do something (a
treatment) to people, animals, or objects (the experimental units) to
observe the response. Here is the basic vocabulary of experiments.
An experimental unit is the smallest entity to which a treatment is
applied. When the units are human beings, they are often called
subjects.
The explanatory variables in an experiment are often called factors.
A specific condition applied to the individuals in an experiment is called
a treatment. If an experiment has several explanatory variables, a
treatment is a combination of specific values of these variables.
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Randomized comparative experiments are designed to give good
evidence that differences in the treatments actually cause the
differences we see in the responses.
Principles of Experimental Design
1.
Control for lurking variables that might affect the response, most
simply by comparing two or more treatments. (Use a placebo, etc)
2.
Randomize: Use chance to assign experimental units to treatments.
3.
Replication: Use enough experimental units in each group to reduce
chance variation in the results.
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
Use a placebo if possible
◦ “placebo effect” is when a patient believes the treatment will work
and this affects the results
 I.E. an improvement in health is due to the patient’s belief, not the
treatment itself. Because they believe they will get better they actually
do!
◦ Using a placebo in a experiment eliminates the placebo effect

Double-blind

Lack of Realism
◦ Neither the subjects nor those who interact with them and measure
the response variable know which treatment a subject received.
◦ The most serious potential weakness of experiments is lack of
realism. The subjects or treatments or setting of an experiment
may not realistically duplicate the conditions we really want to
study.

We will look at 3 Experimental Designs
◦ Completely Randomized Design
◦ Matched-Pairs Design
◦ Block Design
Randomized Comparative
Experiments
In a completely randomized design, the treatments are assigned
to all the experimental units completely by chance.
Some experiments may include a control group that receives an
inactive treatment or an existing baseline treatment. (Control or
Placebo, etc)
Group 1
Experimental
units
Treatment
1
Random
assignment
Compare
results
Group 2
Treatment
2
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Measure weight gain of rats under new diet.
Thirty rats are used.
The simplest randomized comparative design:
assign rats completely randomly to treatments

Only two treatment groups.

Try to pair up individuals that are as similar as possible

Separate each pair, placing one in each group. Done
randomly.
Concept: Try to eliminate differences between the two
groups since each is made up of relatively the “same”
individuals
Ideal: Identical twins

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◦ Sometimes 1 treatment, 1 control
◦ Age, sex, income, etc.
◦ Put one twin in group 1 and one twin in group 2, then any
differences between the two twins should be due to the treatment
alone!
Used when you know that different groups (called blocks) will respond
differently to the treatments
•
Hormone studies—blocks are female and male. Why?
A block is a group of experimental units that are known before the
experiment to be similar in some way that is expected to affect the
response to the treatments.
In a block design, the random assignment of experimental units to
treatments is carried out separately within each block.
Form blocks based on the most important unavoidable sources of variability (lurking
variables) among the experimental units.
Randomization will average out the effects of the remaining lurking variables and allow
an unbiased comparison of the treatments.
26
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The progress of a type of cancer differs in women
and men. A clinical experiment to compare three
therapies for this cancer therefore treats sex as a
blocking variable. Two separate randomizations
are done, one assigning the female subjects to the
treatments and the other assigning the male
subjects.
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Does regularly taking aspirin help protect people against
heart attacks? The Physicians’ Health Study was a
medical experiment that helped answer this question.
The study looked at the effects of two drugs, aspirin and
beta carotene, on 21,996 male physicians. It looked at
all possible combinations of the drugs. An equal number
of subjects was assigned to each treatment.
What are the experimental units?
What are the factors?
What are the treatments? How many are there?
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Many utility companies have programs to encourage their
customers to conserve energy. An electric company is considering
placing electronic meters in households to show what the cost
would be if the electricity use at that moment continued for a month.
Will meters reduce electricity use? Would cheaper methods work
almost as well? The company decides to design an experiment.
One cheaper approach is to give customers a chart and information
about monitoring their electricity use. The experiments compares
these two approaches (meter, chart) with each other and also a
control group of customers who receive no help in monitoring
electricity use. The response variable is total electricity used in a
year. The company finds 60 single-family residences in the same
city willing to participate.
Why is this an experiment?
What type of experiment is this?
Outline the experiment.
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We hope to see a difference between groups
But just any difference is not enough!
An observed effect so large that it would rarely occur by chance is
called statistically significant.
A statistically significant association in data from a well-designed
experiment does imply causation.

Example:
◦ Vitamin C group has an average of 3 colds/yr
◦ Placebo group has an average of 4 colds/yr
◦ Statistical significant?

We will (eventually) learn where this cutoff is
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Population and sample
Voluntary response sample
Simple random sample
Stratified samples
Undercoverage and nonresponse
33
The distinction between population and sample is basic to statistics. To
make sense of any sample result, you must know what population the
sample represents.
The population in a statistical study is the entire group of individuals
about which we want information.
A sample is the part of the population from which we actually collect
information. We use information from a sample to draw conclusions
about the entire population.
Population
Collect data from a
representative Sample...
Sample
Make an Inference about
the Population.
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Random sampling, the use of chance to select a sample, is the
central principle of statistical sampling.
A simple random sample (SRS) of size n consists of n individuals
from the population chosen in such a way that every set of n
individuals has an equal chance to be the sample actually selected.
In practice, people use random numbers generated by a computer
or calculator to choose samples. If you don’t have technology
handy, you can use a table of random digits. More later…
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Sometimes, there are statistical advantages to using more complex
sampling methods. One common alternative to an SRS involves sampling
important groups (called strata) within the population separately. These
“sub-samples” are combined to form one stratified random sample.
To select a stratified random sample, first classify the population into
groups of similar individuals, called strata. Then choose a separate
SRS in each stratum and combine these SRSs to form the full sample.
Other probability sampling:
Another common means of restricting random selection is to choose
the sample in stages. These designs are called multistage designs.
They are still usually random at each stage!
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The design of a sample is biased if it systematically favors certain
outcomes.
Choosing individuals simply because they are easy to reach results
in a convenience sample.
A voluntary response sample consists of people who choose
themselves by responding to a general appeal. Voluntary response
samples often show bias because people with strong opinions
(often in the same direction) may be more likely to respond.
37
Good sampling technique includes the art of reducing all sources of
error.
Undercoverage occurs when some groups in the population
are left out of the process of choosing the sample.
Nonresponse occurs when an individual chosen for the sample
can’t be contacted or refuses to participate.
A systematic pattern of incorrect responses in a sample survey
leads to response bias.
The wording of questions is the most important influence on
the answers given to a sample survey.
38
A table of random digits is a long string of the digits 0, 1, 2, 3, 4, 5, 6,
7, 8, 9 with these properties:
 Each entry in the table is equally likely to be any of the 10 digits
0–9.
 The entries are independent of one another. That is, knowledge
of one part of the table gives no information about any other
part.
How to Choose an SRS Using Table B
Step 1: Label. Give each member of the population a numerical label
of the same length.
Step 2: Table. Read consecutive groups of digits of the appropriate
length from Table B.
Your sample contains the individuals whose labels you find.
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Basic data ethics
Institutional review boards
Informed consent
Confidentiality
Clinical trials
Behavioral and social science experiments
42
The most complex issues of data ethics arise
when we collect data from people.
Basic Data Ethics
The organization that carries out the study must have an
institutional review board that reviews all planned studies in
advance in order to protect the subjects from possible harm.
All individuals who are subjects in a study must give their
informed consent before data are collected.
All individual data must be kept confidential. Only statistical
summaries for groups of subjects may be made public.
43
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The organization that carries out the study must
have an institutional review board that reviews all
planned studies in advance in order to protect the
subjects from possible harm.
The purpose of an institutional review board is “to
protect the rights and welfare of human subjects
(including patients) recruited to participate in
research activities.”
The institutional review board:
 Reviews the plan of study
 Can require changes
 Reviews the consent form
 Monitors progress at least once a year
44
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All subjects must give their informed consent
before data are collected.
Subjects must be informed in advance about the
nature of a study and any risk of harm it might
bring.
Subjects must then consent in writing.
Who can’t give informed consent?
 Prison inmates
 Very young children
 People with mental disorders
45
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All individual data must be kept confidential. Only
statistical summaries may be made public.
Confidentiality is not the same as anonymity.
Anonymity prevents follow-ups to decrease nonresponse or inform subjects of results.
Separate the identity of the subjects from the rest
of the data immediately!
Example: Citizens are required to give information to the
government (tax returns, social security contributions). Some
people feel that individuals should be able to forbid any other use
of their data, even with all identification removed.
46
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Many behavioral experiments rely on hiding the
true purpose of the study.
Subjects would change their behavior if told in
advance what investigators were looking for.
The “Ethical Principles” of the American
Psychological Association require consent, unless
a study merely observes behavior in a public
space.
47
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Parameters and statistics
Sampling variability
Sampling distribution
Bias and variability
Sampling from large populations
48
As we begin to use sample data to draw conclusions about a wider
population, we must be clear about whether a number describes a sample
or a population.
A parameter is a number that describes some characteristic of the
population. In statistical practice, the value of a parameter is not known
because we cannot examine the entire population.
A statistic is a number that describes some characteristic of a sample.
The value of a statistic can be computed directly from the sample data.
We often use a statistic to estimate an unknown parameter.
Remember s and p: statistics come from samples and
parameters come from populations.
49

Example: In a survey of 2500 adults, 1650 admit
shopping is frustrating
◦ Parameter: p = true proportion of adults who find
shopping frustrating
◦ Statistic: p-hat = 1650/2500 = 0.66

Take another such sample, you will get a different
estimate. How variable are they?

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Take more such samples
Observe outcomes
Make histogram to see center and
spread
Good news: very often the distribution of a sample
statistic is well concentrated about its center,
which may happen to be the population parameter.


We take many samples of size n=100 from a
population with parameter p=0.60
The distribution of the sample proportion is
given

What does this say about sample size and the effects on
variation of p-hat?
What does this say about the distribution of sample statistics?
Bias and Variability
We can think of the true value of the population parameter as the bull’s-eye on a target
and of the sample statistic as an arrow fired at the target. Bias and variability describe
what happens when we take many shots at the target.
Bias concerns the center of the
sampling distribution. A statistic used
to estimate a parameter is unbiased
if the mean of its sampling distribution
is equal to the true value of the
parameter being estimated. It is on
target!
The variability of a statistic is
described by the spread of its
sampling distribution. This spread is
determined by the sampling design
and the sample size n. Statistics
from larger probability samples
have smaller spreads.
55
I.
Which is which?
High Bias, High
Variability
II.
High Bias, Low
Variability
III.
Low Bias, High
Variability
IV.
Low Bias, Low
Variability
Managing Bias and Variability
A good sampling scheme must have both small bias and small variability.
To reduce bias, use random sampling.
To reduce variability of a statistic from an SRS, use a larger sample.
Variability of a statistic does not depend
on the size of the population, only on the
size of a sample!!!
The variability of a statistic from a random sample does not depend
on the size of the population, as long as the population is at least 20
times larger than the sample.
57
A study was conducted to compare the overall length
of two species of lake trout, labeled A and B. Ten
fish from each species were randomly selected from
each of three lakes and their respective lengths were
recorded.
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◦
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What is the experimental unit?
What is the population?
What is the sample?
What is the response variable?
Is this an experiment or an observational study?
In a study to assess the impact of aerobic exercise on
weight loss for Lafayette residents, a physical education
student selected the first 20 people who signed up for a
University sponsored Wellness program. At the start of the
study, each of the study participants was weighed. At the
end of the 5 week program, each participant was asked for
the average number of hours per week they did aerobic
exercises. At this time, the 20 people were weighed again.
The amount of weight lost by each person was determined.

◦
◦
◦
◦
◦
What is the experimental unit?
What is the population?
What is the sample?
What is the response variable?
Explain how you would improve the design of this study.

Three methods of group-encounter techniques were compared with
respect to the level of group interaction achieved. Twenty-one group
leaders from around the country were randomly chosen to participate in
the study. The study was conducted at one training camp. Seven group
leaders were randomly assigned to method 1, seven to method 2, and
seven to method 3. After one session, the group leaders were scored
on their respective abilities to achieve meaningful group interaction.
◦
What is the response variable?
◦
What is the experimental unit?
◦
Is this an experiment or observational study?
◦
What type of experimental design is this?
◦
What kind of sample design did they use?

A ________ is a number describing a sample
whereas a ________ is associated with a
population.

“Statistics obtained from relatively large samples
must always be closer to the population
parameter they estimate than statistics obtained
from smaller samples.” True or False? Explain
your answer.