Download Soci708 -- Statistics for Sociologists - Module 5 -

Soci708 – Statistics for Sociologists
Module 5 – Producing Data: Statistical Issues in Research
Design1
François Nielsen
University of North Carolina
Chapel Hill
Fall 2007
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Adapted from slides for the course Quantitative Methods in Sociology
(Sociology 6Z3) taught at McMaster University by Robert Andersen (now at
University of Toronto)
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Goals of This Module
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Discuss various ways of collecting data and their implications
for statistical analysis
Observational studies
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Probability versus nonprobability samples
Experimental studies
Missing data
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Types of missing data and their problems
Strategies for coping with missing data
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Science of Sampling
While the individual man is an absolute puzzle, in the
aggregate he becomes a mathematical certainty. You can
never foretell what any one man will do, but you can say
with precision what an average number will be up to.
Individuals vary, but averages remain constant.
— Sir Arthur Conan Doyle2
Source: Worcester, R. 1991. British Public Opinion.
London: Basil Blackwell, p.151.
2
Author of the Sherlock Holmes series
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Adolphe Quetelet (1796 – 1874)
Inventor of l’homme moyen (Average Man) Concept
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Born in Ghent, died in Brussels
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Essai de physique sociale (1835)
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Invents notion of average person
central to social statistics
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Invents body-mass index (BMI):
BMI =
weight (Kg)
height2 (m2 )
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Normal BMI range: 18.5–25
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Corresponds with Florence
Nightingale (1820 – 1910)
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What is Sampling?
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Process of selecting a small number of cases (sample) in a way
that it will accurately represent a larger number of cases
(population)
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To provide useful descriptions of a population, the sample
must contain essentially the same variations as the population
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If all members of society were identical in all respects, we
would not need to sample
Two Major Types of Samples:
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Non Probability
Probability or Random
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General Sampling Terms (1)
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Population: This is the group of study elements about which
we want to make generalizations
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Finite population: E.g., all eligible voters in Canada
Infinite population: Resulting from a process: e.g., computer
chips made by a certain assembly line; members of species mus
musculus
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Sampling Frame: List of cases in the population. Should
include all elements once and only once: No duplications or
omissions.
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Sampling Pool: List of numbers to choose from in random
digit dialing
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Sampling Element: Each case that is being sampled from the
population
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General Sampling Terms (2)
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Sampling Ratio: Percentage of population that is in the sample
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Sampling ratio = sample/population, e.g.,
sample size = 1000
population = 1, 000, 000
Sampling Ratio = 1000/1, 000, 000
= 0.001 or 0.1%
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Population Parameter: True value of a feature (e.g.
percentage, mean) in the whole population
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Statistic: Value of the feature in the sample data. Statistics are
often used to estimate an unknown population parameter
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Validity & Sampling Bias
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External validity: The degree to which the conclusions of a
study would hold for other persons in other places and at
other times. Does our sample represent the population?
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Sampling Bias: Those selected for the sample are not “typical”
or “representative” of the population.
Two types of sampling bias:
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Noncoverage: Some groups in the population are systematically
left out of the process of choosing the sample
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E.g., homeless people with no telephone
Nonresponse: Associated with survey research – when an
individual chosen for the sample canŠt be reached or refuses
to cooperate
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E.g., Current Population Survey (CPS) has low nonresponse rate
(3%–4%); polls by opinion polling firms may be as high as
50%–60%
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Non-probability Samples
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Haphazard samples (convenience samples; voluntary response
samples)
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No plan – usually not representative
Quota samples
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Match proportion of selected groups to population
Acceptable in exploratory research
Purposive samples (judgmental samples)
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Acceptable for difficult to locate, special populations (e.g.,
homeless people)
Snowball samples (network samples)
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Used in special situations when it is difficult to obtain a list of
the population, but people know one another
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Development of Scientific Sampling (1)
Early Modern Polling (1920-1932)
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Using convenience samples
Literary Digest correctly
predicted all US elections from
1920 to 1932
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Literary Digest gained great
prestige
Disaster in 1936 election
(Predicted Landon over FDR)
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2,000,000 of 10,000,000
questionnaires returned
Biased sampling frame based
on LD subscription list, car &
telephone ownership
Excluded poor & low
response rate
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Development of Scientific Sampling (2)
Era of Quota Sampling (1936-1944)
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Gallup used quota sampling in 1936, correctly predicting
FDR’s win
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Quotas on gender, urban/rural, education, race
Other firms began using quotas
Disaster in 1948 election – wrongly predicted Dewey victory;
Truman won huge victory
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Quotas were not representative of population (they were
based on 1940s census data which under-represented urban
population)
Stopped polling too soon
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Probability or Random Sampling
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A probability or random sample (in general) is one chosen by
chance, so that each possible sample has a known probability
of being chosen
Typically this condition is relaxed somewhat to mean: Each
case has a known probability of being selected
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Outcomes are predictable in the long run over many cases
Selection of cases is “mechanical” and thus rules out bias or
influence by the researcher in the selection process
Random sampling forms the basis of inferential statistics, i.e.
deriving conclusions on a population on the basis of a sample
from that population
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Types of Random Sampling (1)
Simple Random Samples (SRS)
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A simple random sample (SRS) is a probability sample chosen
in such a way that each possible sample has the same
probability of being chosen
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SRS requires a good sampling frame
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Informally, one in which each case has an equal chance of being
selected
It must be possible to reach all cases in the population to do it
properly
Seldom done in practice in social research (especially with
respect to survey research)
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Often cannot get a population list
It is not usually the most efficient method
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Principle of Simple Random Sampling
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Types of Random Sampling (2)
Systematic Samples
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Short-cut form of random sampling (results often nearly
identical to SRS)
1. Obtain a list of the population.
2. Create a sampling interval
Sampling interval =
Population size
Sample size
3. Count cases and select every kth case, where k is the size of
the interval
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Cannot be used when there is a pattern in the cases
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Important to begin with a random start, rather than with the
first case
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Types of Random Sampling (3)
Stratified Sampling
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Stratified, Random Sample
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Divide people or cases into homogeneous groups
Select a random sample from each group
Add the samples together to create a complete sample of the
population
Stratified, Systematic Sample
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Divide people or cases into homogeneous groups
Put the groups together a continuous list
Using a random start, select a systematic sample from the list
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Types of Random Sampling (4)
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Types of Random Sampling (5)
Multistage Cluster Samples
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Used when cases are geographically distant or when
population cannot be easily listed
Steps:
1. Draw a sample from a collection of cases (clusters)
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e.g., select a sample of high schools in the U.S.
2. Sample individual cases from the clusters
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e.g., select samples of 36 students in 10th and 12th grades
Probability Proportionate to Size (PPS)
Used when clusters are of greatly differing sizes
Each cluster is given a chance of selection that is proportionate
to its size
Caution: Multi-stage cluster samples are prone to high
sampling error. Errors are compounded at each stage
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Cluster Sampling
Example (1)
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Goal: A national election study. Want a sample of 3000.
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Problem: No complete population list
Solution: Multi-Stage Cluster Sample
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1. Start with a list of all parliamentary constituencies
2. Randomly select 30 constituencies
3. Randomly select 10 polling stations within each selected
constituency
4. Randomly select 10 people from each selected polling station
area
5. Add all the “clusters” together (N=3000)
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Cluster Sampling
Example (2)
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Goal: Study the attitudes of Catholic women in England.
Want a sample of 1000.
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Problem: No population list
Solution: Multi-Stage Cluster Sample
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1.
2.
3.
4.
5.
Start with a list of all Catholic churches
Randomly select 10 geographic regions
Randomly select 10 churches from each region
Randomly select 10 women from congregation lists
Add all the “clusters” together (N=1000)
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Telephone Sampling
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Most commonly used sampling technique for national surveys
in Canada
We must consider the sampling pool
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The entire set of numbers used to get the desired number of
completions
We always need to select far more numbers to call than we
need because of nonresponse and non-usable numbers
The sampling pool can be determined using list sampling or
random digit dialing
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Random Digit Dialing (1)
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A group of probability sampling techniques
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Does not give equal chance of selection, however
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Some households have more than one phone, but
post-weighting can adjust for this
Limits noncoverage error
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Overcomes problem of unlisted phone numbers when using
directories
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Approx 1/3 are unlisted (Highest in urban areas)
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Random digit dialing (2)
Steps to generating a sampling pool
1. Gather telephone prefixes
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Not always straightforward since areas of interest may not
correspond to prefix boundaries
2. Determine number of lines per prefix and stratify the final
sample by prefix
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Some prefixes have more residential lines
3. Identify non-eligible banks of suffixes
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Reduces interviewer costs
4. Randomly generate a list of numbers
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Computer generated numbers; table of random numbers;
added-digits techniques
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Random digit dialing (3)
Determining the Sampling Pool size
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Size of Sampling Pool =
FSS
(HR)(1 − REC)(1 − LE)
where:
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FSS = Final Sample Size;
HR = Hit Rate (estimate of proportion of numbers not
working residential lines);
REC = Respondent Exclusion Criteria (estimate of proportion
who are not part of target population);
LE=Loss of Eligibles (likely nonresponse rate)
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Sampling Pool Size
An example
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FSS: We want to interview 1000 women
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HR: Estimate that .5 of phones are residential (based on
previous research)
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REC: Estimate that there will be no woman in about .25 of
households
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LE: Estimate the nonresponse rate to be about .3 with 8
callbacks
Size of SP =
=
=
FSS
(HR)(1 − REC)(1 − LE)
1000
(.5)(1 − .25)(1 − .3)
1000
.2625
= 3800
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Telephone sampling:
Selecting within households
1. Uncontrolled selection
1.1 Interviewing the first person who answers the phone
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Will not give a representative sample
Telephone answers are not randomly distributed within the
household unit
1.2 Allowing interviewers to select respondents within the
household
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Typically they will simply choose those who are home
Increases selection bias
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Telephone sampling:
Selecting within households
2. Controlled selection
2.1 Youngest man/youngest woman at home
2.2 Random selection from listing of household members
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All eligible respondents are identified and ordered according to
some criteria (such as age and gender)
Selection table guides choice of respondent
2.3 ‘Next birthday’ methods
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In theory should provide a random selection within the
household
Callbacks and Substitution:
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If the requested respondent is not home, many callback
attempts should be made – there should be no substitution
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Sample Weighting
1. Over-sampling small populations
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Used in stratified samples to ensure representation of a small
group
Before data analysis, correct for the over sampling by
weighting it downwards
2. Known demographic attributes
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Information exists on some demographic variables of interest,
but you canŠt sample them directly
Compare sample to the population along the demographic
lines for which you have information
Post-weight people in the sample upward or downward in the
appropriate direction
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Determining Sample Size
How much sampling error are you willing to accept?
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Sample size needed given desired confidence level & margin
of error
95%
Confidence level
90%
Margin of error
5%
Population size
Required sample size
100
1,000
10,000
100,000
1,000,000
79
278
370
383
384
3%
92
521
982
1,077
1,088
5%
73
216
268
275
275
3%
88
434
711
760
765
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Review of the Sampling Process
1. Decide on the population that you want to study
2. Determine the appropriate sampling method
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Probability samples best (SRS best, Cluster samples for
national surveys)
Nonprobability samples used in special circumstances
(exploratory research, hard to reach populations)
3. Obtain a sampling frame
4. Pick your sample from the sampling frame
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Over-sampling? Stratification?
5. Perform statistical analyses, post-weighting the sample if
necessary
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Finding Evidence of Causation
Does eating hot peppers prevent the flu?
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I eat hot peppers and didn’t get the flu
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My friend eats hot peppers and didn’t get the flu
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Another friend doesn’t eat hot peppers and got the flu
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Last year none of my family ate hot peppers and we all got the
flu; this year we all ate hot peppers and none of us got the flu
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80% of the population who didn’t eat hot peppers got the flu;
Only 40% of those who did eat hot peppers got the flu – Now
we can talk about a relationship, but not causation!
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To be confident we’ve found a causal relationship, we need
before and after measurements and controls – we need to do
an experiment!
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What is an Experiment?
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Key features are manipulation and control
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More specifically, a change in the explanatory variable is
imposed on subjects
Classical experiments meet the three criteria for causation:
1. Empirical relationship
2. Cause precedes effect in time
3. Elimination of rival explanations (i.e., no confounding
variables)
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Three types of Experiments in the Social Sciences:
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Natural Experiments, Field Experiments, Classical Experiments
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Natural Experiments
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A change in environment occurs naturally
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Before and after measures of DV are taken
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Rare, but can be instructive
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Problem: no control group to compare to
Examples
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1. A study assessing the impact of television on violent behaviour
compares the level of violent crime in a particular society
before and after the establishment of television
2. Impact of a new electoral system on social cleavage voting is
assessed by comparing voting patterns before and after the
change
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Field Experiments
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A change in environment is introduced in a natural setting
No before measures on the same people, but rather measure
of how people generally react without the intervention
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e.g., Garfinkel’s breeching experiments
Problems:
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No control group to compare to – we don’t often know what
would happen if the intervention wasn’t introduced
Also difficult to conduct for ethical and practical reasons
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Classical Experiment (1)
(Clinical trials or laboratory experiments)
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Strongest method for showing causation because it easily
meets the requirements for causation
Steps:
1. Randomization of subjects into groups:
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Experimental group, and control group
2. Pre-test
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Measure dependent variable
3. Treatment
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Independent variable
4. Post-test
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Re-measure dependent variable
5. Test for change in dependent variable
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Classical Experiment (2)
Random Assignment
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Most experiments have multiple groups
Subjects are placed into the experimental or control groups
using a true random process
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Each case has an equal chance of ending up in any group (e.g.,
subjects’ names are put into a hat and pulled out randomly)
It is a purely a mechanical process – the researcher has no
control over placement
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Classical Experiment (3)
Overall Structure of a Classical Experimental Design
Randomization
Pretest
Experimental
Group
Treatment
Posttest
Treatment
Compare
Differences
in DV
Random
Allocation
Control Group
No treatment
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Classical Experiments (4)
E.g., Impact of TV violence on attitudes
1. Select a sample of children
2. Randomly divide the children into experimental and control
groups
3. Measure their attitudes at the start
4. Show the children TV programs
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One group gets violent content, the other nonviolent programs
5. Re-test attitudes and examine for changes
6. Changes in attitude, even if temporary, can probably be
attributed to the program (i.e., violent TV –> violent
attitudes)
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Classical Experiment (5)
Other considerations
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Ethics:
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Population:
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Must clearly define the eligible population
Use of a Placebo:
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Often necessary to deceive people
Prevents people from knowing which group they are in (avoids
Hawthorne effect)
Measurement:
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Standardized outcome measures (y) of known reliability and
validity
Double blind: Measurement by staff who do not know who is
in which group
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Assessing Validity
E.g., New teaching method and grades
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We want to improve grades in methods courses
1. We decide extra tutorial consulting with tutors may make a
difference
2. On a test beforehand both the experiment and control group
average 60%
3. I give extra teaching tutorial classes to my class, another
instructor does not
4. We measure performance again: my class has an average of
55%, the other class has an average of 65%
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Can we conclude that the added tutorials had a detrimental
effect?
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Issues of Internal Validity (1)
Was the treatment the true cause of a change in the dependent variable?
1. Unexpected causes
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An unexpected event occurs during the experiment that could
affect the DV
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e.g., New computers make the course easier
2. Selection Bias
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Groups of subjects are not equivalent and there could be
pre-existing differences among them with regard to the
dependent variable
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e.g., Perhaps students chose to take the course from one
instructor over the other
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Issues of Internal Validity (2)
3. Participant Attrition
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People drop out of the experiment
Those who leave may differ from those remaining with regard
to the dependent variable
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e.g., Maybe those who drop out do not like tutorials
4. Instrumentation & Testing
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Pre- and post measurements may not be comparable
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e.g., The two tests measure different things
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Issues of Internal Validity (3)
5. Diffusion of Treatment
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Contamination
Subjects in the different groups communicate with each other
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e.g., control group works harder or control instructor feels sorry
for control group so gives extra office hours to compensate
6. Experimenter Expectancy
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Researcher may want a result, and unintentionally relay this
message to subjects. Subjects then want to please the
researcher.
Double-blind experiments prevent this
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Issues of External Validity
The ability to generalize results to outside of the experimental conditions
1. Experimental Realism
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Can’t always reproduce natural setting
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e.g., Can an experimental study on tactical voting taking place
when there is not an election to tell us how people will really
vote?
2. Participant Reactivity
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Subjects behave differently simply because they know they are
being watched
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e.g., Hawthorne Effect
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Non-experimental Panel Studies
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Explicitly causal aims – the effects of a particular event over
time
Repeat measures on same subjects
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Advantages
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Repeat measures enables one to look at change rather than
difference
Removes danger of recall bias, and gives genuine “before” and
“after” measures
Disadvantages
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Attrition
Conditioning
Birth cohort may lack generalizability
Financial and time costs
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Missing Data
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Types of missing data
1. Missing completely at random (MCAR)
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Probability of response y does not depend on either x(s) or y
2. Missing at random (MAR)
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Probability of response y is related to x(s)
3. Not missing at random (NMAR)
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“Informative” missingness
Probability of (non-)response to y is related to y or to other
variables that were not studied
Statistical inference is unaffected if MAR but less reliable
when the data are MAR or NMAR
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Methods for Handling Missing Data
1. Do nothing
2. Perform a Complete Case Analysis
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Listwise deletion: Use only cases with valid values for all
variables
3. Use a Weighting Adjustment method (e.g., compare
respondents to census data)
4. Imputation Methods (incorporate known auxillary data)
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Mean imputation
Nearest neighbor (especially for panel data)
Regression imputation
Multiple imputation
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Multiple Imputation (1)
How is it done?
1. Note: To use Multiple Imputation, the data must be MAR or
MCAR
2. Impute missing values using a model that incorporates random
variation
3. Do this k times (perhaps 5–10 times), producing k new data
sets that are complete
4. Perform statistical modeling on each data set using standard
complete – data methods
5. Average the values of the parameter estimates across the k
samples to produce a point estimate
6. Standard errors can also be calculated using a more complex
formula
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Statistical Inference
Parameter & Sample
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From before (repeat):
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Population Parameter: True value of a feature (e.g. percentage,
mean) in the whole population – usually unknown
Statistic: Value of the feature in the sample data. Statistics are
often used to estimate an unknown population parameter
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Sampling variability: the value of a statistic varies in repeated
random samples
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The one central idea of statistical inference: to see how
trustworthy a procedure is, ask what would happen if we
repeated it many times
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Sampling Distribution (1)
Sampling Variability
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Suppose that the proportion of people who agree that “The
President is doing a good job with the economy” is p = .6
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If we take a sample of 2500 we might find p̂ = .609
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If we take another sample of 2500 we might have p̂ = .625 . . .
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This is sampling variability
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If we were to sample “many times” with n = 2500 the
resulting distribution of the values of p̂ is called the sampling
distribution of p̂
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We could describe this distribution with a graph (e.g., a
histogram) or with numbers (mean, spread, . . . )
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The idea of the sampling distribution is the most important
idea of statistics
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Sampling Distribution (2)
Computer Simulation of the Sampling Distribution
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The sampling distribution of a statistic is the distribution of
values taken by the statistic in all possible samples of the
same size from the same population
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One can approximate the sampling distribution of p̂ by
simulating the drawing of many samples from a population
where p = .6
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The next two slides demonstrate the sampling distributions of
p̂ in 1000 SRS for p = .6 and n = 100 and n = 2500
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The three slides after that do the same thing in 500 SRS for
p = .3 and n = 10, n = 100, and n = 2500
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Sampling Distribution (3)
Sampling Distribution for p = .6, n = 100
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Sampling Distribution (4)
Sampling Distribution for p = .6, n = 2500
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Sampling Distribution (5)
Sampling Distribution for p = .3, n = 10 (500 Samples)
Histogram of survey10
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2
1
0
Density
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> survey10<-rbinom(500,10,0.3)/10
> survey10[1:5]
[1] 0.4 0.3 0.0 0.2 0.3
> summary(survey10)
Min. 1st Qu. Median
0.00
0.20
0.30
Mean 3rd Qu.
Max.
0.29
0.40
0.70
> hist(survey10, probability=TRUE)
> lines(density(survey10),
col="red", lwd=3)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
survey10
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Sampling Distribution (6)
Sampling Distribution for p = .3, n = 100 (500 Samples)
Histogram of survey100
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2
1
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Density
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> survey100<-rbinom(500,100,0.3)/100
> survey100[1:5]
[1] 0.30 0.28 0.29 0.33 0.30
> summary(survey100)
Min. 1st Qu. Median
0.1500 0.2700 0.3000
Mean 3rd Qu.
Max.
0.3013 0.3300 0.4300
> hist(survey100, probability=TRUE)
> lines(density(survey100),
col="red", lwd=3)
0.15
0.20
0.25
0.30
0.35
0.40
0.45
survey100
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Sampling Distribution (7)
Sampling Distribution for p = .3, n = 2500 (500 Samples)
Histogram of survey2500
0
10
Density
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> survey2500<-rbinom(500,2500,0.3)/2500
> survey2500[1:5]
[1] 0.3196 0.3056 0.2968 0.3052 0.2900
> summary(survey2500)
Min. 1st Qu. Median
0.2732 0.2940 0.2996
Mean
3rd Qu.
Max.
0.3002 0.3064 0.3268
> hist(survey2500, probability=TRUE)
> lines(density(survey2500),
col="red", lwd=3)
0.27
0.28
0.29
0.30
0.31
0.32
0.33
survey2500
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Sampling Distribution (8)
Properties of the Sampling Distribution: Shape, Center & Spread
1. Shape: histogram looks normal; one can confirm this
impression with a normal quantile plot
2. Center: center of the distributions of p̂ tends to be close to p –
mean is .29 for n = 10, .3013 for n = 100 and .3002 for
n = 2500
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In technical terms one says that the p̂ has no bias as an
estimator of p
3. Spread: values of p̂ tend to be less spread out, the larger n –
IQR is .40 − .20 = .20 for n = 10, .330 − .270 = .06 for
n = 100 and .3064 − .2940 = .0124 for n = 2500
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Sampling Distribution (9)
Properties of the Sampling Distribution: Bias & Variability
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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
Variability of a statistic – described by the spread of its
sampling distribution:
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this spread is determined by the sampling design and the
sample size n
statistics from larger probability samples have smaller spreads
Controlling bias & variability:
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To reduce bias: use random sampling – statistic computed from
a SRS is unbiased, neither consistently overestimating or
underestimating the population parameter
To reduce variability: use a larger sample – statistic computed
from a larger sample has smaller spread
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Sampling Distribution (10)
Bias & Variability Illustrated in the Context of Target Shooting (Moore $ McCabe 2006)
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Glimpses Ahead
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The idea of sampling distribution applies to all kinds of
statistics used as estimators of population parameters –
proportion, mean, variance, regression coefficients, etc.
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The next step if to develop principles of probability theory,
and from them develop a mathematical description of the
sampling distribution of a statistic
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The standard deviation of the sampling distribution of the
statistic is called the standard error of the statistic
The standard error is then used to construct
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confidence intervals for the parameter
tests of hypothesis on the value of the parameter
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Constructing confidence intervals & tests of hypothesis is
called statistical inference
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And that’s all there is to it!
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