Probability Sampling
uses random selection
N = number of cases in sampling frame
n = number of cases in the sample
NCn = number of combinations of n
from N
f = n/N = sampling fraction
Variations
Simple random sampling

based on random number generation
Stratified random sampling

divide pop into homogenous subgroups, then simple random
sample w/in
Systematic random sampling

select every kth individual (k = N/n)
Cluster (area) random sampling

randomly select clusters, sample all units w/in cluster
Multistage sampling

combination of methods
Nonprobability sampling
accidental, haphazard, convenience
sampling ...
may or may not represent the
population well
Measurement
... topics in measurement that we don’t
have time to cover ...
Research Design
Elements:





Samples/Groups
Measures
Treatments/Programs
Methods of Assignment
Time
Internal validity
the approximate truth about inferences
regarding cause-effect (causal)
relationships
can observed changes be attributed to
the program or intervention and NOT to
other possible causes (alternative
explanations)?
Establishing a Cause-Effect
Relationship
Temporal precedence
Covariation of cause and effect


if x then y; if not x then not y
if more x then more y; if less x then less y
No plausible alternative explanations
Single Group Example
Single group designs:

Administer treatment -> measure outcome
X -> O
 assumes baseline of “0”

Measure baseline -> treat -> measure outcome
0
X
-> O
 measures change over baseline
Single Group Threats
History threat

a historical event occurs to cause the outcome
Maturation threat

maturation of individual causes the outcome
Testing threat

act of taking the pretest affects the outcome
Instrumentation threat

difference in test from pretest to posttest affects the outcome
Mortality threat

do “drop-outs” occur differentially or randomly across the sample?
Regression threat

statistical phenomenon, nonrandom sample from population and
two imperfectly correlated measures
Addressing these threats
control group + treatment group

both control and treatment groups would
experience same history and maturation
threats, have same testing and
instrumentation issues, similar rates of
mortality and regression to the mean
Multiple-group design
at least two groups
typically:



before-after measurement
treatment group + control group
treatment A group + treatment B group
Multiple-Group Threats
internal validity issue:


degree to which groups are comparable
before the study
“selection bias” or “selection threat”
Multiple-Group Threats
Selection-History Threat

an event occurs between pretest and posttest that groups experience
differently
Selection-Maturation Threat

results from differential rates of normal growth between pretest and
posttest for the groups
Selection-Testing Threat

effect of taking pretest differentially affects posttest outcome of groups
Selection-Instrumentation Threat

test changes differently for the two groups
Selection-Mortality Threat

differential nonrandom dropout between pretest and posttest
Selection-Regression Threat

different rates of regression to the mean in the two groups (if one is more
extreme on the pretest than the other)
Social Interaction Threats
Problem:

social pressures in research context can
lead to posttest differences that are not
directly caused by the treatment
Solution:


isolate the groups
Problem: in many research contexts, hard
to randomly assign and then isolate
Types of Social Interaction
Threats
Diffusion or Imitation of Treatment

control group learns about/imitates experience of treatment
group, decreasing difference in measured effect
Compensatory Rivalry

control group tries to compete w/treatment group, works
harder, decreasing difference in measured effect
Resentful Demoralization

control group discouraged or angry, exaggerates measured
effect
Compensatory Equalization of Treatment

control group compensated in other ways, decreasing
measured effect
Intro to Design/ Design Notation
Observations or Measures
Treatments or Programs
Groups
Assignment to Group
Time
Observations/Measure
Notation: ‘O’

Examples:
 Body weight
 Time to complete
 Number of correct response
Multiple measures: O1, O2, …
Treatments or Programs
Notation: ‘X’


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Use of medication
Use of visualization
Use of audio feedback
Etc.
Sometimes see X+, X-
Groups
Each group is assigned a line in the
design notation
Assignment to Group
R = random
N = non-equivalent groups
C = assignment by cutoff
Time
Moves from left to right in diagram
Types of experiments
True experiment – random assignment
to groups
Quasi experiment – no random
assignment, but has a control group or
multiple measures
Non-experiment – no random
assignment, no control, no multiple
measures
Design Notation Example
R
O1
R
O1
X
O1,2
O1,2
Pretest-posttest treatment versus
comparison group
randomized experimental design
Design Notation Example
N
O
N
O
X
Pretest-posttest
Non-Equivalent Groups
Quasi-experiment
O
O
Design Notation Example
X
Posttest Only
Non-experiment
O
Goals of design ..
Goal:to be able to show causality
First step: internal validity:
If x, then y
AND
 If not X, then not Y

Two-group Designs
Two-group, posttest only, randomized
experiment
R
R
X
O
O
Compare by testing for differences between means
of groups, using t-test or one-way Analysis of
Variance(ANOVA)
Note: 2 groups, post-only measure, two distributions each
with mean and variance, statistical (non-chance) difference
between groups
To analyze …
What do we mean
by a difference?
Possible Outcomes:
Three ways to estimate effect
Independent t-test
One-way Analysis of Variance (ANOVA)
Regression Analysis (most general)
equivalent
The t-test


appropriate for posttest-only two-group
randomized experimental design
See also: paired student t-test for other
situations.
Measuring Differences …
Computing the t-value
Computing standard deviation
• standard deviation is the square root of the sum of the
squared deviations from the mean divided by the number
of scores minus one
•variance is the square of the standard deviation
ANOVA
One-way analysis of variance
ANOVA
Analysis of variance – tests hypotheses
about differences between two or more
means
Could do pairwise comparison using ttests, but can lead to true hypothesis
being rejected (Type I error) (higher
probability than with ANOVA)
Between-subjects design
Example:

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Effect of intensity of background noise on
reading comprehension
Group 1: 30 minutes reading, no
background noise
Group 2: 30 minutes reading, moderate
level of noise
Group 3: 30 minutes reading, loud
background noise
Experimental Design
One factor (noise), three levels(a=3)
Null hypothesis: 1 = 2 = 3
Noise
None
Moderate
High
R
O
O
O
Notation
If all sample sizes same, use n, and
total N = a * n
Else N = n1 + n2 + n3
Assumptions
Normal distributions
Homogeneity of variance

Variance is equal in each of the populations
Random, independent sampling
Still works well when assumptions not
quite true(“robust” to violations)
ANOVA
Compares two estimates of variance


MSE – Mean Square Error, variances within
samples
MSB – Mean Square Between, variance of
the sample means
If null hypothesis


is true, then MSE approx = MSB, since
both are estimates of same quantity
Is false, the MSB sufficiently > MSE
MSE
MSB
Use sample means to calculate sampling
distribution of the mean,
=1
MSB
Sampling distribution of the mean * n
In example, MSB = (n)(sampling dist) =
(4) (1) = 4
Is it significant?
Depends on ratio of MSB to MSE
F = MSB/MSE
Probability value computed based on F value,
F value has sampling distribution based on
degrees of freedom numerator (a-1) and
degrees of freedom denominator (N-a)
Lookup up F-value in table, find p value
For one degree of freedom, F == t^2
Factorial Between-Subjects
ANOVA, Two factors
Three significance tests


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Main factor 1
Main factor 2
interaction
Example Experiment
Two factors (dosage, task)
3 levels of dosage (0, 100, 200 mg)
2 levels of task (simple, complex)
2x3 factorial design, 8 subjects/group
Summary table
SOURCE
Task
Dosage
TD
ERROR
TOTAL
df Sum of Squares
1
47125.3333
2
42.6667
2
1418.6667
42
5152.0000
47
53738.6667
Sources of variation:


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Task
Dosage
Interaction
Error
Mean Square
F
p
47125.3333 384.174 0.000
21.3333
0.174 0.841
709.3333
5.783 0.006
122.6667
Results
Sum of squares (as before)
Mean Squares = (sum of squares) /
degrees of freedom
F ratios = mean square effect / mean
square error
P value : Given F value and degrees of
freedom, look up p value
Results - example
Mean time to complete task was higher
for complex task than for simple
Effect of dosage not significant
Interaction exists between dosage and
task: increase in dosage decreases
performance on complex while
increasing performance on simple
Results
Regression Analysis
Equivalent to t-test and ANOVA for
post-test only two group factorial
design
Regression Analysis
Solve overdetermined system of equations for β0 and
β1, while minimizing sum of e-terms
Regression Analysis
ANOVA
Compares differences within group to
differences between groups
For 2 populations, 1 treatment, same as
t-test
Statistic used is F value, same as square
of t-value from t-test
Other Experimental Designs
Signal enhancers

Factorial designs
Noise reducers

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Covariance designs
Blocking designs
Factorial Designs
Factorial Design
Factor – major independent variable

Setting, time_on_task
Level – subdivision of a factor


Setting= in_class, pull-out
Time_on_task = 1 hour, 4 hours
Factorial Design
Design notation as
shown
2x2 factorial design
(2 levels of one
factor X 2 levels of
second factor)
Outcomes of Factorial Design
Experiments
Null case
Main effect
Interaction Effect
The Null Case
The Null Case
Main Effect - Time
Main Effect - Setting
Main Effect - Both
Interaction effects
Interaction Effects