Download Theories - the Department of Psychology at Illinois State University

Statistics (cont.)
Psych 231: Research
Methods in Psychology
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Quiz 10 (chapter 7) is due on Nov. 13th at midnight
Journal Summary 2 assignment
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Due in class NEXT week (Wednesday, Nov. 18th) <- moved
due date
Group projects
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Plan to have your analyses done before Thanksgiving break,
GAs will be available during lab times to help
Poster sessions are last lab sections of the semester (last week
of classes), so start thinking about your posters. I will lecture
about poster presentations on the Monday before Thanksgiving
break.
Announcements
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2 General kinds of Statistics

Descriptive statistics
• Used to describe, simplify, &
organize data sets
• Describing distributions of
scores

Population
Inferential
statistics
used to
generalize
back
Inferential statistics
• Used to test claims about the
population, based on data
gathered from samples
• Takes sampling error into
account. Are the results above
and beyond what you’d expect
by random chance?
Statistics
Sample

Purpose: To make claims about populations
based on data collected from samples


What’s the big deal?
Example Experiment:
 Group A - gets treatment to improve memory
 Group B - gets no treatment (control)


Population
After treatment period test both groups for
memory
Results:
 Group A’s average memory score is 80%
 Group B’s is 76%

Is the 4% difference a “real” difference
(statistically significant) or is it just sampling
error?
Inferential Statistics
Sample A
Treatment
X = 80%
Sample B
No Treatment
X = 76%
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Step 1: State your hypotheses
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Step 2: Set your decision criteria

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
Null hypothesis (H0)
Alternative hypothesis(ses) (HA)
Type I error (α): concluding that there is an effect (a difference between
groups) when there really isn’t.
Type II error (β): concluding that there isn’t an effect, when there really is.
Step 3: Collect your data from your sample(s)
Step 4: Compute your test statistics
Step 5: Make a decision about your null hypothesis


Reject H0
Fail to reject H0
“statistically significant differences”
“not statistically significant differences”
Testing Hypotheses

Example Experiment:
 Group A - gets treatment to improve memory
 Group B - gets no treatment (control)
 After treatment period test both groups for memory
 Results:
 Group A’s average memory score is 80%
 Group B’s is 76%
 Is the 4% difference a “real” difference (statistically
significant) or is it just sampling error?
Two sample
distributions
Experimenter’s
conclusions
XB
XA
76%
80%
About populations
H0: μA = μB
H0: there is no difference
between Grp A and Grp B
Reject
H0
Fail to
Reject
H0
Summary to this point
Real world (‘truth’)
H0 is
correct
H0 is
wrong
Type I
error
a
Type II
error
b
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“Statistically significant differences”

When you “reject your null hypothesis”
• Essentially this means that the observed difference is
above what you’d expect by chance
• “Chance” is determined by estimating how much
sampling error there is
• Factors affecting “chance”
• Sample size
• Population variability
Statistical significance
Population mean
Population
Distribution
x
n=1
Sampling error
(Pop mean - sample mean)
Sampling error
Population mean
Population
Distribution
Sample mean
x
n=2
x
Sampling error
(Pop mean - sample mean)
Sampling error
 Generally,
as the
sample
Population
mean
size increases, the sampling
error decreases
Sample mean
Population
Distribution
x
x
n = 10
x
x
x x
x
x xx
Sampling error
(Pop mean - sample mean)
Sampling error

Typically the narrower the population distribution, the
narrower the range of possible samples, and the smaller the
“chance”
Small population variability
Sampling error
Large population variability
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These two factors combine to impact the distribution of
sample means.

The distribution of sample means is a distribution of all possible
sample means of a particular sample size that can be drawn
from the population
Population
Distribution of
sample means
Samples
of size = n
XA XB XC XD
“chance”
Avg. Sampling
error
Sampling error
More info

Tests the question:

Are there differences between groups
due to a treatment?
Real world
(‘truth’)
H0 is
correct
Reject
H0
Experimenter’s
conclusions
Fail to
Reject
H0
Two possibilities in the “real world”
H0 is true (no treatment effect)
One
population
Two sample
distributions
XB
XA
76%
80%
“Generic” statistical test
H0 is
wrong
Type I
error
Type II
error

Tests the question:

Real world
(‘truth’)
H0 is
correct
Are there differences between groups
due to a treatment?
Reject
H0
Experimenter’s
conclusions
Fail to
Reject
H0
Two possibilities in the “real world”
H0 is true (no treatment effect)
H0 is
wrong
Type I
error
a
Type II
error
b
H0 is false (is a treatment effect)
Two
populations
XB
XA
XB
XA
76%
80%
76%
80%
People who get the treatment change,
they form a new population
(the “treatment population)
“Generic” statistical test
XA




XB
ER: Random sampling error
ID: Individual differences (if between subjects factor)
TR: The effect of a treatment
Why might the samples be different?
(What is the source of the variability between groups)?
“Generic” statistical test
XA




XB
ER: Random sampling error
ID: Individual differences (if between subjects factor)
TR: The effect of a treatment
The generic test statistic - is a ratio of sources of
variability
Computed
Observed difference
TR + ID + ER
=
=
test statistic
Difference from chance
ID + ER
“Generic” statistical test

The distribution of sample means is a distribution of all
possible sample means of a particular sample size that can be
drawn from the population
Population
Distribution of
sample means
Samples of
size = n
XA XB XC XD
Difference from chance
Sampling error
Avg. Sampling
error

Things that affect the computed test statistic

Size of the treatment effect
• The bigger the effect, the bigger the computed test
statistic
TR
TR + ID + ER
+ ID + ER
ID + ER
ID + ER
XA

XB
XA
XB
Difference expected by chance (sample error)
• Sample size
• Variability in the population
TR + ID + ER
ID + ER
“Generic” statistical test

1 factor with two groups

T-tests
• Between groups: 2-independent samples
• Within groups: Repeated measures samples (matched, related)

1 factor with more than two groups


Analysis of Variance (ANOVA) (either between groups or
repeated measures)
Multi-factorial

Factorial ANOVA
Some inferential statistical tests

Design


2 separate experimental conditions
Degrees of freedom
• Based on the size of the sample and the kind of t-test

Formula:
Observed difference
T=
X1 - X2
Diff by chance
Computation differs for
between and within t-tests
T-test
Based on sample error

Reporting your results

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The observed difference between conditions
Kind of t-test
Computed T-statistic
Degrees of freedom for the test
The “p-value” of the test

“The mean of the treatment group was 12 points higher than the
control group. An independent samples t-test yielded a significant
difference, t(24) = 5.67, p < 0.05.”

“The mean score of the post-test was 12 points higher than the
pre-test. A repeated measures t-test demonstrated that this
difference was significant significant, t(12) = 5.67, p < 0.05.”
T-test
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Designs

XA
XB
XC
More than two groups
• 1 Factor ANOVA, Factorial ANOVA
• Both Within and Between Groups Factors

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Test statistic is an F-ratio
Degrees of freedom
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Several to keep track of
The number of them depends on the design
Analysis of Variance
XA

XB
XC
More than two groups

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Now we can’t just compute a simple difference score since
there are more than one difference
So we use variance instead of simply the difference
• Variance is essentially an average difference
Observed variance
F-ratio =
Variance from chance
Analysis of Variance
XA

XB
XC
1 Factor, with more than two levels

Now we can’t just compute a simple difference score since
there are more than one difference
• A - B, B - C, & A - C
1 factor ANOVA
Null hypothesis:
XA
XB
XC
H0: all the groups are equal
XA = XB = XC
Alternative hypotheses
HA: not all the groups are equal
XA ≠ XB ≠ XC
XA = XB ≠ XC
1 factor ANOVA
The ANOVA
tests this one!!
Do further tests to
pick between these
XA ≠ XB = XC
XA = XC ≠ XB
Planned contrasts and post-hoc tests:
- Further tests used to rule out the different
Alternative hypotheses
XA ≠ XB ≠ XC
Test 1: A ≠ B
Test 2: A ≠ C
Test 3: B = C
XA = XB ≠ XC
XA ≠ XB = XC
XA = XC ≠ XB
1 factor ANOVA
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Reporting your results
The observed differences
 Kind of test
 Computed F-ratio
 Degrees of freedom for the test
 The “p-value” of the test
 Any post-hoc or planned comparison results
“The mean score of Group A was 12, Group B was 25, and
Group C was 27. A 1-way ANOVA was conducted and the
results yielded a significant difference, F(2,25) = 5.67, p < 0.05.
Post hoc tests revealed that the differences between groups A
and B and A and C were statistically reliable (respectively t(1) =
5.67, p < 0.05 & t(1) = 6.02, p < 0.05). Groups B and C did not
differ significantly from one another”


1 factor ANOVA
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
We covered much of this in our experimental design lecture
More than one factor

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Factors may be within or between
Overall design may be entirely within, entirely between, or mixed
Many F-ratios may be computed
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An F-ratio is computed to test the main effect of each factor
An F-ratio is computed to test each of the potential interactions
between the factors
Factorial ANOVAs
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Reporting your results

The observed differences
• Because there may be a lot of these, may present them in a table
instead of directly in the text

Kind of design
• e.g. “2 x 2 completely between factorial design”

Computed F-ratios
• May see separate paragraphs for each factor, and for interactions

Degrees of freedom for the test
• Each F-ratio will have its own set of df’s

The “p-value” of the test
• May want to just say “all tests were tested with an alpha level of
0.05”
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Any post-hoc or planned comparison results
• Typically only the theoretically interesting comparisons are
presented
Factorial ANOVAs
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The following slides are available for a
little more concrete review of distribution
of sample means discussion.
Distribution of sample means

Distribution of sample means is a “theoretical” distribution
between the sample and population
• Mean of a group of scores
– Comparison distribution is
distribution of means
Population
Distribution of sample means
Sample
Distribution of sample means

A simple case

Population:
2
4
6
8
– All possible samples of size n = 2
Assumption: sampling
with replacement
Distribution of sample means

A simple case

Population:
2
4
6
8
– All possible samples of size n = 2
mean
mean
2
2
4
6
2
5
2
4
2
6
2
8
4
2
4
4
3
4
5
4
8
6
2
6
4
3
4
6
6
6
8
6
4
5
There are 16 of them
mean
8 2
5
8
4
8
6
8
8
6
7
Distribution of sample means
6
7
8
5
4
3
2
1
In long run, the random selection of tiles
leads to a predictable pattern
2 3 4 5 6 7 8
means
2
mean
2
2
4
mean
6
5
8
mean
2
5
2
4
3
4
5
4
8
8
4
2
6
6
2
8
6
2
8
6
4
8
8
4
2
3
4
6
6
4
4
6
8
6
4
5
6
7
Distribution of sample means
6
7
8

Shape
If population is Normal, then the dist of sample means
will be Normal
– If the sample size is large (n > 30), regardless of shape of the

population
Population
Distribution of sample means
N > 30
Properties of the distribution of sample means
• Center

The mean of the dist of sample means is equal to the mean of the
population
Population
m
Distribution of sample means
same numeric value
different conceptual values
Properties of the distribution of sample means
Center



The mean of the dist of sample means is equal to the
mean of the population
Consider our earlier example
Population
2
4
6
Distribution of sample means
8
μ= 2+4+6+8
4
=5
5
4
3
2
1
2 3 4 5 6 7 8
means
= 2+3+4+5+3+4+5+6+4+5+6+7+5+6+7+8
16
=5
Properties of the distribution of sample means

Spread
• Standard deviation of the population
• Sample size

Putting them together we get the standard deviation of
the distribution of sample means
sX =
s
n
– Commonly called the standard error
Properties of the distribution of sample means

The standard error is the average amount that
you’d expect a sample (of size n) to deviate from
the population mean

In other words, it is an estimate of the error that you’d
expect by chance (or by sampling)
Standard error

All three of these properties are combined to
form the Central Limit Theorem
– For any population with mean μ and standard
deviation σ, the distribution of sample means for
sample size n will approach a normal distribution
with a mean of m X and a standard deviation of s
n
as n approaches infinity
(good approximation if n > 30).
Central Limit Theorem

Keep your distributions straight by taking care
with your notation
Population
σ
Distribution of sample means
Sample
sX
μ
Distribution of sample means
s
X
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