Transcript Believe it or Not: Stats methods are NOT as bad as you think!

Believe it or Not: Stats methods are NOT
as bad as you think!
Cynthia Wilson Garvan
PhD Statistics, MA Mathematics
College of Nursing
[email protected]
Research starts with problem definition
Based on evidence
from statistical results,
conclusions can be made
The planning phase
includes choosing
measures and study
design
Problem
Conclusion
The analysis phase
is where the
statistical methods
are used
Analysis
Plan
Data
Data is collected as the
study is implemented
Data type determines statistical method
Data Types
Research questions are ultimately
answered using data. Data is
collected through observation and
measurement. Data elements
can be classified into different types.
The choice of statistical procedure
used to analyze data depends on
the data type.
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Level – one value of a data element
Supposed we asked a subject to tell us their
Myers-Briggs personality type. Our data
would be a list of possible responses:
Personality type = {ENFJ, ENFP, ENTJ, …, ISTP}
Each personality type (such as ENFP) in the list
is a “level.”
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Myers-Briggs Personality Type
Myers and Briggs extrapolated
their MBTI theory from Carl Jung's
writings in his book Psychological
Types.
Extroversion (E)/Introversion (I)
Sensing (S)/Intuition (N)
Thinking (T)/Feeling (F)
Judgment (J)/Perception (P)
Classification: Nominal, Ordinal, Interval,
Ratio
Data Element
Nominal:
A scale of
measurement
where levels
are distinct but
do not vary in
magnitude.
Ordinal: A scale of
measurement where
levels vary in order of
magnitude but equal
intervals between
levels cannot be
assumed.
Interval: The interval
level of measurement
has the characteristics
of distinct levels,
ordering in magnitude,
and equal intervals.
Equal intervals are
obtained if equivalent
differences between
measurements
represent the same
amount of difference in
the property being
measured.
Ratio: The ratio level
of measurement has
characteristics of
distinct levels, ordering
in magnitude, equal
intervals, and an
absolute zero.
A measurement has an
absolute zero when a
measurement of zero
represents the
absence of the
property being
measured.
Contrasting Nominal, Ordinal, Interval
and Ratio
categorical
Scale has levels
that are:
Distinctive
Ordered
Equally spaced
Has an
absolute zero
numerical
Nominal
Ordinal
Interval
Ratio
X
X
X
X
X
X
X
X
X
X
Examples
Nominal
County where you live
Race/ethnicity
Favorite flavor of ice cream
Myers-Briggs Personality Type
Ordinal
Favorite size of coffee you order from Starbucks
Birth order
Interval
IQ
Score on Beck’s Depression inventory
Ratio
Number of computers in a household
Temperature in Kelvin
?s
A. Nominal
B. Ordinal
C. Interval
D. Ratio
A. Nominal B. Ordinal C. Interval D. Ratio
A. Nominal B. Ordinal C. Interval D. Ratio
A. Nominal B. Ordinal C. Interval D. Ratio
Finishing order of horse.
A. Nominal B. Ordinal C. Interval D. Ratio
A. Nominal B. Ordinal C. Interval D. Ratio
Number of football player’s jersey.
A. Nominal B. Ordinal C. Interval D. Ratio
Age
C. Garvan
3540 NW 29th Place
Gainesville, FL
32605
Mrs. Susan Plastaras
3 Brookdale Drive
Doylestown, PA
18901
Zip Code
A. Nominal B. Ordinal C. Interval D. Ratio
A. Nominal B. Ordinal C. Interval
D. Ratio
Romatic Quiz
Question: Does love at first sight exist in your
mind?
Definitely.
Maybe.
For some people.
Not at all.
Data Types for our purposes
1. Categorical
2. Binary
3. Numerical
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We need data and we need to
summarize it
Population Data
Sample Data
Summaries of key
features of population
data are called population
parameters.
Summaries of key
features of sample data
are called statistics.
We can summarize data with summary
measures (statisitcs).
Definitions
• A statistic is any quantity that we can
calculate from data.
• Population parameters are quantities that
summarize population data.
Statistics are estimates of population
parameters. The quality of estimation depends
on how fairly the sample represents the
population.
We need data and we need to summarize it
Binary and Categorical
Data
Numerical Data
We can summarize with graphs
Updated: Summaries for each data type
Categorical
Binary
Numerical
Bivariate numerical
We will summarize these
data types using
proportions.
We will summarize this data type in
four ways: with measures of center,
measures of dispersion, measures of
shape, and measures of relative
standing.
We will summarize bivariate
numerical data using
correlations.
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Summarizing Numerical Data
Type Summary Measure
Statistics
Center
Mean
Median
Mode
Dispersion
Range
Variance
Standard deviation
IQR
Shape
Modality
Skewness
Kurtosis
Relative Standing
Percentile
Quartiles
Deciles
Connections
Summary
Population
Parameter
symbol
Statistic
Data type
mean
m
X
NUM
median
h
X˜
NUM
variance
s2
S2
NUM
Standard deviation
s
S
NUM
p

p
BIN & CAT
r
r
BIV NUM
proportion
correlation

Summary
measure
when all
data known
population
parameters
symbol
Population
m
p
r
s2
Summary
measure
when all
data known
statistics
Sample
symbol
Xp
r
S2

BIG PICTURE SUMMARY
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What is the point?
Secret of Statistics: Research
questions are about population
parameters (these are unknown).
Evidence is based on statistics
(known).
?s
A.m(NUM)
B.p(BIN or CAT)
C.r(BIV NUM)
A. m (NUM)
B. p (BIN or CAT)
C. r (BIV NUM)
What is the mean time for a pain killer to start
working?
A. m (NUM)
B. p (BIN or CAT)
C. r (BIV NUM)
What is the percent of patients who
experience POCD (post-operative cognitive
dysfunction)?
A. m (NUM)
B. p (BIN or CAT)
C. r (BIV NUM)
What is the relationship of depression (as
measured by Beck’s Depression Inventory) and
annual income?
Taxonomy of Statistics
Statistics
Descriptive
Summary
measures
Inferential
Graphs
Estimation
Evidence:
Confidence
Interval
Test of Hypothesis
Evidence:
P-value
Estimation
• What is the mean time for a pain killer to start
working?
• What is the percent of patients who
experience POCD (post-operative cognitive
dysfunction)?
• What is the relationship of depression (as
measured by Beck’s Depression Inventory) and
annual income?
Two Ways of Estimating a parameter
Point estimate – this is a single number
m
m
A confidence interval is a plausible range of
values PLUS a measure of success rate for
estimation method (CONFIDENCE).
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Test of Hypothesis (TOH)
Consider the following research questions,
they are examples of test of hypothesis type
inference:
– Is the mean head circumference of newborns exposed
prenatally to cocaine equal to 25 cm?
– Is the percent of homeless cats equal to 40%?
– Is the variance of IQ scores in a certain population
equal to 225?
– Is there a correlation between IQ and newborn head
circumference?
Test of Hypothesis (TOH)
A test of hypothesis (TOH) type of inference is
appropriate when the goal of the research
question is to verify an educated guess,
compare groups, test relationships, or
explain/predict outcomes.
H0: population parameter = hypothesized value
Ha: population parameter  hypothesized value
• The hypothesized value is the value of the educated guess.
• The null hypothesis says that the population parameter is equal to
the hypothesized value.
• The alternative hypothesis says that the population parameter is not
equal to the hypothesized value.
• H0 (read H-NOT) is the symbol for the null hypothesis.
• Ha (read H-A) is the symbol for the alternative hypothesis.
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Example, put the following research question
into a TOH format:
• Is the mean head circumference of exposed
newborns equal to 25 cm?
Hypothesized value
The population parameter (m) is “mean head
circumference”
TOH FORMAT:
H0: m = 25
Ha: m  25
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Example, put the following research question
into a TOH format:
• Is the percent of homeless cats equal to
40%?
Hypothesized value
The population parameter (p) is “percent of
homeless cates”
TOH FORMAT:
H0: p = .40
Ha: p  .40
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Example, put the following research question
into a TOH format:
• Is the variance of IQ in a certain population
equal to 225?
Hypothesized value
The population parameter (s2) is “variance of
IQ”
TOH FORMAT:
H0: s2 = 225
Ha: s2  225
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Formats for TOH
TOH for m
H0: m m0
Ha: m≠ m0
TOH for p
H0: p p0
Ha: p≠ p0
TOH for s2
H0: s2  s20
Ha: s2 ≠ s20
The symbol for hypothesized
proportion is p0
The symbol for
hypothesized mean is m0
The symbol for hypothesized
variance is s20
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Example, put the following research question
into a TOH format:
• Is there a correlation of IQ and newborn
head circumference?
Hypothesized value
The population parameter (r) is “correlation
of IQ and newborn head circumference”
TOH FORMAT:
H0: r = 0
Ha: r  0
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How to conduct a TOH
DATA
How to conduct a TOH
FORMULAS
How to conduct a TOH
How to conduct a TOH
How to conduct a TOH
Null
Alternative
P-value ≤ .05
How to conduct a TOH
Null
P-value > 0.05
Alternative
P-value & Level of Significance
A p-value is the probability of getting the
results you did (or more extreme results)
given that the null hypothesis is true.
The “level of significance” (or alpha or a) is
the value a researcher chooses. It is the
probability that although the null is true, the
p-value supports the alternative hypothesis.
Decision Rule for Test
After a level of significance (a) has been set,
compare the p-value of test:
• If p-value  a then reject null hypothesis.
• If p-value > a then fail to reject null
hypothesis.
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Level of significance
Consider the following TOH
H0:mm0
Ha:m≠ m0
The way this TOH is set up – we will reject H0 if the
value of the test statistic (sample mean) is much bigger
or much smaller than the value of the hypothesized
mean, m0
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This is the distribution if
H0 is true
This is the rejection
region
m0 - 2SE m0 – SE
m0
m0 + SE m0 + 2SE
This is the acceptance
region.
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We can choose any
level of significance
(a) for a TOH by
changing the critical
value = za/2
This is the distribution if
H0 is true
a/
2
m0 – za/2SE m0 – SE
a/2
m0
m0 + SE m0 + za/2SE
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P-value
P-value is perhaps the most important
concept in Statistics. We will motivate the
idea of p-value using a one-sided TOH.
H0:m≤ m0
Ha:m>m0
The way this TOH is set up – we will reject H0 if the value of
the test statistic (sample mean) is much bigger than the value of
the hypothesized mean, m0
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The TS will fall in either the
acceptance region
or
the rejection region
X rv
m0
X


X rv
m0

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P-value is the “observed” level of significance. It is the
probability of the observed TS or data that is more extreme.
For a one-sided test, p-value is the tail area.
When the TS falls in the
acceptance region
the p-value is
large.
m0
This area is the p-value
X rv
X observed

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The level of significance must be specified by the researcher.
P-value is then compared to the level of significance (a) to
determine the TOH conclusion.
When the TS falls in the
rejection region
the p-value is
small.
This area is the p-value
m
0
X rv
X observed

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How to conduct a TOH
Use data to
compute
“p-value”
Compare
“p-value” to “level
of significance
(alpha or a)”
Make Conclusion
Types of Test of Hypothesis
• Univariate
–One variable testing
–Paired variable testing
• Bivariate
–Relationship between two variables
• Multivariate
–Relationships among multiple variables
•
•
•
•
Generally speaking, we can also think about
research questions as:
A research question about change
A research question about comparing groups
A research question about testing
relationships
A research question about
explaining/predicting an outcome
Bivariate Methods
Depend on Data Type
X - explanatory variable Y – outcome
variable
BIN
BIN
CAT
CAT
NUM
NUM
Summary of stats methods for bivariate analysis
Words to describe Y variable: dependent (DV),outcome, response
X
Y
Binary
Binary
Categorical
Numerical
Chi-square test
(c2 test)
T-test
Chi-square test
(c2 test)
Chi-square test
(c2 test)
ANOVA
T-test
ANOVA
Correlation
z-test
(equivalent to chisquare test)
Categorical
Numerical
Words to describe X variable: independent (IV), explanatory,
predictor, factor
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Algorithm for determining stats
method for bivariate analysis
•
•
•
•
•
Identify X
Determine data type of X
Identify Y
Determine data type of Y
Determine stats method
?s
A.
B.
C.
D.
Chi-square
T-test
ANOVA
Correlation
Example
Is there a difference in the mean time for a
pain killer to start working due to hair color
(red, brown, black, blonde)?
X = CAT
Y = NUM
Stat Method: ANOVA
A. Chi-square
B. T-test
C. ANOVA
D. Correlation
Does prenatal cocaine exposure affect IQ?
A. Chi-square
B. T-test
C. ANOVA
D. Correlation
Does incorporating SAS software into
graduate level Statistics course improve
statistics knowledge?
A. Chi-square
B. T-test
C. ANOVA
D. Correlation
Does brand of dog food affect weight gain of
Borzoi puppies?
A. Chi-square
B. T-test
C. ANOVA
D. Correlation
Is favorite T.V. show related to college major?
A. Chi-square
B. T-test
C. ANOVA
D. Correlation
Is there a relationship between “screen time”
and depression (as measured by Beck’s
Depression Inventory)?
A. Chi-square
B. T-test
C. ANOVA
D. Correlation
Is the prevalence of ADHD the same for males
and females?
A. Chi-square
B. T-test
C. ANOVA
D. Correlation
Does education level affect the incidence of
POCD (Post-Operative Cognitive Dysfunction)?
Practical (Clinical) Versus
Statistical Significance
Difference
Meaningful Threshold Value
0
(a)
(b)
Significant
(c)
Definitely Possibly
Not
important important important
(d)
(e)
Not significant
Inconclusive
True negative
result
Type I and Type II Error
Due to the inherent uncertainties of nature, we can never make definite
claims from our experiments
Retain H0
Reject H0
H0 is true
Correct
Type I error
False discovery
H1 is true
Type II error
Failure to
detect true
discovery
Correct
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Experimentwise Error
Experimentwise Error
Probability at least one Type I error in experiment
= 1 – (1 - a)c
a is error rate, c is number of comparisons
Example: a = 0.05, 10 comparisons made (c = 10)
Pr (Type I error in experiment) = .401
Guess the Tables
Let’s Analyze Data
Happy
First
Birthday
Claire!