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Introduction to Basic Statistical Methods
Part 1: “Statistics in a Nutshell”
UWHC Scholarly Forum
March 19, 2014
Ismor Fischer, Ph.D.
UW Dept of Statistics
[email protected]
UWHC Scholarly Forum
March 19, 2014
Ismor Fischer, Ph.D.
UW Dept of Statistics
[email protected]
All slides posted at http://www.stat.wisc.edu/~ifischer/Intro_Stat/UWHC
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“Statistical Inference”
POPULATION
Women in the U.S. who
have given birth
“Statistical Inference”
POPULATION
Study Question:
Has mean (i.e., average) of X = “Age
at First Birth” of women in the U.S.
changed since 2010 (25.4 yrs old)?
Present Day: Assume X = “Age at First
Birth” follows a normal distribution
(i.e., “bell curve”) in the population.
Population
Distribution
X
But what does that mean (at least in principle)?
“Statistical Inference”
POPULATION
Study Question:
Has mean (i.e., average) of X = “Age
at First Birth” of women in the U.S.
changed since 2010 (25.4 yrs old)?
Present Day: Assume X = “Age at First
Birth” follows a normal distribution
(i.e., “bell curve”) in the population.
Population
Distribution
x4
x5
x2
x1
X
x3
Individual ages from the population tend to
collect around a single center with a certain
amount of spread, but occasional “outliers”
are present in left and right symmetric tails.
More precisely…
ad infinitum…
~ The Normal Distribution ~

“population
standard
deviation”

 f ( x) 
 symmetric about its mean
 unimodal (i.e., one peak),
with left and right “tails”
 models many (but not all)
naturally-occurring systems
 useful mathematical
properties…
“population mean”
Example: X = Body Temp (°F)
low
variability
small 
98.6
~ The Normal Distribution ~

“population
standard
deviation”

 f ( x) 
 symmetric about its mean
 unimodal (i.e., one peak),
with left and right “tails”
 models many (but not all)
naturally-occurring systems
 useful mathematical
properties…
“population mean”
Example: X = Body
IQ score
Temp (°F)
low
high
variability
small 
large 
98.6
100
~ The Normal Distribution ~
“population
standard
deviation”

95%
2.5%
≈2σ
2.5%
≈2σ

 f ( x) 
 symmetric about its mean
 unimodal (i.e., one peak),
with left and right “tails”
 models many (but not all)
naturally-occurring systems
 useful mathematical
properties…
“population mean”
Approximately 95% of the population
values are contained between
 – 2σ and  + 2 σ.
95% is called the confidence level.
5% is called the significance level.
POPULATION
Study Question:
Has “Mean (i.e., average) Age at
First Birth” of women in the U.S.
changed since 2010 (25.4 yrs old)?
“Statistical Inference”
via… “Hypothesis Testing”
Present Day: Assume “Mean Age at
First Birth” follows a normal distribution
(i.e., “bell curve”) in the population.
Population
Distribution

X
 cannot be found with 100% certainty,
but can be estimated with high confidence
(e.g., 95%).
H0: pop mean age  = 25.4
(i.e., no change since 2010)
“Null Hypothesis”
POPULATION
Study Question:
Has “Mean (i.e., average) Age at
First Birth” of women in the U.S.
changed since 2010 (25.4 yrs old)?
“Statistical Inference”
via… “Hypothesis Testing”
Present Day: Assume “Mean Age at
First Birth” follows a normal distribution
(i.e., “bell curve”) in the population.
Population
Distribution

X
T-test
x2
“Null Hypothesis”
x4
x1
x3
x5
… etc…
x400
H0: pop mean age  = 25.4
(i.e., no change since 2010)
FORMULA
sample mean age x  25.6
x1  x2 
x
n
 xn
Do the data tend to support or refute the null hypothesis?
Is the difference STATISTICALLY SIGNIFICANT, at the 5% level?
~ The Normal Distribution ~
Population
Distribution
(of ages)
“Sampling
Distribution”
(of mean ages)

X


n
Actually, this is a
special case of…
via
mathematical
proof…
x2
x1
x3
x4
x5
X
… etc…

~ The Normal Distribution ~
Population
Distribution
(of ages)
“Sampling
Distribution”
(of mean ages)

X


n
Actually, this is a
special case of…
… as n gets larger
CENTRAL
LIMIT
THEOREM
x2
x1
x3
x4
x5
X
… etc…

~ The Normal Distribution ~
Population
Distribution
(of ages)
“Sampling
Distribution”
(of mean ages)

X

n

The sample mean values have
much less variability about 
than the population values!
X

~ The Normal Distribution ~
Population
Distribution
(of ages)
“Sampling
Distribution”
(of mean ages)

95%
2.5%
≈2σ
2.5%

n
≈2σ

Approximately 95% of the population
values are contained between
 – 2 σ and  + 2 σ.
Approximately 95% of the sample
mean values are contained between
  2 n and   2 n
X

Approximately 95% of the sample
mean values are contained between
and
  2 n
  2 n
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5
etc…


n is called the 95% margin of error
X
x1


2
x2
x3

x4

x5
Approximately 95% of the sample
mean values are contained between
and
  2 n
  2 n
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5


n is called the 95% margin of error
X
x1


2
x2
x3

x4

x5
Approximately
Approximately95%
95%of
ofthe
theintervals
sample
mean
contained
x  2values
 n arefrom
x  2between
 n
to
and
  2 n
  2 n
contain , and approx 5% do not.
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5


n is called the 95% margin of error
X
x1


2
x2
x3

x4

x5
~ The Normal Distribution ~
Population
Distribution
(of ages)
“Sampling
Distribution”
(of mean ages)

95%
2.5%
≈2σ
2.5%

n
≈2σ

Approximately 95% of the population
values are contained between
 – 2 σ and  + 2 σ.
Approximately 95% of the sample
mean values are contained between
  2 n and   2 n
Approximately 95% of the intervals
x  2 n from x  2 n
to
contain , and approx 5% do not.
X

POPULATION
Study Question:
Has “Mean (i.e., average) Age at
First Birth” of women in the U.S.
changed since 2010 (25.4 yrs old)?
“Statistical Inference”
via… “Hypothesis Testing”
Present Day: Assume “Mean Age at
First Birth” follows a normal distribution
(i.e., “bell curve”) in the population.
“Null Hypothesis”
H0: pop mean age  = 25.4
(i.e., no change since 2010)
FORMULA
SAMPLE
n = 400 ages
sample mean
x
x4
x1
x2
x1  x2 
n
 xn
= 25.6
x3
x5
… etc…
Approximately x95%
PROBLEM!
400 of the intervals
x  2 n from x  2 n σ is unknown the vast
to
majority of the time!
contain , and approx 5% do not.
95% margin of error
2

n
POPULATION
Study Question:
Has “Mean (i.e., average) Age at
First Birth” of women in the U.S.
changed since 2010 (25.4 yrs old)?
“Statistical Inference”
via… “Hypothesis Testing”
Present Day: Assume “Mean Age at
First Birth” follows a normal distribution
(i.e., “bell curve”) in the population.
“Null Hypothesis”
H0: pop mean age  = 25.4
(i.e., no change since 2010)
FORMULA
SAMPLE
n = 400 ages
sample mean
x
x4
x1
x2
x3
x5
… etc…
x400
x1  x2 
n
 xn
= 25.6
sample variance
= modified average of the squared
deviations from the mean
sample standard deviation
95% margin of error
2

n
“Statistical Inference”
via… “Hypothesis Testing”
POPULATION
Study Question:
Has “Mean (i.e., average) Age at
First Birth” of women in the U.S.
changed since 2010 (25.4 yrs old)?
Present Day: Assume “Mean Age at
First Birth” follows a normal distribution
(i.e., “bell curve”) in the population.
“Null Hypothesis”
H0: pop mean age  = 25.4
(i.e., no change since 2010)
FORMULA
SAMPLE
n = 400 ages
sample mean
x
x4
x1
x2
x1  x2 
n
 xn
= 25.6
sample variance
x3
x5
… etc…
x400
( x1  x ) 2  ( x2  x )2 
s 
n 1
 ( xn  x ) 2
2
sample standard deviation
s   s = 1.6
2
95% margin of error
2

n
2
s
= 0.16
n
Approximately 95% of the intervals
x  2 n from x  2 n
to
contain , and approx 5% do not.
x = 25.6
95% margin of error
2
25.44
s
= 0.16
n
2
x = 25.6
s
= 0.16
n
25.76
BASED ON OUR SAMPLE DATA, the true value of μ today is between
25.44 and 25.76 years, with 95% “confidence” (…akin to “probability”).
Two main ways to
conduct a formal
hypothesis test:
95% CONFIDENCE INTERVAL FOR µ
 = 25.4
25.44
x = 25.6
25.76
BASED ON OUR SAMPLE DATA, the true value of μ today is between
25.44 and 25.76 years, with 95% “confidence” (…akin to “probability”).
IF H0 is true, then we would expect a random sample mean x that is at least
0.2 years away from  = 25.4 (as ours was), to occur with probability 1.28%.
“P-VALUE” of our sample
Very informally, the p-value of a sample is the probability (hence a
number between 0 and 1) that it “agrees” with the null hypothesis.
Hence a very small p-value indicates strong evidence against the
null hypothesis. The smaller the p-value, the stronger the evidence,
and the more “statistically significant” the finding (e.g., p < .0001).
25.4 25.6
Two main ways to
conduct a formal
95% CONFIDENCE INTERVAL FOR µ
hypothesis
test: CONCLUSIONS:
FORMAL
 The 95% confidence interval corresponding to our sample mean does not
 =value”
25.4 of25.44
x = 25.6
contain the “null
the population mean,
μ = 25.4 years. 25.76
 The
p-value
ourSAMPLE
sample,DATA,
.0128,the
is less
predetermined
α = .05
BASED
ON of
OUR
truethan
valuethe
of μ
today is between
significance
25.44 andlevel.
25.76 years, with 95% “confidence” (…akin to “probability”).
Based on our sample data, we may (moderately) reject the null hypothesis
is true,
expect a alternative
random sample
mean xH that
is at least
H0: IFμ H=0 25.4
in then
favorwe
of would
the two-sided
hypothesis
A: μ ≠ 25.4,
0.2 αyears
from  =level.
25.4 (as ours was), to occur with probability 1.28%.
at the
= .05away
significance
“P-VALUE” of our sample
INTERPRETATION: According to the results of this study, there exists a
statistically significant difference between the mean ages at first birth in
Very
informally,
theatp-value
a sample islevel.
the probability
2010 (25.4 years
old)
and today,
the 5%of
significance
Moreover,(hence
the a
between
0 and suggest
1) that itthat
“agrees”
with the null
hypothesis.
evidence from number
the sample
data would
the population
mean
age
Henceolder
a very
small
p-value
indicates
evidence
against the
today is significantly
than
in 2010,
rather
thanstrong
significantly
younger.
null hypothesis. The smaller the p-value, the stronger the evidence,
and the more “statistically significant” the finding (e.g., p < .0001).
However, one problem remains…
25.4 25.6
Normal Distribution
Population
Distribution
(of ages)
“Sampling
Distribution”

(mean ages)
95%
2.5%
≈2σ
Normal Distribution
2.5%
≈2σ
s


n
n

Approximately 95% of the population
values are contained between
 – 2 σ and  + 2 σ.
Approximately 95% of the sample
mean values are contained between
  2 n and   2 n
Approximately 95% of the intervals
x  2 n from x  2 n
to
contain , and approx 5% do not.
X

Normal Distribution
Population
Distribution
(of ages)
“Sampling
Distribution”

(mean ages)
95%
2.5%
≈2σ
Normal Distribution
2.5%
≈2σ

Approximately 95% of the population
values are contained between
 – 2 s and  + 2 s.
s


n
n
…IF n is large,
e.g.,  30
Alas, this introduces
“sampling variability.”
Approximately 95% of the sample
mean values are contained between
and   2s n
  2s n
Approximately 95% of the intervals
x  2 s n from x  2 s n
to
contain , and approx 5% do not.
X

Edited R code:
y = rnorm(400, 0, 1)
z = (y - mean(y)) / sd(y)
x = 25.6 + 1.6*z
Generates a normally-distributed random
sample of 400 age values.
sort(round(x, 1))
[1] 19.6 20.2 20.4 20.5 21.2 22.3 22.3 22.4 22.4 22.4 22.6 22.7 22.7 22.7 22.8
[16] 23.0 23.0 23.1 23.1 23.2 23.2 23.2 23.2 23.2 23.3 23.4 23.4 23.4 23.5 23.5
etc...
[391] 28.7 28.7 28.9 29.2 29.3 29.4 29.6 29.7 29.9 30.2
c(mean(x), sd(x))
[1] 25.6
Calculates sample mean and standard deviation.
1.6
t.test(x, mu = 25.4)
One Sample t-test
data: x
t = 2.5, df = 399, p-value = 0.01282
alternative hypothesis: true mean is not equal to 25.4
95 percent confidence interval:
25.44273 25.75727
sample estimates:
mean of x
25.6
Normal Distribution
Population
Distribution
(of ages)
“Sampling
Distribution”

(mean ages)
95%
2.5%
≈2σ
Normal Distribution
2.5%
≈2σ

Approximately 95% of the population
values are contained between
 – 2 s and  + 2 s.
s


n
n
…IF n is large,
e.g.,  30
But if n is
small…
Approximately 95% of the sample
mean values are contained between
and   2s n
  2s n
Approximately 95% of the intervals
x  2 s n from x  2 s n
to
contain , and approx 5% do not.
X

If n is small,
T-score > 2.
… the “T-score" increases (from ≈ 2 to a
max of 12.706 for a 95% confidence level)
as n decreases  larger margin of error
 less power to reject, even if a genuine
statistically significant difference exists!
If n is large,
T-score ≈ 2.
POPULATION
Study Question:
Has “Mean (i.e., average) Age at
First Birth” of women in the U.S.
changed since 2010 (25.4 yrs old)?
“Statistical Inference”
via… “Hypothesis Testing”
Present Day: Assume “Mean Age at
First Birth” follows a normal distribution
(i.e., “bell curve”) in the population.

T-test
x2
“Null Hypothesis”
x4
x1
x3
x5
… etc…
x400
H0: pop mean age  = 25.4
(i.e., no change since 2010)
FORMULA
sample mean age x  25.6
x1  x2 
x
n
 xn
Do the data tend to support or refute the null hypothesis?
Is the difference STATISTICALLY SIGNIFICANT, at the 5% level?
POPULATION
Study Question:
Has “Mean (i.e., average) Age at
First Birth” of women in the U.S.
changed since 2010 (25.4 yrs old)?
“Statistical Inference”
via… “Hypothesis Testing”
Present Day: Assume “Mean Age at
First Birth” follows a normal distribution
(i.e., “bell curve”) in the population.

T-test
H0: pop mean age  = 25.4
(i.e., no change since 2010)
“Null Hypothesis”
Check?
The reasonableness of the normality assumption is empirically verifiable,
and in fact formally testable from the sample data. If violated (e.g.,
skewed) or inconclusive (e.g., small sample size), then “distribution-free”
nonparametric tests can be used instead of the T-test.
Examples: Sign Test, Wilcoxon Signed Rank Test (= Mann-Whitney Test)
POPULATION
Study Question:
Has “Mean (i.e., average) Age at
First Birth” of women in the U.S.
changed since 2010 (25.4 yrs old)?
“Statistical Inference”
via… “Hypothesis Testing”
Present Day: Assume “Mean Age at
First Birth” follows a normal distribution
(i.e., “bell curve”) in the population.

T-test
x2
“Null Hypothesis”
x4
x1
H0: pop mean age  = 25.4
(i.e., no change since 2010)
x3
x5
… etc…
x400
Sample size n partially depends on the
power of the test, i.e., the desired
probability of correctly rejecting a false
null hypothesis (80% or more).
Introduction to Basic Statistical Methods
Part 1: Statistics in a Nutshell
Part 2: Overview of Biostatistics:
“Which Test Do I Use??”
Sincere thanks to…
UWHC Scholarly Forum
March 19, 2014
• Judith Payne
Ismor Fischer, Ph.D.
UW Dept of Statistics
[email protected]
• Samantha Goodrich
• Heidi Miller
• Troy Lawrence
• YOU!