Transcript Lecture 5 Slides (Feb 02)

This Week
• Review of estimation and hypothesis testing
• Reading Le (review)
– Chapter 4: Sections 4.1 – 4.3
– Chapter 5: Sections 5:1 and 5:4
– Chapter 7: Sections 7:1 – 7.3
• Reading C &S
– Chapter 2:A-E
– Chapter 6: A,B,F
Point Estimate
Population
Parameter
m
Point Estimate
Sample mean
p
Sample proportion
r
Sample correlation
m1 - m2
Difference between 2 sample means
p1 - p2
Difference between 2 sample
proportions
s
Sample standard deviation
Sampling error: True value – estimate (unknown)
Statistical Inference
Population
with mean
m=?
The value of x is used to
make inferences about
the value of m.
A simple random sample
of n elements is selected
from the population.
The sample data
provide a value for
the sample mean x.
Interval Estimation
In general, confidence intervals are of the form:
estimate 1.96SE
Estimate = mean, proportion, regression
coefficient, odds ratio...
SE = standard error of your estimate
1.96 = for 95% CI based on normal distribution
Standard normal distribution
2.5% probability
2.5% probability
-1.96
1.96
Estimation for Population Mean
Point estimate:
Estimate of variability
in population
Estimate of variability
in point estimate X
(SE)
95% Confidence Interval

X
m
X
n
s
2
(
X
X
)

n -1
s/ n
X  1.96 SE
A slightly larger number based
on the t-distribution is used
for smaller n
Assumptions
• Data in population follows a normal distribution
or
• Sample size is large enough to apply central limit
theorem (CLT)
• CLT – no matter the shape of the population
distribution of the sample mean approaches a
normal distribution as the sample size gets large
Meaning of Confidence Interval
• There is a 95% chance that your interval
contains m. (That you “captured” the true
value m with your interval)
Example
Suppose sample of n=100 persons
mean = 215 mg/dL, standard deviation = 20
95% CI =
X  1.96s / n
Lower Limit: 215 – 1.96*20/10
Upper Limit: 215 + 1.96*20/10
= (211, 219)
“We are about 95% confident that the interval 211-219
contains m”
We can pretty much rule out that m > 220
Properties of Confidence Intervals
• As sample size increases, CI gets smaller
– Because SE gets smaller;
• Can use different levels of confidence
– 90, 95, 99% common
– More confidence means larger interval; so a 90% CI is
smaller than a 99% CI
– What would a 100% CI look like?
• Changes with population standard deviation
– More variable population means larger interval
Effect of sample size
Suppose we had only 10 observations
What happens to the confidence interval?
X  1.96s / n
For n = 100, 215  1.96(20) / 100  ( 211,219)
For n = 10, 215  1.96(20) / 10  (203,227)
Larger sample size = smaller interval
Effect of confidence level
Suppose we use a 90% interval
What happens to the confidence interval?
X  1.645s / n
90%:
215  1.645(20) / 100  (212,218)
Lower confidence level = smaller interval
(A 99% interval would use 2.58 as multiplier and
the interval would be larger)
Effect of standard deviation
Suppose we had a SD of 40 (instead of 20)
What happens to the confidence interval?
X  1.96s / n
215  1.96(40) / 100  (207,223)
More variation = larger interval
Effect of different sample
Suppose new sample with mean of 212 (but same
standard deviation)
What happens to the confidence interval?
X  1.96s / n
212  1.96(20) / 100  (208,216)
Same size, moves a little
How Big A Sample To Take?
• Depends on the variability in the population
• Depends on how precise an estimate you
want
• Cost - if it doesn’t cost much to sample an
element then sample many
95% Confidence Intervals for m Using
SAS
PROC MEANS DATA = datasetname CLM ;
VAR list of variables
This will display the following statistics
N
Mean
Standard Deviation
Standard Error of Mean
Lower 95% Confidence Limit
Upper 95% Confidence Limit
Confidence Limits
Assessing Normality with Graphs
• Boxplots and stem-and-leaf plots, histograms
• Look for skewness (non-symmetry)
• Hard to get normal looking graphs with small sample
sizes
• Can check effect of transformations
• Normal probability plots
–
–
–
–
x-axis: related to inverse of standard normal distribution
y-axis: actual data
* actual data
+ what we would expect if data were really normal
Assessing normality PROC UNIVARIATE
PROC UNIVARIATE DATA = demo NORMAL PLOT;
VAR ursod;
* Ursod is urinary sodium excretion in 8hours
RUN;
NORMAL and PLOT are two options that test for
normality and display simple graphs
Plots are best - with enough data, tests for normality
almost always reject normality assumption
STEM AND LEAF PLOT
Stem Leaf
# Boxplot
16 6
1
0
15 0
1
0
14 7
1
0
13 6
1
0
12 038
3
0
11 7
1
|
10 49
2
|
9 57
2
|
8 0002
4
|
7 033456
6
|
6 0134568
7 +-----+
5 001347
6 | + |
4 00001123333456777779999
23 *-----*
3 011244455667799
15 +-----+
2 23444556678888999
17
|
1 4677788
7
|
----+----+----+----+--Multiply Stem.Leaf by 10**+1
The UNIVARIATE Procedure
Variable: ursod
Normal Probability Plot
165+
*
|
*
|
*
135+
*
++
|
***
+++
|
*
+++
105+
* +++
|
*++
|
++*
75+
++***
|
++***
|
+++ **
45+
+******
|
*****
|
********
15+*
* ** ** +++
+----+----+----+----+----+----+----+----+----+----+
-2
-1
0
+1
+2
Variable:
lursod
Normal Probability Plot
5.15+
+*
|
*++
|
**++
|
**++
|
** +
4.65+
* ++
|
*++
|
*+
|
***
|
**
4.15+
**
Log transformed value
|
+*
better linear pattern
|
++**
|
+***
|
***
3.65+
**
|
**
|
+*
|
****
|
**
3.15+
**+
|
*+
|
++
|
**+**
|
* +
2.65+*
++
+----+----+----+----+----+----+----+----+----+----+
-2
-1
0
+1
+2
shows a
Hypothesis Testing
Hypothesis: A statement about parameters
of population or of a model (m=200 ?)
Test: Does the data agree with the
hypothesis? (sample mean 220)
Measure the agreement with probability
Steps in hypothesis testing
• State null and alternative hypothesis
(Ho and Ha)
– Ho usually a statement of no effect or no
difference between groups
• Choose α level
– Probability of falsely rejecting Ho (Type I
error)
Steps in hypothesis testing
• Calculate test statistic, find p-value (p)
– Measures how far data are from what you
expect under null hypothesis
• State conclusion:
p < α, reject Ho
p > α, insufficient evidence to reject Ho
Possible results of tests
What we decide
Reject
null
Fail to
reject
null
Null true
Type I
error ()
Correct
Null false
Correct
Type II
error ()
Reality
Details
α related to confidence level
Commonly set at 0.05 or 0.01
β usually predetermined by sample size
One sample t-test;
test for population mean
• Simple random sample from a normal
population (or n large enough for CLT)
• Ho: μ = μo
• Ha : μ  μo , pick α
• test statistic:
x - mo
t
s/ n
Matched pairs data
• Recall independence requirement for CIs
• Similar issue for t-tests
• Observations not independent
Examples; pre and post test, left and right eyes,
brother-sister pairs
• Solution: look at paired differences, do one sample test
on differences
d = X2 - X1
Ho: d = 0, Ha: d  0
PROC TTEST, one sample test
PROC TTEST DATA = DEMO;
VAR age;
RUN;
• Tests if mean age is different than zero. Not
very useful
• Need to be tricky...
• Use a Data step to calculate a new variable
• Subtract value of mean under null hypothesis
•Test new variable for difference from zero
DATA DEMO;
SET DEMO;
dage = age - 25;
RUN;
PROC TTEST DATA=DEMO ;
VAR dage;
RUN;
This tests whether the
mean age is different
from 25
PROC TTEST
one sample output
T-Tests
Variable
DF
t Value
Pr > |t|
dage
11
-0.41
0.6931
Conclusion: We have insufficient evidence to claim that
the mean age is different than 25 (p=0.69)