Transcript Short Course in Statistics

Short Course in
Statistics
Learning Statistics through Computer
 Notice that Microsoft Chinese
Windows is needed in some slides

1
Random Sampling
To obtain information through sampling
 Population and Sample
 Parameter and Statistic

2
Population versus
Sample

Population
– The entire group of
individuals about
which we want
information

Sample
– A part of the
population from
which we actually
collect information,
used to draw
conclusions about
the whole
population.
3
Example

Population = the
measurements of
weights of all
children under 18

Sample = the
measurements of
weights of students
in 20 secondary
and primary
schools
4
Parameter versus
Statistic

Parameter
– A number that
describes the
population.

Statistic
– A number that
describes a sample.
5
Drawing balls from a
box
A box contains 10 balls: 5 red, 5 black
 Population: 10 balls
 Parameter: proportion of red balls
 Draw a random sample of size 3
 Statistic: red balls in the sample
e.g. 2/3

6
Statistical Science

Statistics provides methodology to
estimate the parameter through the
(random) sample
7
How to draw a random
sample
Construct a sampling frame---give a
number (name) to each individual in
the population
 Use “random number table” to draw a
random sample of prescribed size

8
Random Number Table
Imagine that a box containing 10
identical balls with numbers 0, 1, 2, 3,
4, 5, 6, 7, 8 and 9.
 Each time you draw a ball and record
the number before returning it to the
box and draw the next ball --- this list
(record) is the “random number table”

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Example
Objective---draw a sample of size 5
from a class of 30 students
 Sampling frame---label each student
with the numbers 00, 01,…29.
 Read the random number table at line
130 ---- 69051 64817 87174 09517
 69 05 16 48 17 87 17 40 95 17

10
Multiple Label
00=30=60, 01=31=61, 02=32=62, etc.
 Notice 01 will correspond to the
second individual

11
Measurements in the
Laboratory

Each measurement in the physics lab
or chemistry lab can be regarded as
an element in a random sample
12
http://www.cuhk.edu.hk/webct
 User ID & Password
=STA2103(Surname)(Initials)


Go to the above website and learn sample
survey, design of experiment and regression
13

Henry,Chau,STA2103chauh
Ka Ho Enoch,Chan,STA2103chankhe
Jane,Tang,STA2103tangj
Vincent,Pong,STA2103pongv
Clara,Yip,STA2103yipc
14
Why Random Sampling
To be representative
 Some laws governing the statistic--sampling distribution and compute the
 Probability---the chance of the
occurrence of an event in n
independent samplings---can be
computed

15
Not representative
Call in
 Voluntary response on the Web
 Telephone survey asking the
respondents to respond with the
number keys
 Readers’ letters to the newspaper

16
Sampling Distribution
Random sampling  the statistic
would change as the sample varies
 That is, the conclusion might be
changed for different sample
 But, if the samples are randomly
drawn, we can predict the result with
high probability

17
Example
Population: Hong Kong adult residents
 Sample (random): 600 persons
 Parameter: proportion of the
population supporting one more public
holiday
 Statistic; proportion in the sample

18
Consequence of
Random Sampling

If we draw 1000 samples (with each
sample of size 600), and we compute
the statistic for each sample, the
histogram of these 1000 (sample)
proportion is approximately a bellshaped curve---normal density
19
Normal and Probability
Normal density has 2 parameters:
 Mean --- true proportion (p)
 Variance ---var=p(1-p)/n
 Standard deviation (std)=sqrt(var)
 The one sample we draw has
probability .95 in the interval (p-1.96
std, p+1.96 std)

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Mean of normal=true
parameter
If you draw a sample 1000 times, you
have 1000 sample proportions.
 The average of these 1000 sample
proportions would be approximately
the true proportion --- sample
proportion is an unbiased estimate of
the population proportion

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Variance=p(1-p)/n
If it is truly random, we can estimate
the variance of these 1000 sample
proportions using p (parameter) only.
 If I have only one sample with accurate
estimate of p, then the variance of the
1000 sample proportion can be
computed without using the 1000
sample proportions

22
Intuition behind the
formula p(1-p)/n
Symmetric about ½
 It is maximized by p=1/2 (very
uncertain)
 When p is closer to 0 or 1, I.e., things
are more definite, the variance gets
smaller

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Confidence Interval
Conversely, p will be covered by the
interval (p-1.96 std, p+1.96 std) 95
times out of 100 such experiments.
 Notice std=sqrt(p(1-p)/n)

24
95% Confidence
Interval
Use the formula for 100 surveys, we
obtain 100 different interval estimates
 95 out of these 100 intervals would
contain the true p

25
Opinion Polls
People may not give the true
response --- response error
 People may not answer the
questions --- nonresponse error

– Unit nonresponse (the person does not
response at all)
– Item nonresponse (the person does not
respond to some questions)
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Response rate

If the response rate is less than 80%,
we would doubt about the validity of
the inference
27
Election Polls
The respondent may not be voters
 The respondent may not vote even
he/she has registered
 The respondent may lie (response
error)

28
Questionnaire

The way to set questions would affect
the response (well-known)
29
Other Data Collection
Methods
Experimental Design
 Observational Data (e.g. registry Data)

30
How to know the effect
of vaccine in
preventing polio
We cannot apply the vaccine to all
children and compare the results in the
past
 We need two groups:
control group (no “real” treatment)
treatment group (apply the vaccine)

31
We should compare the
two groups under
“equal” conditions
People are different from each other
 By random assignment of participants
into the two groups, we can make the
two groups have almost identical
conditions – e.g., around the same on
average

32
Design of an
Experiment

For comparing one treatment (A) with
the other treatment (B), we need to
randomize the patient into each group
receiving the one of the treatments
33
Some possible
mistakes
Data---from hospital record
 Death rates of surgical patients are
different for operations with different
anesthetics
 Halothane (1.7%), Pentothal (1.7%),
Cyclopropane (3.4%), Ether (1.9%)
 Can we say that cyclopropane is more
dangerous than the other anesthetics?

34
Answer

No! the worst patients were receiving
cyclopropane.
35
The vaccine can
prevent Polio
1956---USA---over two million children
involved
 Should they all receive vaccine?
 Should the male receive vaccine while
the female receive placebo?

36
Placebo
In this case, placebo is another kind of
liquid, which is similar to the vaccine in
its outlook, injected into the children.
 It is used so that all children were
receiving “same” treatment. So that
the difference in the results would not
be explained as psychological effect

37
Data
Polio (after
half year)
No polio (after
half year)
Control
(placebo)
A=115
B=201,114
treatment
C=33
D=200712
38
Analysis
The proportion of control group having
polio after ½ year --- a/(a+b)=0.00057
 The proportion of treatment group
having polio after ½ year--c/(c+d)=0.00016
 The effect of treatment---
– RD (risk difference)=c/(c+d) - a/(a+b)
=0.00041
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Formulation of the
Hypotheses
Null Hypothesis: no difference in the
proportions
 Alternative Hypothesis: the two
proportions are different

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Analysis
We need to compare RD with its
variation
 That is, if we have different
experiments, the results are different.
The variation of these results can be
measured by its variance.
 But we have only one experiment

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Estimate the variation
If there are no effect of the vaccine,
the true risk (probability) of getting
polio is pr=(a+c)/(a+b+c+d)=0.00037
 Under above hypothesis, the variance
of RD is given by
pr(1-pr) / (1/(a+b)+1/(c+d))
 The standard deviation is 0.000061.

42
Contd.
Thus the ratio 0.00041/0.000061=6.76
measures the effect of vaccine.
 Is 6.76 indicates a large or small or no
effect?
 We need a yardstick.

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Intuition
Thus the ratio (RD/std) measures the
effect of the vaccine.
 That is, if it is large in absolute value,
the effect of vaccine is significant
 How large is large?

44
Random assignment of
patients to treatments
If we do the experiment 1000 times
and each time we calculate the ratio
 We also assume that the effect of
vaccine is zero..
 Then we plot the histogram of the
1000 ratios. We find the histogram is
close to a bell-shape curve---normal
density curve.

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Normality
Since we know that the ratio is normal
and we now obtain 6.76.
 We can compute the area to the right
of 6.76----the probability that the ratio
is larger than 6.76 under the
hypothesis of no effect. We find the
area is very small (6.9 x 10^{-12})

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P value
The area correspond to the probability
of the event which is more extreme to
the observed value
 The usual rule --- p-value <0.05 reject
the null hypothesis
 0.05 can be interpreted as 5 wrong
conclusions among 100 experiments

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Chi Square TestAnother approach
We can apply the chi square test to the
same data set.
 The chi square test is used to test
whether the proportion of getting polio
is the same for the two groups
(homogeneity). Equivalently, whether
the occurrence of polio is independent
of the treatment (group)

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Analysis
The chi square test statistic is given by
N(ad - bc)**2/((a+b)(a+c)(b+d)(c+d))
 N=a+b+c+d
 When the statistic is large, the
hypothesis is likely to be wrong

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Statistical Reasoning
The above statistic can be expressed
as the summation of the quantities
 (observed counts-expected counts)**2
divided by the expected counts
 Here expected counts means the
average counts under the hypothesis
that the two groups are the same

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Chi Square distribution
Chi square distribution with one
degree of freedom
 P-value=0.05
 Cutoff point 3.84 I.e., reject if the chi
square statistic is larger than 3.84.
Otherwise, accept the null hypothesis.

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T-test (Two-Sample
unpaired)
Randomize female rats into two
groups (high (low) protein dies)
 Response variables—gain in weight
between the 28th and 84th days of age

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Data

High protein
134 146 104 119
124 161 107 83 113
129 97 123
– Mean=120
– Variance=457.5

Low protein
70 118 101 85 107
132 94
– Mean=101
– Variance=425.3
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Hypotheses
Null hypothesis: no difference in the
two means
 Alternative hypothesis: the means are
different

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Analysis
The difference of the two means
=120 - 101=19
 19 measures the difference in weight
gains between two groups
 Is it large or small? By chance?
 We need to compare with its standard
deviation

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Variance and standard
deviation

Standard deviation=square root of
variance
S S
2
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1
2
S1 
( X 1i  X 1 )

n1  1
2
S2
2
Sp
2
1
2

( X 2i  X 2 )

n2  1
(X


 X 1 )   ( X 2i  X 2 )
2
1i
n1  n2  2
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2
Indicator
X1  X 2
t
Sp
This is a better indicator of the
difference between the two
groups
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Statistical reasoning
Indicator and yardstick
 If we repeat the experiment 1000 times
and compute 1000 t statistics
 Plot the histogram for these 1000 t
statistics
 The histogram is similar to normal but
with heavier tails

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Analysis
We call it a t distribution
 There are many t distribution for
different sample sizes
 The number (the sum of two group
sizes –2) is called the degree of
freedom of the t distribution
(e.g. 12+7-2=17)

60
DF>= 30

When the degrees of freedom is larger
than or equal to 30, the t distribution
would become a normal distribution
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Statistical Reasoning
Given the degree of freedom, we can
find the area (probability)
 If there are no difference between the
two groups, the t distribution would by
symmetric about zero.
 If the data is really arising from two
treatments with same results, the t
statistic should be small

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Statistical Reasoning
If the t-statistics is small, the area
(probability) of observing the actual
statistic or larger must be large.
 Conversely, if the area is small, the
data tells us that the hypothesis is
likely to be wrong

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Statistical Reasoning
In this case, t=1.89
 The area for |t| beyond 1.89 (when
degree of freedom=17) is 0.076.
 This area is called p-value
 Usually, when p-value is lees than 0.05,
we will reject the hypothesis

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1.
2.
Interactive Statistical Pages
Try the t-test ( go to the
procedure)and chi square test (2
x 2 table for sample comparison)
here.
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Regression
Finding the mean of y for each x
To see whether x and y are
associated
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Data
•
國家
•
澳洲
2.5
211
•
奧地利
3.9
167
•
比利時
2.9
131
•
加拿大
2.4
191
•
丹麥
2.9
220
•
芬蘭
0.8
297
•
法國
9.1
71
•
冰島
0.8
211
•
愛爾蘭
0.7
300
•
意大利
7.9
107
酒耗量 心臟病死亡率
國家 酒耗量
荷蘭
1.8
新西蘭
1.9
挪威
0.8
西班牙
6.5
瑞士
5.8
瑞典
1.6
英國
1.3
美國
1.2
西德
2.7
心臟病死亡率
167
266
227
86
115
207
285
199
172
67
散布圖的形狀表示: 酒耗量與心
臟病的死亡率成負相關的關係,
但只反映國家整體數據之間的
關係, 我們不能引申為每個人喝
酒愈多, 其死於心臟病的機會愈
少, 否則便犯上生態的偏差-----Ecologic bias.
1
2
3
4
5
6
7
8
9
68
嚴謹推算與應用分析
300
之間的取捨 !
250
200
150
100
50
69
嚴謹推算與應用分析
300
之間的取捨 !
250
200
150
100
50
70
Analysis
Y (death rate)= 260.56-22.97 x (Alcohol)
The negative sign indicates that Y and x
go in opposite direction.
More Alcohol, less heart disease death
rate?
The result cannot be extended to
individual level --- ecologic bias
71
Analysis
The variance of the error is given by
1434.79
 If we compute the variance of Y, we
find that the variance is given by
4678.05.

72
Questions
Email address:
[email protected]
 Telephone:
 2609-7927

73
Exercises
1.(Sample survey)
 Population=(Adults in Hong Kong)
Sample=(random sample, telephone
survey)
 Parameter=proportion supporting the
government in handling the protest
 Statistic=

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