Short Course in Statistics

1
Short Course in Statistics

• Learning Statistics through Computer
• Notice that Microsoft Chinese Windows is needed
  in some slides

2
Random Sampling

• To obtain information through sampling
• Population and Sample
• Parameter and Statistic

3
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.

4
Example

• Population the measurements of weights of all
  children under 18

• Sample the measurements of weights of students
  in 20 secondary and primary schools

5
Parameter versus Statistic

• Parameter
• A number that describes the population.

• Statistic
• A number that describes a sample.

6
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

7
Statistical Science

• Statistics provides methodology to estimate the
  parameter through the (random) sample

8
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

9
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

10
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

11
Multiple Label

• 003060, 013161, 023262, etc.
• Notice 01 will correspond to the second
  individual

12
Measurements in the Laboratory

• Each measurement in the physics lab or chemistry
  lab can be regarded as an element in a random
  sample

13

• 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

14

• Henry,Chau,STA2103chauhKa Ho Enoch,Chan,STA2103ch
  ankheJane,Tang,STA2103tangjVincent,Pong,STA2103p
  ongvClara,Yip,STA2103yipc

15
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

16
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

17
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

18
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

19
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 bell-shaped
  curve---normal density

20
Normal and Probability

• Normal density has 2 parameters
• Mean --- true proportion (p)
• Variance ---varp(1-p)/n
• Standard deviation (std)sqrt(var)
• The one sample we draw has probability .95 in the
  interval (p-1.96 std, p1.96 std)

21
Mean of normaltrue 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

22
Variancep(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

23
Intuition behind the formula p(1-p)/n

• Symmetric about ½
• It is maximized by p1/2 (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, p1.96 std) 95 times out of 100 such
  experiments.
• Notice stdsqrt(p(1-p)/n)

25
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

26
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)

27
Response rate

• If the response rate is less than 80, we would
  doubt about the validity of the inference

28
Election Polls

• The respondent may not be voters
• The respondent may not vote even he/she has
  registered
• The respondent may lie (response error)

29
Questionnaire

• The way to set questions would affect the
  response (well-known)

30
Other Data Collection Methods

• Experimental Design
• Observational Data (e.g. registry Data)

31
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)

32
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

33
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

34
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?

35
Answer

• No! the worst patients were receiving
  cyclopropane.

36
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?

37
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

38
Data
39
Analysis

• The proportion of control group having polio
  after ½ year --- a/(ab)0.00057
• The proportion of treatment group having polio
  after ½ year---c/(cd)0.00016
• The effect of treatment----
• RD (risk difference)c/(cd) - a/(ab) 0.00041

40
Formulation of the Hypotheses

• Null Hypothesis no difference in the proportions
• Alternative Hypothesis the two proportions are
  different

41
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

42
Estimate the variation

• If there are no effect of the vaccine, the true
  risk (probability) of getting polio is
  pr(ac)/(abcd)0.00037
• Under above hypothesis, the variance of RD is
  given by
• pr(1-pr) / (1/(ab)1/(cd))
• The standard deviation is 0.000061.

43
Contd.

• Thus the ratio 0.00041/0.0000616.76 measures the
  effect of vaccine.
• Is 6.76 indicates a large or small or no effect?
• We need a yardstick.

44
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?

45
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.

46
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)

47
P value

• The area correspond to the probability of the
  event which is more extreme to the observed value
• The usual rule --- p-value lt0.05 reject the null
  hypothesis
• 0.05 can be interpreted as 5 wrong conclusions
  among 100 experiments

48
Chi Square Test-Another 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)

49
Analysis

• The chi square test statistic is given by N(ad -
  bc)2/((ab)(ac)(bd)(cd))
• Nabcd
• When the statistic is large, the hypothesis is
  likely to be wrong

50
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

51
Chi Square distribution

• Chi square distribution with one degree of
  freedom
• P-value0.05
• Cutoff point 3.84 I.e., reject if the chi square
  statistic is larger than 3.84. Otherwise, accept
  the null hypothesis.

52
T-test (Two-Sample unpaired)

• Randomize female rats into two groups (high (low)
  protein dies)
• Response variablesgain in weight between the
  28th and 84th days of age

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Data

• High protein134 146 104 119 124 161 107 83 113
  129 97 123
• Mean120
• Variance457.5

• Low protein70 118 101 85 107 132 94
• Mean101
• Variance425.3

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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 means120 - 10119
• 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

56
Variance and standard deviation

• Standard deviationsquare root of variance

57
(No Transcript)
58
Indicator

• 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

60
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. 127-217)

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DFgt 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

63
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, t1.89
• The area for t beyond 1.89 (when degree of
  freedom17) is 0.076.
• This area is called p-value
• Usually, when p-value is lees than 0.05, we will
  reject the hypothesis

65

• Interactive Statistical Pages
• Try the t-test ( go to the procedure)and chi
  square test (2 x 2 table for sample comparison)
  here.

66
Regression

• Finding the mean of y for each x
• To see whether x and y are associated

67
Data

• ?? ??? ??????
• ?? 2.5 211
• ??? 3.9 167
• ??? 2.9 131
• ??? 2.4 191
• ?? 2.9 220
• ?? 0.8 297
• ?? 9.1 71
• ?? 0.8 211
• ??? 0.7 300
• ??? 7.9 107

• ?? ??? ??????
• ?? 1.8 167
• ??? 1.9 266
• ?? 0.8 227
• ??? 6.5 86
• ?? 5.8 115
• ?? 1.6 207
• ?? 1.3 285
• ?? 1.2 199
• ?? 2.7 172

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??????
???????? ??????????????????, ???????????????,
??????????????, ???????????, ??????????------Ecolo
gic bias.
???
1 2 3 4 5
6 7 8 9
69
??????
????????? ????? !
300 250 200 150 100 50
???(regression line)
???
70
??????
????????? ????? !
300 250 200 150 100 50
???(regression line)
???
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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

72
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.

73
Questions

• Email addresstslau_at_sparc2.sta.cuhk.edu.hk
• Telephone
• 2609-7927

74
Exercises

• 1.(Sample survey)
• Population(Adults in Hong Kong) Sample(random
  sample, telephone survey)
• Parameterproportion supporting the government in
  handling the protest
• Statistic