Chapter 1 Statistical Thinking

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Chapter 1 Statistical Thinking

• What is statistics?
• Why do we study statistics

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Statistical Thinking

• the science of collecting, organizing, and
  analyzing data
• the mathematics of the collection, organization
  and interpretation of numerical data
• The branch of mathematics which is the study of
  the methods of collecting and analyzing data
• a branch of applied mathematics concerned with
  the collection and interpretation of quantitative
  data and the use of probability theory to
  estimate population parameters

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Statistical Thinking

• Statistics is a discipline which is concerned
• with
• designing experiments and other data collection,
• summarizing information to aid understanding,
• drawing conclusions from data, and
• estimating the present or predicting the future.

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Statistical Thinking

• "I like to think of statistics as the science of
  learning from data...." Jon Kettenring, ASA
  President, 1997
• Steps of statistical analysis involve
• collecting information (Data Collection)
• evaluating the information (Data Analysis)
• drawing conclusions (Statistical Inference)

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Statistical Thinking

• What type of information?
• A test group's favorite amount of sweetness in a
  blend of fruit juices
• The number of men and women hired by a city
  government
• The velocity of a burning gas on the sun's
  surface
• Clinical trials to investigate the effectiveness
  of new treatments
• Field experiments to evaluate irrigation methods
  
• Measurements of water quality

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Statistical Thinking

• Problems
• Is a new treatment for heart disease more
  effective than a standard one?
• Is using a high octane gas beneficial to car
  performance?
• Does reading an article in statistics improve
  students statistics grade?

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Statistical Thinking

• Is a new treatment for heart disease more
  effective than a standard one?
• Pick, say, 100 heart patients
• Divide them into two groups, 50 in each group
• Group 1------------New treatment
• Group 2------------Standard treatment

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Statistical Thinking

• Results
• 40 out of 50 of Group 1 patients improved
• 30 out of 50 of Group 2 patients improved
• Conclusion New treatment is more effective!

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Statistical Thinking

• How do you divide the patients?
• Have you controlled other factors? (fitness
  level, life style, age, etc)
• How do you decide who gets what treatment?
  Ethical issues????

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Statistical Thinking

• Comparing Test Scores
• Select 10 students and give them a journal
  article in statistics.
• Test their knowledge about the article and record
  their scores
• Repeat the test after they take STT 231.

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Statistical Thinking

• Result
• 8 out of the 10 students improved their scores.
• Question Can we conclude that reading the
  article has improved students knowledge about
  statistics?

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Statistical Thinking

• Look at worst case scenarios
• Under the assumption that the new
• treatment is no better than the standard one,
• what is the chance that 80 of the patients
• benefit from this treatment?
• Under the assumption that STT 231 brings
• no benefit, how likely is it that we see 80
• of the students improve their scores?

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Statistical Thinking

• Need a model to answer these questions!!
• If STT 231 is not beneficial, then students
• scores may go up or down with 50
• chance.
• This is equivalent to flipping a coin
• 50 chance you get Head
• 50 chance you get Tail

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Statistical Thinking

• Comparing pre and post test scores for 10
  students is equivalent to
• flipping a coin 10 times and calculating the
  chance of observing 8H
• Relevant Questions
• Will the chance of observing 80 of the time H
  depend on the number of students involved in the
  experiment?
• Will this chance go up, down or remain the same
  if you repeat the experiment with 200 students?

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Statistical Thinking

• Suppose the proportion of improvement in 10
  trials is 4.4. What does this mean?
• If STT 231 is not beneficial, then there is a
  4.4chance that we will observe 8 out of 10
  students scores improve.
• There is little hope that 8 students scores will
  improve by just by CHANCE

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Statistical Thinking

• Suppose the proportion of improvement in 10
  trials is 4.4.
• We observed 8 students scores out of 10 improve.
  
• What does this mean?

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Statistical Thinking

• Course is highly effective
• Course is ineffective and we observed an unlikely
  event.
• We do not know which one!

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Statistical Thinking

• Suppose there is a small chance that an event
  happens by CHANCE,
• Then this is an indication for a strong evidence
  that the change that we observe did not happen by
  CHANCE.
• Hence there is a strong evidence for a factor to
  be responsible for this change.

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Statistical Thinking

• The course is highly effective!!
• Reasoning What we observed is very unlikely if
  the course was ineffective. Hence the course is
  effective.
• The 80 score increment is unlikely to be
  achieved if the course was ineffective.

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Statistical Thinking

• Some Remarks
• For questions that involve uncertainty
• Carefully formulate the question you want to
  answer (Modeling)
• Collect Data
• Summarize, analyze and present data
• Draw Conclusions. Conclusions always include
  uncertainty
• Support your conclusions by quantifying how
  confident you are about your conclusions.

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Chapter 2 A Design Example

• The Polio Vaccine Case
• Caused by virus
• Especially deadly in children
• Big problem during the first half of the 20th
  Century
• Develop vaccine to fight the disease
• Jonas Salk (1950)

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A Design Example

• Problem with vaccines
• Are they safe?
• Are they effective?
• Undertake a large scale trial to answer these
  questions

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A Design Example

• Case 1 A Simple Study
• Distribute the vaccine widely (under the
  assumption it is safe)
• Decrease in the number of polio cases after the
  vaccine provides evidence that the vaccine is
  effective
• Problem?????

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A Design Example

• Problems
• Lack of control group
• Is decrease in number of polio due to the vaccine
  or other factors?
• How reliable is the assumption vaccine is safe?

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A Design Example

• Case 2 Adding a Control Group
• Have two groups
• Control group-----gets salt solution
• Treatment group---gets the actual vaccine

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A Design Example

• Example (Observed Control Study)
• Control Group---all 1st and 3rd grade children
• Treatment group---all 2nd graders
• Assumption
• Age difference between control and treatment
  group was felt to be unimportant

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A Design Example

• Potential Problems
• Parents of 2nd graders may not agree to
  vaccinating their kids
• Parents of sicker kids are most likely to accept
  the vaccine
• More educated parents tend to accept the vaccine
• Parents of sick 1st and 3rd graders may object
  that their kids are not getting treatment

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A Design Example

• Difficulty in diagnosing polio
• Extreme case of polio are easy to diagnose
• Less severe cases of polio have symptoms similar
  to other common illnesses

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A Design Example

• Potential Problems
• Physicians are aware of who has received the
  vaccine and who has not
• Less severe case of polio in a 2nd grader (who
  has received the vaccine) may wrongly diagnosed
  as another illness
• Less severe case in a 1st or 3rd grader will most
  likely be diagnosed as polio

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A Design Example

• Case 3 Randomization, Placebo Control, Double
  Blindness
• Random assignment of control and treatment groups
• Select a child
• Flip a coin-------H-------Treatment Group
• T---------Control Group

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Design Example

• Placebo Control
• Kids in the control group receive salt solution
• Double Blind
• Neither the child
• nor the parents
• nor the doctors/nurses
• who make the diagnosis of polio know whether a
• kid receives the vaccine or the placebo

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A Design Example

• Summary
• In designing experiments
• Introduce some sort of control group
• Use randomization to avoid bias in selection and
  assignment of subjects for the study
• Double blind experiments give protection against
  biases, both intentional and unintentional

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A Design Example

• Perform the experiment on a large number of
  subjects (Polio case in millions of kids)
• Repeat the experiment several times before making
  definitive conclusions

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A Design Example

• Basic Principles of Experimental Designs
• Randomization
• Blocking (Treatment/Control Groups)
• Replication