chapter 1 statistical thinking - michigan state university · 2014-08-29 · statistical thinking...

Chapter 1 Statistical Thinking

•What is statistics?

•Why do we study statistics

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


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.


• "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)


• 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


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?


• 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


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!


• 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????


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.


Result

• 8 out of the 10 students improved their

scores.

• Question: Can we conclude that reading

the article has improved students’

knowledge about statistics?


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? “


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


• 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?


• 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

4.4%chance 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


• Suppose the proportion of improvement in

10 trials is 4.4%.

• We observed 8 students’ scores out of 10

improve.

• What does this mean?


• Course is highly effective

• Course is ineffective and we observed an

unlikely event.

• We do not know which one!


• 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.


• 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.


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.

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)

A Design Example

• Problem with vaccines:

– Are they safe?

– Are they effective?

• Undertake a large scale trial to answer

these questions

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?????

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”?

A Design Example

• Case 2: Adding a Control Group

– Have two groups

• Control group-----gets salt solution

• Treatment group---gets the actual vaccine

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

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

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

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

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

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

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

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

A Design Example

Basic Principles of Experimental Designs

• Randomization

• Blocking (Treatment/Control Groups)

• Replication

chapter 1 statistical thinking - michigan state university · 2014-08-29 · statistical thinking...

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