STA6166-1-1

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Introduction and Data Gathering (Chapters 1 2)

• At the end of this lecture, the student should
• Be able to provide a definition of Statistics.
• Discuss the role of statistics in research.
• Be able to state reasons for using statistics.
• Identify the difference between observational and
  experimental studies.
• Be able to organize data into a two-dimensional
  matrix or array.

I hear and I forget I see and I understand
I do and I remember Chinese
Proverb
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A Motivating Example The HIP Trial

• Breast cancer common malignancy among women in
  rich countries.
• Mammography (screening) is today known to lead
  to fewer deaths.
• HIP Trial (1960s). First study to conclusively
  show merits of screening.
• 62,000 women age 40-64 (members of Health
  Insurance Plan, NY).
• Randomized into treatment and control groups
  31,000 in each.
• Treatment an invitation to 4 rounds of annual
  screening.
• Control received usual health care prevalent at
  that time.

If we compare screened (1.1) vs. refused
(1.5), theres hardly a difference? (More later!)
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What Is Statistics?

• Descriptive Statistics. Summary measures, such as
  totals, averages or percentages of measurements,
  counts, or ranks. Graphics used to present,
  organize, and summarize data, e.g. pie-charts,
  histograms, boxplots, scatterplots, etc.
• Inferential Statistics. The analysis and
  interpretation of data. Concerned with the
  extraction of information from data and its use
  in reaching conclusions (inferences) about a
  population from which the data are obtained. E.g.
  confidence intervals, hypothesis tests.
• We will concentrate on (2), although the
  distinction will not always be clear.

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Basic Definitions

• Experimental unit. The basic object on which
  measurements are taken. (May be composed of
  measurement units.)
• Factors. Variables in an experiment that are set
  by the investigator. (Controllable.)
• Response. Variable that is observed in an
  experiment. (Not Controllable.)
• Treatments. Conditions constructed from the
  factors in order to observe the impact on the
  response.
• Control Treatment. Benchmark with respect to
  which the remaining treatments are compared.
• Population. The set of all measurements of
  interest.
• Sample. A subset of measurements taken from the
  population actually measured.
• Statistic. A number calculated from the sample,
  e.g. the sample average, the sample variance.
• Parameter. A number calculated from the entire
  population, e.g. the population average, the
  population variance.

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Population vs. Sample
Using the sample average to make statements about
the population average is an example of
inferential statistics.
Descriptive statistical methods describe the
sample. Inferential statistical methods make
statements about the population based on the
sample.
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First Principle of Statistical Inference

• You make inference about the population from
  which the sample was obtained. (Seems obvious,
  but is often forgotten.)
• In each of the examples below, identify the
  population being sampled and the inference being
  made
• Study cow grazing behavior. One cow (Daisy) in
  pasture (A). Randomly select time intervals for
  observation during month of May.
• Study capital punishment and homicide rates.
  Randomly select 100 US cities. Objective is to
  make causal statements about a process.
• In a pilot study, 20 runs of a manufacturing
  process are carried out in the lab. Objective
  find out how the process will work in large scale
  production.
• Study yield of 3 varieties of winter wheat.
  Randomly sample 30 farms in Kansas, 10 farms grow
  variety A, 10 variety B, and 10 variety C.
  Measure the yield per acre over one growing
  season.

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Scientific Method

• The pursuit of systematic interrelation of facts
  by logical arguments from accepted postulates,
  observation, and experimentation and a
  combination of these three in varying proportions.

Roles of Statistics

• Aid in creating the best' research design with
  which to generate new data.
• Extract the information from the noise or
  variability at the data analysis step.

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Logical Arguments

• Deductive argument Conclusion follows with
  logical necessity or certainty from the premises.
  Nothing new is revealed because we are arguing
  from the general to the specific.
• Specialization Moving from a large set of
  objects, postulates, or events, to consideration
  of a smaller set of objects or events.
• Inductive argument Discovering general laws by
  the observation and combination of particular
  instances. Passing from the specific to the
  general.
• Generalization Passing from the consideration of
  one object, postulate, or occurrence, to the
  consideration of a set of objects, postulates, or
  occurrences.

In statistics we attempt to formalize and use
these concepts in a quantitative way.
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Scientific Progress
We gain knowledge by iterating between models and
data.
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Basic Study Steps

• State the problem. What are the questions?
• Devise a plan of solution. What will I do?
• Implement the plan. This is how I do it?
• Analysis of data. What happened?
• Interpretation of results. What does this mean?
• Reexamination. Is my logic correct? What next?

Study design and study implementation may require
iteration.
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Graphical Depiction of Scientific Study
Problem
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Research Design Categories

• Census (Complete Enumeration) Every individual
  in the population of interest is observed. In a
  census, the sample equals the population.
• Observational Studies (Mensurative Experiments)
  Populations to be compared are defined, and
  individuals are randomly selected from these
  populations for measurement. This involves mere
  data collection no interference with the
  processes generating the data.
• Experimental Studies (Manipulative Experiments)
  Individuals in one or more populations are
  carefully chosen or created to test specific
  manipulations under highly controlled conditions.
  Explanatory variables are manipulated their
  effect on the response variable(s) is then
  observed.

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Observational Study Design

• Observational studies are of 3 varieties
• Sample survey studies a population at a
  particular point in time.
• Prospective study observes a population in the
  present using a sample survey, and proceeds to
  follow subjects into the future.
• Retrospective study observes a population in the
  present using a sample survey, and collects data
  about the subjects on events in the past.
• The possible presence of confounding variables
  poses a severe limitation in observational
  studies.
• Confounder. A (non-measured) variable, other than
  the explanatory variable, that affects the
  response variable. Confounders may affect both
  response and explanatory variables, and are
  outside the control of the researcher.

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Observational Study Design

• Example Study lung cancer rates among smokers
  and non-smokers.
• What are populations of interest?
• How will individuals be selected for measurement?
• What will be measured?
• Which analyses will be performed?
• How many individuals are needed?
• How large an effect will be considered important?
• Are available resources adequate for this study?

Many of these questions are answered by subject
matter experts, some can be answered by a
statistical analysis.
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Observational Study( Mensuration Experiment)
What is measured?
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How are individuals selected?

• Individually identified (the sample unit).
• Randomly chosen (no biases introduced in
  selection).
• Each possible set of individuals has the same
  probability of selection (Simple Random Sampling).

Special situations allow for increased efficacy
of selection.

• Stratification (account for an extraneous
  factor)
• Clusters (select natural groups of sample units)
• Multi-stage (select large units then parts of
  units)
• Systematic (set pattern)

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Simple Random Sampling Example
A researcher wishes to determine the prevalence
of a disease in a greenhouse of tomato seedlings.
Each seedling tested for the disease is destroyed
in the process, hence only a minimal number
should be tested. Expectations are that only
about .01 of the roughly 50,000 seedlings in the
greenhouse have the disease.
How to select a simple random sample?

1. Number each pot. Use a random number table (or
   spreadsheet random number generator) to produce a
   list of numbers, in random order from 1 to the
   total number of pots. Measure plants in pots
   whose numbers are selected (difficult).
2. Align pots in rows and columns. Use random number
   table to select a list of row and column number
   pairs. Measure plant in pots located in the (row,
   column) pair selected (easier).

Table 13 in Ott and Longnecker.
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Table 13 in Ott Longnecker
Random number tables are constructed in such a
way that, no matter where you start and no matter
in which direction you move, the digits occur
randomly with equal probability. These numbers
can also be generated with statistical software
packages.
Ex Greenhouse seedlings

1. Use random number table to select a list of row
   and column number pairs. Have a total of 100 rows
   and 500 columns.
2. First two blocks of numbers in Table 13 are
   10480 15011.
3. Moving in 2-digit and 3-digit increments we get
   10 and 480. So we select the pot at intersection
   of row 10 column 480.
4. The next pot would be at intersection of row 15
   column 11.

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Simple Random Sample
Textbook definition.
A simple random sample of n units is defined such
that each possible sample of size n is equally
likely to be drawn.
Practical definition.
This sampling principle assures that each unit in
the population has the same probability
(likelihood) of being selected in the sample.
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Stratified Sampling
Allows us to take into account a factor we
already know affects the response of interest. To
remove a source of known variability.
16 years healthy
20 years diseased
22 years healthy
Pine forest Estimate expected yield from plot.
Individuals selected at random within each
strata. Variability in diseased subpopulation
expected to be much greater than in healthy area.
Mean yield greater at 22y than 16y.
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Cluster Sampling
Estimate the average sponge size on natural reefs.
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REEF
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Number of sponges on reef
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7
5
Selecting sponges at random would be very
resource inefficient. Cheaper to select reefs
(sponge clusters) at random with probability
proportional to size. All sponges on selected
reefs are measured (a cheap thing to do that
increases the sample size easily).
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MultiStage Sampling
Typically large areas or large complex
populations can be more effectively sampled in
stages. At the first stage, natural or synthetic
clusters are selected. At subsequent stages the
selected clusters are subdivided into units and
samples of these are selected.
Example National crop yield survey.
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Greenhouse Example
Stratification Maybe we have observed that
plants near the door seem less healthy than those
further into greenhouse. Divide room into plants
near door and plants inside. Random samples
from each stratum. Cluster Suppose plants are
arranged on tables. We could select tables at
random then examine all plants on each table
selected. Note that if one plant on a table is
diseased, all plants on table have an increased
probability of also being diseased. Multi-Stage
Again suppose plants are on tables. Select some
tables at random. Next select a few plants from
each selected table for testing. First stage unit
is the table. Second stage unit is the plant.
Third stage unit could be the leaf on the plant,
etc. Systematic Imagine plants arranged on a
large table. Randomly pick a row and column to
start. Then, following a systematic route, pick,
say, every 10th plant.
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What is measured?
Variable Apt or liable to vary or change from
individual to individual, capable of being varied
or changed (factor), alterable, inconsistent,
having much variation or diversity, a quantity
that may assume any given value from a set of
values (the variables range).
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Types of Variables Categorical

• Categorical, classification, or qualitative
  variable
• Discrete essentially describes some
  characteristic of a sample unit. E.g. color,
  gender, grade, health status, treatment group.
  Further subdivided into
• nominal (think name) arithmetic doesnt make
  sense, e.g. gender M,F even if coded 0,1
• ordinal (think order) nominal data with order,
  e.g. grades A,B,C,D,F, strength of agreement
  1strongly agree, 2agree, 3neutral,
  4disagree, 5strongly disagree.
• In ordinal data the order is meaningful, but the
  difference between responses isnt. Also,
  arithmetic is sometimes done, but its meaning is
  debatable.

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Types of Variables Quantitative

• Quantitative or amount variable
• Can be either discrete or continuous measures
  the amount or level of a characteristic of a
  sample unit. For example age, weight, height,
  temperature, biomass, volume. Further subdivided
  into
• interval - differences between values have
  meaning but there is no definite or meaningful
  zero point, e.g. GPA, SAT scores, temperature
• ratio like interval but with a meaningful zero
  point, e.g. weight, money, yield.

In this course we will deal primarily with
quantitative variables (ratio).
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Study Design Questions

• How is the response (effect) to be measured?
• What characteristics of the response are to be
  analyzed?
• What factors influence the characteristics to be
  analyzed?
• Which of these factors will be studied in this
  investigation?
• How many times should the basic experiment be
  performed?
• What should be the form of the analysis?
• How large an effect (effect size) will be
  considered important?
• What resources are available for this study? Are
  they adequate?

It is important to be able to define the
underlined words.
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Terminology

• The response typically refers to the measured
  variable(s) of primary interest (e.g. weight,
  health status, growth, etc).
• Characteristics Is it change in the average
  response, the spread of responses, the maximum
  response, etc, that will be examined? These
  characteristics typically refer to some
  statistical aspect of effects measured among
  individuals in the populations being studied.
• A factor refers to the characteristic(s) that
  primarily differ among the populations being
  studied (compared). Some factors we cannot
  manipulate (I.e. such as descriptors like gender,
  geographic location, genetic makeup). Other
  factors identify characteristics we have caused
  to be different between the two populations (as
  in an experiment where we manipulate the
  populations by giving them different
  treatments).
• Basic Experiment The selecting of an individual
  for measurement. In an observational study, the
  basic experiment is the selection and measurement
  of an individual from the population. In an
  Experimental Study, the basic experiment is the
  selection of an individual from the pool, the
  application of a treatment, and the measurement
  of responses.

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Terminology (Cont)

• By the form of the analysis, we refer to the
  statistical procedure(s) that match the
  characteristics of the study design, the
  characteristics of the responses measured and the
  estimates and hypothesis tests needed to answer
  the questions of interest. So, when someone asks
  What form will your analysis take? you might
  answer with something like I will be using
  regression analysis (the statistical method) to
  explore associations between fat intake and
  cholesterol level (the hypotheses of interest)
  between two populations identified geographically
  and by gender (study design factors).
• The size of the effect of interest refers to how
  big of a difference must there be before we would
  conclude that there is a real difference.
  Typically we are interested in specifying this at
  the design phase of a study since the size of the
  effect of interest drives the sample size
  question. Thus if you say a difference of less
  than 2 points in cholesterol level between gender
  groups would not be important, but anything
  greater than 2 is large enough to be noteworthy,
  you could use this to set the study sample size.
  If the difference were raised to 10 points, a
  much smaller sample size would be needed.
• Resources Money, personnel, time, access,
  material.

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Experimental Study

• Manipulation Experiment A research design in
  which the researcher deliberately introduces
  certain changes in the levels of factors that are
  hypothesized as affecting the process of
  interest, and then makes observations to
  determine the effect of these changes.
• Experimental Design A study plan which assures
  that measurements will be relevant to the problem
  under study.
• Treatments Changes to those factors which are
  suspected of affecting the process under study.

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Ex Factorial Experiment
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Standard Form for a Data Set
Observation Number
CATEGORIES
AMOUNTS
1 1 F RED x x ... 10.2 x x ... 2 1 F
WHITE x x ... 12.9 x x ... 3 1 M BLUE
x x ... 20.1 x x ... . . . . . . n 1
F BLUE x x ... 16.0 x x ...
strata
Other quantitative variable
Other categorical variable
gender
weight
color
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Example Data Set in Spreadsheet Format
Indicator of missing data
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Inventor's Paradox
The more ambitious the plan, the more chances of
success, and the more opportunity for failure.
How does one decide on what to do?
Are there open questions ? Are there available
resources? Does someone really want the
answer? Can a study be done? Will the study be
able to answer the question?
Statistics may help answer the last question!
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The HIP Trial Revisited

• Seems natural to compare screened (cancer
  rate1.1) vs. refused (cancer rate1.5), in the
  treatment group hardly a difference!
• But realize that this is an observational
  comparison (in an experimental study), and hence
  is prone to confounding.
• Social status is a confounder. Richer and better
  educated women were more likely to accept the
  screening, and breast cancer hits the richer
  harder than the poorer. (Pregnancy, esp. early
  pregnancy, is now known to protect against breast
  cancer.)
• So the analysis by treatment received is biased.
  But the analysis by intention-to-treat is
  appropriate.

• Intention to screen cancer rate (1.3).
• Control cancer rate (2.0).
• A sizeable difference.
• Five-year cancer rate ratio (treat/control) is
  39/6362.