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), there’s
hardly a difference…?
(More later!)
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What Is Statistics?
1. 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.
2. 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.
STA6166-1-4
Population vs. Sample
Using the sample average to make statements about the
population average is an example of inferential statistics.
Sample
Population
Inference
Descriptive statistical methods: describe the sample.
Inferential statistical methods: make statements about the
population based on the sample.
STA6166-1-5
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:
1. Study cow grazing behavior. One cow (Daisy) in pasture (A).
Randomly select time intervals for observation during month of May.
2. Study capital punishment and homicide rates. Randomly select 100
US cities. Objective is to make causal statements about a process.
3. 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.
4. 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.
STA6166-1-6
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.
Hypothesis
Model, Conjecture
New Hypothesis, New Model
Progress
and
Understanding
Data, Measurements
New 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.
STA6166-1-10
Graphical Depiction of Scientific Study
Knowledge
Base
Problem
Constraints
Objectives &
Hypotheses
Experiment
Sample
DESIGN
How to measure?
DATA
Interpretation
STATISTICAL ANALYSIS
Graphics & Visualization
•Modeling
•Estimates and Confidence Intervals
•Formal Statistical Tests
Conclusions
STA6166-1-11
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.
STA6166-1-12
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.
STA6166-1-13
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)
Population 1
Population 2
Sample 1
Sample 2
What is measured?
Characteristics
How Selected?
1 1
2 1
3 1
…
n 1
x x x x x…
x x x x x…
x x x x x…
x x x x x ...
1 2 x x x x x…
2 2 x x x x x…
3 2 x x x x x…
…
m 2 x x x x x ...
STA6166-1-15
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)
STA6166-1-16
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.
STA6166-1-17
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.
STA6166-1-18
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.
STA6166-1-19
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
22 years
healthy
20 years
diseased
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.
STA6166-1-20
Cluster Sampling
Estimate the average sponge size on natural reefs.
REEF
9
25
12
Number of
sponges on
reef
21
5
14
7
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).
STA6166-1-21
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.
a. Random Selection
b. Systematic Selection
random
starting
point
randomly
located
grid
c. Multi-Stage Selection
Second stage
unit
Measurement
units
First-stage
unit
Example: National crop yield survey.
STA6166-1-22
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.
STA6166-1-23
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 variable’s range).
Examples:
•
•
•
Plant biomass – varies from plant to plant.
Blood arsenic level – varies from person to person.
Gender – we are not all male or all female.
STA6166-1-24
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 doesn’t 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 {1=strongly
agree, 2=agree, 3=neutral, 4=disagree, 5=strongly
disagree}.
In ordinal data the order is meaningful, but the difference
between responses isn’t. Also, arithmetic is sometimes
done, but it’s meaning is debatable.
STA6166-1-25
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).
STA6166-1-26
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.
STA6166-1-27
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.
STA6166-1-28
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.
STA6166-1-29
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.
STA6166-1-30
Ex: Factorial Experiment
Nitrogen Level
FACTORS
LEVELS
Phosphorus
Level
0 kg/ha
10 kg/ha
20 kg/ha
0 kg/ha
0/0
10 / 0
20 / 0
10 kg/ha
0 / 10
10 / 10
20 / 10
EXPERIMENTAL
UNIT (PLOT)
TREATMENTS
SITE 1
(block 1)
0 / 10
10 / 0
20 / 10
10 / 10
20 / 0
0/0
SITE 2
(block 2)
10 / 10
20 / 10
10 / 0
0/0
0 / 10
20 / 0
BLOCKED LAYOUT
(complete block - all treatments in each block)
STA6166-1-31
Standard Form for a Data Set
Observation
Number
1
2
3
.
.
.
n
1
1
1
.
.
.
1
CATEGORIES
AMOUNTS
F
F
M
RED
WHITE
BLUE
x
x
x
x ... 10.2
x ... 12.9
x ... 20.1
x
x
x
x ...
x ...
x ...
F
BLUE
x
x ... 16.0
x
x ...
strata
gender
color
Other
categorical
variable
weight
Other
quantitative
variable
STA6166-1-32
Example Data Set in Spreadsheet Format
OBS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
ITEMP
IRH
24.47
24.47
24.47
24.45
24.45
24.45
24.68
24.68
24.68
24.79
24.79
24.79
25.03
25.03
25.03
24.44
24.44
24.44
24.43
24.43
24.43
25.24
25.24
25.24
25.35
25.35
25.35
IWB
64
64
64
50
50
50
50
50
50
51
51
51
74
74
74
74
74
74
73
73
73
78
78
78
89
89
89
20.2
20.2
20.2
18.55
18.55
18.55
18.45
18.45
18.45
18.57
18.57
18.57
21.6
21.6
21.6
21.22
21.22
21.22
21.2
21.2
21.2
21.91
21.91
21.91
23.78
23.78
23.78
FWB
REP
20.25
20.25
20.25
18.6
18.6
18.6
19.52
19.52
19.52
18.2
18.2
18.2
21.8
21.8
21.8
21.5
21.5
21.5
21.76
21.76
21.76
22.06
22.06
22.06
24.01
24.01
24.01
BIRD
1
1
1
2
2
2
3
3
3
4
4
4
1
1
1
2
2
2
3
3
3
4
4
4
1
1
1
BN
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
IBT
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
ATBT
40.6
40.6
40.9
40.3
40.4
40.1
41.1
41.2
40.9
39.8
39.6
39.8
39.8
39.8
39.4
40.1
40.1
40
39.4
39.8
39.5
.
.
.
.
.
.
Indicator of missing data
39.7
40.2
39.4
40.1
39.4
39.2
40.5
40.8
40.9
39.4
39.4
39.8
38.9
38.7
39.4
39.6
39.8
39.6
39.9
40.2
39.2
.
.
.
.
.
.
WEIGHT
2.21
2.265
2.185
2.275
2.264
2.205
2.343
2.193
2.238
2.32
2.298
2.31
2.212
2.21
2.198
2.235
2.257
2.284
2.33
2.314
2.295
2.149
2.12
2.127
2.213
2.216
2.36
SATBT
SITEMP SIWB
-1.24351 -1.28723 -1.27434
-0.69343 -1.28723 -1.27434
-1.57355 -1.28723 -1.27434
-0.80345 -1.29196 -1.67386
-1.57355 -1.29196 -1.67386
-1.79358 -1.29196 -1.67386
-0.36338 -1.23754 -1.69807
-0.03334 -1.23754 -1.69807
0.07668 -1.23754 -1.69807
-1.57355 -1.21151 -1.66902
-1.57355 -1.21151 -1.66902
-1.13349 -1.21151 -1.66902
-2.12363 -1.15472 -0.93536
-2.34366 -1.15472 -0.93536
-1.57355 -1.15472 -0.93536
-1.35352 -1.29433 -1.02737
-1.13349 -1.29433 -1.02737
-1.35352 -1.29433 -1.02737
-1.02348 -1.29669 -1.03221
-0.69343 -1.29669 -1.03221
-1.79358 -1.29669 -1.03221
.
-1.10503
-0.8603
.
-1.10503
-0.8603
.
-1.10503
-0.8603
.
-1.079 -0.40751
.
-1.079 -0.40751
.
-1.079 -0.40751
STA6166-1-33
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!
STA6166-1-34
The HIP Trial Revisited
•
•
•
•
Seems natural to compare “screened” (cancer rate=1.1) vs. “refused”
(cancer rate=1.5), 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 39/63=62%.
STA6166-1-35