Lesson2

Outline
Comparative Studies
The Role of Probability

Lesson 2
Chapter 1: Basic Statistical Concepts

Michael Akritas

Department of Statistics
The Pennsylvania State University

Michael Akritas Lesson 2 Chapter 1: Basic Statistical Concepts

Outline
Comparative Studies

1 Comparative Studies
Terminology and Comparative Graphics
Randomization, Confounding and Simpson’s Paradox
Causation: Experiments and Observational Studies
Factorial Experiments

2 The Role of Probability


Outline
Comparative Studies

• Comparative studies aim at discerning and explaining
differences between two or more populations. Examples
include:

The comparison of two methods of cloud seeding for hail
and fog suppression at international airports,
the comparison of two or more cement mixtures in terms of
compressive strength,
the comparison the survival times of a type of root system
under different watering regimens,
the comparison of the effectiveness of three cleaning
products in removing four different types of stains.


Outline
Comparative Studies

Outline

1 Comparative Studies

2 The Role of Probability


Outline
Comparative Studies

Jargon used in comparative studies

One-factor studies.
Factor levels; treatments; populations
Response variable

Example
In the comparison the survival times of a type of root system
under different watering regimens,
Watering is the factor.
The different watering regimens are called factor levels or
treatments. Treatments correspond to populations.
The survival time is the response variable.


Outline
Comparative Studies

More jargon

Experimental units: These are the subjects or objects on
which measurements are made.
In previous example, the roots are the experimental units
Multi-factor studies.
Factor levels combinations; treatments; populations

Factor B
Factor A 1 2 3 4

1 Tr11 Tr12 Tr13 Tr14

2 Tr21 Tr22 Tr23 Tr24


Outline
Comparative Studies

Example
To study the effect of ﬁve different temperature levels and ﬁve
different humidity levels affect the yield of a chemical reaction:
Factors are temperature and humidity, with 5 levels each.
Treatments are the different factor level combinations,
which, again, correspond to the different populations.
Response is the yield of the chemical reaction.
Experimental units is the set of materials used for the
chemical reaction.


Outline
Comparative Studies

Example
A study will compare the level of radiation emitted by ﬁve kinds
of cell phones at each of three volume settings. State the
factors involved in this study, the number of levels for each
factor, the total number of populations or treatments, the
response variable and the experimental units.


Outline
Comparative Studies

Comparisons typically focus on differences (e.g. of means,
or proportions, or medians), or ratios (e.g. ratios of
variances).
Differences are also called contrasts.
The comparison of two different cloud seeding methods
may focus on the contrast µ1 − µ2 .
In studies involving more than two populations a number of
different contrasts may be of interest.
For example, in a study aimed at comparing the mean
tread life of four types of high performance tires designed
for use at higher speeds, possible sets of contrasts of
interest are
1 µ1 − µ2 , µ1 − µ3 , µ1 − µ4 (control vs treatment)
µ1 + µ2 µ3 + µ4
2 − (brand A vs brand B)
2 2
3 µ1 − µ, µ2 − µ, µ3 − µ, µ4 − µ (tire effects)

Outline
Comparative Studies

The Comparative Boxplot

The comparative boxplot consists of side-by-side individual
boxplots for the data sets from each population.
It is useful for providing a visual impression of differences
in the median and percentiles.

Example
Iron concentration measurements from four different iron ore
formations are given in http://www.stat.psu.edu/˜mga/
401/Data/anova.fe.data.txt. The comparative boxplot
can be seen in http://www.stat.psu.edu/˜mga/401/
fig/BoxplotComp_Fe.pdf


Outline
Comparative Studies

The Comparative Bar Graph

The comparative bar graph generalizes the bar graph in
that for each category it plots several bars represents the
category’s proportion in each of the populations being
compared; different colors are used to distinguish bars that
correspond to different populations.

Example
The light vehicle market share of car companies for the month
of November in 2010 and 2011 is given in
http://www.stat.psu.edu/˜mga/401/Data/
MarketShareLightVehComp.txt. The comparative bar
graph can be seen in http:
//stat.psu.edu/˜mga/401/fig/LvMsBarComp.pdf


Outline
Comparative Studies

The Stacked Bar Graph

Example
The site http:
//stat.psu.edu/˜mga/401/Data/QsalesIphone.txt
shows worldwide iPhone sales data, in thousands of units,
categorized by year and quarter. The stacked (or segmented)
bar graph can be seen in http://sites.stat.psu.edu/
˜mga/401/fig/QsIphones.pdf.


Outline
Comparative Studies

To avoid comparing apples with oranges, the experimental units
for the different treatments must be homogenous.

If fabric age affects the effectiveness of cleaning products
then, unless the fabrics used in different treatments are
age- homogenous, the comparison of treatments will be
distorted.
If the meditation group in the diet study consists mainly of
those subjects who had practiced meditation before, the
comparison will be distorted.
To mitigate the distorting effects, called confounding of other
possible factors, called lurking variables, it is recommended
that the allocation of units to treatments be randomized.


Outline
Comparative Studies

Randomizing the allocation of fabric pieces to the different
treatments (cleaning product and stain) avoids
confounding with the factor age of fabric.
Randomizing the allocation of subjects to the control (diet
alone) and treatment (diet plus meditation) groups avoids
confounding with the experience factor.


Outline
Comparative Studies

The distortion caused by lurking variables in the comparison of
proportions is called Simpson’s Paradox.
Example
The success rates of two treatments, Treatments A and B, for
kidney stones are:

Treatment A Treatment B
78% (273/350) 83% (289/350)

The obvious conclusion is that Treatment B is more effective.
The lurking variable here is the size of the kidney stone.


Outline
Comparative Studies

Example (Kidney Stone Example Continued)
When the size of the treated kidney stone is taken into
consideration, the success rates are as follows:
Small Large Combined
Tr.A 81/87 or .93 192/263 or .73 273/350 or .78
Tr.B 234/270 or .87 55/80 or .69 289/350 or .83

Now we see that Treatment A has higher success rate for both
small and large stones.


Outline
Comparative Studies

Batting Averages

Example
The overall batting average of baseball players Derek Jeter and
David Justice during the years 1995 and 1996 were 0.310 and
0.270, respectively. But looking at each year separately we get
a different picture:

1995 1996 Combined
Jeter 12/48 or .250 183/582 or .314 195/630 or .310
Justice 104/411 or .253 45/140 or .321 149/551 or .270

Justice had a higher batting average than Jeter in both 1995
and 1996.


Outline
Comparative Studies

Deﬁnition
A study is called a statistical experiment if the investigator
controls the allocation of units to treatments or factor-level
combinations, and this allocation is done in a randomized
fashion. Otherwise the study is called observational.

• Causation can only be established via a statistical
experiment. Thus, a relation between salary increase and
productivity does not imply that salary increases cause
increased productivity.
• Observational studies cannot establish causation, unless
there is additional corroborating evidence. Thus, the link
between smoking and health has been established through
observational studies.


Outline
Comparative Studies

A statistical experiment involving several factors is called a
factorial experiment if all factor-level combinations are
considered. Thus,

Factor B
Factor A 1 2 3 4

1 Tr11 Tr12 Tr13 Tr14

2 Tr21 Tr22 Tr23 Tr24

is a factorial experiment if all 8 treatments are included in
the study.
Of interest in factorial experiments is the comparison of the
levels within each factor.


Outline
Comparative Studies

Main Effects and Interactions

Deﬁnition
Synergistic effects among the levels of two different factors, i.e.,
when a change in the level of factor A has different effects on
the levels of factor B, we say that there is interaction between
the two factors. The absence of interaction is called additivity.

Example
An experiment considers two types of corn, used for bio-fuel,
and two types of fertilizer. The following two tables give
possible population mean yields for the four combinations of
seed type and fertilizer type.


Outline
Comparative Studies

Fertilizer Row Main
I II Averages Row Effects

Seed A µ11 = 107 µ12 = 111 µ1· = 109 α1 = −0.25

Seed B µ21 = 109 µ22 = 110 µ2· = 109.5 α2 = 0.25
Column
Averages µ·1 = 108 µ·2 = 110.5 µ·· = 109.25
Main
Column β1 = −1.25 β2 = 1.25
Effects

Here the factors interact.


Outline
Comparative Studies

Fertilizer Row Main Row
I II Averages Effects

Seed A µ11 = 107 µ12 = 111 µ1· = 109 α1 = −1

Seed B µ21 = 109 µ22 = 113 µ2· = 111 α2 = 1
Column
Averages µ·1 = 108 µ·2 = 112 µ·· = 110
Main
Column β1 = −2 β2 = 2
Effects

Here the factors do not interact.


Outline
Comparative Studies

Under additivity:
There is an indisputably best level for each factor, and
The best factor level combination is that of the best level of
factor A with the best level of factor B.
What is the best level of each factor in the above design?
Under additivity, the comparison of the levels of each factor
are based on the main effects:
αi = µi· − µ·· , βj = µ·j − µ··

See the main effects in the above two designs.
Under additivity,
µij = µ·· + αi + βj

See the above design.


Outline
Comparative Studies

When the factors interact, the cell means are not given in
terms of the main effects as above.
The difference

γij = µij − (µ·· + αi + βj )

quantiﬁes the interaction effect.
For example, in the above non-additive design,

γ11 = µ11 − µ·· − α1 − β1
= 107 − 109.25 + 0.25 + 1.25
= −0.75.


Outline
Comparative Studies

Data Versions of Main Effects and Interactions

Data from a two-factor factorial experiment use three
subscripts:

Factor B
Factor A 1 2 3

1 x11k , x12k , x13k ,
k = 1, . . . , n11 k = 1, . . . , n12 k = 1, . . . , n13

2 x21k , x22k , x23k ,
k = 1, . . . , n21 k = 1, . . . , n22 k = 1, . . . , n23


Outline
Comparative Studies

Sample versions of main effects and interactions are
deﬁned using
nij
1
x ij = xijk ,
nij
k=1

instead of µij :

Sample Main Row
αi = x i· − x ·· , βj = x ·j − x ··
and Column Effects

Sample Interaction
γij = x ij − x ·· + αi + βj
Effects


Outline
Comparative Studies

Sample versions of main effects and interactions estimate
their population counterparts but, in general, they are not
equal to them.
Thus, even if the data has come from an additive design,
the sample interaction effects will not be zero.
The interaction plot is a graphical technique that can help
assess whether the sample interaction effects are
signiﬁcantly different from zero.
For each level of, say, factor B, the interaction plot traces
the cell means along the levels of factor A. See
http://stat.psu.edu/˜mga/401/fig/
CloudSeedInterPlot.pdf for an example.
For data coming from additive designs, these traces (or
proﬁles) should be approximately parallel.


Outline
Comparative Studies

Probability and Statistics

Probability plays a central role in statistics, but the two differ:

In a probability problem, the properties of the population of
interest are assumed known, whereas statistics is
concerned with learning those properties.
Thus probability uses properties of the population to infer
those of the sample, while statistical inference does the
opposite.


Outline
Comparative Studies

Example (Examples of Probability Questions)
If 5% of electrical components have a certain defect, what
are the chances that a batch of 500 such components will
contain less than 20 defective ones?
60% of all batteries last more than 1500 hours of operation,
what are the chances that in a sample of 100 batteries
there will be at least 80 that last more than 1500 hours?
If the highway mileage achieved by the 2011 Toyota Prius
cars has population mean and standard deviation of 51
and 1.5 miles per gallon, respectively, what are the
changes that in a sample of size 10 cars the average
highway mileage is lass than 50 miles per gallon?


Outline
Comparative Studies

Figure: The reverse actions of Probability and Statistics

In spite of this difference, statistical inference itself would not be
possible without probability. Read also Example 1.9.3, p. 41,
and the paragraph above it.


Outline
Comparative Studies

Go to previous lesson http://www.stat.psu.edu/
˜mga/401/course.info/lesson1.pdf
Go to next lesson http://www.stat.psu.edu/˜mga/
401/course.info/lesson3.pdf
Go to the Stat 401 home page http:
//www.stat.psu.edu/˜mga/401/course.info/


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