Introduction the Development of PhilosophySocrates ( the unexam

Introduction the Development of Philosophy
Socrates ( "the unexamined life is not worth living"
Demonstrate knowledge on:
What is Philosophy?
The noun philosophy means the study of proper behavior, and
the search for wisdom. The original meaning of the
word philosophy comes from the Greek roots philo-
meaning "love" and -sophos, or "wisdom." ... In other words,
they want to know the meaning of life.
Watch Video: What is
Philosophy https://www.youtube.com/watch?v=nRG-rV8hhpU
What is Ethics?
Ethics or moral philosophy is a branch of philosophy that
involves systematizing, defending, and recommending concepts
of right and wrong conduct. ... Ethics seeks to resolve questions
of human morality by defining concepts such as good and evil,
right and wrong, virtue and vice, justice and crime.
View Video: What is Ethics
https://www.youtube.com/watch?v=3_t4obUc51A
4,200 religions
According to some estimates, there are roughly
4,200 religions in the world. The word religion is sometimes
used interchangeably with "faith" or "belief system",
but religion differs from private belief in that it has a public
aspect.
List of Religions and Spiritual Traditions -
https://en.wikipedia.org › wiki ›
List_of_religions_and_spiritual_traditions
Forms of Religious Belief : Monotheism, Atheism, Polytheism,
Agnostic
A. Monotheism The term monotheism comes from the

Greek monos, (one) and theos (god). Thus, monotheism is the
belief in the existence of a single god.
B. Polytheism which is a belief in many gods
C. Atheism An atheist doesn't believe in a god or divine
being. ...
D. Agnostic an agnostic neither believes nor disbelieves in a
god or religious doctrine. Agnostics assert that it's impossible
for human beings to know anything about how the universe was
created and if divine beings exist. They are open to the
possibility of a divine being an atheist is not open to such a
possibility.
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Two Types of Religions/Historical and Mythological Religions
Mythological Religion Mythology is the main
component of Religion. It refers to systems of legends and
stories and concepts that are of high importance to a certain
community, making statements concerning
the supernatural or sacred. Religion is the broader term, besides
mythological system, it includes ritual. A given mythology is
almost always associated with a certain religion such as Greek
mythology with Ancient Greek religion. Disconnected from its
religious system, a myth may lose its immediate relevance to
the community and evolve—away from sacred importance—into
a legend or folktale.
Historical Religions can be traced back in history to actual
people, places and events which are documented in history and
archeology. Information about the teachings and life situation of
Jesus, Mohammed, Moses, The Jewish Prophets can be found in
historical records.
Religious Theory Philosophy( Ethics based on a Religious
teaching) Religious philosophy is philosophical thinking that is
inspired and directed by a particular religion. It can be done
objectively, but may also be done as a persuasion tool by
believers in that faith.It guides a personson's action and choices
based on religious teachings.
A. Mythology and legends as the basis of truth in the ancient

world (watch the video The Gallery of the Gods)
Zeus: the King of the Greek Gods
Hera: the Queen of the Greek Gods married to Zeus
Predestination: Ancient Greek religion taught that human lives
were at the mercy of fate which is determined by the Gods)
People were subject to their destiny with no chance to change
their lives
Socrates (the Father of Classical Philosophy )
At the time of Sacrates, Ehics were based upon Greek
Mythology Religion
A. The great question of Socrates? Can I know GOOD apart
from GOD/RELIGION
B. The Great Statement of Socrates : The Unexamined Life is
not Worth Living
THis semester we will be analyzing Ethical Questions from two
opposing philosophical positions
A. Judeo-ChristianTradition
Teaches that there is an Objective Truthrevealed by God to
human beings on how to live a good life. It is contained in the
teachings of the Old and New Testament)
B. Secular Humanism
everything is subjective it teaches that there No God and that
human beings have no spiritual nature. They are just a physical
body. Human beings create their own ideas of right and wrong
subject to their own will. Human beings become their own
God.)
What is Revelation
The word “revelation” comes from the word “reveal.”
Revelation is “something that is revealed.” Biblically, the word
“revelation” refers to something revealed by a spiritual source,
which may be God, the Lord Jesus Christ. The “book of
Revelation” is so called because its contents were revealed by
God to Jesus, who revealed it to an angel, who revealed it to the
Apostle John (Rev. 1:1). For Christians it is the teachings in the
Old and New Testaments . For Jews it teachings in the Old
testament

What is the Covenant in Judaeo Christian Tradition?
Covenant. Literally, a contract. In the Bible (see also Bible), an
agreement between God and his people, in which God makes
promises to his people and, usually, requires certain conduct
from them. In the Old Testament, God made agreements with
Noah, Abraham, and Moses.
What are the Ten Commandments
Also known as the Decalogue, the Ten Commandments come
from the Old Testament of the Bible, where they are revealed to
Moses on Mt. Sinai and carved into two stone tablets. The
commandments are mentioned as laws in Exodus 24:12-13 and
named as the Ten Commandments in Exodus 34:28. The phrase
appears in English as early as 1280. The seminal 1611 King
James Version of the Bible renders the commandments in the
now familiar and widely quoted Thou shalt not formula and are
summarized as follows:
1. Thou shalt have no other gods before me.
2. Thou shalt not make unto thee any graven image.
3. Thou shalt not take the name of the Lord thy God in vain.
4. Remember the sabbath day, to keep it holy.
5. Honour thy father and thy mother
6. Thou shalt not kill.
7. Thou shalt not commit adultery.
8. Thou shalt not steal.
9. Thou shalt not bear false witness against thy neighbour.
10. Thou shalt not covet thy neighbor’s house (wife, servants,
and animals).
Forming the basis of Judeo-Christian morality and
ethics, theTen Commandments are widely taught, memorized,
cited, and displayed by Jews and Christians, referenced in
everything from Sunday School to bumper stickers.
Judeo-Christian( definition) In this philosophy GOD is GOD
and reveals right and wrong action in His Commandmen ts it is
a term that groups Judaism and Christianity, either in reference
to Christianity's derivation from Judaism, both religions'
common use of the Bible, or due to perceived parallels or

commonalities shared values between those two religions, which
has become part of the development of laws and civilization
of Western culture in Europe and the Americas
The term became prevalent towards the middle of the 20th
century in the United States to link broader principles of Judeo-
Christian ethics such as the dignity of human life, adherence to
the Abrahamic covenant, common decency, and support of
traditional family values.[1]
The concept of "Judeo-Christian values" in an ethical (rather
than theological or liturgical) sense was used by George
Orwell in 1939, with the phrase "the Judaeo-Christian scheme
of morals."[2] It has become part of the American civil
religion since the 1940s.
Secular Humanism (definition) In this form of philosophy MAN
IS HIS OWN GOD
Secular humanism, or simply humanism, is a philosophy or life
stance that embraces human reason, ethics, and philosophical
naturalism while specifically rejecting religious dogma,
supernaturalism, pseudoscience, and superstition as the basis of
morality and decision making.
Belief in Deity
Not considered important. Most Humanists are atheists or
agnostics.
•Incarnations
Same as above.
•Origin of Universe and Life
The scientific method is most respected as the means for
revealing the mysteries of the origins of the universe and life.
•After Death
An afterlife or spiritual existence after death is not recognized.
•Why Evil?
No concept of “evil.” Reasons for wrongdoing are explored

through scientific methods, e.g. through study of sociology,
psychology, criminology.
•Salvation
No concept of afterlife or spiritual liberation or salvation.
Realizing ones personal potential and working for the
betterment of humanity through ethical consciousness and social
works are considered paramount, but from a naturalistic rather
than supernatural standpoint.
•Undeserved Suffering
No spiritual reasons but rather a matter of human vulnerability
to misfortune, illness, and victimization.
•Contemporary Issues
The American Humanist Association endorses elective abortion.
Other contemporary views include working for equality for
homosexuals, gender equality, a secular approach to divorce and
remarriage, working to end poverty, promoting peace and
nonviolence, and environmental protection.
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Introductory Philosophical Terms and Concepts in Ethics
Philosophy The study of the Love of Wisdom from the past
used in the present
Ethics the study of morality what is considered "good" and
"bad" right and wrong human conduct and behavior
Moral what is considered good or right. Should be
characterized by pleasure, happiness and harmony in one's life
Immoral what is considered bad or wrong characterized by
unhappiness and disharmony in one's life
Amoral A person having no moral sense or being indifferent to
right and wrong actions. Survival of self the only value.
Sometimes the result of physical/psychological imbalance in the
brain.

Nonmoral usually involves objects and their use. Involves
objects and how they are used to give them a moral quality
For example, A Gun or Knife if used to hunt food for family it
is a GOOD, if used to kill neighbor in an argument it is BAD
Four Aspects of Morality
Religious Morality concerned with human beings in relationship
to a supernatural being
Morality and Nature human beings in relationship and respectful
of the natural world of God's creation
Individual Morality human beings in relation to themselves, self
respect important value
Social Morality human beings in relation to other human beings,
a social consciousness is a value
Morality is Objective moral laws are revealed by a divine being
to human beings thus must be followed as written
Morality is Subjective human beings create their own moral
laws
Customary/Traditional Morality moral laws are based on
inherited custom and tradition often accepted without thought or
reflection. A person is born into a religion in one's family and
identifies with it even though they may not practice the
religion.
Reflective Morality critical evaluation of all moral issues
whether or not they are based on religion, custom or tradition
Morality and the Law what is legal may not always be moral.
Legal means I may do something without fear of punishment.
Moral asks the question: Should I do this action??
Morality and Religion/The Question of Truth human beings
believe that their religion teaches them the truth about right and
wrong actions and choices.
Revelation God reveals his will for good and bad in the Old and
New Testament Example: Ten Commandments
Covenant A Free Will agreement between God and and the
individual person to accept Revelation(the teachings in the
Bible) as a rule for life's ethical choicesYouTube Videos
https://youtu.be/nRG-rV8hhpU

https://youtu.be/GmHAdgDkcCw
https://youtu.be/3_t4obUc51A
https://youtu.be/J7hQempjqpQ
https://youtu.be/ytdMUQNaBG0
YouTube Videos 2
https://youtu.be/-WgPp6xvRZs
https://youtu.be/Ut-UOhY0s8E
https://youtu.be/dFVcdcLX1zk
https://slideplayer.com/slide/10599268/
Discussion Board #1 Secular Humanism vs Judeo Christian
Tradition
Part I What did you learn from Week I ideas on development of
Philosophy material? ( at least one full paragraph)
Part II Today we will also be reflecting on different Ethical
Philosophy and Ethical Language Definitions
For this Discussion Board question, I would like for you to
reflect upon the different philosophies called
Secular Humanism and Judeo Christian Tradition as discussed
in the material for Module One November 2, 2021
Watch the videos and examine carefully their
definitions.Compare both of them and what they teach as moral.
What is your reaction to these different ways of looking at the
world?
RESPOND to ANOTHER STUDENT'S POSTING
Correlation vs Causality in Linear Regression Analysis
Chapter 6
© 2019 McGraw-Hill Education. All rights reserved. Authorized
only for instructor use in the classroom. No reproduction or
distribution without the prior written consent of McGraw -Hill
Education

Learning Objectives
Differentiate between correlation and causality in general and in
the regression environment
Calculate partial and semi partial correlation
Execute inference for correlation regression analysis
Execute passive prediction using regression analysis
Execute inference for determining functions
Execute active prediction using regression analysis
Distinguish the relevance of model fit between active and
passive prediction
‹#›
© 2019 McGraw-Hill Education.
The Difference Between Correlation and Causality
Yi = fi(X1i, X2i, …, XKi) + Ui
We define as the determining function, since it comprises the
part of the outcome that we can explicitly determine
Ui can only be inferred by solving Yi – fi(X1i, X2i, …, XKi)
Data-generating process as a framework for modeling causality
The reasoning established to measure an average treatment
effect using sample means easily maps to this framework
Easily extends into modeling causality for multi-level
treatments and multiple-treatments
‹#›

A causal relationship between two variables clearly implies co-
movement.
If X casually impacts Y, then when X changes, we expect a
change in Y
However, variables often move together even when there is no
casual relationship between them
For example, height of two different children of ages 5 and 10.
Since both the children are growing during these ages, their
heights will generally move together. this co-movement is not
due to causality – an increase in height by one child will not
change in the height for the other.
‹#›
Measurement of the co-movement between two variables in a
dataset is captured through sample covariance or correlation:
Covariance: sCov(X,Y) =
Correlation: sCorr(X,Y) =
‹#›
When there are more than two variables, e.g., Y, X1, X2, we
can also measure partial correlation between two of the
variables

Partial correlation between two variables is their correlation
after holding one or more other variables fixed
‹#›
Causality implies that a change in one variable or variables
causes a change in another
Data analysis attempting to measure causality generally
involves an attempt to measure the determining function within
the data-generating process
Correlation implies that variables move together
Data analysis attempting to measure correlation is not
concerned about the data-generating process and determining
function, it uses standard statistical formulas (sample
correlation, partial correlation) to assess how variables move
together
‹#›
The dataset is a cross-section of 230 grocery stores
AvgPrice = Average Price
AvgHHSize = Average Size of Households of Customers at that
Grocery Store.
Regression Analysis for Correlation

‹#›
Sales = b + m1AvgPrice + m2AvgHHSize
Solving b, m1, m2:
Sales = 1591.54 – 181.66 × AvgPrice + 128.09 × AvgHHSize
This equation provides us information about how the variables
in the equation are correlated within our sample.
‹#›
Unconditional correlation is the standard measure of correl ation
between two variables X and Y
Corr(X,Y) =
Sx = Sample standard deviation for X and
SY = Sample standard deviation for Y
Partial correlation between X and Y is a measure of the
relationship between these two variables, holding at least one
other variable fixed
Semi-partial correlation between X and Y is a measure of the
relationship between these two variables, holding at least one
other variable fixed for only X or Y
Different Ways to Measure Correlation Between Two Variables

‹#›
For the general regression equation: Y = b + m1X1 + …
+mKXK the solutions for m1 through mk when solving the
sample moment equations are proportional to the partial and
semi-partial correlation between Y and the respective Xs
‹#›
Suppose we have the data for the entire population for our
grocery store data, then, we have:
Sales = B + M1AvgPrice + M2AvgHHSize
Capital letters are used to indicate that these are the intercept
and slopes for the population, rather than the sample
Solve for B, M1, and M2 by solving the sample moment
equations using the entire population of data
Regression and Population Correlation
‹#›
Regression and Population Criteria
We do not have the data for the entire population, but for a
sample dataset for the population whose regression line is:
Sales = b + m1AvgPrice + m2AvgHHSi ze
Solve for b, m1 and m2

The intercept and slope(s) of the regression equation describing
a sample are estimators for the intercept and slope(s) of the
corresponding regression equation describing the population.
‹#›
Consistent estimator is an estimator whose realized value gets
close to its corresponding population parameter as the sample
size gets large.
Regression and Population Correlation
‹#›
Regression Line for Full Population
‹#›
Regression Lines for Three Samples of Size 10
‹#›

Regression Lines for Three Samples of Size 30
‹#›
In order to conduct hypothesis testing or building confidence
intervals for the population parameters of a regression equation,
we need to know the distribution of the estimators
Each estimator becomes very close to its corresponding
population parameters for a large sample
For a large sample, these estimators are normally distributed
Confidence Interval and Hypothesis Testing for the Population
Parameters
‹#›
A large random sample implies that:
b~N(B,σB)
m1~N(M1,σm1)
mk~N(MK,σmk)
If we write each element in the population as:
Yi = B + M1X1i + … + MKXK + Ei
, where Ei is the residual, then Var(Y|X) is equal to Var(E|X)
Common assumption that this variance is constant across all
values of X , so Var(Y|X) = Var(E|X) = Var(E) = σ2
This consistency of variance is called homoscedasticity
Confidence Interval and Hypothesis Testing for the Population
Parameters

‹#›
Sales = 1591.54 – 181.66 × AvgPrice + 128.09 × AvgHHSize
If Store A has an average price of $0.50 higher than Store B,
and Store A has an average household size that is 0.40 less than
Store B, then:
= -181.66 × 0.50 + 128.09 × (-0.4) = -142
We predict Store A has 143 fewer sales than Store B
When using correlational regression analysis to make
predictions, we must be considering a population that spans
across time and we assume that the population regression
equation best describes the future population
Prediction Using Regression
‹#›
Regression and Causation
Data-generating process of an outcome Y can be written as:
Yi = fi(X1i, X2i, …, XKi) + Ui
We assume the determining function can be written as:
fi(X1i, X2i, …, XKi) = α + β1X1i + β2X2i +… βKXKi
Combining these assumptions into a single assumption, the
data-generating process can be written as:
Yi = α + β1X1i + β2X2i +… βKXKi + Ui
Error term represents unobserved factors that determine the
outcome

‹#›
Yi = B + M1X1i + … +MKXK + Ei (Correlation model)
Yi = α + β1X1i + … βKXKi + Ui (Causality model)
Correlational model residuals (Ei) have a mean of zero and are
uncorrelated with each of Xs. For this model, we simply plot
all the data points in the population and write each observation
in terms of equation that best describes these points.
For the causality model, the data-generating process is the
process that actually generating the data we observe and
determining function need not be the equation that best describe
the data.
‹#›
CONSIDERING THESE DATA FOR Y, X, AND U ARE FOR
THE ENTIRE POPULATION:
THESE DATA WERE GENERATED USING THE DATA-
GENERATING PROCESS: Yi = 5 + 3.2Xi + Ui
MEANING WE HAVE A DETERMING
FUNCTION : f(X) = 5 + 3.2X
The Difference Between the Correlation Model and the
Causality Model: An Example

‹#›
Scatterplot, Regression Line, and Determining Function of X
and Y
IN THIS FIGURE, WE PLOT Y AND X ALONG WITH THE
DETERMING FUNCTION (BLUE LINE) AND THE
POPULATION REGRESSION EQUATION (RED LINE).
‹#›
The correlation model describes the data best but need not
coincide with the causal mechanism generating the data
The causality model provides the casual mechanism but need
not describe the data best
‹#›
The Relevance of Model Fit for Passive and Active Prediction
Total sum of squares (TSS): The sum of the squared difference
between each observation of Y and the average value of Yi
TSS = Yi – )2
Sum of squared residuals (SSRes): The sum of the squared
residuals.
SSRes = i

R-squared: The fraction of the total variance in Y that can be
attributed to variation in the Xs
R2 = 1 – SSRes/TSS
‹#›
The Relevance of Model Fit for Passive and Active Prediction
A high R-squared implies a good fit, meaning the points on the
regression equation tend to be close to the actual Y values
R-squared for passive prediction (correlation) : Finding a high
R-squared implies the prediction is close to reality
R-squared for active prediction (causality): R-squared is not a
primary consideration when evaluating predictions
‹#›
Linear Regression as a Fundamental Descriptive Tool
Chapter 5
© 2019 McGraw-Hill Education. All rights reserved. Authorized
only for instructor use in the classroom. No reproduction or
distribution without the prior written consent of McGraw -Hill
Education
Learning Objectives

Construct a regression line for a dichotomous treatment
Construct a regression line for a multi-level treatment
Explain both intuitively and formerly the formulas generating a
regression line for a single treatment
Distinguish the use of sample moment equations from
estimation via least squares
Distinguish regression equations for single and multiple
treatments
Describe a dataset with multiple treatments using multiple
regression
Explain the difference between linear regression and a
regression line
‹#›
Scatterplot of Price and Sales
How do we summarize the relationship between these two
variables?
‹#›
The Regression Line for a Dichotomous Treatment
Dichotomous treatment
Two treatment statuses—treated and untreated
Regression analysis
The process of using a function to describe the relationship
among variables

‹#›
The Regression Line for a Dichotomous Treatment : An
Intuitive Approach

Draw a line through these data that will best describe the
relationship between Price and Treatment
‹#›
The Regression Line for a Dichotomous Treatment: An Intuitive
Approach
In general, the formula for a line is: Y = f(X) = b + mX,
where b is the intercept and m is the slope of the line
‹#›
Line Describing the Relationship Between Profits and
Treatment

What is the equation for the line shown here?
Profits = 208.33 – 20 × Treatment
‹#›
Line Describing the Relationship Between Profits and Price
Knowing the two point on the Profits/Price line, solve for slope
and intercept
Profits = 248.33 – 40 × Price
‹#›
The Regression Line for a Dichotomous Treatment
Whenever there is a dichotomous treatment, a line can be built
describing the relationship between the treatment and outcome
by using the means for each treatment status called the

regression line for a dichotomous treatment
Set f(0) = and f(1)
The equation for the line is:
Outcome = +
(- ) × Treatment
‹#›
The Regression Line for a Dichotomous Treatment: A Formal
Approach
Observed outcomes in terms of two points on a line
Profiti = f(1.00) + ei if Pricei = 1.00
Profiti = f(1.50) + ei if Pricei = 1.50
i delineates between different observations, (i
ei is the residual for the observation i.
The residual is the difference between the observed outcome
and the corresponding point on the regression line for a given
observation
ei = Yi – f(xi)
‹#›
Scatterplot of Residuals for Price of $1.00 when f(1.00) = $220
FIRST RESIDUAL IS 20. THIS MEANS THE ACTUAL
PROFIT WE OBSERVE (240) IS 20 HIGHER THAN WHAT
WE OBSERVE (220).

SECOND RESIDUAL IS -20. THIS MEANS THE ACTUAL
WE OBSERVE (220).
THIRD RESIDUAL IS -35. THIS MEANS THE ACTUAL
WE OBSERVE (185).
‹#›
Approach
Residuals for price of $1.00 when f(1.00) = $220
The average residual is [20 + (-20) + (-35)]/3 = -11.67
A choice for f(1.00) is best if it tends to neither overshoot nor
undershoot the observed outcomes. That means, a choice for
f(1.00) is best if the corresponding residuals are on average,
zero.
‹#›

Approach
For the residuals to average zero means:
THE RESIDUALS TO AVERAGE ZERO, BEST CHOICE FOR
f(1.00):
Similarly, when price is $1.50, the best choice is the average of
profits when the price is $1.50 = (205 + 170 + 190)/3 = 188.33
‹#›
The Regression Line for a Multi-Level Treatment: An Intuitive
Approach
Multi-level treatment is a treatment that can be administered in
more than one quantity
HERE, PRICES ARE, $1.00, $1.50, $2.00. PRICE OF $1.00 IS
UNTREATED AND A $0.50 PRICE INCREASE IS THE
TREATMENT.
‹#›
Approach
The approach we used for the dichotomous treatment generally
does not work for a multi-level treatment
The problem is that when three or more points are plotted on a
graph, it is generally the case that they might not fall on the

same line
‹#›
Approach
Line attempting to connect average profits to the following
price levels:
f(1.00) = 208.33
f(1.50) = 188.33
f(2.00) = 160
‹#›
Approach
Using the average outcome to plot the points for each treatment
level generally will result in not being able to connect three
points on a single line when there more than two treatment
levels
f(1.00) = b + m × 1.00 208.33 = b + m × 1.00
f(1.50) = b + m × 1.50 188.33 = b + m × 1.50
f(2.00) = b + m × 2.00 160 = b + m × 2.00
We cannot solve for m and b as there are three equations to
solve but only two unknowns

‹#›
Approach
Rather than plot an “ideal” point for each treatment level and
then solve for the corresponding slope and intercept, try to
directly solve for the slope and intercept of the line believed to
best describe the describes the data
It should not generally overshoot or undershoot the data
Its tendency to over or undershoot the data across specific price
levels should not depend on the price level
‹#›
Two Candidate Lines for Describing Profits and Price Data
‹#›
The Regression Line for a Multi-Level Treatment: A Formal
Approach
For our example, we have three levels and nine points.

Expressing them in terms of intercept and slope:
Profiti = b + m × 1.00 + ei, if Pricei = 1.00
Here i takes on the values one through nine, since there are nine
points. Residuals, ei, are the difference between the observed
profit and the corresponding point on the line for a given
observation.
Ei = Profiti – b – m × Pricei
‹#›
Approach
Applying the same approach used for a dichotomous treatment,
solve for the “best” line by finding a slope and intercept that
makes the residuals average zero for each price point.
THIS AGAIN GIVES US THREE EQUATIONS AND TWO
UNKNOWNS.
‹#›
Approach
Alternative way of defining what makes a line the best to
describe the data. Criteria includes:

It should not generally overshoot or undershoot the data
Its tendency to over or undershoot the data across specific price
levels should not depend on the price level
‹#›
Approach
Translating these criteria in terms of residuals:
The residuals for all data points average to zero
The size of the residuals is not correlated with the treatment
level
Expressing these two criteria in equation form:
‹#›
Approach
The first equation ensures that the residual average zero across
all observations, and the second equation ensures that the size
of the residuals is not related to Price level
Solving these two equations yields:
m = -48.33
b = 258.06
The line that best fits the data, where “best” implies residuals
that average zero and are not correlated with the treatment:
Profit = 258.06 – 48.33 × Price

‹#›
Approach
Simple regression line
The slope is the sample covariance of the treatment and
outcome divided by the sample variance of the treatment
The intercept is the mean value of the outcome minus the slope
times the mean value of the treatment
Y = b + mX
Solving for m and b yields the following formulas for the slope
and intercept of a simple regression line:
m =
b = – m
‹#›
Approach
Applying these generalized formulas for our dichotomo us
price/profit example:
USING THE FORMULAS FOR VARIANCE AND
COVRIANCE:

sCov (Profit, Price) = -3,
sVar (Price) = 0.075,
= 1.25 and = 198.33
Plugging these into our formulas,
m = -3/0.075 = -40, and
b = 198.33 – (-40)1.25= 248.33.
‹#›
Sample Moments and Least Squares
Sample moment
The mean of a function of a random variable(s) for a given
sample
For example, for a sample size 20 that contains information on
salaries, is a sample moment, where Salaryi is the random
variable and the function is defined as f(a) = a3
Ordinary least squares
The process of solving for the slope and intercept that minimize
the sum of the squared residuals
Minb,m =Yi – b – mXi)2
‹#›
Sample Moments and Least Square
Objective function
A function ultimately wished to be maximized or minimized
For ordinary least squares, the objective function is the sum of

squared residuals ()
Least absolute deviations (LAD)
Use the sum of the absolute value of the residuals as the
objective function and solve for the slope and intercept that
minimize it
‹#›
Ordinary Least Square vs Least Absolute Deviation for
Describing a Dataset
LINE A IS CLOSER TO THE OUTLIER, SO IT IS COMING
FROM OLS AND LINE B IS COMING FROM LAD.
‹#›
Regression for Multiple Treatments
CHOLESTEROL LEVEL AND DRUG DOSES FOR 15
INDIVIDUALS.
‹#›
Regression for Multiple Treatments
Single vs. Multiple Treatments
Cholesterol = 235.17 – 0.997 × Drug A
Cholesterol = 205.83 – 0.107 × Drug B

Cholesterol outcome as follows:
Cholesteroli = b + m1DrugAi + m2DrugBi + ei
Expressing the OLS criteria in equation form:
‹#›
Regression Output in Excel for Cholesterol Regressed on Drug
A and Drug B
HERE WE HAVE THE VALUES FOR:
b = 256.20,
m1 = -1.259, AND
m2 = -0.514.
‹#›
Regression Plane for Cholesterol Regressed on Drug A and
Drug B
‹#›
Multiple regression
Solving for a function that best describes the data the implies

the use of OLS (or equivalently, the sample moment equations)
Single regression the process that produces the simple
regression line for a single treatment
Multiple Regression
‹#›
Multiple Regression
For a sample size of N with K treatments, the associated
equations are:
‹#›
What Makes Regression Linear?
Linear regression is the process of fitting a function that is
linear in its parameters to a given dataset
Y = b + m1X1 + m2X2 + … + mKXK
Here {b, m1, …, mK} are the parameters for this function
The use of linear regression does not at all imply construction
of a line to fit the data
Linear regression is linear in the parameters but not necessarily
the treatment(s)
It allows for an unlimited number of possible “shapes” for the
relationship between the outcome and any particular treatment

‹#›

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