Roger B. Hammer Assistant Professor Department of Sociology Oregon State University Conducting...

Roger B. HammerAssistant Professor

Department of SociologyOregon State University

Conducting Social ResearchConducting Social Research

Statistical Principles and Statistical Principles and An Overview of An Overview of

Regression AnalysisRegression AnalysisUnivariate, Bivariate, and Univariate, Bivariate, and

Multivariate StatisticsMultivariate Statistics


Basic Notation Basic Notation Y A random variable (data vector) that we want A random variable (data vector) that we want

to model.to model.

iY The The iithth observation in our data vector. observation in our data vector.

5

8,7,6,5,4

2

Y

Y

Notation Notation: It varies, so be flexible. Notation Notation: It varies, so be flexible.


Basic Notation Basic Notation

iY The observed value of The observed value of iithth observation. observation.

Y Y-Bar is the mean of Y.Y-Bar is the mean of Y.

iY Y-Hat is the estimated or predicted Y-Hat is the estimated or predicted

value of value of iithth observation. observation.

Random VariableRandom Variable• A variableA variable whose numerical value whose numerical value

is determined by chance, the is determined by chance, the outcome of a random phenomenon.outcome of a random phenomenon.• DiscreteDiscrete has a countable number has a countable number

of values. of values. • ContinuousContinuous can take on any value can take on any value

in an interval.in an interval.


Is “statistical anxiety” continuous or discrete?Is “statistical anxiety” continuous or discrete?

ProbabilityProbability• ProbabilityProbability is the likelihood or chance that is the likelihood or chance that

something (an event) is the case or will happen.* something (an event) is the case or will happen.* • The probability of an event is represented by a real The probability of an event is represented by a real

number in the range from 0 to 1.*number in the range from 0 to 1.*• An impossible event has a probability of 0, and a An impossible event has a probability of 0, and a

certain event has a probability of 1.*certain event has a probability of 1.*


*Wikipedia*Wikipedia **Studenmund**Studenmund

**

****P[X]

P(A), p(A) or Pr(A)

Probability DistributionProbability Distribution• Assigns probabilities to the possible Assigns probabilities to the possible

values of a discrete variable.values of a discrete variable.


P[X] + P[Not X] = 1

P[Not X] = 1 - P[X] In the Statistical Anxiety Survey data, what In the Statistical Anxiety Survey data, what

is the probability of having taken a is the probability of having taken a previous statistics course? previous statistics course?

Of not having taken one?Of not having taken one?

Conducting Social Research

Normal (Gaussian) DistributionNormal (Gaussian) DistributionThe Bell CurveThe Bell Curve


Law of Large NumbersLaw of Large Numbers• The first theorem of probability that The first theorem of probability that

describes the long-term stability of a describes the long-term stability of a random variable. random variable.

• Given a sample of independent and Given a sample of independent and identically distributed (iid) random identically distributed (iid) random variables with a finite population mean, variables with a finite population mean, the average of these observations will the average of these observations will eventually approach and stay close to eventually approach and stay close to the population mean.the population mean.


The Central Limit TheoremThe Central Limit Theorem• The second theorem of probability that The second theorem of probability that

describes the distribution of a random describes the distribution of a random variable. variable.

• Given a sample of independent and Given a sample of independent and identically distributed (iid) random identically distributed (iid) random variables with a finite, nonzero standard variables with a finite, nonzero standard deviation, the probability distribution deviation, the probability distribution approaches the normal distribution as approaches the normal distribution as the sample size increases.the sample size increases.

SamplingSampling• PopulationPopulation is the entire is the entire

group of items of interest.group of items of interest.• SampleSample is the observed is the observed

part of the population.part of the population.


Is the Statistical Anxiety Survey data sample Is the Statistical Anxiety Survey data sample or population based?or population based?

Statistical InferenceStatistical Inference• The use of a sample to draw The use of a sample to draw

conclusions about the population conclusions about the population from which the sample came.from which the sample came.

• Inference is necessary because it is Inference is necessary because it is often impractical to “scrutinize” the often impractical to “scrutinize” the entire population.entire population.


Are medical blood tests based on inference?Are medical blood tests based on inference?

Is the U.S. Census based on inference?Is the U.S. Census based on inference?

Random SamplingRandom Sampling• The use of a sample to draw The use of a sample to draw

conclusions about the population conclusions about the population from which the sample came.from which the sample came.

• Inference is necessary because it is Inference is necessary because it is often impractical to “scrutinize” the often impractical to “scrutinize” the entire population.entire population.


Are medical blood tests based on inference?Are medical blood tests based on inference?

Is the U.S. Census based on inference?Is the U.S. Census based on inference?

Selection BiasSelection Bias• The exclusion or under-The exclusion or under-

representation of certain types of representation of certain types of respondents/observations in a respondents/observations in a sample, resulting in a non-sample, resulting in a non-representative sample.representative sample.


Can you give an example of selection bias Can you give an example of selection bias highlighted recently in the media?highlighted recently in the media?

Is the Statistical Anxiety Survey data sample Is the Statistical Anxiety Survey data sample biased? Why or Why not?biased? Why or Why not?

The Expected Value of a The Expected Value of a Random VariableRandom Variable


][PX

][E

i ii

X

X

• A weighted average of all the A weighted average of all the possible values of the random possible values of the random variable (population mean).variable (population mean).

Notation Notation: The italics don’t exactly conform to Studenmund. Remember to be flexible. Notation Notation: The italics don’t exactly conform to Studenmund. Remember to be flexible.

The Variance of a The Variance of a Random VariableRandom Variable


][P)(

])[(E2

22

ii

iXX

X

• The extent to which the values may The extent to which the values may differ from the expected value.differ from the expected value.

• The expected value of the difference.The expected value of the difference.

Similarity of Mean and Similarity of Mean and Variance FormulasVariance Formulas


• Substitution of the squared difference Substitution of the squared difference for the value.for the value.

][P)(

][P22

ii

i

ii

i

XX

XX

The Standard Deviation of The Standard Deviation of a Random Variablea Random Variable


][P)(

])[(E2

2

ii

iXX

X

• The square root of the variance.The square root of the variance.• Absolute value of the difference.Absolute value of the difference.• Residuals.Residuals.

Population Parameters and Population Parameters and Sample StatisticsSample Statistics

ConceptConceptSample Statistic Population Parameter

Mean

Variance

Standard Deviation

Y2ys

ys

]Y[E]Y[2

y Var]Y[y Var


We have obtained a sample of 40 housing sales We have obtained a sample of 40 housing sales that took place somewhere in some year. The that took place somewhere in some year. The data contains two variables, price (in $’s) and size data contains two variables, price (in $’s) and size (total above grade finished area in feet(total above grade finished area in feet22).).


Sample Statistics ExampleSample Statistics Example

Do you think that price and size would be Do you think that price and size would be related to each other?related to each other?

Would one “cause” the other?Would one “cause” the other?

Which variable would you consider to be Which variable would you consider to be independent (X) and which dependent (Y)? independent (X) and which dependent (Y)?

Why?Why?


Price and SizePrice and Size

• X=X= Size and Size and Y =Y = Price Price • For a buyer the price that they are willing to For a buyer the price that they are willing to

pay is a function of the size of the house, along pay is a function of the size of the house, along with other factors. with other factors.

• X=X= Price and Price and Y =Y = Size Size• For a builder the price that they want to receive For a builder the price that they want to receive

for a home will determine its size, along with for a home will determine its size, along with other factors.other factors.


Independent and Independent and Dependent VariablesDependent Variables


Univariate StatisticsUnivariate Statistics

The Sample Mean of The Sample Mean of PricePriceConducting Social Research

n/Y...YYYY n321

030,87$

40/200,481,3$

n/YYn

1ii

If we drew a second sample of 40 housing sales If we drew a second sample of 40 housing sales would the mean be exactly the same as the would the mean be exactly the same as the mean of the first sample?mean of the first sample?

Is the sample mean exactly the same as the Is the sample mean exactly the same as the population mean?population mean?


Population Mean and Population Mean and Sample MeansSample Means


The expectation of the The expectation of the Sample MeansSample Means

• The Law of Large numbers.The Law of Large numbers.

][E][E XX

• The Central Limit Theorem.The Central Limit Theorem.

),(][E 2NX

The Sample Mean of The Sample Mean of SizeSize

n/X...XXXX n321

427,4

40/097,177

n/XXn

1ii


The Sum of the DeviationsThe Sum of the DeviationsThe Zero-sum PropertyThe Zero-sum Property

0)XX()XX(E ii

0)YY()YY(E ii


The Sum of the Squared The Sum of the Squared DeviationsDeviations

Total Sum of SquaresTotal Sum of Squares

9405,415,59)XX( 2i

84,000$114,245,0 )YY( 2i


The Sample VarianceThe Sample Variance

10,395,271

9/39405,415,59

)n/()XX(s iX

122

128361929

122

,,2,

4,000/39114,245,08

)n/()YY(s iY


Sample Standard DeviationSample Standard Deviation

3,224ss 2XX

123,54$ss 2YY



Bivariate StatisticsBivariate Statistics(Skipping Ahead to Chapter 2)(Skipping Ahead to Chapter 2)

Covariance of Covariance of XX and and YY

972,356,731

/39922,921,7606,

)1n/()YY)(XX(s iXY


Covariance of Covariance of YY and and YYis the Variance ofis the Variance of Y Y

128,361,9292,

4,000/39114,245,08

s

)1n/()YY)(YY(s2Y

iiYY


Correlation of Correlation of XX and and YY

9934.

4,1245*3,224

972,356,731

ss

sr

yx

xy


Regression AnalysisRegression Analysis• Econometricians use regression analysis to make Econometricians use regression analysis to make

quantitative estimates of economic relationships quantitative estimates of economic relationships that previously have been completely theoretical in that previously have been completely theoretical in nature.nature.

• Sociologists use regression analysis to make Sociologists use regression analysis to make quantitative estimates of social relationships that quantitative estimates of social relationships that previously have been completely theoretical in previously have been completely theoretical in nature.nature.

• Political scientists use regression analysis to make Political scientists use regression analysis to make quantitative estimates of political relationships that quantitative estimates of political relationships that previously have been completely theoretical in previously have been completely theoretical in naturenature


The Basic (Theoretical) The Basic (Theoretical) Linear ModelLinear Model

)size(fprice.)x(fy g.e

Xxf10

)( XY

10

• β0 is the Y-intercept, the point at which the regression line crosses the vertical axis.

• β1 is the slope of the regression line, a 1 unit change in Xi results in a β1 unit change in Yi.


Change in the Expected Value of Change in the Expected Value of YY

iiXYE

10][

Other determinants of Other determinants of YY

iiiiiXYYEY

10][

Change in the Observed Value of Change in the Observed Value of YY

iiiiiXYEY

10][


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Roger B. Hammer Assistant Professor Department of Sociology Oregon State University Conducting...

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