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Simple Linear Regression
Saturday, February 13, 2016

1:19 PM

Probabilistic Models

• If we were to construct a model that hypothesized an exact relationship between varia
deterministic model
○ I
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y = 15x
○ Implies that y can always be determined exactly when the value of x is unknown
○ There is no allowance for error in this prediction
• The probabilistic model includes both a deterministic component and a random error c
○ Y = Deterministic Component + Random Error
○ We always assume that the mean value of the random error equals 0
○ E(y) = Deterministic Component
§ E(y) is the mean value of y
• A First-Order (Straight-Line) Probabilistic Model: y = ℬ0 + ℬ1(x) + ⍷
○ Y = Dependent or response variable (variable to be modeled)
○ X = Independent or predictor variable (variable used as a predictor of y)
§ E(y) = ℬ0 + ℬ1(x) = Deterministic component
○ ⍷ = Random error component
○ ℬ0 = y-intercept of the line, that is, the point at which the line intercepts or cuts
○ ℬ1 = Slope of the line, that is, the change (amount of increase or decrease) in the
for every 1-unit increase in x
§ A positive slope implies that E(y) increases by the amount ℬ1 for each unit
§ A negative slope implies that E(y) decreases by the amount ℬ1
• In the probabilistic model, the deterministic component is referred to as the line of me
is equal to the straight-line component of the model
○ E(y) = ℬ0 + ℬ1(x)
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E(y) = ℬ0 + ℬ1(x)
2
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Helpful to plot the sample data in a scatterplot
b
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Specify the probability distribution of the random error term and estimate the standar
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When satisfied that the model is useful, use it for prediction, estimation, and other pur

Fitting the Model: the Least Squares Approach

• One way to decide quantitatively hoe well a straight line fits a set of data is to note the
deviate from the line

ables, it would be called a

n

component

through the y-axis
e deterministic component of y

t increase in x

eans because the mean of y, E(y),

the independent variable x

rd deviation of this distribution

rposes

e extent to which the data points

3
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Statistically evaluate the usefulness of the model
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