# Linear regression models

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```					Linear regression models
Simple Linear Regression
History
• Developed by Sir Francis Galton (1822-
1911) in his article “Regression towards
mediocrity in hereditary structure”
Purposes:
• To describe the linear relationship between two
continuous variables, the response variable (y-
axis) and a single predictor variable (x-axis)
• To determine how much of the variation in Y can
be explained by the linear relationship with X
and how much of this relationship remains
unexplained
• To predict new values of Y from new values of X
The linear regression model is:

• Xi and Yi are paired observations (i = 1 to n)
• β0 = population intercept (when Xi =0)
• β1 = population slope (measures the change in Yi
per unit change in Xi)
• εi = the random or unexplained error associated
with the i th observation. The εi are assumed to be
independent and distributed as N(0, σ2).
Linear relationship

ß1
Y

1.0

ß0

X
Linear models approximate non-linear functions
over a limited domain

extrapolation   interpolation   extrapolation
• For a given value of X, the sampled Y
values are independent with normally
distributed errors:   Y = β + β *X + ε
i   o       1   i   i
ε ~ N(0,σ2) à E(εi) = 0
E(Yi ) = βo + β1*Xi

Y

E(Y2)

E(Y1)

X
X1          X2
Fitting data to a linear model:

Yi
Yi – Ŷi = εi (residual)
Ŷi

Xi
The residual

The residual sum of squares
Estimating Regression Parameters
• The “best fit” estimates for the regression
population parameters (β0 and β1) are the
values that minimize the residual sum of
squares (SSresidual) between each
observed value and the predicted value of
the model:
Sum of squares

Sum of cross products
Least-squares parameter estimates

where
Sample variance of X:

Sample covariance:
Solving for the intercept:

Thus, our estimated regression
equation is:
Hypothesis Tests with Regression
• Null hypothesis is that there is no linear
relationship between X and Y:

H 0: β 1 = 0 à Y i = β 0 + ε i

H A: β 1 ≠ 0 à Y i = β 0 + β 1 X i + ε i

• We can use an F-ratio (i.e., the ratio of
variances) to test these hypotheses
Variance of the error of regression:

NOTE: this is also referred to as residual
variance, mean squared error (MSE) or
residual mean square (MSresidual)
Mean square of regression:

The F-ratio is: (MSRegression)/(MSResidual)

This ratio follows the F-distribution with (1, n
-2) degrees of freedom
Variance components and
Coefficient of determination
Coefficient of determination
ANOVA table for regression
Source       Degrees    Sum of squares   Mean     Expected      F
of freedom                  square   mean square   ratio

Regression       1

Residual        n-2

Total           n-1
Product-moment correlation
coefficient
Parametric Confidence Intervals
•   If we assume our parameter of interest has a particular sampling
distribution and we have estimated its expected value and variance,
we can construct a confidence interval for a given percentile.
•   Example: if we assume Y is a normal random variable with unknown
mean μ and variance σ2, then               is distributed as a
standard normal variable. But, since we don’t know σ, we must
divide by the standard error instead:              , giving us a t-
distribution with (n-1) degrees of freedom.
•   The 100(1-α)% confidence interval for μ is then given by:

•   IMPORTANT: this does not mean “There is a 100(1-α)% chance
that the true population mean μ occurs inside this interval.” It
means that if we were to repeatedly sample the population in
the same way, 100(1-α)% of the confidence intervals would
contain the true population mean μ.
Publication form of ANOVA table
for regression
Sum of              Mean
Source        Squares   df       Square    F       Sig.
Regression     11.479        1   11.479   21.044   0.00035

Residual
8.182   15         .545

Total          19.661   16
Variance of estimated intercept
Variance of the slope estimator
Variance of the fitted value
Variance of the predicted value
(Ỹ):
Regression
Assumptions of regression
• The linear model correctly describes the
functional relationship between X and Y
• The X variable is measured without error
• For a given value of X, the sampled Y
values are independent with normally
distributed errors
• Variances are constant along the
regression line
Residual plot for species-area
relationship

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 views: 0 posted: 6/9/2014 language: English pages: 31