STA 302/1001-Methods of Data Analysis I
Sections L0101/2001, L0201 & L0301
Shivon Sue-Chee
Module 2
Shivon Sue-Chee Simple Linear Regression 1
Module 2 - Simple Linear Regression
I 2.1. The SLR Model
I 2.2. Estimating regression parameters
I 2.3. Properties of LS estimators
I 2.4. Statistical Assumptions of SLR
I 2.5. SLR In R: Data Example
Shivon Sue-Chee Simple Linear Regression 2
Cartoon of the week
Shivon Sue-Chee Simple Linear Regression 3
2.1. The SLR Model
I What is a linear model?
I Examples of linear and non-linear models
I What is an SLR model?
Shivon Sue-Chee Simple Linear Regression 4
General form of models
General form of a model for Y in terms of three predictors:
Y = f (X1,X2,X3) + e
I f is some unknown function
I e is the error not accounted for in f
I Issue: If f is a smooth, continuous function, then there
are many possibilities for f . Also, we would need infinite
data to estimate f directly.
I A fix: Restrict f to a linear form
Shivon Sue-Chee Simple Linear Regression 5
General form of models
General form of a model for Y in terms of three predictors:
Y = f (X1,X2,X3) + e
I f is some unknown function
I e is the error not accounted for in f
I Issue: If f is a smooth, continuous function, then there
are many possibilities for f . Also, we would need infinite
data to estimate f directly.
I A fix: Restrict f to a linear form
Shivon Sue-Chee Simple Linear Regression 6
General form of models
General form of a model for Y in terms of three predictors:
Y = f (X1,X2,X3) + e
I f is some unknown function
I e is the error not accounted for in f
I Issue: If f is a smooth, continuous function, then there
are many possibilities for f . Also, we would need infinite
data to estimate f directly.
I A fix: Restrict f to a linear form
Shivon Sue-Chee Simple Linear Regression 7
What is a Linear Model?
Definition (Linear Model)
In a linear model for Y , the parameters enter linearly or Y
is linear in terms of the parameters.
Examples of linear models:
I Y = β0 + β1x + e
I Y = β0 + β1 log x + e
I Y = β0 + β1x + β2x2 + e
I Y = β0 + β1 log x1 + β2x2 + β3x1x2 + e
I Y = β0xβ1e
I Y = exp (β0 + β1x + e)
I Tip: apply a suitable transform to Y to see that the
model is linear.
Shivon Sue-Chee Simple Linear Regression 8
What is a Linear Model?
Definition (Linear Model)
In a linear model for Y , the parameters enter linearly or Y
is linear in terms of the parameters.
Examples of linear models:
I Y = β0 + β1x + e
I Y = β0 + β1 log x + e
I Y = β0 + β1x + β2x2 + e
I Y = β0 + β1 log x1 + β2x2 + β3x1x2 + e
I Y = β0xβ1e
I Y = exp (β0 + β1x + e)
I Tip: apply a suitable transform to Y to see that the
model is linear.
Shivon Sue-Chee Simple Linear Regression 9
Examples of Non-Linear Models
I Y = β0 + exp(β1x) + e
I Y = exp (β0 + exp β1x) + e
I Y = β0 + β1xβ2 + e
I Y = β0 + β1x − exp(β2 + β3x) + e
Shivon Sue-Chee Simple Linear Regression 10
Linear and Non-linear Models
I True non-linear models are rare.
I Linear models can handle complex datasets.
I Because predictors can be transformed and combined in
many ways, linear models are very flexible.
I All straight lines are linear models but all linear models
are not just straight lines.
Shivon Sue-Chee Simple Linear Regression 11
Linear and Non-linear Models
I True non-linear models are rare.
I Linear models can handle complex datasets.
I Because predictors can be transformed and combined in
many ways, linear models are very flexible.
I All straight lines are linear models but all linear models
are not just straight lines.
Shivon Sue-Chee Simple Linear Regression 12
Linear and Non-linear Models
I True non-linear models are rare.
I Linear models can handle complex datasets.
I Because predictors can be transformed and combined in
many ways, linear models are very flexible.
I All straight lines are linear models but all linear models
are not just straight lines.
Shivon Sue-Chee Simple Linear Regression 13
Linear and Non-linear Models
I True non-linear models are rare.
I Linear models can handle complex datasets.
I Because predictors can be transformed and combined in
many ways, linear models are very flexible.
I All straight lines are linear models but all linear models
are not just straight lines.
Shivon Sue-Chee Simple Linear Regression 14
Simple Linear Regression (SLR) Models
Y=β0 + β1X+ e
I Y - dependent or response or output variable
I X - independent or explanatory or predictor or input
variable
I β0 - intercept parameter
I β1 - slope parameter
I e - random error/noise, variation in measures that we
cannot account for.
Q: Why is this model ‘simple’? How useful is this simple
model?
Shivon Sue-Chee Simple Linear Regression 15
Simple Linear Regression (SLR) Models
Y=β0 + β1X+ e
I Y - dependent or response or output variable
I X - independent or explanatory or predictor or input
variable
I β0 - intercept parameter
I β1 - slope parameter
I e - random error/noise, variation in measures that we
cannot account for.
Q: Why is this model ‘simple’? How useful is this simple
model?
Shivon Sue-Chee Simple Linear Regression 16
Simple Linear Regression (SLR) Models
Y=β0 + β1X+ e
I Y - dependent or response or output variable
I X - independent or explanatory or predictor or input
variable
I β0 - intercept parameter
I β1 - slope parameter
I e - random error/noise, variation in measures that we
cannot account for.
Q: Why is this model ‘simple’? How useful is this simple
model?
Shivon Sue-Chee Simple Linear Regression 17
Simple Linear Regression (SLR) Models
Y=β0 + β1X+ e
I Y - dependent or response or output variable
I X - independent or explanatory or predictor or input
variable
I β0 - intercept parameter
I β1 - slope parameter
I e - random error/noise, variation in measures that we
cannot account for.
Q: Why is this model ‘simple’? How useful is this simple
model?
Shivon Sue-Chee Simple Linear Regression 18
Simple Linear Regression (SLR) Models
Y=β0 + β1X+ e
I Y - dependent or response or output variable
I X - independent or explanatory or predictor or input
variable
I β0 - intercept parameter
I β1 - slope parameter
I e - random error/noise, variation in measures that we
cannot account for.
Q: Why is this model ‘simple’? How useful is this simple
model?
Shivon Sue-Chee Simple Linear Regression 19
Shivon Sue-Chee Simple Linear Regression 20
Estimating β in SLR
Shivon Sue-Chee Simple Linear Regression 21
2.2. Estimating the regression parameters
I The Least Squares (LS) Method
I The Maximum Likelihood Estimator (MLE)
I Bayesian Approach
Shivon Sue-Chee Simple Linear Regression 22
Fitting an SLR model
MODEL:
Y=β0 + β1X+ e
AIM: Given a specific value of X, that is, X = x , find the
expected value of Y , that is,
E (Y |X = x)
I need estimates of the regression parameters β0, β1
I need to assess the fit
Shivon Sue-Chee Simple Linear Regression 23
Fitting an SLR model
MODEL:
Y=β0 + β1X+ e
AIM: Given a specific value of X, that is, X = x , find the
expected value of Y , that is,
E (Y |X = x)
I need estimates of the regression parameters β0, β1
I need to assess the fit
Shivon Sue-Chee Simple Linear Regression 24
Estimating β in SLR
I Get data (observational or experimental):
I n pairs
I bivariate data:
(x1, y1), (x2, y2), (x3, y3), . . . , (xn, yn)
I Notation:
I Estimators: β̂0, β̂1
I Estimates: b0, b1
Shivon Sue-Chee Simple Linear Regression 25
Geometrical representation of estimating β
(Figure 2.1. Faraway, 2005)
I The response Y is in an n-dimensional space, Y ∈ Rn
I The regression parameters are in a p + 1-dimensional
space, β ∈ Rp+1
I where p is the number of predictors, so p + 1 is the
number of regression parameters; p < n
Shivon Sue-Chee Simple Linear Regression 26
(i) The Least Squares (LS) Method
I Consider
RSS =
n∑
i=1
[yi − (b0 + b1xi)]2
-‘least squares criterion’
I Method: LEAST SQUARES METHOD
-Find the estimators, b0, b1 that minimize the criterion,
RSS
Shivon Sue-Chee Simple Linear Regression 27
The LS Method: Fitted line and Residuals
I Predicted or Fitted Value for each xi :
yˆi = b0 + b1xi
I Residuals:
eˆi = yi − yˆi
Shivon Sue-Chee Simple Linear Regression 28
The LS Method: Why vertical distances?
I Want to predict Y from X and so we want yˆi to be as
close as possible to yi .
I If we minimize the horizontal distances, we will get a
different answer for b0 and b1.
I Regression is not symmetric!
I It matters which variable is dependent and which is
independent.
Shivon Sue-Chee Simple Linear Regression 29
The LS Method: Why vertical distances?
I Want to predict Y from X and so we want yˆi to be as
close as possible to yi .
I If we minimize the horizontal distances, we will get a
different answer for b0 and b1.
I Regression is not symmetric!
I It matters which variable is dependent and which is
independent.
Shivon Sue-Chee Simple Linear Regression 30
The LS Method: Why vertical distances?
I Want to predict Y from X and so we want yˆi to be as
close as possible to yi .
I If we minimize the horizontal distances, we will get a
different answer for b0 and b1.
I Regression is not symmetric!
I It matters which variable is dependent and which is
independent.
Shivon Sue-Chee Simple Linear Regression 31
The LS Method:
Why squared deviations?
I makes no statistical assumptions
I mean square error is the most common way to measure
error in Statistics
I LS estimators have ‘good’ properties
Shivon Sue-Chee Simple Linear Regression 32
The LS Method: Analytical Derivations
I l.s. criterion: RSS =
∑n
i=1[yi − (b0 + b1xi)]2
I using calculus to minimize the criterion.
I get the NORMAL EQUATIONS.
wrt b1 wrt b0
Shivon Sue-Chee Simple Linear Regression 33
(ii) Maximum Likelihood Estimation (MLE)
I Parameter θ, Estimator θ̂MLE
I MLE steps:
1. define the likelihood function as a function of the
parameter(s) θ;
L(θ) = Distribution(Y |θ)
considered a working model of the parameter given the specific
data
2. find the value of the parameter that maximizes the
likelihood function; that is, the estimator that gives the
highest probability density to the observed data
θ̂MLE = arg max
θ
L(θ)
Shivon Sue-Chee Simple Linear Regression 34
(ii) Maximum Likelihood Estimation (MLE)
I Parameter θ, Estimator θ̂MLE
I MLE steps:
1. define the likelihood function as a function of the
parameter(s) θ;
L(θ) = Distribution(Y |θ)
considered a working model of the parameter given the specific
data
2. find the value of the parameter that maximizes the
likelihood function; that is, the estimator that gives the
highest probability density to the observed data
θ̂MLE = arg max
θ
L(θ)
Shivon Sue-Chee Simple Linear Regression 35
MLE Properties
I regularity conditions are needed to derive the asymptotic
distribution of the MLE
I inference follows the frequentist paradigm
I MLE’s have nice properties:
I asymptotically unbiased,
I consistent,
I sufficient,
I have minimum variance,
I invariance principle holds.
Shivon Sue-Chee Simple Linear Regression 36
MLE Example
1. Consider a normal likelihood for Y in terms of the
parameters β, σ2
L(β, σ2) ∼ Nn(xβ, σ2I n)
2. Using calculus we get:
β̂0,MLE =
β̂1,MLE =
σ̂2MLE =
Shivon Sue-Chee Simple Linear Regression 37
(iii) Bayesian Approach to estimating β
I The parameters are considered random- not fixed
constants.
p(β), p(σ2)
1. Hence, the parameters have a prior (ie, before observing the
data) distribution. Priors could be proper or improper.
pi(β, σ2)
2. Assume a likelihood for Y, as a function of the parameters,
L(β, σ2) = Distribution(Y |β, σ2)
3. Derive the posterior distribution of the parameters given the
data,
p(β, σ2|y) ∝ L(β, σ2)× pi(β, σ2)
Shivon Sue-Chee Simple Linear Regression 38
(iii) Bayesian Approach to estimating β
I The parameters are considered random- not fixed
constants.
p(β), p(σ2)
1. Hence, the parameters have a prior (ie, before observing the
data) distribution. Priors could be proper or improper.
pi(β, σ2)
2. Assume a likelihood for Y, as a function of the parameters,
L(β, σ2) = Distribution(Y |β, σ2)
3. Derive the posterior distribution of the parameters given the
data,
p(β, σ2|y) ∝ L(β, σ2)× pi(β, σ2)
Shivon Sue-Chee Simple Linear Regression 39
(iii) Bayesian Approach to estimating β
I The parameters are considered random- not fixed
constants.
p(β), p(σ2)
1. Hence, the parameters have a prior (ie, before observing the
data) distribution. Priors could be proper or improper.
pi(β, σ2)
2. Assume a likelihood for Y, as a function of the parameters,
L(β, σ2) = Distribution(Y |β, σ2)
3. Derive the posterior distribution of the parameters given the
data,
p(β, σ2|y) ∝ L(β, σ2)× pi(β, σ2)
Shivon Sue-Chee Simple Linear Regression 40
Bayesian Approach to estimating β
I Obtain credible (rather than confidence) intervals for β
where the interpretation differs!
I With a credible interval, we speak about the probability
that the unknown parameter falls into the interval.
I often more computationally challenging than LS/ML
approaches
Shivon Sue-Chee Simple Linear Regression 41
Bayesian Approach: Example
1. Choose improper prior
pi(β, σ2) = p(β)× p(σ2) ∝ σ2
2. Assume likelihood of Y is
L(β, σ2) ∼ Nn(xβ, σ2I n)
3. The posterior distribution of β given the data is the kernel
of a (k + 1)- dimensional t distribution.
Results:
I posterior mean results are identical to LS approach, ML
approach under normality
I 100(1− α)% credible intervals yield the same results as
the 100(1− α)% confidence intervals but the
interpretations are different.
Shivon Sue-Chee Simple Linear Regression 42
Bayesian Approach: Example
1. Choose improper prior
pi(β, σ2) = p(β)× p(σ2) ∝ σ2
2. Assume likelihood of Y is
L(β, σ2) ∼ Nn(xβ, σ2I n)
3. The posterior distribution of β given the data is the kernel
of a (k + 1)- dimensional t distribution.
Results:
I posterior mean results are identical to LS approach, ML
approach under normality
I 100(1− α)% credible intervals yield the same results as
the 100(1− α)% confidence intervals but the
interpretations are different.
Shivon Sue-Chee Simple Linear Regression 43
Properties of LS Estimators
Shivon Sue-Chee Simple Linear Regression 44
2.3. Properties of LS estimators
I Properties of the fitted line
I Properties of regression parameter estimators
I Gauss Markov Theorem
Shivon Sue-Chee Simple Linear Regression 45
Least Squares Regression Parameter Estimates
I Intercept parameter estimate
b0 = y¯ − b1x¯ (2.3)
I Slope parameter estimate
b1 =
∑n
i=1 xiyi − nxy∑n
i=1 x
2 − nx¯2 =
∑n
i=1(xi − x¯)(yi − y¯)∑n
i=1(xi − x¯)2
=
SXY
SXX
(2.4)
Exercise: Show 2.4.
Shivon Sue-Chee Simple Linear Regression 46
Showing Equation 2.4 (SJS)
Shivon Sue-Chee Simple Linear Regression 47
Interpreting Regression Parameter Estimates
I Slope, b1: When x changes by 1 unit, the corresponding
average change in y is the slope.
I Intercept, b0: The average value of y when x = 0.
(No practical interpretation unless 0 is within the range of
the predictor (x) values.)
Shivon Sue-Chee Simple Linear Regression 48
Properties of Fitted LS Regression Line
I Fitted Line:
yˆ = b0 + b1x
Show.
1. The Average of the Residuals is always 0.
n∑
i=1
eˆi ≡ 0
Shivon Sue-Chee Simple Linear Regression 49
Properties of Fitted Regression Line
2. The Sum of Squares of Residuals is NOT 0; unless the fit to
the data is perfect!
RSS =
n∑
i=1
eˆ2i 6= 0
Shivon Sue-Chee Simple Linear Regression 50
Properties of Fitted LS Regression Line
3.
∑n
i=1 eˆixi = 0
4.
∑n
i=1 eˆi yˆi = 0
5.
∑n
i=1 yˆi =
∑n
i=1 yi
Shivon Sue-Chee Simple Linear Regression 51
Gauss-Markov Theorem
Theorem (Gauss-Markov Theorem)
Under the conditions of the simple linear regression model, the
least-squares parameter estimators are BLUE (“Best Linear
Unbiased Estimators”).
I parameter, θ; estimator, θˆ
I Unbiased, E (θˆ) = θ
- i.e., does not overestimate or underestimate
systematically
I Linear- linear in the parameters
I “Best”- obtain minimum variance among all unbiased
linear estimators
Shivon Sue-Chee Simple Linear Regression 52
Rules of expectation
I E(a) = a, a ∈ R
I E(aY ) = aE(Y )
I E(X ± Y ) = E(X )± E(Y )
I E(XY ) = E(X )E(Y ), if X and Y are independent
I Tower rule: E(Y ) = E[E(Y |X )]
Shivon Sue-Chee Simple Linear Regression 53
Properties of Slope Estimator: Expectation
Recall:
b1 =
∑n
i=1 xiyi − nx¯ y¯∑n
i=1 x
2 − nx¯2 =
∑n
i=1(xi − x¯)(yi − y¯)∑n
i=1(xi − x¯)2
=
SXY
SXX
(2.4)
Since
∑n
i=1(xi − x¯) = 0,
n∑
i=1
(xi−x¯)(yi−y¯) =
n∑
i=1
(xi−x¯)yi−y¯
n∑
i=1
(xi−x¯) =
n∑
i=1
(xi−x¯)yi
I Let ci =
xi − x¯
SXX
.
I Then rewrite b1 as
b1 =
∑n
i=1 ciyi
Shivon Sue-Chee Simple Linear Regression 54
Properties of Slope Estimator: Expectation
I Treat X ’s as fixed
I Mean of slope estimate, b1
E (b1|X ) = E
[
n∑
i=1
ciyi |X = xi
]
Shivon Sue-Chee Simple Linear Regression 55
Properties of Intercept Estimator: Expectation
I Recall: b0 = y¯ − b1x¯
I Mean of intercept estimate, b0
E (b0|X ) = E [(y¯ − b1x¯)|X = xi ]
Shivon Sue-Chee Simple Linear Regression 56
Variance and Covariance
I V(a) = 0, a ∈ R
I V(aY ) = a2V(Y )
I Cov(X ,Y ) = E{(X − E(X ))(Y − E(Y ))} =
E(XY )− E(X )E(Y )
I Cov(Y ,Y ) = V(Y )
I V(Y ) = V[E(Y |X )] + E[V(Y |X )]
I V(X ± Y ) = V(X ) + V(Y )± 2Cov(X ,Y )
I Cov(X ,Y ) = 0, if X and Y are independent
I Cov(aX + bY , cU + dW ) = acCov(X ,U) +
adCov(X ,W ) + bcCov(Y ,U) + bdCov(Y ,W )
I Correlation: ρXY =
Cov(X ,Y )√
V(X )V(Y )
Shivon Sue-Chee Simple Linear Regression 57
Properties of Slope Estimator: Variance
I Variance of slope estimate, b1
Var(b1|X ) = Var
[
n∑
i=1
ciyi |X = xi
]
Shivon Sue-Chee Simple Linear Regression 58
Properties of Intercept Estimator: Variance
I Variance of intercept estimate, b0
Var(b0|X ) = Var [(y¯ − b1x¯)|X = xi ]
Shivon Sue-Chee Simple Linear Regression 59
Statistical Assumptions of SLR
Shivon Sue-Chee Simple Linear Regression 60
2.4. Statistical Assumptions of SLR
I SLR Assumptions
I Estimating σ2
I Sampling distributions of slope and intercept
estimators
Shivon Sue-Chee Simple Linear Regression 61
SLR Assumptions
1. We assumed that Y is related to x by the SLR model
Yi = β0 + β1xi + ei , i = 1, . . . , n
or
E (Y |X = xi) = β0 + β1xi .
In other words, the linear model is appropriate.
And the following three Gauss-Markov conditions :
2. The errors e1, e2, . . . , en are have mean of 0, i.e.,
E (ei) = 0.
3. The errors e1, e2, . . . , en have a common variance σ2, i.e.,
Var(ei) = σ
2. The variation is the same for all
observations, i.e., homoscedastic.
4. The errors e1, e2, . . . , en are uncorrelated, i.e.
Cov(ei , ej) = 0, i 6= j .
Shivon Sue-Chee Simple Linear Regression 62
SLR Assumptions
1. We assumed that Y is related to x by the SLR model
Yi = β0 + β1xi + ei , i = 1, . . . , n
or
E (Y |X = xi) = β0 + β1xi .
In other words, the linear model is appropriate.
And the following three Gauss-Markov conditions :
2. The errors e1, e2, . . . , en are have mean of 0, i.e.,
E (ei) = 0.
3. The errors e1, e2, . . . , en have a common variance σ2, i.e.,
Var(ei) = σ
2. The variation is the same for all
observations, i.e., homoscedastic.
4. The errors e1, e2, . . . , en are uncorrelated, i.e.
Cov(ei , ej) = 0, i 6= j .
Shivon Sue-Chee Simple Linear Regression 63
SLR Assumptions
1. We assumed that Y is related to x by the SLR model
Yi = β0 + β1xi + ei , i = 1, . . . , n
or
E (Y |X = xi) = β0 + β1xi .
In other words, the linear model is appropriate.
And the following three Gauss-Markov conditions :
2. The errors e1, e2, . . . , en are have mean of 0, i.e.,
E (ei) = 0.
3. The errors e1, e2, . . . , en have a common variance σ2, i.e.,
Var(ei) = σ
2. The variation is the same for all
observations, i.e., homoscedastic.
4. The errors e1, e2, . . . , en are uncorrelated, i.e.
Cov(ei , ej) = 0, i 6= j .
Shivon Sue-Chee Simple Linear Regression 64
SLR Assumptions
1. We assumed that Y is related to x by the SLR model
Yi = β0 + β1xi + ei , i = 1, . . . , n
or
E (Y |X = xi) = β0 + β1xi .
In other words, the linear model is appropriate.
And the following three Gauss-Markov conditions :
2. The errors e1, e2, . . . , en are have mean of 0, i.e.,
E (ei) = 0.
3. The errors e1, e2, . . . , en have a common variance σ2, i.e.,
Var(ei) = σ
2. The variation is the same for all
observations, i.e., homoscedastic.
4. The errors e1, e2, . . . , en are uncorrelated, i.e.
Cov(ei , ej) = 0, i 6= j .
Shivon Sue-Chee Simple Linear Regression 65
Estimating σ2- variance of the random error term
I The random error ei has mean 0 and variance σ2.
I The variance σ2 is another parameter of the SLR model.
I Aim: estimate σ2
I Why: to measure the variability of our estimates of Y ,
carry out inference on our model
Shivon Sue-Chee Simple Linear Regression 66
Estimating σ2- variance of the random error term
I Notice that:
ei = Yi − (β0 + β1xi) = Yi − unknown regression line at xi
I Replacing β0 and β1 by their respective least squares
estimates, we estimate the errors by
eˆi = Y − (b0 + b1xi) = Yi − estimated regression line at xi
I Using the estimated errors, we can show that an unbiased
estimate of σ2 is
S2 =
∑n
i=1 eˆ
2
i
n − 2 =
RSS
n − 2
Shivon Sue-Chee Simple Linear Regression 67
Statistical assumption for inference
I In order to make inferences, we need one more
assumption about the errors, ei ’s
I Assume: The errors are Normally distributed, i.e.,
ei ∼ N (0, σ2)
or
e ∼ Nn(0, σ2I n)
Shivon Sue-Chee Simple Linear Regression 68
Statistical assumption for inference
Implications:
1. The normality assumption implies that the errors are
independent (since they are uncorrelated).
2. Since yi = β0 + β1xi + ei , i = 1, . . . , n, Yi |xi is normally
distributed.
3. The LS estimates of β0 and β1 are equivalent to their
MLE’s.
Shivon Sue-Chee Simple Linear Regression 69
Normal Error Regression Model
Shivon Sue-Chee Simple Linear Regression 70
Sampling distributions of Slope and Intercept
Estimators
I Slope: Since b1 =
∑n
i=1 ciyi is a linear combination of
the yi ’s, b1|x is also normally distributed, i.e.,
βˆ1 ∼ N
(
β1,
σ2
SXX
)
I Intercept: Since b1|X is normally distributed, y¯ is
normally distributed and b0/x is a linear
combination of b1|X and y¯ , we have that
βˆ0 ∼ N
[
β0, σ
2
(1
n
+
x¯2
SXX
)]
Shivon Sue-Chee Simple Linear Regression 71
SLR in R:
Old Faithful data models
Shivon Sue-Chee Simple Linear Regression 72

欢迎咨询51作业君