7 regression nc

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Linear Regression

Linear re ressionis a statistical rocedure that

uses relationships to predict unknown Yscores

based on the Xscores from a correlated

.

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Predicted YScores

Y The symbol stands for a predicted Y

r

ac s our es pre c on o escore at a correspondingX, based on the

linear relationship that is summarized by the

re ression line

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Linear Regression Line

The linear regression line is the straight linethat summarizes the linear relationship in the

scatter lot b on avera e assin throu h

the center of the Y scores at each X.

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Slope and Intercept

The slope is a number that indicates howslanted the regression line is and the direction

in which it slants.

The Y-intercept is the value ofYat the point

where the regression line intercepts, or

crosses, the Yaxis that is, when Xe uals 0 .

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The Linear Regression Equation

This equation indicates that the predicted Yvalues are equal to the slope (b) times a given

Xvalue and that this roduct then is added to

the Y-intercept (a)

=

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Computing the Slope

The formula for the slope (b) is

22

))(()( YXXYNb

=

Since we usually first compute r, the values of

the elements of this formula alread areknown

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Computing the Y-Intercept

The formula for the Y-intercept (a) is

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Example 1

1 8

2 6

,

calculate the linear

3 6

4 5

regress on equat on.

5 1

6 3

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Plotting the Regression Line

We compute predicted scores of Y from ourX scores using the regression equation and

lot them accordin l .

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Predicted Y alue

Using the linear regression equation fromexamp e 1, eterm ne t e pre cte score

for X= 4.

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Errors in Prediction

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Variance

Y The variance of the Yscores around ise average square erence e ween e

actual Yscores and their corresponding

predicted scores.Y

Ys one way o escr e e average error

when using linear regression to predict Yscores.

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rYY =


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Estimate

The standard error of the estimate is similar tothe standard deviation of the Yscores around

describe the average error when using to Y

predict Yscores.

21 rSSYY

=

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Example 3

X Y

Using the same data set,2 6

3 6

calculate the standard error of

the estimate.4 5

5 1

6 3

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The Stren th of a Relationshi

and Prediction Error

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The Strength of a Relationship

As the strength of the relationshipand theabsolute value ofrincreases, the actual Y

scores, producing less prediction error andY

smaller values of and .2Y

S Y

S

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Relationshi

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Scatterplot of a Weak Relationship

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ssum tions of Linear

Regression

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Assumption 1

The first assum tion of linear re ression is

that the data are homoscedastic

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Homoscedasticity

Homoscedasticity occurs when the Y scores are spread

out to the same de ree at ever X.

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Heteroscedasticity

Heteroscedasticity occurs when the spread inY

isnot equa t roug out t e re at ons p.

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Assumption 2

The second assum tion of linear re ression is

that the Yscores at each X form an

approximately normal distribution

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Scatterplot Showing Normal

Distribution of Y Scores at Each

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The Proportion of Variance

ccounte or

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Accounted For

The ro ortion o variance accounted oris the

proportional improvement in predictions

achieved by using a relationship to predict,

relationship.

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Accounted For

When we do not use the relationship,

we use the overall mean of the Yscores)(Yas everyones predicted Y.

Y

the actual Yscores and the that we

When we do not use the relationshi to

pre c ey go .)( YY

2

YSpredict scores, our error is .

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Pro ortion of Variance

Accounted For

Y

en we o use e re a ons p, we

use the corresponding asetermne y t e near regress onequation as our predicted value

The error here is the difference between

)( YY

predict they got

2S

When we do use the relationship toredict scores our error is

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Accounted For

The computational formula for the proportionof variance in Y that isaccounted for by a

2 .

that the formula for computingr is

2222YYNXXN

r

=

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Not Accounted For

The com utational formula for the ro ortionof variance in Y that is notaccounted for by a

linear relationship with X is2

1 r

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Example 4

X Y

1 8 Using the same data set,2 6

3 6

calculate the proportion of

variance accounted for and

4 5

5 1the proportion of variance

6 3.

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Example 5

A researcher measures how positive a personsmood is and how Participant Mood (X) Creativity (Y)

1 10 7

is, obtaining the2 8 63 9 11

interval scores on 5 5 5

e a e:7 7 4

8 2 5

349 4 6

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Example 5

A) Compute theParticipant Mood (X) Creativity (Y)

1 10 7 s a s c a

summarizes this2 8 6

3 9 11

relationship5 5 56 3 7

7 7 4

8 2 59 4 6

10 1 4

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Example 5

B) What is theParticipant Mood (X) Creativity (Y)

1 10 7 pre c e crea v y

score for anyone2 8 6

3 9 11

scoring 3 onmood?5 5 56 3 7

7 7 4

8 2 59 4 6

10 1 4

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Example 5

C) If your predictionParticipant Mood (X) Creativity (Y)

1 10 7 s n error, w a s

the amount of2 8 6

3 9 11

error you expect tohave?5 5 56 3 7

7 7 4

8 2 59 4 6

10 1 4

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