Regression

Regression

• What is regression to the mean?

• Suppose the mean temperature in November is 5 degrees

• What’s your best guess for tomorrow’s temperature?

1. exactly 5?

2. warmer than 5?

3. colder than 5?

Regression


• Suppose the mean temperature in November is 5 degrees and today the temperature is 15

• What’s your best guess for tomorrow’s temperature?1. exactly 15 again?

2. exactly 5?

3. warmer than 15?

4. something between 5 and 15?

Regression


• Regression to the mean is the fact that scores tend to be closer to the mean than the values they are paired with

– e.g. Daughters tend to be shorter than mothers if the mothers are taller than the mean and taller than mothers if the mothers are shorter than the mean

– e.g. Parents with high IQs tend to have kids with lower IQs, parents with low IQs tend to have kids with higher IQs

Regression


• The strength of the correlation between two variables tells you the degree to which regression to the mean affects scores

– strong correlation means little regression to the mean

– weak correlation means strong regression to the mean

– no correlation means that one variable has no influence on values of the other - the mean is always your best guess

Regression

• Suppose you measured workload and credit hours for 8 students

Could you predict the number of homework hours from credit hours?

Regression

• Suppose you measured workload and credit hours for 8 students

Your first guess might be to pick the mean number of homework hours which is 12.9

Regression

• Sum of Squares

•Adding up the squared deviation scores gives you a measure of the total error of your estimate

Regression

• Sum of Squares

•ideally you would pick an equation that minimized the sum of the squared deviations

•You would need a line is as close as possible to each point

Regression

• The regression line

•That line is called the regression line

•The sum of squared deviations from it is called the sum of squared error or SSE

Regression

• The regression line

•That line is called the regression line

•its equation is:

€

′ y i = rxy

Sy

Sx

x i + y − rxy

Sy

Sx

x

Regression

€

′ y i = rxy

Sy

Sx

x i + y − rxy

Sy

Sx

x

predicted y

remember: y = ax + b

ax b+

• What happens if you had transformed all the scores to z scores and were trying to predict a z score?

Regression

€

′ y i = rxy

Sy

Sx

x i + y − rxy

Sy

Sx

x

Regression

€

′ y i = rxy

Sy

Sx

x i + y − rxy

Sy

Sx

x

Sy = Sx = 1

€

y = x = 0

€

′ z yi= rxyzxi

So….

but…

• What happens if you had transformed all the scores to z scores and were trying to predict a z score?

The Regression Line

• The regression line is a linear function that generates a y for a given x

The Regression Line


• What should its slope and y-intercept be to be the best predictor?

The Regression Line



• What does best predictor mean? It means least distance between the predicted y and an actual y for a given x

The Regression Line



• What does best predictor mean? It means least distance between the predicted y and an actual y for a given x

• in other words, how much variability is residual after using the correlation to explain the y scores

Mean Square Residual

• Recall that

€

Sy2 =

(y i − y )2∑n


• The variance of Zy is the average squared distance of each point from the x axis (note that the mean of Zy = 0)

Regression

-3.0

0.0

3.0

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0Actual Scores


• Some of the variance in the Zy scores is due to the correlation with x• Some of the variance in the Zy scores is due to other (probably

random) factors

Regression

-3.0

0.0

3.0

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0Actual Scores


• the variance due to other factors is called “residual” because it is “leftover” after fitting a regression line

• The best predictor should minimize this residual variance


€

MSres =(y i − y'i )∑

2

n

MSres is the average squared deviation of the actual scores from the regression line

Minimizing MSres

• the regression line (the best predictor of y) is the line with a slope and y intercept such that MSres is minimized

Minimizing MSres• What will be its y intercept?

– if there was no correlation at all, your best guess for y at any x would be the mean of y

– if there was a strong correlation between x and y, your best guess for the y that matches the mean x would be the mean y

– the mean of Zx is zero so the best guess for the Zy that goes with it will be zero (the mean of the Zy’s)

Minimizing MSres

• In other words, the regression line will predict zero when Zx is zero so the y intercept of the regression line will be zero (only so for Z scores !)

Minimizing MSres

• y intercept is zero

Regression

-3.0

0.0

3.0

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0Actual Scores

Minimizing MSres

• what is the slope?

Regression

-3.0

0.0

3.0

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0Actual Scores

Minimizing MSres

• what is the slope? consider the extremes:• Do the slopes look familiar?

Z scores

-3.0

0.0

3.0

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

Zy = ZxZy’=Zxslope = 1

Z scores

-3.0

0.0

3.0

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

Zy=-ZxZy’=-Zxslope = -1

Z scores

-3.0

0.0

3.0

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

Zy is random with respect to ZxZy’=mean Zy=0slope = 0

Minimizing MSres

• a line (regression of Zy on Zx) that has a slope of rxy and a y intercept of zero minimizes MSres

Predicting raw scores

• we have a regression line in z scores:

• can we predict a raw-score y from a raw-score x?

€

zy = rxyzx


• recall that:

€

zyi=

y i − y

Sy

€

zxi=

x i − x

Sxand


• by substituting we get:

€

y i = rxy

Sy

Sxx i + y − rxy

Sy

Sxx


• by substituting we get:

• note that this is still of the form:

• note that the slope still depends on r and the intercept still depends on the mean of y

€

y i = rxy

Sy

Sxx i + y − rxy

Sy

Sxx

y = ax + b

a

+ b

Interpreting rxy in terms of variance

• Recall that rxy is the slope of the regression line that minimizes MSres


• Recall that rxy is the slope of the regression line that minimizes MSres

€

MSres =(y i − ′ y )2∑

n= Sy− ′ y

2


• MSres can be simplified to:

€

Sy− ′ y 2 = Sy

2(1− rxy2 )


• Thus:

€

rxy2 =

Sy2 − Sy− ′ y

2

Sy2


• Thus:

• So can be thought of as the proportion of original variance accounted for by the regression line

€

rxy2 =

Sy2 − Sy− ′ y

2

Sy2

€

rxy2


Regression Line

Observed y

Predicted y

What % of this distance

is this distance

Mean of y

Subtract this distance


• it follows that 1 - is the proportion of

variance not accounted for by the regression

line - this is the residual variance

€

rxy2


• this can be thought of as a partitioning of variance into the variance accounted for by the regression and the variance unaccounted for

€

Sy2 = S ′ y

2 + Sy− ′ y 2


• this can be thought of as a partitioning of variance into the variance accounted for by the regression and the variance unaccounted for

€

(y i − y )2∑n

=(y '−y )2∑

n+

(y i − ′ y )2∑n


• often written in terms of sums of squares:

• or simply

€

(y i − y )2∑ = (y '−y )2∑ + (y i − ′ y )2∑

SStotal = SSregression + SSresidual

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Regression