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Lecture 12

# PSYC 2002 Lecture Notes - Lecture 12: Simple Linear Regression, Linear Regression, Standard Score

by OC504149

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PSYC 2002 – Regression

Lecture Outline:

- Regression as prediction

- Relationship b/w correlation & regression

- Conceptual basis of regression

- Regression lines on scatterplots

- Multiple regression

Readings:

- Regression – NH Chapter 14

Regression and correlation defined:

- Correlation: measure of degree of relationship b/w 2 variables

- Regression: how well variation in 1 variable can predict variation in another variable

Linear Regression:

- If we know value for X, can we predict what corresponding value of Y will be?

- Ex. Does knowing how many hrs person studies predict final exam grades? Does lots of

study predict higher grades?

Equation for a graphing a line:

-Ŷ = bX + a (aka Y = mX + b)

-X = variable used to do prediction (predictor variable)

-b = slope of line

oSlope of regression line = rise/run

oRise/run = Y distance / X distance

-a = y-intercept

-Ŷ = variable that’s predicted (dependent variable)

- *graph in lecture slide*

Relationship between regression and correlation:

- Regression & correlation related to each other

- Conceptually: if 2 variables related to each other, then knowing value of 1 variable

should allow you to make prediction about value of other variable

- Mathematically: r & Z-scores

Mathematical relationship between correlation and regression:

-ẐY = rxyZX

orxy = coefficient for correlation b/w X & Y

Same “r” we calculated for correlations

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Note: only true for simple linear regression (i.e., Pearson r coefficient isn’t

regression coefficient for multiple regression…)

oẐY = predicted Z score

- Collectively known as prediction model or standardized regression equation

Prediction using rxy & X-scores:

-ẐY = rxyZX

- Prediction using this formula is in terms of Z-scores

oFor given Z-score for X, you can predict Z-score for Y

oRemember that we’re standardizing (using Z-scores) in case X & Y values use

different metrics

Regression calculations: Example *in lecture slide*

Regression calculations:

- Step 1: calculate r

- Step 2: select 2 values of X (i.e., predictor variable) & transform them into their

corresponding Z-scores

- Step 3: insert Z-scores for predictor variable (separately) in standardized regression

equation (multiply by r)

oCalculating 2 separate values of ẐY

- Step 4: take resulting ẐY & transform into Ŷ

oTransforming ẐY “back” into raw (predicted) score of Y

- Step 5: add X & Ŷ coordinates (2 pairs) to scatterplot

Regression calculations: Example

- Step 1: calculate r

oAlready done: rxy = -0.962

- Step 2: select 2 X values to transform to Z values; use ZX = (X-MX)/SDX

oOriginal Z-score formula from way back when…

oX = 1 & X = 20

o1 → (1 – 6.3)/4.9

ZX1 = -1.082

o20 → (20 – 6.3)/4.9

ZX2 = 2.796

- Insert ZX-values into standardized regression equation: ẐY = rxyZX (where rxy = -.0962)

oẐY = (-0.962)(-1.082)

oFor X =1, ẐY = 1.040

oẐY = (-0.962)(2.796)

oFor X =20, ẐY = -2.689

- These ẐY values need to be transformed back into predicted raw scores

- Use Ŷ = [(ẐY)(SDY)] + MY

oŶ1 = [(1.040)(16.907)] + 62.6

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= 80.189

oŶ2 = [(-2.689)(16.907)] + 62.6

= 17.135

- Thus, X-Y pairs required to plot the line are: (1, 80.189), (20, 17.135)

- Recap: how did we get these values?

- Chose 2 values of X (preferably at “opposite” ends of range of X) & converted these

values to Z-scores

oX1 = 1, X2 = 20

oZX = (X-MX)/SDX

- Plugged those Z-scores into standard regression equation & calculated 2 values of ẐY

oẐY = rxyZX

- Converted both values of ẐY back to raw scores:

oŶ = [(ẐY)(SDY)] + MY

- This provides us w/ predicted values of Y (80.189 & 17.135) that we can plot w/ their

corresponding Z values (1 & 20)

Regression Line:

- Once prediction model has been generated, regression line can be added to scatter plot

oRegression line based on X & Y pairs (selected values of X & their corresponding

predicted values of Y, Ŷ) – line can simply be drawn “connecting the dots”

oNeed minimum of two “sample” X values (& corresponding predicted Y values) to

determine line

- *graph in lecture slide*

- Regression line = “line of best fit”

oLine that best represents pattern of relationship b/w X & Y

- Regression line may or may not correspond to actual values; predicted range of values

based on what’s known about actual X & Y values

Simple linear regression equation:

- Can now calculate equation for regression line (given our 2 points)

- Simple linear regression equation, Ŷ = bX + a, can be obtained as 2 parts: slope & Y-

intercept

- Slope can be calculated using X-Ŷ pairs used to draw regression line on scatterplot

Regression Line: b (slope)

- Slope = “rise/run” = Y-distance/X-distance

- Slope of regression line can be calculated by taking ratio of difference b/w 2 X-values &

2 Ŷ-values

Regression Line: a (Y-intercept)

- Same process as finding values of Ŷ but we use X = 0

- Insert X-value for zero (0) into standardized regression equation: ẐY = rxyZX

oUse ZX = (X-MX)/SDX

(0 - 6.3)/4.9

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