EXAMINING RELATIONSHIPS

Residuals and Residual Plots

FACTS ABOUT LEAST SQUARES LINE

Must be clear on Explanatory & Response Variables Switching the variables changes your equation

Line always passes through the point (x-bar,y-bar) This always gives us a point to start w/ or use

during graphing Correlation is closely related to slope

Smaller r = smaller effect of x on predictions r and r2 help define the strength of a straight

line relationship between the variables Higher values = stronger relationship

RESIDUALS Press the Button and I will use Linear Regression to tell your Future!! (or at least something close to

it!!)Just like our friend ZOLTAR we can make predictions using our Line of Best Fit.

However, do we know just how good our predictions are? Would we be willing to put a lot of CASH MONEY down to back them up?

Luckily, we have an indicator in statistics that can help us decide the strength of our predictions AND

tell us if a line is the “Best Fit”.

RESIDUALS

Unless your r value is perfect, your predictions won’t be

A residual is the difference between the actual value and your predicted value

Each value observed value has a residual

The sum of the residuals is always 0 (or really, really close)

Should be… If not, that equation might not be the best fit!

-roundoff error – when earlier values are rounded, the sum may not equal exacty 0

Residual =^

yy

GRAPHING THE LSL ON YOUR SCATTER PLOT

Using the Bone Data, Let’s look at how we get the residuals (and how your calculator does it)

Femur Humerus

38 41

56 63

59 70

64 72

74 84

y = -3.659486682 + 1.196900115x

Plug in all your x values into the equation and get a predicted y-hat

Femur Predicted Humerus (y-hat)

38 -3.659486682 + 1.196900115(38)

56 -3.659486682 + 1.196900115(56)

59

64

74

Femur Residual (y – y-hat)

38 41 – 41.82271769

56 56 – 63.3669976

59

64

74

Now, subtract the PREDICTED value from ACTUAL value.

RESIDUAL PLOT

Scatterplot of the residuals against the explanatory variable (x). Assess the fit of the regression line

Does your plot show the line fits?

Residuals Fit

No pattern Good Fit

Curve Non Linear

Increasing spread

Worse predictions for larger x

Decreasing Spread

Smaller x, worse predictions

o Individual Points w/ Large Residuals = Outliers in y

o Individual Points extreme in x = Influential Points

Why use Residuals?

The residual plot describes how well a

LINEAR model fits our data

RESIDUAL PLOT ON CALCULATOR

Plot the scatterplot of the data Find the least squares equation (LinReg y=a+bx) Put the equation into Y1 and graph it

In L3, You need to get the residuals (quickly)

Go to the top of L3 – 2nd Stat - RESID

Press enter (*Your calculator finds them for you!! YIPPEEE!!)

You have to have STAT: CALC: 8; 1st, before you run the Residuals… You’re calculator has to have an equation to plug into to find the Residuals

Now do a scatterplot with Xlist = L1 and Ylist = L3 (residuals)

The line in the middle is the least squares line.

You can do 1 Variable Stats

your RESID list to find out if the

residual sum is 0.

RESIDUALS ON CALCULATOR (SCREENSHOTS) – BY HAND PRACTICE? Run the GESSEL program

ScatterplotCalc

FunctionRegression

Stats Plot w/ EQ

Residual List Function

Residual Plot

INFLUENTIAL POINT VS. OUTLIER

Outlier – observation that is outside overall pattern (out of whack in the Y direction)

Influential Point – observation that IF removed would dramatically change the result of least squares line and/or predictions (way out in the X direction)

INFLUENTIAL POINT VS. OUTLIER

Let’s Change Child 19’s test score from 121 to 85 and see what happens to the EQ and Graph

ORIGINAL NEW

Notice the minimal change in the

equation and graph… This is an example of

why Child 19 is considered an outlier. An “outlier” in y has a minimal effect on the

equation and subsequent predicted

values.The change here is in the R values.

INFLUENTIAL POINT VS. OUTLIER

Let’s Change Child 18’s test score from 57 to 85 and see what happens to the EQ and Graph

ORIGINAL NEW

Notice the dramatic change in the equation and graph… This is an example of why Child 18 is considered an

influential point. A point in the extreme x can

dramatically effect the position of the least

squares line.

HOMEWORK

Anscombe Discovery#46