E XAMINING R ELATIONSHIPS Residuals and Residual Plots

F ACTS A BOUT L EAST S QUARES L INE 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 r 2 help define the strength of a straight line relationship between the variables Higher values = stronger relationship

R ESIDUALS 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”.

R ESIDUALS 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 =

G RAPHING THE LSL ON YOUR S CATTER P LOT Using the Bone Data, Let’s look at how we get the residuals (and how your calculator does it) FemurHumerus y = x Plug in all your x values into the equation and get a predicted y-hat FemurPredicted Humerus (y-hat) ( 38 ) ( 56 ) FemurResidual (y – y-hat) 3841 – – Now, subtract the PREDICTED value from ACTUAL value.

R ESIDUAL P LOT Scatterplot of the residuals against the explanatory variable (x). Assess the fit of the regression line Does your plot show the line fits? ResidualsFit No patternGood Fit CurveNon Linear Increasing spread Worse predictions for larger x Decreasing Spread Smaller x, worse predictions oIndividual Points w/ Large Residuals = Outliers in y oIndividual Points extreme in x = Influential Points Why use Residuals? The residual plot describes how well a LINEAR model fits our data

R ESIDUAL P LOT ON C ALCULATOR 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 – 2 nd Stat - RESID Press enter (*Your calculator finds them for you!! YIPPEEE!!) You have to have STAT: CALC: 8; 1 st, 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.

R ESIDUALS O N C ALCULATOR (S CREENSHOTS ) – B Y HAND PRACTICE ? Run the GESSEL program Scatterplot Calc Function Regression Stats Plot w/ EQ Residual List Function Residual Plot

I NFLUENTIAL P OINT VS. O UTLIER 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)

I NFLUENTIAL P OINT VS. O UTLIER Let’s Change Child 19’s test score from 121 to 85 and see what happens to the EQ and Graph ORIGINALNEW 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.

I NFLUENTIAL P OINT VS. O UTLIER Let’s Change Child 18’s test score from 57 to 85 and see what happens to the EQ and Graph ORIGINALNEW 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.

H OMEWORK Anscombe Discovery #46