PowerPoint Presentation

Simple & Multiple Regression1: Simple Regression- Prediction models1

r = .8168???Regression techniques allow us to do thisSuppose we wanted to predict the weight of a person who was 68in tall?Lets take our scatterplot as a start1

r = .81We use a method of least squares estimation (cue statistical hocus pocus music)And we generate a line through the data so that all deviations (vertical) between the line and the data points are minimized12

r = .81This line will have a certain slopebrings a change in weightA change in heightSLOPEAnd it will have a value on the y-axis for the zero value of the x-axis-234INTERCEPT123

The intercept can be seen more clearly if we redraw the graph with appropriate axes-234lbs1

2

68r = .81From the line, we can predict that an increase in height of 1 inch should be accompanied by a rise in weight of 5.434lbs. We can also find the expected weight for a person of 68in height.135lbsUsing regression to make predictions1234

From this data fileWhere is this in SPSS, and what is this going to look like elsewhere?1

Choose this analysisWhere is this in SPSS, and what is this going to look like elsewhere?1

Specify dependent and independent variablesWhere is this in SPSS, and what is this going to look like elsewhere?1

SPSS output:SLOPEINTERCEPTWhere is this in SPSS, and what is this going to look like elsewhere?1

And how about Excel?Excels regression function can be accessed via the wizard, but it still needs some extra knowledge to get it to work, so Im just going to show you the muggle (non-wizard) way1

1. Select a 2 (columns) by 5 (rows) arrayAnd how about Excel?1

2. Use the linest (linear estimate) function3. The first array is the dependent variable4. The second array is the independent variable5. After 2 commas, true means you want all the statsExcel1

6. Hit [CTRL_SHIFT_ENTER] at end of function NOT enterand heres all the stuffslopeinterceptR2FExcel1

General form of equation:Y = a + bXSLOPEINTERCEPTWeight = -234 + 5.434 (Height)Predicted values of the d.v.values of the i.v. (predictor)The regression equation1234

A note on the equation and errorHere is another general form of the equation from a text book:

Dont be confused by thisits obvious really. Its the error term. Note actual y, rather than predicted y, is on the leftFor an actual value y1

A note on the equation and error

The least squares method used in regression just minimizes the sum of these squared vertical distancese1e2e3e4e5e6e71

234

How good, generally, is the fit?R2Coefficient of determinationStandard error of the estimateThe average size of the error in predicting any value of YThe standard deviation of actual Ys about predicted YsOr, the SD of the es (residuals)Critically related to R21234

56

r = .81More on the SE of estimateAt any point of X, the various Ys are expected to be normally distributed about the regression line123

Height = 63

More on the SE of estimateThat means that you can set up expected margins of error of Y about YE.G. What proportion of Y would fit within 2 standard errors of the estimate???All depends upon key assumptionsHomoscedasticityLinear relationship between X and YY normally distributed about Y

123

Time for a breakKNR 445Regression: Deep stuff - slide 211