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Everyday is a new beginning in life. Every moment is a time for self

vigilance.

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Simple Linear Regression

Regression modelGoodness of fitModel diagnosis

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Humm… How long is my

armspan?

Goal: to predict the length of Armspan for a given Height

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Armspan DataHEIGHT ARMSPAN68.75 64.2575.75 70.2545.75 43.0066.75 66.2566.50 66.7572.25 71.2548.25 47.25…75.50 70.0075.00 77.2564.00 65.2568.50 67.50

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Review: Math Equation for a Line

Y: the response variable X: the explanatory variable

X

Y YX

}

} 1

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Regression Model

The regression line models the relationship between X and Y on average.– Population regression line – Least squared regression line

The math equation of a regression line is called regression equation.

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The Predicted Y Value

We use the regression line to estimate the average Y value for a specified X value and use this Y value to predict what Y value we might observe at this X value in the near future.

This predicted Y value, denoted as and pronounced as “y hat,” is the Y value on the regression line. So,

XY ˆˆˆ

Y

Regression equation

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The Usage of Regression Equation

Predict the value of Y for a given X valueEg. Wish to predict a lady’s weight by her height.** What is X? Y?** Suppose are estimated as -205 and 5: ** For ladies with HT of 60”, their WT will be

predicted as x60=95 pounds, the (estimated) average WT of all ladies with HT of 60’’.

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• The predicted WT of a given HT

• The predicted armspan of a given height

Examples of the Predicted Y

XY 5205ˆ

XY 04.173.3ˆ

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The Limitation of the Regression Equation

The regression equation cannot be used to predict Y value for the X values which are (far) beyond the range in which data are observed.

Eg. Given HT of 40”, the regression equation will give us WT of -205+5x40 = -5 pounds!!

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The Unpredicted Part

The value is the part the regression equation (model) cannot catch, and it is called “residual,” denoted as e, an estimate of “error” at this observation

YY ˆ

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residual {

Least Square Method

The regression line is the line which minimizes the sum of squares of residuals (SSE) and so the formulas for intercept and slope on the regression line are:

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n

ii

n

iii

xx

yyxx

1

2

1

)(

))(( xy ˆˆ

Inference for Regression Slope

Standard error of

Confidence interval

Hypothesis test

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Goodness of Fit

For each observation: residuals For the whole data set: the coefficient of

determination R2, which measures the proportion of variability in Y explained by the model (the linear regression of Y on X);

For simple linear regression (only one predictor) R2 = r2

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Model Assumptions and Diagnosis

1. Independent observations2. Y|X=x follows a normal distribution with a

common standard deviation , independent of x value

Diagnosis: Residual Plot, residual vs. fitted value

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Residual Plot: Is the spread level of residuals more or less the same over fitted value?

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Minitab: Stat>>Regression>> regression …

Select the response and predictors accordingly

Click “graphs” for residual plots

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Residual Plots

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Click “residuals versus fits”

Minitab Output

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Regression Analysis: ARMSPAN versus HEIGHT

The regression equation isARMSPAN = - 3.73 + 1.04 HEIGHT

Predictor Coef SE Coef T PConstant -3.728 2.660 -1.40 0.169HEIGHT 1.03655 0.04082 25.39 0.000

S = 2.12905 R-Sq = 94.4% R-Sq(adj) = 94.3%

Analysis of Variance

Source DF SS MS F PRegression 1 2922.8 2922.8 644.81 0.000Residual Error 38 172.2 4.5Total 39 3095.1