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Chapter 8: Regression Analysis

PowerPoint Slides Prepared By:Alan Olinsky Bryant University

Management Science: The Art of Modeling with Spreadsheets, 2e

S.G. Powell

K.R. Baker

© John Wiley and Sons, Inc.

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Modeling Relationships

In some circumstances, data can be valuable in helping to determine the parameters in a relationship or its structural form.

The process of using data to formulate relationships is known as regression analysis.

In this approach, we identify one variable as the response variable, which means that it can be predicted from the values of other variables.

Those other variables are called explanatory variables.

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Types of Regression Models

Regression models that involve one explanatory variable are called simple regressions

When two or more explanatory variables are involved, the relationships are called multiple regressions.

Regression models are also divided into linear and nonlinear models, depending on whether the relationship between the response and explanatory variables is linear or nonlinear.

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Estimating Relationships

Scatter plot – visualize association Correlation:

n – number of pairs of observations for x, y sx, sy – standard deviations of x, y r – measures strength of linear relationship between

x and y

r 1

(n 1)

x i x sx

i1

n

(yi y )

sy

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r-statistic

Independent of units of measurement Lies in range [-1, 1] r > 0 – positive association r < 0 – negative association r close to 1 (or –1) implies a strong association r close to 0 implies a weak association Excel function: CORREL(xrange,yrange)

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Simple Linear Regression

y = a + bx + e

y - dependent variable x - independent variable e - an “error” term. Constants a and b represent the intercept and

slope, respectively, of the regression line.

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Error Term in Regression

Unexplained “noise” in the relationship May represent limitations of knowledge Or may represent random deviations of the

dependent variable from its mean, y

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Regression Goal

Want to find line to most closely match the observed relationship between x and y

Define “most closely” as minimizing sum of squared differences between observed and model values Minimizing sum of differences would set y equal to its

mean Penalizes large differences more than small differences

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Performing Regression

Residuals:

ei = yi – y = yi – (a + bxi)

Sum of squared differences between observations and model :

SS =

The regression problem: choose a and b to minimize SS

ei2

i 1

n

(yii 1

n

a bx1)2

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Regression Analysis

Assumes residuals are normally distributed with mean 0

Regression parameters can be calculated directly from the data

Simpler to use Excel’s regression tool (Under Data Analysis menu)

b n x iyi x i

i1

n

yii1

n

i 1

n

n x i2 ( xi

i1

n

)2

i1

n

a y bx

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Goodness of Fit

Coefficient of determination: R2

Lies in range [0, 1] Closer to one – better fit Measures how much of the variation in y-

values is explained by model 1 – perfect match to model 0 – equation explains none of observed variation

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Regression Window

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Regression Output

R Squared

Degree of significance (under 0.1 is significant)

Estimate for a Estimate for b

P values of under 0.1 are statistically significant

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Regression Statistics

Four measures are used to judge the statistical qualities of a regression:

R2: Measures the percent of variation in the explanatory variable accounted for by the regression model.

F-statistic (Significance F): Measures the probability of observing the given R2 (or higher) when all the true regression coefficients are zero.

p-value: Measures the probability of observing the given estimate of the regression coefficient (or a larger value, positive or negative) when the true coefficient is zero.

Confidence interval: Gives a range within which the true regression coefficient lies with given probability.

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Simple Nonlinear Regression

A straight line may not be the most plausible description of dependency, e.g., y = axb .

Can follow previous ideas to minimize sum of squared differences No Excel functions or simple formulas

Or can transform non-linear relationship into linear one, e.g., log y = log a + b log x Give up some intuition for convenience

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Multiple Linear Regression

Multiple independent variables

y = a0 + a1x1 + a2x2 + … + amxm + e

Work with n observations – each has: One observation of dependent variable One observation each of the m independent variables

Seek to minimize the sum of squared differences Put all independent variables into x-range in Excel’s

regression tool

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Regression Output

Coefficient of multiple determination

Coefficients of regression equation

P values of under 0.1 are statistically significant

Square root of R square

Accounts for presence of multiple variables

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Values to Include in Regression

Ideally pick values that can be justified based on practical or theoretical grounds

Could choose set that generates largest value of adjusted R2

Also could choose based on those with significant p-values for coefficients

Remember that good models require good forecasts for the independent variables.

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Regression Assumptions

Errors in the regression model: Follow a Normal distribution Are mutually independent Have the same variance

Linearity is assumed to hold

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*Using the Excel Tools Trendline and LINEST

Excel provides several alternative methods for performing regression analysis.

Trendline is a charting option that allows the user to fit one of six families of curves to a set of data and to add the resulting regression line to the plot.

LINEST is an array function that can be used to compute regression statistics and use them directly as parameters in a model.

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*Trendline

The Trendline option appears in Excel only when a chart has been selected.

Trendline offers the option to fit any one of the following six families of curves:

Linear: y = a + bx Logarithmic: y = a+ b×ln(x) Polynomial: y = a + bx + cx2+ dx3+ …(the user selects the

Order of the polynomial, which is the largest exponent of x). Power: y = axb

Exponential: y = aebx

Moving average: y = average of previous n y-values (the user selects the Period for the moving average, which is n, the number of previous values used to calculate the result).

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Trendline Window

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Linear Trendline

y = 61873x + 443600

R2 = 0.976

-

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1 2 3 4 5 6 7 8 9 10

Year

Dem

and

Linear trendline for BPI data

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Power Trendline

Chart Title

y = 450609x0.3475

R2 = 0.9656

-

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1 2 3 4 5 6 7 8 9 10

Year

Dem

and

Power trendline for BPI data

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*LINEST

LINEST is an Excel function that calculates regression parameters and measures of goodness-of-fit for simple or multiple regressions.

It is one of a set of functions including SLOPE, INTERCEPT, and TREND, which can be used as alternatives to the Data Analysis add-in.

LINEST is an array function, which means that it physically occupies more than one cell in the spreadsheet.

Like all Excel functions, it is linked to the underlying data, so if the data change the regression parameters calculated by LINEST change automatically.

This is not true of regression results calculated using the Analysis Toolpak: if the underlying data change, the user must be careful to re-run the Regression procedure.

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Forecasting Model Using LINEST.

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Function Wizard for LINEST

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Summary

Modeling is the central task for the analyst and data collection and statistical analysis support the modeling task where appropriate.

When sensitivity testing indicates that certain parameters or relationships must be determined precisely, we often collect data and perform statistical analysis to refine the parameters and relations in our models.

Regression analysis is a means for using data to help formulate relationships among variables.

All regression methods are based on the idea of fitting a family of curves to data by choosing parameters that minimize the sum of squared residuals.

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Summary

The simplest regression model is a linear relationship with one explanatory variable, although regression can also be applied in cases where there are multiple explanatory variables and nonlinear relationships. .

The most complete method in Excel is the Regression option within the Analysis Toolpak add-in.

Other useful methods include Trendline, which can be used to fit any one of six families of curves to plotted data, and LINEST, which can be used to calculate regression estimates dynamically.

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