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# sm339-multregr - PowerPoint PPT Presentation

Multiple Regression. In multiple regression, we consider the response , y, to be a function of more than one predictor variable, x1, …, xk Easiest to express in terms of matrices. Multiple Regression. Let Y be a col vector whose rows are the observations of the response Let X be a matrix.

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• In multiple regression, we consider the response, y, to be a function of more than one predictor variable, x1, …, xk

• Easiest to express in terms of matrices

SM339 Mult Regr - Spring 2007

• Let Y be a col vector whose rows are the observations of the response

• Let X be a matrix.

• First col of X contains all 1’s

• Other cols contain the observations of the other predictor vars

SM339 Mult Regr - Spring 2007

• Y = X * b + e

• B is a col vector of coefficients

• E is a col vector of the normal errors

• Matlab does matrices very well, but you MUST watch the sizes and orders when you multiply

SM339 Mult Regr - Spring 2007

• A lot of MR is similar to simple regression

• B=X\Y gives coefficients

• YH=X*B gives fitted values

• SSE = (y-yh)’ *(y-yh)

• SSR = (yavg-yh)’ * (yavg-yh)

• SST = (y-yavg)’ * (y-yavg)

SM339 Mult Regr - Spring 2007

• We can set up the ANOVA table

• Df for Regr = # vars

• (So df=1 for simple regr)

• F test is as before

• R^2 is as before, with same interpretation

SM339 Mult Regr - Spring 2007

• Instead of X\Y, we can solve the equations for B

• B = (X’*X)-1 X’*Y

• We saw things like X’*X before as sum of squares

• Because of the shape of X, X’*X is square, so it makes sense to use its inverse

• (Actually, X’*X is always square)

SM339 Mult Regr - Spring 2007

• When we consider the coefficients, we not only have variances (SDs), but the relationship between coeffs

• The sd of a coefficient is the corresponding diagonal element of est(SD) (X’*X)-1

• We can use this to get conf int for coefficients (and other information)

SM339 Mult Regr - Spring 2007

• Exercises

• 1. Compute coeffs, ANOVA and SD(coeff) for Fig13.11, p 608 where time = f(vol, wt, shift). Find PV for testing B1=0. Find 95% confidence interval for B1.

• 2. Repeat for Fig 13.17, p. 613 where y=run time

• 3. Repeat for DS13.2.1, p 614 where y= sales volume

SM339 Mult Regr - Spring 2007

• Suppose we have two models in mind

• #1 uses a set of predictors

• #2 includes #1, but has extra variables

• SSE for #2 is never greater than SSE for #1

• We can always consider a model for #2 which has zeros for the new coefficients

SM339 Mult Regr - Spring 2007

• As always, we have to ask “Is the decrease in SSE unusually large?”

• Suppose that model1 has p variables and model2 has p+k models

• SSE1 is Chi^2 with df=N-p

• SSE2 is Chi^2 with df=N-(p+k)

• Then SSE1-SSE2 is Chi^2 with df=k

SM339 Mult Regr - Spring 2007

• Partial F = (SSE1-SSE2)/k / MSE2

• Note that numerator is Chi^2 divided by df

• Denominator is MS for model with more variables

• Note that subtraction is “larger – smaller”

SM339 Mult Regr - Spring 2007

• Consider Fig 13.11 on p. 608

• Model1: X=volume

• F=203, PV very small

• Model2: X=volume and wt

• F=96 and PV still very small

SM339 Mult Regr - Spring 2007

• SSE1=215.6991

• SSE2=215.1867

• So Model2 is better (smaller SSE), but only trivially

• MSE2=12.6580

• Partial F = 0.0405

• So the decrease in SSE is not significant at all, even though Model2 is significant

• (In part because Model2 includes Model1)

SM339 Mult Regr - Spring 2007

• Recall that the SD of the coefficients can be found

• The sd of a coefficient is the corresponding diagonal element of est(SD) (X’*X)-1

• In Model2 of the example, the 3rd diagonal element = 0.0030

• SD of coeff3 = 0.1961

• Coeff3/SD = 0.2012 = # SD the coefficient is away from zero

• (0.2012)2 = 0.0405 = Partial F

SM339 Mult Regr - Spring 2007

• Suppose we have a number of variables to choose from

• What set of variables should we use?

• Several approaches

• Stepwise regression

• Step up or step down

SM339 Mult Regr - Spring 2007

• Step up method

• Fit regression using each variable on its own

• If the best model (smallest SSE or largest F) is significant, then continue

• Using the variable identified at step1, add all other variables one at a time

• Of all these models, consider the one with smallest SSE (or largest F)

• Compute partial F to see if this model is better than the single variable

SM339 Mult Regr - Spring 2007

• We can continue until the best variable to add does not have a significant partial F

• To be complete, after we have added a variable, we should check to be sure that all the variables in the model are still needed

SM339 Mult Regr - Spring 2007

• One by one, drop each other variable from the model

• Compute partial F

• If partial F is small, then we can drop this variable

• After all variables have been dropped that can be, we can resume adding variables

SM339 Mult Regr - Spring 2007

• Recall that when adding single variables, we can find partial F by squaring coeff/SD(coeff)

• So, for single variables, we don’t need to compute a large number of models because the partial F’s can be computed in one step from the larger model

SM339 Mult Regr - Spring 2007

• Because it allows for multiple “predictors”, MLR is very flexible

• We can fit polynomials by including not only X, but other columns for powers of X

SM339 Mult Regr - Spring 2007

• Consider Fig13.7, p 605

• Yield is a fn of Temp

• Model1: Temp only

• F=162, highly significant

• Model2: Temp and Temp^2

• F=326

• Partial F = 29.46

• Conclude that the quadratic model is significantly better than the linear model

SM339 Mult Regr - Spring 2007

• Would a cubic model work better?

• Partial F = 4.5*e-4

• So cubic model is NOT preferred

SM339 Mult Regr - Spring 2007

• Taylor’s Theorem

• “Continuous functions are approximately polynomials”

• In Calc, we started with the function and used the fact that the coefficients are related to the derivatives

• Here, we do not know the function, but can find (estimate) the coefficients

SM339 Mult Regr - Spring 2007

• Consider Fig13.15, p. 611

• If y=f(water, fertilizer), then F<1

• Plot y vs each variable

• VERY linear fn of water

• Somewhat quadratic fn of fertilizer

• Consider a quadratic fn of both (and product)

SM339 Mult Regr - Spring 2007

• F is about 17 and pv is near 0

• All partial F’s are large, so should keep all terms in model

• Look at coeff’s

• Quadratics are neg, so the surface has a local max

SM339 Mult Regr - Spring 2007

• Solve for max response

• Water=6.3753, Fert=11.1667

• Which is within the range of values, but in the lower left corner

• (1) We can find a confidence interval on where the max occurs

• (2) Because of the cross product term, the optimal fertilizer varies with water

SM339 Mult Regr - Spring 2007

• Exercises

• 1. Consider Fig13.48, p 721. (Fix the line that starts 24.2. The 3rd col should be 10.6.) Is there any evidence of a quadratic relation?

• 2. Consider Fig13.49, p. 721. Fit the response model. Comment. Plot y vs yh. What is the est SD of the residuals?

SM339 Mult Regr - Spring 2007

• For simple regression, if we used an indicator variable, we were doing a 2 sample t test

• We can use indicator variables and multiple regression to do ANOVA

SM339 Mult Regr - Spring 2007

• Do indicators by

• for i=1:max(ndx),

• y(:,i)=(ndx==i);end

• VERY IMPORTANT

• If you are going to use the intercept, then you must leave out one column of the indicators (usually the last col)

SM339 Mult Regr - Spring 2007

• F is the same for regression as for ANOVA

• The intercept is the avg of the group that was left out of indicators

• The other coefficients are the differences between their avg and the intercept

SM339 Mult Regr - Spring 2007

• Exercises

• Compare the sumstats approach and the regression approach for Fig11.4, Fig11.5 on p. 488, 489

SM339 Mult Regr - Spring 2007

• Why bother with a second way to solve a problem we already can solve?

• The regression approach works easily for other problems

• But note that we cannot use regression approach on summary stats

SM339 Mult Regr - Spring 2007

• Two-way ANOVA

• Want to compare Treatments, but the data has another component that we want to control for

• Called “Blocks” from the origin in agriculture testing

SM339 Mult Regr - Spring 2007

• So we have 2 category variables, one for Treatment and one for Blocks

• Set up indicators for both and use all these for X

• Omit one column from each set

SM339 Mult Regr - Spring 2007

• We would like to separate the Treatment effect from the Block effect

• Use partial F

• ANOVA table often includes the change in SS separately for Treatment and Blocks

SM339 Mult Regr - Spring 2007

• Consider Fig 14.4 on p 640

• 3 machines and 4 solder methods

• Problem doesn’t tell us which is Treatment and which is Blocks, so we’ll let machines be Treatments

SM339 Mult Regr - Spring 2007

• >> x=[i1 i2];

• >> [b,f,pv,aov,invxtx]=multregr(x,y);aov

• aov =

• 60.6610 5.0000 12.1322 13.9598

• 26.0725 30.0000 0.8691 0.2425

• This is for both sets of indicators

SM339 Mult Regr - Spring 2007

• For just machine

• >> x=[i1];

• >> [b,f,pv,aov,invxtx]=multregr(x,y);aov

• aov =

• 1.8145 2.0000 0.9073 0.3526

• 84.9189 33.0000 2.5733 0.5805

• Change in SS is 60.6610- 1.8145 when we use Solder as well

SM339 Mult Regr - Spring 2007

• For just solder

• >> x=[i2];

• >> [b,f,pv,aov,invxtx]=multregr(x,y);aov

• aov =

• 58.8465 3.0000 19.6155 22.5085

• 27.8870 32.0000 0.8715 0.1986

• SSR is the same as the previous difference

• If we list them separately, then we use SSE for model with both vars so that it will properly add up to SSTotal

SM339 Mult Regr - Spring 2007

• The effect of Solder may not be the same for each Machine

• This is called “interaction” where a combination may not be the sum of the parts

• We can measure interaction by using a product of the indicator variables

• Need all possible products (2*3 in this case)

SM339 Mult Regr - Spring 2007

• Including interaction

• >> [b,f,pv,aov,invxtx]=multregr(x,y);aov

• aov =

• 64.6193 11.0000 5.8745 6.3754

• 22.1142 24.0000 0.9214 0.3144

• We can subtract to get the SS for Interaction

• 64.6193 - 60.6610

• See ANOVA table on p 654

SM339 Mult Regr - Spring 2007

• We can do interaction between categorical variables and quantitative variables

• Allows for different slopes for different categories

• Can also add an indicator, which allows for different intercepts for different categories

• With this approach, we are assuming a single SD for the e’s in all the models

• May or may not be a good idea

SM339 Mult Regr - Spring 2007

• We can do regression with a combination of categorical and quantitative variables

• The quantitative variable is sometimes called a co-variate

• Suppose we want to see if test scores vary among different groups

• But the diff groups may come from diff backgrounds which would affect their scores

• Use some measure of background (quantitative) in the regression

SM339 Mult Regr - Spring 2007

• Then the partial F for the category variable after starting with the quan variable will measure the diff among groups after correcting for background

SM339 Mult Regr - Spring 2007

• Suppose we want to know if mercury levels in fish vary among 4 locations

• We catch some fish in each location and measure Hg

• But the amount of Hg could depend on size (which indicates age), so we also measure that

• Then we regress on both Size and the indicators for Location

• If partial F for Location is large then we say that Location matters, after correcting for Size

SM339 Mult Regr - Spring 2007