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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

SM339 Mult Regr - Spring 2007

Multiple Regression

- 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

Multiple Regression

- 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

Multiple Regression

- 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

Multiple Regression

- 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

Matrix Formulation

- 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

Matrix Formulation

- 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

Matrix Formulation

- 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

Comparing Models

- 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

Comparing Models

- 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

Comparing Models

- 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

Comparing Models

- Consider Fig 13.11 on p. 608
- Y=unloading time
- 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

Comparing Models

- 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

Comparing Models

- 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

Comparing Models

- 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
- Either start from scratch and add variables or start with all variables and delete variables

SM339 Mult Regr - Spring 2007

Comparing Models

- 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

Comparing Models

- 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

Comparing Models

- 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

Comparing Models

- 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

Other Models

- 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

Other Models

- 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

Other Models

- Would a cubic model work better?
- Partial F = 4.5*e-4
- So cubic model is NOT preferred

SM339 Mult Regr - Spring 2007

Other Models

- 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

Other Models

- 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

Other Models

- 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

Other Models

- 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

Other Models

- 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

Indicator Variables

- 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

Indicator Variables

- Return to Fig11.4 on blood flow
- 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

Indicator Variables

- 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

Indicator Variables

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

SM339 Mult Regr - Spring 2007

Other ANOVA

- 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

Other ANOVA

- 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

Other ANOVA

- 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

Other ANOVA

- 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

Other ANOVA

- 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

Other ANOVA

- >> 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

Other ANOVA

- 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

Other ANOVA

- 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

Interaction

- 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

Interaction

- 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

Interaction

- 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

ANACOVA

- 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

ANACOVA

- 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

ANACOVA

- 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

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