Multiple Linear Regression

Multiple Linear Regression
AMS 572 Group #2

Outline Jinmiao Fu—Introduction and History
Ning Ma—Establish and Fitting of the model Ruoyu Zhou—Multiple Regression Model in Matrix Notation Dawei Xu and Yuan Shang—Statistical Inference for Multiple Regression Yu Mu—Regression Diagnostics Chen Wang and Tianyu Lu—Topics in Regression Modeling Tian Feng—Variable Selection Methods Hua Mo—Chapter Summary and modern application

Introduction Multiple linear regression attempts to model the relationship between two or more explanatory variables and a response variable by fitting a linear equation to observed data. Every value of the independent variable x is associated with a value of the dependent variable

Example: The relationship between an adult’s health and his/her daily eating amount of wheat, vegetable and meat.

History

Karl Pearson (1857–1936) Lawyer, Germanist, eugenicist, mathematician and statistician Correlation coefficient Method of moments Pearson's system of continuous curves. Chi distance, P-value Statistical hypothesis testing theory, statistical decision theory. Pearson's chi-square test, Principal component analysis.

Sir Francis Galton FRS (16 February 1822 – 17 January 1911)
Anthropology and polymathy Doctoral students Karl Pearson In the late 1860s, Galton conceived the standard deviation. He created the statistical concept of correlation and also discovered the properties of the bivariate normal distribution and its relationship to regression analysis

Galton invented the use of the regression line (Bulmer 2003, p
Galton invented the use of the regression line (Bulmer 2003, p. 184), and was the first to describe and explain the common phenomenon of regression toward the mean, which he first observed in his experiments on the size of the seeds of successive generations of sweet peas.

The publication by his cousin Charles Darwin of The Origin of Species in 1859 was an event that changed Galton's life. He came to be gripped by the work, especially the first chapter on "Variation under Domestication" concerning the breeding of domestic animals.

Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French mathematician. He made important contributions to statistics, number theory, abstract algebra and mathematical analysis. He developed the least squares method, which has broad application in linear regression, signal processing, statistics, and curve fitting.

Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.

Gauss, who was 23 at the time, heard about the problem and tackled it
Gauss, who was 23 at the time, heard about the problem and tackled it. After three months of intense work, he predicted a position for Ceres in December 1801—just about a year after its first sighting—and this turned out to be accurate within a half-degree. In the process, he so streamlined the cumbersome mathematics of 18th century orbital prediction that his work—published a few years later as Theory of Celestial Movement—remains a cornerstone of astronomical computation.

It introduced the Gaussian gravitational constant, and contained an influential treatment of the method of least squares, a procedure used in all sciences to this day to minimize the impact of measurement error. Gauss was able to prove the method in 1809 under the assumption of normally distributed errors (see Gauss–Markov theorem; see also Gaussian). The method had been described earlier by Adrien-Marie Legendre in 1805, but Gauss claimed that he had been using it since 1795.

Sir Ronald Aylmer Fisher FRS (17 February 1890 – 29 July 1962) was an English statistician, evolutionary biologist, eugenicist and geneticist. He was described by Anders Hald as "a genius who almost single-handedly created the foundations for modern statistical science," and Richard Dawkins described him as "the greatest of Darwin's successors".

In addition to "analysis of variance", Fisher invented the technique of maximum likelihood and originated the concepts of sufficiency, ancillarity, Fisher's linear discriminator and Fisher information.

Establish and Fitting of the Model

Probabilistic Model : the observed value of the random variable(r.v.)
depends on fixed predictor values ,i=1,2,3,…,n unknown model parameters n is the number of observations. ~ N (0, ) i.i.d

Fitting the model LS provides estimates of the unknown model parameters, which minimizes Q (j=1,2,…,k)

Tire tread wear vs. mileage (example11.1 in textbook)
Mileage (in 1000 miles) Groove Depth (in mils) 394.33 4 329.50 8 291.00 12 255.17 16 229.33 20 204.83 24 179.00 28 163.83 32 150.33 The table gives the measurements on the groove of one tire after every 4000 miles. Our Goal: to build a model to find the relation between the mileage and groove depth of the tire.

SAS code----fitting the model
Data example； Input mile depth Sqmile=mile*mile； Datalines; ; run; Proc reg data=example; Model Depth= mile sqmile; Run;

Depth= mile+0.172sqmile

Goodness of Fit of the Model
Residuals are the fitted values An overall measure of the goodness of fit Error sum of squares (SSE): total sum of squares (SST): SST is the SSE obtained when fitting the model Yi = B0 + ei, which ignores all the x’s R^2 = 0.5 means 50% of the variation in y is accounted for by x, in this case, all x’s regression sum of squares (SSR):

Multiple Regression Model
In Matrix Notation

1. Transform the Formulas to Matrix Notation

𝑋= 1 1 𝑥 11 𝑥 12 𝑥 21 𝑥 22 ⋯ 𝑥 1𝑘 ⋯ 𝑥 2𝑘 ⋮ ⋮ ⋮ ⋱ ⋮ 1 𝑥 𝑛1 𝑥 𝑛2 ⋯ 𝑥 𝑛𝑘
𝑋→𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑜𝑟 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 The first column of X 1 1 ⋮ 1 denotes the constant term 𝛽 0 (We can treat this as 𝛽 0 𝑥 𝑖0 with 𝑥 𝑖0 =1.)

𝛽→ the (k+1)×1 vectors of unknown parameters
Finally let 𝛽= 𝛽 0 𝛽 1 ⋮ 𝛽 𝑘 and 𝛽 = 𝛽 𝛽 1 ⋮ 𝛽 𝑘 where 𝛽→ the (k+1)×1 vectors of unknown parameters 𝛽 → 𝛽 ′ 𝑠 LS estimates

𝛽 0 𝑛+ 𝛽 1 𝑖=1 𝑛 𝑥 𝑖1 +…+ 𝛽 𝑘 𝑖=1 𝑛 𝑥 𝑖𝑘 = 𝑖=1 𝑛 𝑦 𝑖
Formula 𝑌 𝑖 = 𝛽 0 + 𝛽 1 𝑥 𝑖1 + 𝛽 2 𝑥 𝑖2 +⋯+ 𝛽 𝑘 𝑥 𝑖𝑘 + 𝜖 𝑖 becomes 𝑌=𝑋𝛽+𝜖 Simultaneously, the linear equation 𝛽 0 𝑛+ 𝛽 1 𝑖=1 𝑛 𝑥 𝑖1 +…+ 𝛽 𝑘 𝑖=1 𝑛 𝑥 𝑖𝑘 = 𝑖=1 𝑛 𝑦 𝑖 are changed to 𝑋 ′ 𝑋𝛽= 𝑋 ′ 𝑦 Solve this equation respect to 𝛽 and we get 𝛽 = 𝑋 ′ 𝑋 𝑋 ′ 𝑦 (if the inverse of the matrix 𝑋 ′ 𝑋 exists.) -1

2. Example 11.2 (Tire Wear Data: Quadratic Fit Using Hand Calculations)
We will do Example 11.1 again in this part using the matrix approach. For the quadratic model to be fitted 𝑋= 𝑎𝑛𝑑 𝑦=

According to formula 𝛽 = 𝑋 ′ 𝑋 𝑋 ′ 𝑦
𝛽 = 𝑋 ′ 𝑋 𝑋 ′ 𝑦 we need to calculate 𝑋 ′ 𝑋 first and then invert it and get ( 𝑋 ′ 𝑋) −1 𝑋 ′ 𝑋= , ,944 2,245,632 ( 𝑋 ′ 𝑋) −1 = − − − − -1

Finally, we calculate the vector of LS estimates 𝛽
= 𝛽 𝛽 𝛽 2 = ( 𝑋 ′ 𝑋) −1 𝑋 ′ 𝑦 = − − − − = −

Therefore, the LS quadratic model is 𝑦 =386.265−12.772𝑥+0.172 𝑥 2 .
This model is the same as we obtained in Example 11.1.

Statistical Inference for Multiple Regression

Statistical Inference for Multiple Regression
Determine which predictor variables have statistically significant effects We test the hypotheses: If we can’t reject H0j, then xj is not a significant predictor of y.

Statistical Inference on
Review statistical inference for Simple Linear Regression

What about Multiple Regression? The steps are similar

What’s Vjj? Why ? 1. Mean Recall from simple linear regression, the least squares estimators for the regression parameters and are unbiased. Here, of least squares estimators is also unbiased.

2.Variance Constant Variance assumption:

Let Vjj be the jth diagonal of the matrix

Therefore,

Derivation of confidence interval of The 100(1-α)% confidence interval for is

Rejects H0j if

Prediction of Future Observation
Having fitted a multiple regression model, suppose we wish to predict the future value of Y for a specified vector of predictor variables x*=(x0*,x1*,…,xk*) One way is to estimate E(Y*) by a confidence interval(CI).

Prediction of Future Observation

F-Test for Consider: Here is the overall null hypothesis, which states that none of the variables are related to . The alternative one shows at least one is related.

How to Build a F-Test…… The test statistic F=MSR/MSE follows F-distribution with k and n-(k+1) d.f. The α -level test rejects if recall that MSE(error mean square) with n-(k+1) degrees of freedom.

The relation between F and r
F can be written as a function of r. By using the formula: F can be as: We see that F is an increasing function of r ² and test the significance of it.

Analysis of Variance (ANOVA)
The relation between SST, SSR and SSE: where they are respectively equals to: The corresponding degrees of freedom(d.f.) is:

ANOVA Table for Multiple Regression
Source of Variation (source) Sum of Squares (SS) Degrees of Freedom (d.f.) Mean Square (MS) F Regression Error SSR SSE k n-(k+1) Total SST n-1 This table gives us a clear view of analysis of variance of Multiple Regression.

Extra Sum of Squares Method for Testing Subsets of Parameters
Before, we consider the full model with k parameters. Now we consider the partial model: while the rest m coefficients are set to zero. And we could test these m coefficients to check out the significance:

Building F-test by Using Extra Sum of Squares Method
Let and be the regression and error sums of squares for the partial model. Since SST Is fixed regardless of the particular model, so: then, we have: The α-level F-test rejects null hypothesis if

Remarks on the F-test The numerator d.f. is m which is the number of coefficients set to zero. While the denominator d.f. is n-(k+1) which is the error d.f. for the full model. The MSE in the denominator is the normalizing factor, which is an estimate of σ² for the full model. If the ratio is large, we reject .

Links between ANOVA and Extra Sum of Squares Method
Let m=1 and m=k respectively, we have: From above we can derive: Hence, the F-ratio equals: with k and n-(k+1) d.f.

Regression Diagnostics

5 Regression Diagnostics
5.1 Checking the Model Assumptions Plots of the residuals against individual predictor variables: check for linearity A plot of the residuals against fitted values: check for constant variance A normal plot of the residuals: check for normality

A run chart of the residuals: check if the random errors are auto correlated.
Plots of the residuals against any omitted predictor variables: check if any of the omitted predictor variables should be included in the model.

Example: Plots of the residuals against individual predictor variables

SAS code

Example: plot of the residuals against fitted values

SAS code

Example: normal plot of the residuals

SAS code

5.2 Checking for Outliers and Influential Observations
Standardized residuals Large values indicate outlier observation. Hat matrix If the Hat matrix diagonal , then ith observation is influential.

Example: graphical exploration of outliers

Example: leverage plot

5.3 Data transformation Transformations of the variables(both y and the x’s) are often necessary to satisfy the assumptions of linearity, normality, and constant error variance. Many seemingly nonlinear models can be written in the multiple linear regression model form after making a suitable transformation. For example, after transformation: or

Topics in Regression Modeling

Multicollinearity Multicollinearity occurs when two or more predictors in the model are correlated and provide redundant information about the response. Example of multicollinear predictors are height and weight of a person, years of education and income, and assessed value and square footage of a home. Consequences of high multicollinearity: a. Increased standard error of estimates of the β ’s b. Often confused and misled results.

Detecting Multicollinearity
Easy way: compute correlations between all pairs of predictors. If some r are close to 1 or -1, remove one of the two correlated predictors from the model. X1 colinear X2 independent X3 X2 Equal to 1 Correlations

Detecting Multicollinearity
Another way: calculate the variance inflation factors for each predictor xj: where is the coefficient of determination of the model that includes all predictors except the jth predictor. If VIFj≥10, then there is a problem of multicollinearity.

Muticollinearity-Example
See Example11.5 on Page 416, Response is the heat of cement on a per gram basis (y) and predictors are tricalcium aluminate(x1), tricalcium silicate(x2), tetracalcium alumino ferrite(x3) and dicalcium silicate(x4).

Estimated parameters in first order model: ˆy = x x x x4. F = with p−value below Individual t−statistics and p−values: 2.08 (0.071), 0.7 (0.501) and 0.14 (0.896), (0.844). Note that sign on β4 is opposite of what is expected. And very high F would suggest more than just one significant predictor.

Correlations Correlations were r13 = , r24 = Also the VIF were all greater than 10. So there is a multicollinearity problem in such model and we need to choose the optimal algorithm to help us select the variables necessary.

Muticollinearity-Subsets Selection
Algorithms for Selecting Subsets All possible subsets Only feasible with small number of potential predictors (maybe 10 or less) Then can use one or more of possible numerical criteria to find overall best Leaps and bounds method Identifies best subsets for each value of p Requires fewer variables than observations Can be quite effective for medium-sized data sets Advantage to have several slightly different models to compare

Muticollinearity-Subsets Selectioin
Forward stepwise regression Start with no predictors First include predictor with highest correlation with response In subsequent steps add predictors with highest partial correlation with response controlling for variables already in equations Stop when numerical criterion signals maximum (minimum) Sometimes eliminate variables when t value gets too small Only possible method for very large predictor pools Local optimization at each step, no guarantee of finding overall optimum Backward elimination Start with all predictors in equation Remove predictor with smallest t value Continue until numerical criterion signals maximum (minimum) Often produces different final model than forward stepwise method

Muticollinearity-Best Subsets Criteria
Numerical Criteria for Choosing Best Subsets No single generally accepted criterion Should not be followed too mindlessly Most common criteria combine measures of with add penalties for increasing complexity (number of predictors) Coefficient of determination Ordinary multiple R-square Always increases with increasing number of predictors, so not very good for comparing models with different numbers of predictors Adjusted R-Square Will decrease if increase in R-Square with increasing p is small

Muticollinearity-Best Subsets Criteria
Residual mean square (MSEp) Equivalent to adjusted r-square except look for minimum Minimum occurs when added variable doesn't decrease error sum of squares enough to offset loss of error degree of freedom Mallows' Cp statistic Should be about equal to p and look for small values near p Need to estimate overall error variance PRESS statistic The one associated with the minimum value of PRESSp is chosen Intuitively easier to grasp than the Cp-criterion.

Muticollinearity-Forward Stepwise
First include predictor with highest correlation with response >FIN=4

In subsequent steps add predictors with highest partial correlation with response controlling for variables already in equations. (if Fi>FIN=4, enter the Xi and Fi<FOUT=4, remove the Xi) >FIN=4

>FIN=4 <FOUT=4

Summarize the stepwise algorithms Therefore our “Best Model” should only include x1 and x2, which is y= x x2

Check the significance of the model and individual parameter again. We find p value are all small and each VIF is far less than 10.

Muticollinearity-Best Subsets
Also we can stop when numerical criterion signals maximum (minimum) and sometimes eliminate variables when t value gets too small.

Muticollinearity-Best Subsets
The largest R squared value is associated with the full model. The best subset which minimizes the Cp-criterion includes x1,x2 The subset which maximizes Adjusted R squared or equivalently minimizes MSEp is x1,x2,x4. And the Adjusted R squared increases only from to by the addition of x4to the model already containing x1 and x2. Thus the simpler model chosen by the Cp-criterion is preferred, which the fitted model is y= x x2

Polynomial model Polynomial models are useful in situations where the analyst knows that curvilinear effects are present in the true response function. We can do this with more than one explanatory variable using Polynomial regression model:

Multicollinearity-Polynomial Models
Multicollinearity is a problem in polynomial regression (with terms of second and higher order): x and x2 tend to be highly correlated. A special solution in polynomial models is to use zi = xi − ¯xi instead of just xi. That is, first subtract each predictor from its mean and then use the deviations in the model.

Multicollinearity – Polynomial model
Example: x = 2, 3, 4, 5, 6 and x2 = 4, 9, 16, 25, 36. As x increases, so does x2. rx,x2 = 0.98. = 4 then z = −2,−1, 0, 1, 2 and z2 = 4, 1, 0, 1, 4. Thus, z and z2 are no longer correlated. rz,z2 = 0. We can get the estimates of the β’s from the estimates of the γ ’s. Since

Dummy Predictor Variable
The dummy variable is a simple and useful method of introducing into a regression analysis information contained in variables that are not conventionally measured on a numerical scale, e.g., race, gender, region, etc.

The categories of an ordinal variable could be assigned suitable numerical scores. A nominal variable with c≥2 categories can be coded using c – 1 indicator variables, X1,…,Xc-1, called dummy variables. Xi=1, for ith category and 0 otherwise X1=,…,=Xc-1=0, for the cth category

If y is a worker’s salary and Di = 1 if a non-smoker Di = 0 if a smoker We can model this in the following way:

Equally we could have used the dummy variable in a model with other explanatory variables. In addition to the dummy variable we could also add years of experience (x), to give: For smoker For non-smoker

α α+β y x Non-smoker Smoker

We can also add the interaction to between smoking and experience with respect to their effects on salary. For non-smoker For smoker

α α+β y x Non-smoker Smoker

Standardized Regression Coefficients
We typically wants to compare predictors in terms of the magnitudes of their effects on response variable. We use standardized regression coefficients to judge the effects of predictors with different units

They are the LS parameter estimates obtained by running a regression on standardized variables, defined as follows: Where and are sample SD’s of and

Let And The magnitudes of can be directly compared to judge the relative effects of on y.

Since , the constant can be dropped from the model. Let be the vector of the and be the matrix of

So we can get This method of computing is numerically more stable than computing directly, because all entries of R and r are between -1 and 1.

Example (Given in page 424) From the calculation, we can obtain that And sample standard deviations of x1,x2 and are Then we have Note that ,although Thus x1 has a larger effect than x2 on y.

We can also use the matrix method to compute standardized regression coefficients. First we compute the correlation matrix between x1 ,x2 and y Then we have Next calculate Hence Which is as same result as before

Variable Selection Methods

How to decide their salaries?
23 32 Attacker Defender 5 years 11 years more than 20 goals per year less than 1 goals per year Lionel Messi 10,000,000 EURO/yr Carles Puyol 5,000,000 EURO/yr

How to select variables?
1) Stepwise Regression 2)Best Subset Regression

Stepwise Regression Partial F-test Partial Correlation Coefficients
How to do it by SAS? Drawbacks

Partial F-test (p-1)-Variable Model: p-Variable Model:

How to do the test? We reject in favor of at level α if

Another way to interpret the test:
test statistics: We reject at level α if

Partial Correlation Coeffientients
test statistics: *Add to the regression equation that includes only if is large enough.

How to do it by SAS? (EX9 Continuity of Ex5)
The table shows data on the heat evolved in calories during the hardening of cement on a per gram basis (y) along with the percentages of four ingredients: tricalcium aluminate (x1), tricalcium silicate (x2), tetracalcium alumino ferrite (x3), and dicalcium silicate (x4). No. X1 X2 X3 X4 Y 1 7 26 6 60 78.5 2 29 15 52 74.3 3 11 56 8 20 104.3 4 31 47 87.6 5 33 95.9 55 9 22 109.2 71 17 102.7 44 72.5 54 18 93.1 10 21 1159 40 23 34 83.8 12 66 113.3 13 68 109.4

SAS Code ; proc reg data=example1;
input x1 x2 x3 x4 y; datalines; ; Run; proc reg data=example1; model y= x1 x2 x3 x4 /selection=stepwise; run;

SAS output

Interpretation At the first step, x4 is chosen into the equation as it has the largest correlation with y among the 4 predictors; At the second step, we choose x1 into the equation for it has the highest partial correlation with y controlling for x4; At the third step, since is greater than , x2 is chosen into the equation rather than x3.

Interpretation At the 4th step, we removed x4 from the model since its partial F-statistics is too small. From Ex11.5, we know that x4 is highly correlated with x2. Note that in Step4, the R-Square is , which is slightly higher that , the R-Square of Step 2. It indicates that even x4 is the best predictor of y, the pair (x1,x2) is a better predictor than the predictor (x1,x4).

Drawbacks The final model is not guaranteed to be optimal in any specified case. It yields a single final model while in practice there are often several equally good model.

Best Subset Regression
Comparison to Stepwise Method Optimality Criteria How to do it by SAS?

Comparison to Stepwise Regression
In best subsets regression, a subset of variables is chosen from that optimizes a well-defined objective criterion. The best regression algorithm permits determination of a specified number of best subsets from which the choice of the final model can be made by the investigator.

Optimality Criteria

Optimality Criteria Standardized mean square error of prediction:
involves unknown parameters such as ‘s, so minimize a sample estimate of Mallows’ :

Optimality Criteria It practice, we use the because of its ease of computation and its ability to judge the predictive power of a model.

How to do it by SAS?(Ex11.9) proc reg data=example1;
model y= x1 x2 x3 x4 /selection=adjrsq mse cp; run;

SAS output

Interpretation The best subset which minimizes the
is x1, x2 which is the same model selected using stepwise regression in the former example. The subset which maximizes is x1, x2, x4. However, increases only from to by the addition of x4 to the model which already contains x1 and x2. Thus, the model chosen by the is preferred.

and Modern Application
Chapter Summary and Modern Application

Multiple Regression Model Fitting the MLR Model
MLR Model in Matrix Notation Model (Extension of Simple Regression): are unknown parameters Least squares method: Goodness of fit of the model:

Statistical Inference for Multiple Regression Regression Diagnostics Statistical Inference on Hypotheses: vs. Test statistic: Hypotheses: vs. At least one Test statistic: Residual Analysis Data Transformation

The General Hypothesis Test:
the full model: the partial model: Compare Hypotheses: vs. Test statistic: RejectH0 when Estimating and Predicting Future Observations: Let and Test statistic: CI for the estimated mean *: PI for the estimated Y*:

Topics in regression modeling Variable Selection Methods
Multicollinearity Polynomial Regression Dummy Predictor Variables Logistic egression Model Variable Selection Methods Stepwise Regression: Stepwise Regression Algorithm Best Subsets Regression Strategy for building a MLR model partial F-test partial Correlation Coefficient

Application of the MLR model
Linear regression is widely used in biological, chemistry, finance and social sciences to describe possible relationships between variables. It ranks as one of the most important tools used in these disciplines.

Multiple linear regression
Financial market biology Housing price heredity Chemistry

Example Broadly speaking, an asset pricing model can be expressed as:
Where , and k denote the expected return on asset i, the kth risk factor and the number of risk factors, respectively. denotes the specific return on asset i.

The equation can also be expressed in the matrix notation:
is called the factor loading

Inflation rate GDP What’s the most important factors? Interest rate Rate of return on the market portfolio Employment rate Government policies

Method Step 1: Find the efficient factors (EM algorithms, maximum likelihood) Step 2: Fit the model and estimate the factor loading (Multiple linear regression)

According to the multiple linear regression and run data on SAS, we can get the factor loading and the coefficient of multiple determination We can ensure the factors that mostly effect the return in term of SAS output and then build the appropriate multiple factor models We can use the model to predict the future return and make a good choice!

Questions Thank you

Multiple Linear Regression

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