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Published on February 29, 2008

Author: Alohomora

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Multiple linear regression:  Multiple linear regression Tron Anders Moger 11.10.2006 Example::  Example: Repetition: Simple linear regression:  Repetition: Simple linear regression We define a model where are independent, normally distributed, with equal variance Wish to fit a line as close to the observed data (two normally distributed varaibles) as possible Example: Birth weight=β0+β1*mother’s weight Least squares regression:  Least squares regression How to compute the line fit with the least squares method? :  How to compute the line fit with the least squares method? Let (x1, y1), (x2, y2),...,(xn, yn) denote the points in the plane. Find a and b so that y=a+bx fit the points by minimizing Solution: where and all sums are done for i=1,...,n. How do you get this answer? :  How do you get this answer? Differentiate S with respect to a og b, and set the result to 0 We get: This is two equations with two unknowns, and the solution of these give the answer. How close are the data to the fitted line? R2:  How close are the data to the fitted line? R2 Define SSE: Error sum of squares SSR: Regression sum of squares SST: Total sum of squares We can show that SST = SSR + SSE Define R2 is the ”coefficient of determination” What is the logic behind R2?:  What is the logic behind R2? xi Example: Regression of birth weight with mother’s weight as independent variable:  Example: Regression of birth weight with mother’s weight as independent variable Interpretation::  Interpretation: Have fitted the line Birth weight=2369.672+4.429*mother’s weight If mother’s weight increases by 20 pounds, what is the predicted impact on infant’s birth weight? 4.429*20=89 grams What’s the predicted birth weight of an infant with a 150 pound mother? 2369.672+4.429*150=3034 grams But how to answer questions like: :  But how to answer questions like: Given that a positive slope (b) has been estimated: Does it give a reproducible indication that there is a positive trend, or is it a result of random variation? What is a confidence interval for the estimated slope? What is the prediction, with uncertainty, at a new x value? Confidence intervals for simple regression:  Confidence intervals for simple regression In a simple regression model, a estimates b estimates estimates Also, where estimates variance of b So a confidence interval for is given by Hypothesis testing for simple regression:  Hypothesis testing for simple regression Choose hypotheses: Test statistic: Reject H0 if or Prediction from a simple regression model:  Prediction from a simple regression model A regression model can be used to predict the response at a new value xn+1 The uncertainty in this prediction comes from two sources: The uncertainty in the regression line The uncertainty of any response, given the regression line A confidence interval for the prediction: Example: The confidence interval of the predicted birth weight of an infant with a 150 pound mother:  Example: The confidence interval of the predicted birth weight of an infant with a 150 pound mother Found that the predicted weight was 3034 grams The confidence interval for the prediction is: 2369.67+4.43*150±t187,0.025*1.71*√(1+1/189+(150-129.81)2/(175798.52)) Which becomes (3030.8, 3037.5) =1.96 More than one independent variable: Multiple regression:  More than one independent variable: Multiple regression Assume we have data of the type (x11, x12, x13, y1), (x21, x22, x23, y2), ... We want to ”explain” y from the x-values by fitting the following model: Just like before, one can produce formulas for a,b,c,d minimizing the sum of the squares of the ”errors”. x1,x2,x3 can be transformations of different variables, or transformations of the same variable Multiple regression model:  Multiple regression model The errors are independent random (normal) variables with expectation zero and variance The explanatory variables x1i, x2i, …, xni cannot be linearily related New example: Traffic deaths in 1976 (from file crash on textbook CD):  New example: Traffic deaths in 1976 (from file crash on textbook CD) Want to find if there is any relationship between highway death rate (deaths per 1000 per state) in the U.S. and the following variables: Average car age (in months) Average car weight (in 1000 pounds) Percentage light trucks Percentage imported cars All data are per state First: Scatter plots::  First: Scatter plots: Univariate effects (one independent variable at a time!)::  Univariate effects (one independent variable at a time!): Hence: If all else is equal, if average car age increases by one month, you get 0.062 fewer deaths per 1000 inhabitants; increase age by 12 months, you get 12*0.062=0.74 fewer deaths per 1000 inhabitants Deaths per 1000=a+b*car age (in months) Deaths per 1000=a+b*car weight (in pounds) Univariate effects cont’d (one independent variable at a time!)::  Univariate effects cont’d (one independent variable at a time!): Hence: Increase prop. light trucks by 20 means 20*0.007=0.14 more deaths per 1000 inhabitants Predicted number of deaths per 1000 if prop. Imported cars is 10%: 0.206-0.004*10=0.17 Building a multiple regression model::  Building a multiple regression model: Forward regression: Try all independent variables, one at a time, keep the variable with the lowest p-value Repeat step 1, with the independent variable from the first round now included in the model Repeat until no more variables can be added to the model (no more significant variables) Backward regression: Include all independent variables in the model, remove the variable with the highest p-value Continue until only significant variables are left However: These methods are not always correct to use in practice! For the traffic deaths, end up with::  For the traffic deaths, end up with: Deaths per 1000=2.7-0.037*car age +0.006*perc. light trucks Conclusion: Did a multiple linear regression on traffic deaths, with car age, car weight, prop. light trucks and prop. imported cars as independent variables. Car age (in months, β=-0.037, 95% CI=(-0.063, -0.012)) and prop. light trucks (β=0.006, 95% CI=(0.004, 0.009)) were significant on 5%-level Check of assumptions::  Check of assumptions: Check of assumptions cont’d::  Check of assumptions cont’d: Least squares estimation:  Least squares estimation The least squares estimates of are the values b1, b2, …, bK minimizing They can be computed with similar but more complex formulas as with simple regression Explanatory power:  Explanatory power Defining We get as before We define We also get that Coefficient of determination Adjusted coefficient of determination:  Adjusted coefficient of determination Adding more independent variables will generally increase SSR and decrease SSE Thus the coefficient of determination will tend to indicate that models with many variables always fit better. To avoid this effect, the adjusted coefficient of determination may be used: Drawing inference about the model parameters:  Drawing inference about the model parameters Similar to simple regression, we get that the following statistic has a t distribution with n-K-1 degrees of freedom: where bj is the least squares estimate for and sbj is its estimated standard deviation sbj is computed from SSE and the correlation between independent variables Confidence intervals and hypothesis tests:  Confidence intervals and hypothesis tests A confidence interval for becomes Testing the hypothesis vs Reject if or Testing sets of parameters:  Testing sets of parameters We can also test the null hypothesis that a specific set of the betas are simultaneously zero. The alternative hypothesis is that at least one beta in the set is nonzero. The test statistic has an F distribution, and is computed by comparing the SSE in the full model, and the SSE when setting the parameters in the set to zero. Making predictions from the model:  Making predictions from the model As in simple regression, we can use the estimated coefficients to make predictions As in simple regression, the uncertainty in the predictions has two sources: The variance around the regression estimate The variance of the estimated regression model What if the relationship is non-linear: Transformed variables:  What if the relationship is non-linear: Transformed variables The relationship between variables may not be linear Example: The natural model may be We want to find a and b so that the line approximates the points as well as possible Example (cont.):  Example (cont.) When then Use standard formulas on the pairs (x1,log(y1)), (x2, log(y2)), ..., (xn, log(yn)) We get estimates for log(a) and b, and thus a and b Another example of transformed variables:  Another example of transformed variables Another natural model may be We get that Use standard formulas on the pairs (log(x1), log(y1)), (log(x2), log(y2)), ...,(log(xn),log(yn)) Note: In this model, the curve goes through (0,0) A third example::  A third example: Assume data (x1,y1),..., (xn,yn) seem to follow a third degree polynomial We use multivariate regression on (x1, x12, x13, y1), (x2, x22, x23, y2),... We get estimated a,b,c,d, in a third degree polynomial curve Doing a regression analysis:  Doing a regression analysis Plot the data first, to investigate whether there is a natural relationship Linear or transformed model? Are there outliers which will unduly affect the result? Fit a model. Different models with same number of parameters may be compared with R2 Check the assumptions! Make tests / confidence intervals for parameters A lot of practice is needed! Conclusion and further options:  Conclusion and further options Regression versus correlation: Can include more independent variables in regression Gets a more detailed picture on the effect a independent variable has on the dependent variable What if the dependent variable only has two possible values? Logistic regression Similar to linear regression But the interpretations of the β’s are different: They are interpreted as odds-ratios instead of the slope of a line

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