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Error Variance In Multiple Regression

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Define "regression coefficient" Define "beta weight" Explain what R is and how it is related to r Explain why a regression weight is called a "partial slope" Explain why the sum of squares explained in a multiple regression model is usually less than the sum of variance in multiple regression analysis the sums of squares in simple regression Define R2 in terms of proportion explained Test R2

Unique Variance Multiple Regression

for significance Test the difference between a complete and reduced model for significance State the assumptions of multiple regression and specify which aspects unique variance multiple regression spss of the analysis require assumptions In simple linear regression, a criterion variable is predicted from one predictor variable. In multiple regression, the criterion is predicted by two or more variables. For example, in the SAT case study, you might

Multiple Regression Variance Explained

want to predict a student's university grade point average on the basis of their High-School GPA (HSGPA) and their total SAT score (verbal + math). The basic idea is to find a linear combination of HSGPA and SAT that best predicts University GPA (UGPA). That is, the problem is to find the values of b1 and b2 in the equation shown below that give the best predictions of UGPA. As in the case of simple linear regression, we define multiple regression variance explained by each variable the best predictions as the predictions that minimize the squared errors of prediction. UGPA' = b1HSGPA + b2SAT + A where UGPA' is the predicted value of University GPA and A is a constant. For these data, the best prediction equation is shown below: UGPA' = 0.541 x HSGPA + 0.008 x SAT + 0.540 In other words, to compute the prediction of a student's University GPA, you add up (a) their High-School GPA multiplied by 0.541, (b) their SAT multiplied by 0.008, and (c) 0.540. Table 1 shows the data and predictions for the first five students in the dataset. Table 1. Data and Predictions. HSGPA SAT UGPA' 3.45 1232 3.38 2.78 1070 2.89 2.52 1086 2.76 3.67 1287 3.55 3.24 1130 3.19 The values of b (b1 and b2) are sometimes called "regression coefficients" and sometimes called "regression weights." These two terms are synonymous. The multiple correlation (R) is equal to the correlation between the predicted scores and the actual scores. In this example, it is the correlation between UGPA' and UGPA, which turns out to be 0.79. That is, R = 0.79. Note that R will never be negative since if there are negative correlations between the predictor variables and the criterion, the regression weights will be negative so that the correlation between the predicted and actual scores will be positive. Interpretation of Regression Coefficients A regression coefficient in multiple

not going to use total because it's just the sum of snatch and clean. Data The heaviest weights (in kg) that men who weigh more than 105 kg were able to lift are given in the table. Data

Variance Covariance Matrix Multiple Regression

Dictionary Age The age the competitor will be on their birthday in 2004. Body The

Variance Logistic Regression

weight (kg) of the competitor Snatch The maximum weight (kg) lifted during the three attempts at a snatch lift Clean The maximum weight (kg) standard error multiple regression lifted during the three attempts at a clean and jerk lift Total The total weight (kg) lifted by the competitor Age Body Snatch Clean Total 26 163.0 210.0 262.5 472.5 30 140.7 205.0 250.0 455.0 22 161.3 207.5 240.0 http://onlinestatbook.com/2/regression/multiple_regression.html 447.5 27 118.4 200.0 240.0 440.0 23 125.1 195.0 242.5 437.5 31 140.4 190.0 240.0 430.0 32 158.9 192.5 237.5 430.0 22 136.9 202.5 225.0 427.5 32 145.3 187.5 232.5 420.0 27 124.3 190.0 225.0 415.0 20 142.7 185.0 220.0 405.0 29 127.7 170.0 215.0 385.0 23 134.3 160.0 210.0 370.0 18 137.7 155.0 192.5 347.5 Regression Model If there are k predictor variables, then the regression equation model is y = β0 + β1x1 + β2x2 + ... https://people.richland.edu/james/ictcm/2004/multiple.html + βkxk + ε. The x1, x2, ..., xk represent the k predictor variables. Those parameters are the same as before, β0 is the y-intercept or constant, β1 is the coefficient on the first predictor variable, β2 is the coefficient on the second predictor variable, and so on. ε is the error term or the residual that can't be explained by the model. Those parameters are estimated by b0, b1, b2, ..., bk. This gives us a regression equation used for prediction of y = b0 + b1x1 + b2x2 + ...+ bkxk. Basically, everything we did with simple linear regression will just be extended to involve k predictor variables instead of just one. Regression Analysis Explained Round 1: All Predictor Variables Included Minitab was used to perform the regression analysis. This is not really something you want to try by hand. Response Variable: clean Predictor Variables: age, body, snatch Regression Equation The regression equation isclean = 32.9 + 1.03 age + 0.106 body + 0.828 snatch There's the regression equation. You can use it for estimation purposes, but you really should look further down the page to see if the equation is a good predictor or not. Table of Coefficients Predictor Coef SE Coef T PConstant 32.88 28.33 1.16 0.273age 1.0257 0.4809 2.13 0.059body 0.1057 0.1624 0.65 0.530snatch 0.8279 0.1371 6.04 0.000 Notice how the coefficients column (labeled "Coef") are again the coe

i = 1, 2, . . . , n where ui are values of http://www.unesco.org/webworld/idams/advguide/Chapt5_2.htm an unobserved error term, u, and. the unknown parameters are constants. Assumptions The error terms ui are mutually independent and identically distributed, with mean = 0 and constant variances E [ui] = 0 V [ui] = This is so, because the observations y1, y2, . . . ,yn are a random sample, they are mutually independent and hence the error terms are also mutually independent The distribution of the error term multiple regression is independent of the joint distribution of x i, x 2, . . . , x p The unknown parameters b 0, b 1, b 2, . . . ,b p are constants. Equations relating the n observations can be written as: The parameters b 0, b 1, . . . b p can be estimated using the least squares procedure, which minimizes the sum of squares of errors.             variance in multiple               Minimizing the sum of squares leads to the following equations, from which the values of b can be computed: Geometrical Representation The problem of multiple regression can be geometrically represented as follows. We can visualize that n observations (xi1, xi2, …..xip, yi) i = 1, 2, ….n are represented as points in a (p+1) - dimensional space. The regression problem is to determine the possible hyper-planes in the p - dimensional space, which will be the best- fit. We use the least squares criterion and locate the hyper-plane that minimizes the sum of squares of the errors, i.e., the distances from the points around the plane (observations) and the point on the plane. (i.e. the estimate ŷ). ŷ = a+b1x1+b2x2+…+bpxp Standard error of the estimate Se = where yi = the sample value of the dependent variable ŷi = corresponding value estimated from the regression equation n = number observations p = number of predictors or independent variable The denominator of the equation indicates that in multiple regression with p independent variables, the standard error has n-p-1 degrees of freedom. This happens because the degrees of freedom are reduced from n by p+1 numerical constants a, b1, b2, …..bp, that have been estimated from the sample. Fit of

 

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