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### 26 Cards in this Set

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 Define Multiple Regression Analysis to analyze the relationship between 2 or more variables Formula for multiple regression y = A + Bx + Cz + Dt How is multiple and simple regression similar 1. the least-squares criterion is the same 2. the residuals should be random Measure of closeness for Multiple Regression is .. the adjusted correlation coefficient: R-bar-squared Describe R-bar-squared is based on R-squared but adjusted to make allowance for the number of right-hand side variables included in the regression Describe the problem of collinearity when tow (or more) of the x variables are highly correlated. Difficult to discriminate between the effects of the two variables. aka multi-collinearity. What are the remedies for collinearity? 1. Use only one of the variables (subjective)O 2. Amalgamate the variables (if it would have meaning) 3. Substitute one of the variables with a new variable with has similar meaning How to test for collinearity? Inspect the correlation coefficient for each pair of x variables. Highly coefficient pairs are collinear. Process for multiple regression 1. Inspect scatter diagrams to confirm approx. linear relationships. 2. Complete regression analysis by computer 3. Check residuals 4. Check for collinearity. Describe method of carrying out a non-linear regression. relies on transforming the non-linear regression so that it can be treated as if it were linear. Variable is transformed when ... a variable x is turned into it square, its reciprocal, or its logarithm Closeness of Fit can be measured by ... the ratio of the mean squares of the two sources of variation is a F ratio if there is no linear relationship. How to test for randomness of residuals? The Runs Test What is the limit to the number of variables that can be included in Regression analysis? Any number, but need computing power. Important criterion - variables must have significant effect on value of y. Significance test for the coefficients based on what distributions? Observations < 30 - t-distribution Observations > 30 - normal distribution What are the steps for significance test for any right-hand side variable: 1. Hypothesis is true pop. coefficient for variable is 0. 2. Evidence is just the set of observations from which regression coefficients have been estimated. Also coefficient of the variable in question, standard error must be calculated. 3. Significance level - 5% 4. Degrees of freedom are: n - k - 1 (k = number of variables) 5. If t Obs exceeds t.025 hypothesis rejected, is significant. less than .025 then not significant. How is the accuracy of a prediction calculated? By calculating the standard error of the residuals. Uncertainty in a prediction comes from: 1. What regression doesn't deal with - residuals. 2. What the regression does deal with - error in the estimation of regression model coefficients. How are confidence levels for predictions calculated? By calculating the Standard Error of Predicted Values (SE(Pred)). What are the criterion for SE(Pred)? 1. More than 30 data points 2. 95% of future values of the variable are likely to fall with +-2SE(Pred) of the point estimate. 3. If fewer than 30 data points, then t-distribution applies. The 95% confidence level for predictions is called? The forecast interval. What is the structured approach to Regression problem? 1. Propose a tentative model (scatter diagrams or prior knowledge) - what equation might be like 2. Run the regression and check closeness of fit 3. Check the residuals 4. Decide if any x variables should be discarded 5. Check for collinearity 6. Decide if regression estimates are accurate enough 7. If necessary, formulate a new regression model. Similarities of simple and multiple regressioni 1. Substitution of x values in regression to make predictions 2. F test to measure closeness of fit 3. Checking of residuals for randomness 4. Use of SE(Pred) for accuracy Differences between simple and multiple regression 1. Adjustment of correlation coefficient to allow for degrees of freedom 2. t test to determine variables to take out 3. Check for collinearity What is the Degrees of Freedom for Rthe sum of squares (regression) in a multiple regression? k, the number of x variables What are the degrees of freedom for the sum of squares (residuals)? n - k - 1