SM222_2

SM222_2 - Measuring the Relationship between 2 Variables...

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Measuring the Relationship between 2 Variables Correlation: quantifying the relationship between 2 variables Correlation does not imply causality Causation: moving x leads to a change in y Linear Regression Predicted: Ŷ i = a + bX i LHS variable= intercept + coefficient*RHS variable Actual/Observed: Y i = a + b X i + error Estimating Relationships with Randomized Experiments The p-value of the test tells you the following: If the null hypothesis is true (there’s no effect), what is the probability you would have seen a difference between conditions at least this extreme…just by chance If the p-value is low enough, we can reject the null hypothesis If the p-value higher, we cannot reject the null hypothesis It doesn’t mean the alternative hypothesis is true If p-value < 5%, the difference is significant at the 5% level SEE is a measure of how much the data varies around the regression line We can use it to compare regressions with the same dependent variable The smaller the SEE, the better the fit Confidence intervals 68% of the time, our prediction for Y at a given X will be within ± 1 standard error of the actual value It means, “We are 68% certain that the actual Y lies within ± 1 standard error of the Y predicted 95% of the time, our prediction will be within ± 2 (1.96) standard errors of the actual value The Standard Error of the Coefficient 68% of the time, the true relationship between Y and X will be within ± 1 standard error of the coefficient 95% of the time, the true relationship between Y and X will be within ± 1.96 standard errors of the coefficient T-Statistic/p-value t = b - b H /S.E.(b) t=the number of standard errors away the coefficient b is from the hypothesized value 0 If t-stat is close to 1, p-value is .32 If t-stat is close to 2, p-value is .05 Tstat=the coefficient / std error of coefficient If b is 1 std error away from 0, the t-stat is 1, and we can be 68% certain that the true relationship is not 0. If b is 1.96 std errors away from 0, the t-stat is 1.96, and we can be 95% certain that the true relationship is not 0. If absolute value of t > 1.96, p-value < 0.05 If absolute value of t < 1.96, p-value > 0.05 p-value < 5% statistically significant at the 5% level The p-value is the probability we will be wrong when we reject the hypothesis that the true
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value of the coefficient is 0 If the p-value is less than .05, we say that we can reject the hypothesis of no relationship between X and Y at the 5% level of significance, or that the coefficient is statistically significant at the 5% level R-squared, Adjusted R-squared, and Choosing the Right Multiple Regression SST= SSR + SSE R 2 = SSR / SST or R 2 = 1- SSE/SST R 2 in a simple regression= square of the correlation between x and y SSR=SST-SSE Total variation=explained variation + unexplained variation SSR is a measure of the predicted Y’s variation around the average of actual Y’s
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This note was uploaded on 02/21/2012 for the course SMG SM 222 taught by Professor Kahn during the Spring '08 term at BU.

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SM222_2 - Measuring the Relationship between 2 Variables...

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