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The Difference between bivariate and multiple (VERY IMPORTANT TO KNOW THE DIFFERENCE!):(There will obviously be more differences as you learn about them through the different interpretations, calculations, etc.)Multiple regression extends bivariate regression by including more than one independent variable.A multiple regression model incorporates the effects of more than one independent variable on Y. Multiple regression helps determine whether the relationship between X and Y is spurious by adding other independent variables that may influence both X and Y. An example of Bivariate:Unit of analysis: the local-government employee of “Riverside,” a city of 306 employees.Theory: an increase in education causes an increase in income.Since both variables are continuous, bivariate regression is appropriate.Bivariate regression does not help establish that the relationship between X and Y is not spurious.Even when X has no effect on Y, if X is strongly but spuriously related to Y, a bivariate regression would yield a slope coefficient indicating a strong effect of X on Y.Multiple regression can help demonstrate that a relationship between X and Y is nonspurious.When X has no effect on Y, multiple regression with both X and Z as independent variables should produce a slope coefficient for X equal to 0, correctly indicating no effect of X on YBiased Slope CoefficientDefinition: A sample slope coefficient (β) is biased if, on average, it does not equal the population slope coefficient (β).Put differently, β is biased if the mean of the sampling distribution for β does not equal the population slope coefficient, β.Calculating a predicted value of the dependent variable for specified values of the independent variable(s) (in both bivariate regression model and a multiple regression model) Row Attitude about divorce Men Women Total Views divorce as acceptable 56 (56%) 91 (45.5%) 147 (49%) Views divorce as not acceptable 44 (44%) 109 (54.5%) 153 (51%) Column total 100 200 300 (100%)Percentage difference = 56% – 45.5% = 54.5% – 44% = 10.5%X (dependent variable) = acceptable/unacceptableY (independent variable) = genderConstructing and interpreting a confidence interval for a slope coefficient (in both bivariate regression model and a multiple regression model)Three components:A point estimate (which serves as the center of the confidence interval), The degree of confidence desired, andAn interval (extending out from both sides of the point estimate)The point estimate is β.A 95% confidence level is typical.The width of the interval, extending in both directions, is 2 standard errors.Thus, the 95% confidence interval is β ±2 se(β), where se(β) is the standard error of βTo be technically correct, the width of the confidence interval is not exactly 2 standard errors.For a large sample (n > 100), the width is approximately 2 standard errors.For small samples, the width is wider than 2 standard errors, and depends on the degrees of freedom.For example, when df = 10, the 95% confidence interval is β ±2.23 se(β)The 95% confidence interval is approximately β ±2 se(β)In our example: 732 ± 2(118) = 732 ± 236, i.e., 496 to 968.Interpretation: There is a 95% chance that the population slope coefficient is between 496 and 968.The Connection between Confidence Intervals and Hypothesis TestingConstructing a 95% CI for a statistic and then determining whether the interval contains zero is the equivalent of testing whether the statistic is statistically significant at the .05 level.If the interval does not contain zero, the statistic is statistically significant.If the interval does contain zero, the statistic is statistically insignificant.Dichotomous Variabledichotomous variable = binary variable = dummy variableA dichotomous DV violates the assumptions of regression. With a dichotomous DV, logit and probit are the most common techniques.A linear probability model can also be used.Dummy Variabledichotomous variable = binary variable = dummy variableDummy-variable TrapY = α + β1X + β2D + u ,Where X and Y are continuous variables, and D is a dichotomous variable (= 0 or 1)β2 represents the expected change in Y when D increases by one unit and X is held constant. The only one-unit increase in D that’s possible is from 0 to 1.β2 represents the expected change in Y when D increases by one unit and X is held constant. Hypothesis testing about a slope coefficient (in both bivariate regression model and a multiple regression model)The most appropriate test is determined by the level of measurement of the independent and dependent variables. Type of Independent Variable Categorical/Ordinal ContinuousType Categorical/ Tabular analysis Probit/Logitof Ordinal (Cross-tabulation)DV Continuous Difference of means Correlation coeff. Bivariate regressionThis chart is on slide 3 of chapter 8 class slidesSteps:Develop a research hypothesis: Education and income are positively related in the population (i.e., β > 0)Establish the null hypothesis: There is no relationship between education and income in the population (i.e., β = 0).Draw a random sample from the population: n=32Measure X and Y in the sample: Use years of schooling and annual income.Calculate a statistic measuring the relationship between X and Y in the sample: Bivariate regression yields a sample slope coefficient, β , of 732.Determine the p-valueDecide whether to reject the null hypothesisEstablish the significance level for the testAssess whether the relationship is significantThe correlation as a descriptive statisticHow strong is the relationship between X and Y in a set of casesHypothesis testing about the correlationWhat can we learn about the correlation between X and Y in a population from knowing the correlation between X and Y in a random sampleCorrelation CoefficientA measure of the extent to which the relationship between two variables approximates a linear relationshipHas a sign (negative, positive, 0), andA numerical magnitude between 0 and 1The magnitude of a correlation ranges between 0 and 1:1=Perfect linear relationship0=no linear relationshipThe greater the

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