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 Y Biased Slope Coefficient Definition A sample slope coefficient the population slope coefficient is biased if on average it does not equal Put differently is biased if the mean of the sampling distribution for does not equal the population slope coefficient 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 Attitude about divorce Men Women Total Row 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 unacceptable Y independent variable gender Constructing 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 and An interval extending out from both sides of the point estimate The width of the interval extending in both directions is 2 standard errors The point estimate is A 95 confidence level is typical Thus the 95 confidence interval is 2 se where se is the standard error of To 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 2 23 se The 95 confidence interval is approximately 2 se 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 Testing Constructing 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 Variable dichotomous variable binary variable dummy variable A 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 Variable dichotomous variable binary variable dummy variable Dummy variable Trap Y 1X 2D u Where X and Y are continuous variables and D is a dichotomous variable 0 or 1 2 represents the expected change in Y when D increases by one unit and X is held The only one unit increase in D that s possible is from 0 to 1 2 represents the expected change in Y when D increases by one unit and X is held constant 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 Continuous Type Categorical Tabular analysis Probit Logit of Ordinal Cross tabulation Continuous Difference of means Correlation coeff Bivariate regression This chart is on slide 3 of chapter 8 class slides DV Steps Develop a research hypothesis Education and income are positively related in the population i e 0 Establish the null hypothesis There is no relationship between education and income in the population i e 0 Draw a random sample from the population n 32 Measure 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 of 732 Determine the p value Decide whether to reject the null hypothesis Establish the significance level for the test Assess whether the relationship is significant The correlation as a descriptive statistic How strong is the relationship between X and Y in a set of cases Hypothesis testing about the correlation What can we learn about the correlation between X and Y in a population from knowing the correlation between X and Y in a random sample Correlation Coefficient A measure of the extent to which the relationship between two variables approximates a linear relationship Has a sign negative positive 0 and A numerical magnitude between 0 and 1 The magnitude of a correlation ranges between 0 and 1 1 Perfect linear relationship 0 no linear relationship The greater the numerical magnitude the stronger the relationship Interpretation of the intercept in both bivariate regression model and a multiple regression model In the bivariate regression model denotes the expected value of Y when X 0 In the multiple regression model with two independent variables X and Z the intercept is the expected or average or predicted value of Y when X Z zero In the multiple regression model with any number of independent variables the intercept is the expected value of Y when all independent variables are zero In terms of the graph the intercept is the value
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