Roger B. Hammer Assistant Professor Department of Sociology Oregon State University Conducting Social Research The Classical Model and Hypothesis Testing

Conducting Social Research Lecture Outline Review of uni-, bi-, and multi-variate statistics. Regression and Hypothesis testing. Sampling Distribution of Regression Coefficients. T-Tests. Confidence Intervals. Model Assumptions Properties of OLS Estimators.

Review of Statistics Univariate to Bivariate to Multivariate Conducting Social Research The observations make the mean.

Conducting Social Research Review of Statistics Univariate to Bivariate to Multivariate The mean makes the variance.

Conducting Social Research Review of Statistics Univariate to Bivariate to Multivariate The variance makes the standard deviation.

Conducting Social Research Review of Statistics Univariate to Bivariate to Multivariate The means make the covariance.

Conducting Social Research Review of Statistics Univariate to Bivariate to Multivariate The covariance makes the correlation.

Conducting Social Research Review of Statistics Univariate to Bivariate to Multivariate The correlation makes the slope.

Conducting Social Research Review of Statistics Univariate to Bivariate to Multivariate The slope makes the intercept.

Conducting Social Research Review of Statistics Univariate to Bivariate to Multivariate The coefficients make the predicted.

Conducting Social Research Review of Statistics Univariate to Bivariate to Multivariate The coefficients make the predictions.

Conducting Social Research Review of Statistics Univariate to Bivariate to Multivariate The predictions make the explained sum of squared deviaitons.

Conducting Social Research Review of Statistics Univariate to Bivariate to Multivariate They also make the residuals.

Conducting Social Research Review of Statistics Univariate to Bivariate to Multivariate The residuals make the residual sum of squared deviations.

Conducting Social Research Review of Statistics Univariate to Bivariate to Multivariate The RSS makes the total sum of squared deviations.

Conducting Social Research The Statistics in the

Conducting Social Research The Statistics in the

Conducting Social Research The Statistics in the

Conducting Social Research Housing Sales Data We have obtained a sample of 40 housing sales that took place somewhere in some year. The data contains two variables, price (in $’s) and size (total above grade finished area in feet 2 ).

Conducting Social Research Housing Sales Data

Conducting Social Research The Residual Standard Deviation

Conducting Social Research The Standard Error of a Bivariate Regression Coefficient (Slope)

Conducting Social Research The Standard Error of a Bivariate Regression Intercept

Conducting Social Research Hypothesis Testing From last week, what were some of our hypotheses concerning “Statistics Class Anxiety”? How would you state that hypothesis in terms of regression analysis? How would the null hypothesis be stated in terms of regression analysis?

Conducting Social Research Null Hypothesis The result that is not expected. Either rejected or not rejected but never accepted. General includes the equals sign in some form. Can estimate the probability of rejecting a null hypothesis that is true.

Conducting Social Research Alternative Hypothesis The result that is expected. Can never be accepted or proven but can be rejected. Can estimate the probability of accepting an alternative hypothesis that is false.

Conducting Social Research Types of Errors Not to be confused with residuals Type I: We reject a true null hypothesis. Type II: We fail to reject a false null hypothesis.

Conducting Social Research Americans find type II errors disturbing but not as horrifying as type I errors. A type I error means that not only has an innocent person been sent to jail but the truly guilty person has also gone free. In a sense, a type I error is twice as bad as a type II error. Needless to say, the American justice system puts a lot of emphasis on avoiding type I errors. This emphasis on avoiding type I errors, however, is not true in all cases where hypothesis testing is done. Justice System - Trial Defendan t Innocent Defendan t Guilty Reject Presumpti on of Innocenc e (Guilty Verdict) Type I Error Correct Fail to Reject Presumpti on of Innocenc e (Not Guilty Verdict) Correct Type II Error Statisti cs - Hypothesis Test Null Hypoth True Null Hypoth False Reject Null Hypothes is Type I Error Correct Fail to Reject Null Hypothes is Correc t Type II Error Defendant Innocent Defendant Guilty Reject Presumption of Innocence (Guilty Verdict) Type I Error Correct Fail to Reject Presumption of Innocence (Not Guilty Verdict) Correct Type II Error Hypothesis Testing Criminal Trial

Conducting Social Research Americans find type II errors disturbing but not as horrifying as type I errors. A type I error means that not only has an innocent person been sent to jail but the truly guilty person has also gone free. In a sense, a type I error is twice as bad as a type II error. Needless to say, the American justice system puts a lot of emphasis on avoiding type I errors. This emphasis on avoiding type I errors, however, is not true in all cases where hypothesis testing is done. Justice System - Trial Defendan t Innocent Defendan t Guilty Reject Presumpti on of Innocenc e (Guilty Verdict) Type I Error Correct Fail to Reject Presumpti on of Innocenc e (Not Guilty Verdict) Correct Type II Error Statisti cs - Hypothesis Test Null Hypoth True Null Hypoth False Reject Null Hypothes is Type I Error Correct Fail to Reject Null Hypothes is Correc t Type II Error Hypothesis Testing Criminal Trial

Conducting Social Research Americans find type II errors disturbing but not as horrifying as type I errors. A type I error means that not only has an innocent person been sent to jail but the truly guilty person has also gone free. In a sense, a type I error is twice as bad as a type II error. Needless to say, the American justice system puts a lot of emphasis on avoiding type I errors. This emphasis on avoiding type I errors, however, is not true in all cases where hypothesis testing is done. Justice System - Trial Defendan t Innocent Defendan t Guilty Reject Presumpti on of Innocenc e (Guilty Verdict) Type I Error Correct Fail to Reject Presumpti on of Innocenc e (Not Guilty Verdict) Correct Type II Error Statisti cs - Hypothesis Test Null Hypoth True Null Hypoth False Reject Null Hypothes is Type I Error Correct Fail to Reject Null Hypothes is Correc t Type II Error Hypothesis Testing Criminal Trial

Conducting Social Research Americans find type II errors disturbing but not as horrifying as type I errors. A type I error means that not only has an innocent person been sent to jail but the truly guilty person has also gone free. In a sense, a type I error is twice as bad as a type II error. Needless to say, the American justice system puts a lot of emphasis on avoiding type I errors. This emphasis on avoiding type I errors, however, is not true in all cases where hypothesis testing is done. Justice System - Trial Defendan t Innocent Defendan t Guilty Reject Presumpti on of Innocenc e (Guilty Verdict) Type I Error Correct Fail to Reject Presumpti on of Innocenc e (Not Guilty Verdict) Correct Type II Error Statisti cs - Hypothesis Test Null Hypoth True Null Hypoth False Reject Null Hypothes is Type I Error Correct Fail to Reject Null Hypothes is Correc t Type II Error Null Hypothesis True Null Hypothesis False Reject Null Hypothesis Type I Error Correct Fail to Reject Null Hypothesis Correct Type II Error Hypothesis Testing Research

Conducting Social Research Sampling Distribution of Regression Coefficients Given a sample of independent and identically distributed (iid) random variables with a finite population mean and a finite, nonzero standard deviation, as the sample size increases: the average of these observations will approach the population mean andthe average of these observations will approach the population mean and the probability distribution will approaches the normal distribution.the probability distribution will approaches the normal distribution. Are regression coefficients iid random variables with the requisite mean and standard deviation properties?

Conducting Social Research Sampling Distribution of Regression Coefficients What is the sampling distribution of regression coefficients?

Conducting Social Research Hypothesis Testing Two-tailed Test

Conducting Social Research Hypothesis Testing One-tailed Test

Conducting Social Research The “Significance” of Regression Coefficients (t-statistic)

Conducting Social Research Confidence Intervals of Regression Coefficients

Conducting Social Research The Regression Model F-statistic The Constrained Model F-statistic

Conducting Social Research Classical Assumptions of OLS I.The regression model is linear, is correctly specified, and has an additive error term. II.The error term has a zero population mean. III.All explanatory variables are uncorrelated with the error term. IV.Observations of the error term are uncorrelated with each other. V.The error term has a constant variance (homoskedasticity). VI.No explanatory variable is a perfect linear function of any other explanatory variable(s). VII.The error term is normally distributed.

Common Assumptions 1)Errors are independent, that is not correlated with X or the errors of other cases. 2)Errors have identical distributions (iid), with mean zero and equal variance (Homoscedasticity) for every value of X. 3)Errors are normally distributed. Conducting Social Research

The Gauss-Markov Theorem If the first six of the classical assumptions are met, the OLS estimators are the Best Linear Unbiased Estimators (BLUE): Minimum Variance (Best) - distribution around the true parameter values as narrow as possible.Minimum Variance (Best) - distribution around the true parameter values as narrow as possible. Unbiased-centered on the population values.Unbiased-centered on the population values. Consistent - As the sample size increases, the estimates converge toward the population value for each coefficient being estimated.Consistent - As the sample size increases, the estimates converge toward the population value for each coefficient being estimated. Normally Distributed.Normally Distributed.