Statistics for Social and Behavioral Sciences Part IV: Causality Comparison of two groups Chapter 7 Prof. Amine Ouazad

Statistics Course Outline P ART I. I NTRODUCTION AND R ESEARCH D ESIGN P ART II. D ESCRIBING DATA P ART III. D RAWING CONCLUSIONS FROM DATA : I NFERENTIAL S TATISTICS P ART IV. : C ORRELATION AND C AUSATION : T WO G ROUPS, R EGRESSION A NALYSIS Week 1 Weeks 2-4 Weeks 5-9 Weeks This is where we talk about Zmapp and Ebola! Estimating a parameter using sample statistics. Confidence Interval at 90%, 95%, 99% Testing a hypothesis using the CI method and the t method. Sample statistics: Mean, Median, SD, Variance, Percentiles, IQR, Empirical Rule Bivariate sample statistics: Correlation, Slope Four Steps of “Thinking Like a Statistician” Study Design: Simple Random Sampling, Cluster Sampling, Stratified Sampling Biases: Nonresponse bias, Response bias, Sampling bias

Coming up “Comparison of Two Groups” This Session. “Univariate Regression Analysis” Next Session Saturday. “Association and Causality” Tuesday, Thursday and Extra Session. “Randomized Experiments (Cted), ANOVA”. Last Tuesday and Extra Session. “Robustness Checks and Wrap Up”. Last Thursday.

Outline 1.Randomized controlled trials 2.t test for equality of means Next time:Inference in Univariate Regressions

Do U.S. Employers Discriminate? Employers post job ads. Sometimes mentioning they are an “Equal Opportunity Employer.” Some Employers are federal contractors. Lots of anecdotal evidence… – “In hiring, racial bias is still a problem.” Forbes. – “Protesters allege hiring discrimination by Ferrara Candy”, Chicago Tribune, October 28, But we can’t trust stories… Very very tough question. Should be extra careful. What about causal evidence from statistical data?

Outline 1.Randomized controlled trials 2.t test for equality of means Next time:Inference in Univariate Regressions

Difference of means Two groups: White and African American. – m 1 : sample mean in first group. X 1 i : observation of individual i in group 1. – m 2 : sample mean in the second group. X 2 i : observation of individual i in group 2. The expected value of the difference m 1 -m 2 ? Sampling distribution of the difference m 1 -m 2 ? Standard error of m 1 -m 2 : Standard deviation of the sampling distribution of m 1 -m 2.

Very very similar to the one group. t is also chosen from Table 5.1. Degrees of freedom df given either by the Welch approximation or the Satterthwaite approximation (see end of handout). In general: use of t distribution makes normality assumption on X. Robustness to violations of the normality assumptions, esp. for proportions. Confidence Interval for the Difference of Means

t statistic for the difference of means Built similarly as in the one group case. Can also subtract numerator by v when testing for the equality of the mean to a number v. Methods for hypothesis testing H 0 :  1 =  2 H a :  1 different from  2. Reject the null hypothesis if either: The confidence interval does not include 0 The t statistic is above the t score in absolute value Methods for hypothesis testing H 0 :  1 =  2 H a :  1 different from  2. Reject the null hypothesis if either: The confidence interval does not include 0 The t statistic is above the t score in absolute value

Application Compute the t statistics of the difference for each city. Can you recover the p values using Stata? Can you reject the null hypothesis that the call back rates for White and African American names are equal?

t statisticReject? Can you recover the p values? Using display 2*ttail in Stata. And df approximately n 1 +n 2 – 2. Fill in the t statistics here. Can we reject the null hypothesis?

Resume Quality And Callback Rates

A refinement for sample proportions When X 1 i and X 2 i are variables that take only two values 0 or 1. m 1 and m 2 are sample proportions  1 and  2. Group 1 size : n 1. Group 2 size: n 2. H 0 : “  1 =  2 ” Under the null, the standard deviations of the two samples are equal (  ), and thus: df = n 1 + n 2 – 2 and

Back to Café Firenze? What we did before: – Confidence interval around the mean m 1 of Café Firenze and the mean m 2 of Lebanese Express. – We showed that: Exercise at home: – Can we reject the null hypothesis that the true mean rating  1 of Café Firenze is different from the true mean rating  2 of Lebanese Express?

“In many studies, one group of volunteers will be given an experimental or "test" drug or treatment, while the control group is given either a standard treatment for the illness or an inactive pill, liquid, or powder that has no treatment value (placebo). This control group provides a basis for comparison for assessing effects of the test treatment. In some studies, the control group will receive a placebo instead of an active drug or treatment. In other cases, it is considered unethical to use placebos, particularly if an effective treatment is available. Withholding treatment (even for a short time) would subject research participants to unreasonable risks.” Another application of t tests for the equality of means

Coming up: Reading : Chapter on “Comparing Two Groups”. Next chapter 9 with t tests for slope coefficients. Online quiz this weekend on this material. Session on Saturday at in the same room -> catch up for National Day. Make sure you come to sessions and recitations. For help: Amine Ouazad Office 1135, Social Science building Office hour: Tuesday from 5 to 6.30pm. GAF: Irene Paneda Sunday recitations. At the Academic Resource Center, Monday from 2 to 4pm.

Read only if interested: Degrees of freedom for two groups x and y Satterthwaite’s approximate formula: Welch’s approximate formula: