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Statistics for Social and Behavioral Sciences Part IV: Causality Randomized Experiments, ANOVA Chapter 12, Section 12.1 Prof. Amine Ouazad

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Some of their claims Does the data back these claims??

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Weight Loss Diet Are weight loss diets effective at reducing weight? 1.Dr Atkins’ new diet revolution. 2.Slim Fast Plan. 3.Weight Watchers. 4.Rosemary Conley’s eat yourself slim and fitness plan.

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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 Randomized Experiments, ANOVA 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

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Coming up “Comparison of Two Groups” Last week. “Univariate Regression Analysis” Last Saturday, Section 9.5. “Association and Causality: Multivariate Regression” Last Saturday, Chapter 10. Monday, Tuesday, Chapter 11 – R Squared, F test. “Randomized Experiments and ANOVA”. Today. Chapter 12. “Wrap Up”. Tomorrow.

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Measuring the Effectiveness of Weight Loss Programs Observational data: – Collect data on a sample of N individuals. – Ask the following questions: Did you follow a weight loss program in the last 6 months? Yes/No What was your weight (kg) 6 months ago? What is your weight (kg) now? – Compute the weight loss for each individual i in the sample. t test of the difference of means.

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Outline 1.BBC Diet Program 2.Results: Are Diets Effective? 3.ANOVA: Are All Diets Equally Effective? Next time:Wrap Up

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Experimental Data BBC Diet Program Identified potential participants via a BBC advertising campaign (television and other form of media). – Participants had a body mass index between 27 and 40. – Lives within 30 miles of a test center. – Aged between 18 and 65. If you take part in the experiment, we will randomly assign you a diet and you will follow that diet that we randomly chose for you. Sample Size N = 210. Started in July 2002.

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Outline 1.BBC Diet Program 2.Results: Are Diets Effective? 3.ANOVA: Are All Diets Equally Effective? Next time:Wrap Up

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Are Diets Effective? Number of individuals in each group

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Is the Atkins Diet Ineffective: “H 0 : Atkins =0” Mean weight loss in the Atkins Diet group: 8.5 kg. Number of participants in the Atkins Diet group: 29 Standard deviation of weight loss in the Atkins Diet group: 6.0 kg 95% confidence interval for mean weight loss? [,] t statistic for mean weight loss? Can we reject the null hypothesis at 95% confidence?

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Is the Atkins Diet as Effective as the Weight Watchers Diet? H 0 : “ Atkins = Weight Watchers ” H a : “ Atkins ≠ Weight Watchers ” See session on Employment Discrimination. Test of the difference of means.

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se 1 = se 2 = se = m 1 -m 2 = t = Can we reject the null hypothesis that the Atkins diet is as effective as the Weight Watchers diet? Is the Atkins Diet (1) as Effective as the Weight Watchers Diet (2) ?

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Outline 1.BBC Diet Program 2.Results: Are Diets Effective? 3.ANOVA: Are All Diets Equally Effective? Next time:Wrap Up

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ANalysis Of VAriance (ANOVA) Question: Are all diets equally effective? H 0 : “ Atkins = Weight Watchers = Slim Fast = Conley ” H a : “At least one diet is different from the others” – Could be for instance that Slim Fast is less effective than another diet (e.g. Slim Fast < Conley ). – Or for instance Conley is more effective than other diets (e.g. Conley < Atkins

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Formally, for K means: H0 : “ 1 = 2 =…= K ” H a : “For at least one pair k,k’ k ≠ k’ ” As usual we need: – The statistic: an F statistic – The sampling distribution of the statistic under the H 0 : and its degrees of freedom. – The p value of the statistic. ANalysis Of VAriance (ANOVA)

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F Statistic Reminder: Variance is the square of the standard deviation. Follows an F distribution with df 1 =G-1 and df 2 =N-G degrees of freedom. Reject the H 0 at 95% confidence level, i.e. 5% significance level, if the p value is lower than The p value is given by ‘display Ftail(df 1,df 2,F)’ in Stata.

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Within group variance We know the standard deviation for each group: 6.0, 7.9, 5.7, 5.5, Therefore we know the variance in each group: The within group is approximately the average of those variances: See section 12.1 of Agresti and Finlay if interested

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Between group variance Take the mean of each diet group: Consider the overall mean across all diet groups: Compute the variance of those means, weighted by the number ng of observations in each diet group: See section 12.1 of Agresti and Finlay if interested

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Applying t tests and ANOVA to Experimental Data We used three different methods: – Confidence interval on a mean. – t test for the equality of a mean to 0. – t test for the equality of two means. – ANOVA for the equality of multiple means. Are diets effective? Yes !!! But not much difference between the 4 different ones we saw.

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Coming up: Wrap Up Coverage for the final just ends right after the F test. Chapter on “Association and Causality”, and “Multivariate Regression”. Make sure you come to sessions and the last recitation. ANOVA is in chapter 12. Important for your future career !!! SundayMonday Multivariate Regression Tuesday Multivariate Regression The F test Wednesday Randomized Experiments and ANOVA Thursday Wrap up RecitationEvening session 7.30pm West Administration 002 Usual class 12.45pm Usual room Evening session 7.30pm West Administration 001 Usual class 12.45pm Usual room

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