Last time: One-way Analysis of Variance

Example List of 50 spoken words 3 x 10 Subjects (split among I=3 groups) Group 1: (Fast sound) Person in movie reads list, but sounds precede lip movement slightly Group 2: (Slow sound) Person in movie reads list, but sounds lag behind lip movement slightly Group 3: (Synchrony) Person in movie reads list with auditory and visual stimuli in synchrony Memory Task: Subjects are asked to recall as many items as possible.

One-way Analysis of Variance Model Assumptions: I many Independent Groups Data … … Popu lation Sample Size … …

One-way Analysis of Variance

Similar recipe as in Linear Regression! Sum Squares Total (SST) Sum Squares Error (SSE) Sum Squares Groups (SSG) Degrees of Freedom DFT = N-1 Degrees of Freedom DFG = I-1 Degrees of Freedom DFE=N-I = + MSG

Let’s grind it out for our example… MSG Large MSG leads to significant F statistic. Reject Null Hypothesis! Conclusion: The population means are not identical across groups

What if I=2? Remember: The Square of a t Random Variable with n-2 degrees of freedom is an F Random Variable with 1 degree of freedom in the numerator and with n-2 degrees of freedom in the denominator. Thus, the one-way analysis of variance is a natural extension of the comparison of two means from independent samples (with equal population variances).

Robustness If the samples sizes are equal, then the assumption of equal variance (equal standard deviation) is not crucial. CLT helps with violations of normality, i.e. as long as sample sizes are large, we do not need normality of the X variables.

Today: Wrap up “Loose Ends” An Illustrating Example on Simple Regression Typo Correction One last quiz…

etc.

(Rent per square foot) (Square-footage)

Is there significant evidence for a linear relationship? Test using the correlation Test using the slope Test using the ANOVA table

Sample correlation R n n-2 t-stat

Sample correlation R t-stat

Sample correlation R t-stat The correlation is significant at 5% significance level. Yes, significant evidence for a linear relationship.

Observed t-statistics for * * p-value = 95% CIs

Observed t-statistic for * * p-value <.001 Yes, significant evidence for linear relationship 95% CI

p-value <.001 Yes, significant evidence for linear relationship

What is the best fitting regression equation?

“I bet the population intercept is more then 900” This would mean that you pay a fixed minimum flat amount of $900, plus whatever rent you need to pay based on square footage.

I bet, for every additional 10 Square Feet, you have to pay more than an extra $4 Rent! That would mean more than $.4 extra rent per extra square foot. That would mean the slope is >.4.

Significant at 2% significance level. Yes, significant evidence that we pay over $4 extra per 10sqft extra.

For every additional 1,000 Square Feet, how much extra Rent do you have to pay? Give a 95% Confidence Interval

This is our 95% CI for the extra Rent per extra Square Foot. Thus: 95% CI for extra Rent per 1,000 Square Feet: [$407, $496]

What is our best guess at the standard deviation of the Error Term? What percentage of the variance are we able to explain with this model?

SSR = SST-SSE

Prediction Region

Slide Typo Correction: 2x2 Contingency Tables

Special Case: 2x2 Tables This typo occurred in several slides due to cut and pasting.

Last (and special) Quiz Counts as 5 Bonus Points in Grand Total Regression