Class 4: Tues., Sept. 21

1. Class 4: Tues., Sept. 21 • External/Internal Reliability Clarification • Regression Analysis Examples: • Appropriate Dating Ages • Father’s and son’s heights • Variability of Y given X in the Simple Linear Regression Model

2. Reliability • In general, a measurement is reliable if it gives consistent results. • My distinction between internal/external reliability of a measurement (e.g., a test) was not very precise. Here’s a better categorization. • Four types of reliability for a measurement (degree of reliability can be measured by correlation): • Inter-observer: Different measurements of the same object/information give consistent results (e.g., two psychiatrists rate the behavior of a patient similarly; two Olympic judges score a gymnastics contestant similarly).

3. Types of Reliability Continued 2. Test-retest: Measurements taken at two different times are similar (e.g., a person’s pulse is similar for two different readings) • Parallel form: Two tests of different forms that supposedly test the same material give similar results (e.g., a person’s SAT scores are similar for two forms of the test). • Split-half: If the items on a test are divided in half (e.g., odd vs. even), the scores on the two halves are similar.

4. Examples of Reliability

5. Regression Analysis • Provides a model for the mean of Y given X=X0, E(Y|X=X0) and the variability of Y given X=X0. Useful for understanding the association between Y and X and for predicting Y based on X. • Simple linear regression model: • has a normal distribution with mean 0 and standard deviation

7. Example: What age is too young? • In U.S. culture, an older man dating a younger woman is not uncommon but when the age difference becomes too large, it may seem to some be unacceptable. • A survey was taken of ten people whom were each asked the minimum acceptable age for a woman to be dating a man of a certain age for a range of ages. • Y=minimum acceptable age of woman dating man of X years of age. X=age of man • What is the mean of people’s minimum acceptable for a woman to be dating a man of X years of age, i.e., what is E(Y|X=X0)?

9. Linear Fit Minimum Woman's Age = 5.472037 + 0.5753518 Man's Age • Estimated Mean (among survey population) Minimum Acceptable Age for a Woman dating a man who is • 20 years old: 5.47+0.58*20 = 17.07 • 30 years old: 5.47+0.58*30 = 22.87 • 40 years old: 5.47+0.58*40 = 28.67 • 50 years old: 5.47+0.58*50 =34.47 • 60 years olds: 5.47+0.58*60=40.27 • 70 years old: 5.47+0.58*70 = 46.07

10. Father and Son’s Height • Y=Son’s Height, X=Father’s Height (Galton’s Data from 19th century England)

11. Simple Linear Regression Model for Height Data

12. Estimated regression model: E(Son’s height | Father’s Height ) = 33.89 + 0.51 *Father’s height • Estimated slope = 0.51. For each additional inch of father’s height, the mean son’s height increases by 0.51 inches. • Predicted son’s heights: • Father’s height = 60 inches. Predicted son’s height = 33.89 + 0.51 * 60 = 64.5 inches • Father’s height = 72 inches. Predicted son’s height = 33.89 + 0.51 * 72 = 70.6 inches

13. Variability of Y given X • The simple linear regression model tells us more than the mean of Y given X=X0, it tells us about the variability and distribution of Y given X=X0. • Simple linear regression model: • has a normal distribution with mean 0 and standard deviation (SD) • The subpopulation of Y with corresponding X=X0 has a normal distribution with mean and SD

14. Residuals and Estimating • Estimating • Use least squares to estimate the slope and intercept of the simple linear regression model. Denote the slope estimates by and the intercept estimate by • Predicted value of Yi for observation i based on Xi and regression model estimate: • Residual for observation i: Prediction error of using least squares line to predict Yi for observation i • Root mean square error = (approximately) standard deviation of residuals. Root mean square error is an estimate of • For father-son height data, root mean square error = 2.4. This means that, according to the simple linear regression model, a son whose father is 72 inches has a mean height of 33.89 + .51*72 = 70.6 inches with a standard deviation of 2.4 inches.

15. Normal Distribution • About 68% of the observations from a normal distribution will fall within one standard deviation ( ) of the mean ( ) • About 95% of the observations from a normal distribution will fall within two standard deviations of the mean. • About 99% of the observations will fall within three standard deviations of the mean.

16. Variability of Y given X • According to the estimated regression model, the distribution of heights for sons whose father are 72 inches is a normal distribution with a mean of 70.6 inches and a standard deviation of 2.4 inches. • If a son’s father’s height is 72 inches, • 68% of the time the son’s height will be between inches • 95% of the time, the son’s height will be between inches 99% of the time, the son’s height will be between inches.

17. Summary • Regression model provides information about both the mean of Y given X and the variability of Y given X. • For the simple linear regression model, the standard deviation of Y given X is estimated by the root mean square error. • For the simple linear regression model, approximately 68% of the time, Y given X will be within one root mean square error of the estimated mean of Y given X ( ), approximately 95% of the time, Y given X will be within two root mean square errors of the mean of Y given X.