Building Statistical Models

1. Building Statistical Models Lesson 4

2. Theories & Models • Theories • Describe, explain, & predict real-world events/objects • Models • Replicas of real-world events/objects • Can test predictions ~

3. Models & Fit • Model not exact replica • Smaller, simulated • Sample • Model of population • Introduces error • Fit • How well does model represent population? • Good fit  more useful ~

4. Models in Psychology • My research model • Domestic chicks • Effects of pre-/postnatal drug use • Addiction & its consequences • Who/What do most psychologists study? • Rats, pigeons, intro. psych. students • External validity • Good fit with real-world populations? ~

5. The General Linear Model • Relationship b/n predictor & outcome variables form straight line • Correlation, regression, analysis of variance • Other more complex models ~

6. The Mean as a Statistical Model • Very simple model • 1 number represents all the observations • Often hypothetical value e.g., mean # friends = 2.6 • Error introduced • Actual # friends = mean + error • Deviation (deviance) • ~

7. Assessing the Fit of the Mean • How well does it represent all observations? • On average near or far from mean? Distance from mean • Or width of distribution

8. 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 90 90 Mean Daily Temperature m For which group is the mean a better fit for the data? m

9. Measures of Variability • Deviation: for a single score • Range • Highest value – lowest value + 1 • Standard deviation • Conceptually: mean of all deviation scores • average distance of scores from mean • Variance • Used to calculate standard deviation • Also used in analysis of variance ~

10. Calculating the Standard Deviation Xi -m • Why only conceptually mean of deviation scores? • If • What is mean deviation? • S(Xi – m) = 0 ~ Xi 1 2 3 4 5

11. Variability: Notation & Formulas • 3 steps to standard deviation • Sums of squares (squared deviations) • SS = S(Xi – m)2 • Variance = mean of squared deviations (MS) • square root of variance = standard deviation ~

12. Standard Deviation (SD) • Conceptually mean deviation score for all data • Gives width (dispersion) of distribution • Describing a distribution • Report mean & standard deviation • m, s ~

13. Samples & Variability • Usually study samples • to learn about populations • Sampling introduces error • Change symbols & formula

14. Samples: Degrees of Freedom (df) • df = N – 1 • For a single sample (or group) • s tends to underestimate s • Fewer Xi used to calculate • Dividing by N-1 boosts value of s • Also used for • Confidence intervals for sample means • Critical values in hypothesis testing ~

15. Degrees of Freedom: Extra • Don’t lose any sleep over this • df theory • If n= 4 & sample mean = 10 • 3 of Xi can be any value, 4th can be only one value • See Jane Superbrain 2.2 (pg 37) ~

16. Level Of Measurement & Variability • Which can be used? • nominal • none • ordinal • range only • interval/ratio • all 3 OK • range, standard deviation, & variance ~

17. Statistical Models • Representation of the population • We will focus on linear models • Mean is a simple model • One number represents all data • Both • Standard deviation • measures fit of model • Better fit  more useful • Smaller ~