Building Statistical Models Lesson 4 Theories Models Theories

Building Statistical Models Lesson 4

Theories & Models Theories l Describe, explain, & predict real-world events/objects n Models l Replicas of real-world events/objects l Can test predictions ~ n

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

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

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

The Mean as a Statistical Model n Very simple model l 1 number represents all the observations l Often hypothetical value e. g. , mean # friends = 2. 6 Error introduced l Actual # friends = mean + error n Deviation (deviance) l ~ n

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

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

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

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

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

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

Samples & Variability n Usually study samples u l l to learn about populations Sampling introduces error Change symbols & formula

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

Degrees of Freedom: Extra Don’t lose any sleep over this n df theory l If n= 4 & sample mean = 10 th can be only l 3 of Xi can be any value, 4 one value n See Jane Superbrain 2. 2 (pg 37) ~ n

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

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