Statistics Continued

Purpose of Inferential Statistics Try to reach conclusions that extend beyond the immediate data Make judgments about whether an observed difference between conditions is a dependable one or one that might have happened by chance Differs from descriptive statistics where we simply describe what's going on in the data

Types of Inferential Statistics Parametric tests 3 assumptions underlying the use of parametric statistics: –scores have been sampled randomly from the population –scores are normally distributed –within-groups variance is homogenous Non-Parametric tests –Tests used when any of the above assumptions are not met

Types of Parametric Tests T-test (t-value) –Use when you have 2 levels of an IV. Versions for between subjects (independent samples) and within subjects (correlated or paired samples) designs ANOVA (F-ratio) –1-way ANOVA –2-way ANOVA (repeated measures; between S’s; mixed) –3-way ANOVA or above ANCOVA (analysis of covariance) MANOVA (multiple DV’s)

Types of Nonparametric Tests Chi-Square –Use when you have a dichotomous variable Mann-Whitney U Test –Use when you have ordinal data. Alternative to a t-test when assumptions underlying t-test are not met.

Parametric vs. Nonparametric Tests Parametric tests are more powerful Nonparametrics should be used when data do not meet assumptions of parametric tests Nonparametrics are not always available for complex designs

Statistical vs. Practical Significance The fact that the means across conditions differ significantly may or may not be important Sometimes statistical and practical significance overlap, sometimes not Alpha levels must be chosen with the goal of the research in mind Replication is sometimes a substitute for statistics Effect sizes versus significance testing?

Bayesian Statistics Is based on probability of outcomes The probability of a hypothesis is determined by a combination of the inherent likeliness of a hypothesis (the prior) and the compatibility of the observed evidence with the hypothesis (the likelihood)

Baye’s Theorem If the evidence doesn't match up with a hypothesis, you’re unlikely to believe it. But if a hypothesis is extremely unlikely a priori, you’re also unlikely to believe it even if the evidence does appear to match up. Provides a way of combining new evidence with prior beliefs.

Example 1 Hypotheses about the nature of a newborn baby of a friend, including: –the baby is a brown-haired boy –the baby is a blond-haired girl –the baby is a dog

Example 2 Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent years, it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. When it doesn't rain, he incorrectly forecasts rain 10% of the time. What is the probability that it will rain on the day of Marie's wedding?

Solution: The sample space is defined by two mutually-exclusive events - it rains or it does not rain. Additionally, a third event occurs when the weatherman predicts rain. Notation for these events appears below. Event A 1. It rains on Marie's wedding. Event A 2. It does not rain on Marie's wedding. Event B. The weatherman predicts rain.

In terms of probabilities, we know the following: P( A 1 ) = 5/365 =0.014 [It rains 5 days out of the year.] P( A 2 ) = 360/365 = [It does not rain 360 days out of the year.] P( B | A 1 ) = 0.9 [When it rains, the weatherman predicts rain 90% of the time.] P( B | A 2 ) = 0.1 [When it does not rain, the weatherman predicts rain 10% of the time.]

We want to know P( A 1 | B ), the probability it will rain on the day of Marie's wedding, given a forecast for rain by the weatherman. The answer can be determined from Bayes' theorem, as shown below. P( A 1 | B ) = P( A 1 ) P( B | A 1 ) / P( A 1 ) P( B | A 1 ) + P( A 2 ) P( B | A 2 ) P( A 1 | B ) = (0.014)(0.9) / [ (0.014)(0.9) + (0.986)(0.1) ] P( A 1 | B ) = = 11% chance of rain Despite the weatherman's gloomy prediction, there is a good chance that Marie will not get rained on at her wedding.

Visualizing Data

Fisher (1925) “A well constructed graph can be worth a thousand p values.” Friedman (2008) “The main goal of data visualization is to communicate information clearly and effectively through graphical means.” visual data

netflix

Multidimensional Scaling (MDS) Statistical technique for studying the structural attributes of objects or people. Data consists of measures of proximity between pairs of objects “A family of geometric models for multidimensional representation of data and a corresponding set of methods for fitting such models to actual data” (Carroll & Arabic, 1980)

Typically, Ps asked to rate on a scale how similar they believe two items in a pair are Can rate pictures of things, visual stimuli, auditory stimuli, semantic meanings of words, etc., etc. e.g., “rate on 1-7 scale how similar the words “angry” and “sad” are

Graphical Techniques Begin with numerical data or image data Extract data dimensions (1-D, 2-D, 3-D or more) Plot and map the data Color image and volume rendering Animation Virtual/augmented reality medical data sci datamedical datasci data pesudo-tactile

Dynamic Visualization ocean currents data over time animated charts misc data