Experimental Design, Statistical Analysis CSCI 4800/6800 University of Georgia March 7, 2002 Eileen Kraemer

Research Design Elements: Observations/Measures Treatments/Programs Groups Assignment to Group Time

Observations/Measure Notation: ‘O’ Examples:  Body weight  Time to complete  Number of correct response Multiple measures: O 1, O 2, …

Treatments or Programs Notation: ‘X’ Use of medication Use of visualization Use of audio feedback Etc. Sometimes see X+, X-

Groups Each group is assigned a line in the design notation

Assignment to Group R = random N = non-equivalent groups C = assignment by cutoffs

Time Moves from left to right in diagram

Types of experiments True experiment – random assignment to groups Quasi experiment – no random assignment, but has a control group or multiple measures Non-experiment – no random assignment, no control, no multiple measures

Design Notation Example RO1O1 XO 1,2 RO1O1 Pretest-posttest treatment comparison group randomized experiment

Design Notation Example NOXO NOO Pretest-posttest Non-Equivalent Groups Quasi-experiment

Design Notation Example XO Posttest Only Non-experiment

Goals of design.. Goal:to be able to show causality First step: internal validity: If x, then y AND If not X, then not Y

Two-group Designs Two-group, posttest only, randomized experiment RXO RO Compare by testing for differences between means of groups, using t-test or one-way Analysis of Variance(ANOVA) Note: 2 groups, post-only measure, two distributions each with mean and variance, statistical (non-chance) difference between groups

To analyze … What do we mean by a difference?

Possible Outcomes:

Measuring Differences …

Three ways to estimate effect Independent t-test One-way Analysis of Variance (ANOVA) Regression Analysis (most general) equivalent

Computing the t-value

Computing the variance

Regression Analysis Solve overdetermined system of equations for β 0 and β 1, while minimizing sum of e-terms

Regression Analysis

ANOVA Compares differences within group to differences between groups For 2 populations, 1 treatment, same as t-test Statistic used is F value, same as square of t-value from t-test

Other Experimental Designs Signal enhancers Factorial designs Noise reducers Covariance designs Blocking designs

Factorial Designs

Factorial Design Factor – major independent variable Setting, time_on_task Level – subdivision of a factor Setting= in_class, pull-out Time_on_task = 1 hour, 4 hours

Factorial Design Design notation as shown 2x2 factorial design (2 levels of one factor X 2 levels of second factor)

Outcomes of Factorial Design Experiments Null case Main effect Interaction Effect

The Null Case

Main Effect - Time

Main Effect - Setting

Main Effect - Both

Interaction effects

Interaction Effects

Statistical Methods for Factorial Design Regression Analysis ANOVA

Analysis of variance – tests hypotheses about differences between two or more means Could do pairwise comparison using t- tests, but can lead to true hypothesis being rejected (Type I error) (higher probability than with ANOVA)

Between-subjects design Example: Effect of intensity of background noise on reading comprehension Group 1: 30 minutes reading, no background noise Group 2: 30 minutes reading, moderate level of noise Group 3: 30 minutes reading, loud background noise

Experimental Design One factor (noise), three levels(a=3) Null hypothesis:  1 =  2 =  3 NoiseNoneModerateHigh ROOO

Notation If all sample sizes same, use n, and total N = a * n Else N = n 1 + n 2 + n 3

Assumptions Normal distributions Homogeneity of variance Variance is equal in each of the populations Random, independent sampling Still works well when assumptions not quite true(“robust” to violations)

ANOVA Compares two estimates of variance MSE – Mean Square Error, variances within samples MSB – Mean Square Between, variance of the sample means If null hypothesis is true, then MSE approx = MSB, since both are estimates of same quantity Is false, the MSB sufficiently > MSE

MSE

MSB Use sample means to calculate sampling distribution of the mean, = 1

MSB Sampling distribution of the mean * n In example, MSB = (n)(sampling dist) = (4) (1) = 4

Is it significant? Depends on ratio of MSB to MSE F = MSB/MSE Probability value computed based on F value, F value has sampling distribution based on degrees of freedom numerator (a-1) and degrees of freedom denominator (N-a) Lookup up F-value in table, find p value For one degree of freedom, F == t^2

Factorial Between-Subjects ANOVA, Two factors Three significance tests Main factor 1 Main factor 2 interaction

Example Experiment Two factors (dosage, task) 3 levels of dosage (0, 100, 200 mg) 2 levels of task (simple, complex) 2x3 factorial design, 8 subjects/group

Summary table SOURCE df Sum of Squares Mean Square F p Task Dosage TD ERROR TOTAL Sources of variation: Task Dosage Interaction Error

Results Sum of squares (as before) Mean Squares = (sum of squares) / degrees of freedom F ratios = mean square effect / mean square error P value : Given F value and degrees of freedom, look up p value

Results - example Mean time to complete task was higher for complex task than for simple Effect of dosage not significant Interaction exists between dosage and task: increase in dosage decreases performance on complex while increasing performance on simple

Results