Probability Sampling uses random selection N = number of cases in sampling frame n = number of cases in the sample NCn = number of combinations of n from N f = n/N = sampling fraction

Variations Simple random sampling Stratified random sampling based on random number generation Stratified random sampling divide pop into homogenous subgroups, then simple random sample w/in Systematic random sampling select every kth individual (k = N/n) Cluster (area) random sampling randomly select clusters, sample all units w/in cluster Multistage sampling combination of methods

Nonprobability sampling accidental, haphazard, convenience sampling ... may or may not represent the population well

Measurement ... topics in measurement that we don’t have time to cover ...

Research Design Elements: Samples/Groups Measures Treatments/Programs Methods of Assignment Time

Internal validity the approximate truth about inferences regarding cause-effect (causal) relationships can observed changes be attributed to the program or intervention and NOT to other possible causes (alternative explanations)?

Establishing a Cause-Effect Relationship Temporal precedence Covariation of cause and effect if x then y; if not x then not y if more x then more y; if less x then less y No plausible alternative explanations

Single Group Example Single group designs: Administer treatment -> measure outcome X -> O assumes baseline of “0” Measure baseline -> treat -> measure outcome 0 X -> O measures change over baseline

Single Group Threats History threat Maturation threat Testing threat a historical event occurs to cause the outcome Maturation threat maturation of individual causes the outcome Testing threat act of taking the pretest affects the outcome Instrumentation threat difference in test from pretest to posttest affects the outcome Mortality threat do “drop-outs” occur differentially or randomly across the sample? Regression threat statistical phenomenon, nonrandom sample from population and two imperfectly correlated measures

Addressing these threats control group + treatment group both control and treatment groups would experience same history and maturation threats, have same testing and instrumentation issues, similar rates of mortality and regression to the mean

Multiple-group design at least two groups typically: before-after measurement treatment group + control group treatment A group + treatment B group

Multiple-Group Threats internal validity issue: degree to which groups are comparable before the study “selection bias” or “selection threat”

Multiple-Group Threats Selection-History Threat an event occurs between pretest and posttest that groups experience differently Selection-Maturation Threat results from differential rates of normal growth between pretest and posttest for the groups Selection-Testing Threat effect of taking pretest differentially affects posttest outcome of groups Selection-Instrumentation Threat test changes differently for the two groups Selection-Mortality Threat differential nonrandom dropout between pretest and posttest Selection-Regression Threat different rates of regression to the mean in the two groups (if one is more extreme on the pretest than the other)

Social Interaction Threats Problem: social pressures in research context can lead to posttest differences that are not directly caused by the treatment Solution: isolate the groups Problem: in many research contexts, hard to randomly assign and then isolate

Types of Social Interaction Threats Diffusion or Imitation of Treatment control group learns about/imitates experience of treatment group, decreasing difference in measured effect Compensatory Rivalry control group tries to compete w/treatment group, works harder, decreasing difference in measured effect Resentful Demoralization control group discouraged or angry, exaggerates measured effect Compensatory Equalization of Treatment control group compensated in other ways, decreasing measured effect

Intro to Design/ Design Notation Observations or Measures Treatments or Programs Groups Assignment to Group Time

Observations/Measure Notation: ‘O’ Examples: Body weight Time to complete Number of correct response Multiple measures: O1, O2, …

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 cutoff

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 R O1 X O1,2 Pretest-posttest treatment versus comparison group randomized experimental design

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

Design Notation Example 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 R X 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:

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

The t-test appropriate for posttest-only two-group randomized experimental design See also: paired student t-test for other situations.

Measuring Differences …

Computing the t-value

Computing standard deviation standard deviation is the square root of the sum of the squared deviations from the mean divided by the number of scores minus one variance is the square of the standard deviation

ANOVA One-way analysis of variance

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 Noise None Moderate High R O

Notation If all sample sizes same, use n, and total N = a * n Else N = n1 + n2 + n3

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 If null hypothesis 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 Sources of variation: Task Dosage Interaction Error SOURCE df Sum of Squares Mean Square F p Task 1 47125.3333 47125.3333 384.174 0.000 Dosage 2 42.6667 21.3333 0.174 0.841 TD 2 1418.6667 709.3333 5.783 0.006 ERROR 42 5152.0000 122.6667 TOTAL 47 53738.6667 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

Regression Analysis Equivalent to t-test and ANOVA for post-test only two group factorial design

Regression Analysis Solve overdetermined system of equations for β0 and β1, 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

The Null Case

Main Effect - Time

Main Effect - Setting

Main Effect - Both

Interaction effects

Interaction Effects