Probability Sampling

Probability Sampling

uses random selection N = number of cases in sampling framen = number of cases in the sample

NCn = number of combinations of n from Nf = n/N = sampling fraction

Variations

Simple 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 precedenceCovariation 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

outcome0 X -> O

measures change over baseline

Single Group Threats

History 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

outcomeMortality 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 groupstypically: 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 differentlySelection-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

groupsSelection-Instrumentation Threat

test changes differently for the two groupsSelection-Mortality Threat

differential nonrandom dropout between pretest and posttestSelection-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 MeasuresTreatments or ProgramsGroupsAssignment to GroupTime

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 = randomN = non-equivalent groupsC = assignment by cutoff

Time

Moves from left to right in diagram

Types of experiments

True experiment – random assignment to groupsQuasi experiment – no random assignment, but has a control group or multiple measuresNon-experiment – no random assignment, no control, no multiple measures

Design Notation ExampleR O1 X O1,2

R O1 O1,2

Pretest-posttest treatment versus

comparison group

randomized experimental design

Design Notation Example

N O X O

N O O

Pretest-posttest

Non-Equivalent Groups

Quasi-experiment

Design Notation ExampleX O

Posttest Only

Non-experiment

Goals of design ..

Goal:to be able to show causalityFirst 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 O

R O

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-testOne-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 meansCould 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 O O

Notation

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

Assumptions

Normal distributions

Homogeneity of variance Variance is equal in each of the

populations

Random, independent samplingStill 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 * nIn example, MSB = (n)(sampling dist) = (4) (1) = 4

Is it significant?

Depends on ratio of MSB to MSEF = MSB/MSEProbability 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 valueFor 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 tableSOURCE df Sum of Squares Mean Square F pTask 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 freedomF ratios = mean square effect / mean square errorP 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 simpleEffect of dosage not significantInteraction 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 groupsFor 2 populations, 1 treatment, same as t-testStatistic 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 shown2x2 factorial design (2 levels of one factor X 2 levels of second factor)

Outcomes of Factorial Design Experiments

Null caseMain effectInteraction Effect

The Null Case

The Null Case

Main Effect - Time

Main Effect - Setting

Main Effect - Both

Interaction effects

Interaction Effects