Independent Measures T-Test

Independent Measures T-Test Quantitative Methods in HPELS 440:210

Agenda • Introduction • The t Statistic for Independent-Measures • Hypothesis Tests with Independent-Measures t-Test • Instat • Assumptions

Introduction • Recall  Single-Sample t-Test: • Collect data from one sample • Compare to population with: • Known µ • Unknown  • This scenario is rare: • Often researchers must collect data from two samples • There are two possible scenarios

Introduction • Scenario #1: • Data from 1st sample are INDEPENDENT from data from 2nd • AKA: • Independent-measures design • Between-subjects design • Scenario #2: • Data from 1st sample are RELATED or DEPENDENT on data from 2nd • AKA: • Correlated-samples design • Within-subjects design

Independent-Measures t-Test • Statistical Notation: • µ1 + µ2: Population means for group 1 and group 2 • M1 + M2: Sample means for group 1 and group 2 • n1 + n2: Sample size for group 1 and group 2 • SS1 + SS2: Sum of squares for group 1 and group 2 • df1 + df2: Degrees of freedom for group 1 and group 2 • Note: Total df = (n1 – 1) + (n2 – 1) • s(M1-M2): Estimated SEM

Independent-Measures t-Test • Formula Considerations: • t = (M1-M2) – (µ1-µ2) / s(M1-M2) • Recall  Estimated SEM (s(M1-M2)): • Sample estimate of a population  always error • SEM measures ability to estimate the population • Independent-Measures t-test uses two samples therefore: • Two sources of error • SEM estimation must consider both • Pooled variance (s2p) • SEM (s(M1-M2)): • s(M1-M2) = √s2p/n1 + s2p/n2 where: • s2p = SS1+SS2 / df1+df2

Independent-Measures Designs • Static-Group Comparison Design: • Administer treatment to one group and perform posttest • Perform posttest to control group • Compare groups X O O

Independent-Measures Designs • Quasi-Experimental Pretest Posttest Control Group Design: • Perform pretest on both groups • Administer treatment to treatment group • Perform posttests on both groups • Compare delta (Δ) scores O X O  Δ O O  Δ

Independent-Measures Designs • Randomized Pretest Posttest Control Group Design: • Randomly select subjects from two populations • Perform pretest on both groups • Administer treatment to treatment group • Perform posttests on both groups • Compare delta (Δ) scores R O X O  Δ R O O  Δ

Hypothesis Test: Independent-Measures t-Test • Recall  General Process: • State hypotheses • State relative to the two samples • No effect  samples will be equal • Set criteria for decision making • Sample data and calculate statistic • Make decision

Hypothesis Test: Independent-Measures t-Test • Example 10.1 (p 317) • Overview: • Researchers are interested in determining the effect of mental images on memory • The researcher prepares 40 pairs of nouns (dog/bicycle, lamp/piano . . .) • Two separate groups (n1=10, n2=10) of people are obtained • n1 Provided 5-minutes to memorize the list with instructions to use mental images • n2 Provided 5-minutes to memorize the list

Hypothesis Test: Independent-Measures t-Test • Researchers provide the first noun and ask subjects to recall second noun • Number of correct answers recorded • Questions: • What is the experimental design? • What is the independent variable? • What is the dependent variable?

Step 1: State Hypotheses Non-Directional H0: µ1 = µ2 H1: µ1≠ µ2 Directional H0: µ1≤ µ2 H1: µ1 > µ2 Degrees of Freedom: df = (n1 – 1) + (n2 – 1) df = (10 – 1) + (10 – 1) = 18 Critical Values: Non-Directional  2.101 Directional  1.734 Step 2: Set Criteria Alpha (a) = 0.05 1.734

Step 3: Collect Data and Calculate Statistic Pooled Variance (s2p) s2p = SS1 + SS2 / df1 + df2 s2p = 200 + 160 / 9 + 9 s2p = 360 / 18 s2p = 20 SEM (s(M1-M2)) s(M1-M2) = √s2p / n1 + s2p / n2 s(M1-M2) = √20 / 10 + 20 / 10 s(M1-M2) = √2 +2 s(M1-M2) = 2 t-test: t = (M1-M2) – (µ1-µ2) / s(M1-M2) t = (25-19) – (0-0) / 2 t = 6 / 2 = 3 Step 4: Make Decision Accept or Reject?

Instat • Type data from sample into a column. • Label column appropriately. • Choose “Manage” • Choose “Column Properties” • Choose “Name” • Choose “Statistics” • Choose “Simple Models” • Choose “Normal, Two Samples” • Layout Menu: • Choose “Two Data Columns”

Instat • Data Column Menu: • Choose variable of interest • Parameter Menu: • Choose “Mean (t-interval)” • Confidence Level: • 90% = alpha 0.10 • 95% = alpha 0.05

Instat • Check “Significance Test” box: • Check “Two-Sided” if using non-directional hypothesis. • Enter value from null hypothesis. • If variances are unequal, check appropriate box • More on this later • Click OK. • Interpret the p-value!!!

Reporting t-Test Results • How to report the results of a t-test: • Information to include: • Value of the t statistic • Degrees of freedom (n – 1) • p-value • Examples: • Girls scored significantly higher than boys (t(25) = 2.34, p = 0.001). • There was no significant difference between boys and girls (t(25) = 0.45, p = 0.34).

Assumptions of Independent-Measures t-Test • Independent Observations • Normal Distribution • Scale of Measurement • Interval or ratio • Equal variances (homogeneity): • Violated if one variance twice as large as the other • Can still use parametric  with penalty

Violation of Assumptions • Nonparametric Version  Mann-Whitney U (Chapter 17) • When to use the Mann-Whitney U Test: • Independent-Measures design • Scale of measurement assumption violation: • Ordinal data • Normality assumption violation: • Regardless of scale of measurement

Textbook Assignment • Problems: 3, 11, 19

Independent Measures T-Test