Formula for the t-test

Transcription

1 Formula for the t-test: How the t-test Relates to the Distribution of the Data for the Groups

2 Formula for the t-test: Formula for the Standard Error of the Difference Between the Means

3 Formula for the t-test

4 The t-test (Cont d) Determination of whether or not a t- value is large enough to be significant can be determined from a standard table of significance given: t-value alpha level (most commonly.05) df (number of persons in both groups minus 2)

5 The Regression Formula for the t-test (and Also the Post-Test Only Two-Group One-Way Analysis of Variance or ANOVA Model) y i = β 0 + β 1 z i + e i where: y i = Outcome score for the ith unit β 0 = Coefficient for the intercept β 1 = Coefficient for the slope z i = 1 if ith unit is in the treatment group 0 if ith unit is in the control group e i = Residual for the ith unit

6 The Elements of the Equation in the Previous Slide in Graphic Form

7 The t-test: ANOVA and Regression Analysis Analysis of variance (ANOVA) an analysis that estimates the difference between groups on a posttest; can examine main and interaction effects in a factorial design. Yields an F statistic. With two groups, one posttest variable: t 2 = F The t-test, one-way ANOVA, and regression analysis are mathematically equivalent and yield the same results The regression analysis method of the t-test utilizes a dummy variable (z) for treatment t-value for the B 1 coefficient is the same number you would get if you did a t-test for independent groups

8 Factorial Design Analysis The simplest factorial design is the 2 x 2, i.e., two factors, each of which has two levels A dummy variable (z) is used for each factor Results in two main effects and one interaction effect

9 Regression Model for a 2 x 2 Factorial Design y i = β 0 + β 1 z 1i + β 2 z 2i + β 3 z 1i z 2i + e i where: y i = Outcome score for the ith unit β 0 = Coefficient for the intercept β 1 = Mean difference on factor 1 β 2 = Mean difference on factor 2 β 3 = Interaction of factor 1 and factor 2 z 1i = Dummy variable for factor 1 (0 = 1 hour/week, 1 = 4 hours/week) z 2i = Dummy variable for factor 2 (0 = in-class, 1 = pull-out) e i = Residual for the ith unit

10 Regression Model for a Randomized Block Design y i = β 0 + β 1 z 1i + β 2 z 2i + β 3 z 3i + β 4 z 4i + e i where: y i = Outcome score for the ith unit β 0 = Coefficient for the intercept β 1 = Mean difference for treatment β 2 = Blocking coefficient for block 2 β 3 = Blocking coefficient for block 3 β 4 = Blocking coefficient for block 4 z 1i = Dummy variable for treatment (0 = 1 control, 1 = treatment) z 2i = 1 if block 2, 0 otherwise z 3i = 1 if block 3, 0 otherwise z 4i = 1 if block 4, 0 otherwise e i = Residual for the ith unit

11 Regression Model for the ANCOVA y i = β 0 + β 1 x i + β 2 z i + e i where: y i = Outcome score for the ith unit β 0 = Coefficient for the intercept β 1 = Pretest coefficient x i = Covariate β 2 = Mean difference for treatment z i = Dummy variable for treatment (0 = control, 1 = treatment) e i = Residual for the ith unit

12 Quasi-Experimental Analysis Principles are the same. The general linear model is still used. Analyses become more complex because of adjustment for nonrandom assignment to groups.

13 Nonequivalent Groups Analysis Bias in ANCOVA with nonrandom assignment is due to: Pretest measurement error Pretest nonequivalence of groups Requires adjustment of the pretest scores for the amount of measurement error

14 Formula for Adjusting Pretest Values for Unreliability in the Reliability-Corrected ANCOVA x adj = x + r (x x) where: x adj = Adjusted pretest value x = Original pretest value r = Reliability

15 The Regression Model for the Reliability-Corrected ANCOVA for the Nonequivalent Groups Design (NEGD) where: y i = β 0 + β 1 x adj + β 2 z i + e i y i = Outcome score for the ith unit β 0 = Coefficient for the intercept β 1 = Pretest coefficient β 2 = Mean difference for treatment x adj = Transformed pretest z i = Dummy variable for treatment (0 = control, 1 = treatment) e i = Residual for the ith unit

16 Regression-Discontinuity Analysis Includes a term for the pretest, one for the posttest, and a dummy-coded variable to represent the program Analysis complication has to do with the cutoff point Requires adjustment of the pretest scores in order to center the regression model on the pretest cutoff

17 Adjusting the Pretest by Subtracting the Cutoff in the Regression-Discontinuity (RD) Analysis Model x i = x i x c x i = Pretest cutoff value x i = Pretest x c = Cutoff value

18 The Regression Model for the Basic Regression-Discontinuity Design where: y i = β 0 + β 1 x i + β 2 z i + e i y i = Outcome score for the ith unit β 0 = Coefficient for the intercept β 1 = Pretest coefficient β 2 = Mean difference for treatment x i = Pretest cutoff value z i = Dummy variable for treatment (0 = control, 1 = treatment) e i = Residual for the ith unit

19 The Regression Model for the RPD Design, Assuming a Linear Pre-Post Relationship y i = β 0 + β 1 x i + β 2 z i + e i where: y i = Outcome score for the ith unit β 0 = Coefficient for the intercept β 1 = Pretest coefficient β 2 = Mean difference for treatment x i = Covariate z i = Dummy variable for treatment (0 = control, 1 = treatment [n = 1]) e i = Residual for the ith unit