The t Test for Two Related Samples

Why Might We Have Related Samples? Repeated Measures Repeated Measures A study in which a single sample of individuals is measured more than once on the same dependent variable. The same subjects are used in all of the treatment conditions. A study in which a single sample of individuals is measured more than once on the same dependent variable. The same subjects are used in all of the treatment conditions. Matched Subjects Matched Subjects A study in which each individual in one sample is matched with a subject in the other sample. The matching is done so that the two individuals are equivalent (or nearly equivalent) with respect to a specific variable. A study in which each individual in one sample is matched with a subject in the other sample. The matching is done so that the two individuals are equivalent (or nearly equivalent) with respect to a specific variable.

What Might Our Hypotheses Be About In These Cases? Difference Scores Difference Scores Here we would like to see if the treatment causes a difference in the scores. Here we would like to see if the treatment causes a difference in the scores. Therefore, we subtract the score before the treatment from the score after the treatment. Therefore, we subtract the score before the treatment from the score after the treatment. This yields a difference score. This yields a difference score.

How Do We Calculate The Statistics? We use D as our X: We use D as our X: t = (M D - µ D )/s M D t = (M D - µ D )/s M D s = [SS/(n-1)] = (SS/df) s = [SS/(n-1)] = (SS/df) s M D = s/n = (s 2 /n) s M D = s/n = (s 2 /n)

A Quick Note Have we been using parameters or statistics to state our hypotheses? Have we been using parameters or statistics to state our hypotheses? Why is this? Why is this?

What Are The Degrees of Freedom (df)? As usual, n - 1 As usual, n - 1

Lets Do An Example ID X1X1X1X1 X2X2X2X2D H 0 : µ D = 0 H 1 : µ D 0 df: 6 α =.05 t crit = ±2.447

Assumptions Within treatment scores should be independent Within treatment scores should be independent D scores must be distributed normally in the population D scores must be distributed normally in the population

Threats To Validity What might mess up the validity of this type of design? What might mess up the validity of this type of design? Time related factors Time related factors Counterbalancing Counterbalancing

Effect Size Cohens d = mean difference/standard deviation Cohens d = mean difference/standard deviation 5/3.43 = /3.43 = 1.46 r 2 = Variability accounted for / total variability r 2 = Variability accounted for / total variability r 2 = t 2 /(t 2 + df) r 2 = t 2 /(t 2 + df)

Confidence Intervals Point Estimate Point Estimate Interval Estimate Interval Estimate