1. PY 427 Statistics 1Fall 2006 Kin Ching Kong, Ph.D Lecture 9 Chicago School of Professional Psychology

2. Agenda The t Test for Two Related Samples Two Types of Related-Samples The Data for a Related-Samples Study The Hypotheses for a Related-Samples Test Variance and Estimated Standard Error The t Statistic for Related-Samples Hypothesis Testing Directional Tests Measuring Effect Size Assumptions for the Related-Sample t test Introduction to Analysis of Variance (ANOVA)

3. Two Types of Related-Samples (Correlated-Samples) Repeated-Measures or Within-Subjects Design: a single sample is measured more than once on the dependent variable (DV) e.g. depression score before & after a new therapy Matched-Subjects Design: two samples are used, but each individual in one sample is matched with an individual in the other sample based on variables important to the study. e.g. in the above example, two groups of clients, one group selected to receive the new therapy, one group selected to receive the usual treatment/no treatment, are matched one- to-one in terms of age, depression scores at intake, gender, and any other variables important to the study.

4. The Data in a Related-Samples Study The Raw Data: consists of two sets of scores (X 1 and X 2 ). Data for Hypothesis Test: The Difference Scores (D) D = X 2 – X 1 e.g. reaction time before and after taking a cold medication  D = 64 M D = 64/4 = 16 personBefore (X 1 ) after (X 2 ) Difference D A215210-5 B22124221 C19621923 D20322825

5. The Hypotheses for a Related-Samples Test The Null Hypothesis: H 0 :  D = 0 (the mean difference for the population is 0; the difference scores are not consistently positive or negative) The Alternative Hypothesis: H1:  D = 0 (the mean difference for the population is not equal to 0; the different scores are consistently positive or negative)

6. The t Statistic for Related-Samples The basic structure of the t Statistic: t = sample statistic – hypothesized population parameter estimated standard error t = data – hypothesis error The single-sample t t = M –  s M The related-samples t t = M D –  D s M D

7. The Standard Error Estimated standard error measures how accurately the sample statistic represents the population parameter. In other words, how much difference to expect between statistic and parameter by chance alone. s statistic For the matched-samples t, the estimated standard error measures the amount of error that is expected when you use M D to represent  D s M D is calculated the same way as s M, except you use D scores instead of X scores s D 2 = SS/(n-1) =SS/df or s D = s M D = or s M D = s D /

8. Hypothesis Testing (the experiment) Research question: Does relaxation training alters the severity of asthma symptoms? Experiment: Participants: 5 patients suffering from asthma Treatment (IV): All participants received relaxation training. Dependent Variable (DV): Before treatment: how many doses of asthma med. used After treatment: how many doses of asthma med. used

9. Hypothesis Testing with Repeated-Measures (Data)  D = -20  D 2 = 96 M D =  D/n = -20/5 = -4.00 SS =  D 2 – (  D) 2 /n = 96 – (-20) 2 /5 =96 – 80 = 16 PatientBeforeAfterDD2D2 A93-636 B41-39 C50-525 D431 E72-525

10. Hypothesis Testing with Repeated-Measures Step 1: State the Hypotheses: H 0 :  D = 0 (there is no change in symptoms) H 1 :  D = 0 (there is a change) The level of significant is set at  =.05, 2-tailed Step 2: Locate the Critical Region: df = n-1 = 4, t critical = + 2.776 Step 3: Calculate the test statistic: t = M D –  D s M D = s 2 = SS/(n-1) = 16/4 = 4.0 s M D = = 0.894 = -4.00 – 0 = - 4.47 0.894 Step 4: Make a decision: Since the test statistic, t = -4.47 falls in the critical region, reject H 0 and concludes that relaxation training does affect the amount of asthma medication used.

11. Directional Test (One-Tailed Tests) In the previous example, the researcher predicts that people will need less medication after relaxation training. Use  =.05 to test the researcher’s prediction Answer

12. Effect Size for the matched-samples t Cohen’s d = mean difference standard deviation d = M D s For the previous example d = 4.00 = 2.00 2.00 Note: we ignore the negative sign because d measures the effect size, not direction r 2 : percentage of variance explained by the treatment. r 2 = t 2 = (4.47) 2 = 19.98 = 0.833 t 2 + df (4.47) 2 + 4 23.98 So 83.3% of the variance in the difference scores are explained by the treatment effect (i.e. relaxation training).

13. Advantages of the Repeated-Measures Design Fewer participants need to achieve the same power Especially suited to study changes over time Reduces problems caused by individual differences Eliminate systematic difference that can exist between groups of participants (e.g. group 1 has higher IQ than group 2) Reduce error variance, thus increase the chance of finding a significant result. Example Table 11.3 of your bookTable 11.3 of your book The two data sets contain exactly the same numerical scores and show the same 5-point difference between the treatment conditions. For the independent-measures study: the standard error= 5.77, t = 0.87, thus no significant difference between the treatment conditions. For the repeated-measures study: the standard error = 1.15, t = 4.35, thus significant difference between the treatment conditions.

14. Disadvantages of the Repeated-Measures Design Carryover effect A participant’s response in the 2 nd treatment is altered by lingering aftereffects from the first treatment. Example 1: compare 2 drugs: the 2 nd is given when the 1 st is still in the participant’s system. Example 2: compare 2 tasks: if the 1 st one is very difficult, participants might lose motivation, so performance suffers on the 2 nd task. Progressive Error A participant’s performance or response changes consistently over time. Example 1: performance decline over time due to fatigue. Example 2: performance improve over time due to practice. Solutions Counterbalance Matched-samples

15. Assumptions Assumptions underlying the related- samples t: The observations within each treatment condition must be independent. The population of difference scores (D values) must be normally distributed.

16. Analysis of Variance (ANOVA) ANOVA is used to compare two or more means The Question: Does the mean differences observed among the samples reflect mean differences among the populations? Two Possibilities: There really are no differences in the population means, the observed differences are due to chance (sampling error). The population means are truly different, and are partly responsible for differences in the sample means. Figure 13.1

17. Analysis of Variance, Terminology Factor: the variable that designates the groups. Examples: the effectiveness of 3 teaching methods. Factor: teaching methods the STM of 5-year-olds, 10-year-olds, 15-year-olds, 25- year-olds and 40-year-olds Factor: age Levels: the individual conditions or values that make up a factor. Example 1 above has 3 levels Example 2 above has 5 levels