Dependent-Samples t Test Lecture PDF

Dependent-Samples t-Test Analyzing Two-sample Matched-Groups and Within-Subjects Designs  Comparison of Between-Subjects Designs: Independent and Matched Samples Independent Samples Design Matched Samples Design  Different groups of  Different groups of...

Dependent-Samples t-Test Analyzing Two-sample Matched-Groups and Within-Subjects Designs  Comparison of Between-Subjects Designs: Independent and Matched Samples Independent Samples Design Matched Samples Design  Different groups of  Different groups of participants receive different participants receive different levels of the IV levels of the IV  Each participant serves in  Each participant serves in only one condition only one condition  Independent samples are  Dependent samples are used in each condition used in each condition  Participants are selected  Participants are matched without regard to who is in to someone in the other the other condition condition on variable(s) correlated with the DV  Comparison of Between-Subjects and Within-Subjects Design Between-Subjects Design Within-Subjects Design  Different groups of  One group of participants participants receive receives every level of different levels of the IV the IV  Each participant serves in  Each participant serves only one condition of the IV in all conditions of the IV  Independent or matched  The same sample is samples are used in each used in each condition condition Appropriate Statistics?  Independent-Samples Design  independent-samples t test  Matched-Groups Design  dependent-samples t test  Within-Subjects Design  dependent-samples t test Dependent-Samples t Tests  Independent-Sample Dependent-Samples t-Test t-Test  Randomly selected  Randomly selected samples samples  DV normally distributed  DV normally distributed  DV measured using  DV measured using ratio or interval scale ratio or interval scale  Homogeneity of  Homogeneity of variance variance  Requires dependent samples (and equal n’s)  General Model for z-Test and Single-Sample t-Test Original H0 Population Sample HA Treated Population General Model for Independent-Samples t-Test Population A Sample A (Control/ H0 Original) Population B Sample B (Experimental/ Treated) H 0 : 1 - 2 = 0 General Model for Independent-Samples t-Test Population A Sample A (Control/ Original) HA Population B Sample B (Experimental/ Treated) H A : 1 - 2  0  General Model for Dependent-Samples t-Test Independent-Samples t-Test:  compare the mean of Group 1 with the mean of Group 2 to determine if they are equal (H0) or different (HA) Dependent-Samples t-Test:  1. find the difference for each pair of scores (D = X – X ) 1 2  2. calculate the mean of the difference (D )  3. determine if the mean difference differs from 0 General Model for Dependent-Samples t-Test Difference Scores of Population A (Control/ H 0: μ D = 0 Original) D Difference Scores HA : μD ≠ 0 of Population B (Experimental/ Treated)  Definitional Formulas Single-Sample Dependent-Samples t-Test t-Test X  D  D tobt  tobt  sX sD  t-Tests Formulas Single-Sample t-Test sample mean  population mean tobt  estimated standard error Dependent-Samples t-Test sample mean difference  population mean difference tobt  estimated standard error of the mean difference  Definitional Formulas Single-Sample Independent-Samples t-Test t-Test X  ( X 1  X 2 )  ( 1   2 ) tobt  tobt  sX s X1  X 2  Single-Sample Dependent- Samples t-Test t-Test s 2   (X  X) 2 s 2   (D  D ) 2 D Step 1: X n 1 n 1 2 2 Step 2: s s sX  X sD  D n n Step 3: X   D  D t obt  t obt  sX sD Dependent-Samples t-Test s 2   (D  D ) 2 Calculate the estimated variance of the population of difference D Step 1: n 1 scores 2 Step 2: s sD  D Calculate the estimated standard error of the mean difference n Step 3: D  D t obt  Calculate t-obtained sD Hypothesis Testing 1. State the hypotheses. 2. Set the significance level   =.05. Determine tcrit. 3. Select and compute the appropriate statistic. 4. Make a decision. 5. Report the statistical results. 6. Write a conclusion. Hypothesis Testing with Dependent Samples An Example  Research Question: Which reinforcement schedule elicits more correct responses from pigeons?  A total of 16 pigeons from 8 clutches (2 pigeons from each clutch)  From each clutch, 1 pigeon is assigned to reinforcement schedule A and 1 is assigned to schedule B Step 1. State the hypotheses. A. Is it a one-tailed or two-tailed test?  Two-tailed B. Research hypotheses  Alternative hypothesis: Pigeons in Condition A will differ in the number of correct responses from pigeons in Condition B.  Null hypothesis: Pigeons in Condition A will not differ in the number of correct responses from pigeons in Condition B. C. Statistical hypotheses:  HA: D ≠ 0  H0: D = 0 Step 2. Set the significance level   =.05. Determine tcrit. Factors That Must Be Known to Find tcrit 1. Is it a one-tailed or a two-tailed test?  two-tailed 2. What is the alpha level? .05 3. What are the degrees of freedom?  df = ? Degrees of Freedom Single-Sample Dependent-Samples t-Test t-Test  df = n – 1  df = n – 1  n = # of scores  n = # of pairs Degrees of Freedom Dependent-Samples t-Test Example: df = n – 1 =8–1 =7  Step 3. Select and compute the appropriate statistic. Dependent-Samples t-Test 2 s   (D  D ) 2 Calculate the estimated variance of the population. Step 1: D n 1 2 Step 2: s Calculate the estimated standard sD  D error of the mean difference n Step 3: D  D Calculate t-obtained t obt  sD Schedule Schedule D A B D  D (D  D )2 6 4 8 6 5 2 7 6 5 3 7 6 5 6 7 7 ΣX = ΣX = X = X = D=  (D  D ) 2  Schedule Schedule D D  D (D  D )2 A B 6 4 2.75.5625 8 6 2.75.5625 5 2 3 1.75 3.0625 7 6 1 -.25.0625 5 3 2.75.5625 7 6 1 -.25.0625 5 6 -1 -2.25 5.0625 7 7 0 -1.25 1.5625 ΣX = 50 ΣX = 40  (D  D ) 2 11.5 X = 6.25 X = 5 D = 1.25   ( D  D ) 2 s 2 D  D 2 s  D n 1 sD  D tobt  n sD 11.5  1.643 1.25  0 7   8 0.453 1.643 .205 tobt 2.76 .453 Step 4. Make a decision.  Determine whether the value of the test statistic is in the critical region. Draw a picture. tcrit = ??? tcrit = ??? Step 4. Make a decision.  Determine whether the value of the test statistic is in the critical region. Draw a picture. tcrit = -2.365 tcrit = +2.365 tobt = 2.76  Decision? +tobt > +tcrit  Reject Ho Step 5. Report the statistical results. t(7) = 2.76, p <.05 Does this indicate that you retain or reject the null hypothesis? What does it mean to say that p <.05? Step 6: Write a conclusion.  State the relationship between the IV and the DV in words:  Pigeons in Condition A (M = 6.25) made significantly more correct responses than pigeons in Condition B (M = 5), t(7) = 2.76, p <.05. Step 6: Write a conclusion.  State the relationship between the IV and the DV in words:  Schedule A resulted in an average of M = 1.25 more correct responses than schedule B. There was a significant difference in performance between the two schedules, t(7) = 2.76, p <.05. Step 7. Compute the estimated d. D estimated d  sD Step 7. Compute the estimated d. D estimated d  sD 1.25  1.282 .98 Effect Size d Effect Size 0.2 Small effect 0.5 Medium effect 0.8 Large effect Step 8. Compute r2 and write a conclusion. 2 2 t r  2 t  df Step 8. Compute r2 and write a conclusion. 2 2 t r  2 t  df 2 (2.76) 7.618 7.618  2   0.5211 (2.76)  7 7.618  7 14.618 Step 8. Compute r2 and write a conclusion.  The reinforcement schedule can account for 52.11% of the variance in the difference in number of correct responses between the two conditions. Simpler:  The reinforcement schedule can account for 52.11% of the variance in the difference scores. Percentage of Variance Explained (r2) r2 Percentage of Variance Explained 0.01 Small effect 0.09 Medium effect 0.25 Large effect