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Two types of t -tests:
Independent Measures (last week)
& Repeated Measures
POPULATION Random factors determine which group n subjects Group 1: N 1 subject Group 2: N 2 subject Level 1 of independent variable given to all A subjects (N 1 ) Level 2 of independent variable given to all A subjects (N 2 ) Group 1 subjects measured on dependent variable B Group 2 subjects measured on dependent variable B Compute ŝ 1 ! X 1 ! X (^) 2 Compute ŝ 2 Test your hypothesis H 0 : μ 1 – μ 2 = 0 H 1 : μ 1 – μ 2 ≠ 0 Independent Samples experiment. To examine the effect of variable A on variable B n subjects are selected from the population and split into two groups N 1 and N 2 N 1 + N 2 = n Subjects in each group receive identical treatment except different levels of independent variable A are given to each group Subjects in each group are measured in the same way on the dependent variable B Statistics are computed and hypothesis test carried out to decide if the difference between and is due to sampling variability or effect of A on B. ! X 1 ! X (^) 2 POPULATION N subjects Level 1 of independent variable A administered subjects measured on dependent variable B Compute Test your hypothesis H 0 : μ 1 – μ 2 = 0 H 1 : μ 1 – μ 2 ≠ 0 Repeated measures experiment. To examine the effect of variable A on variable B N subjects are selected from the population Statistics are computed and hypothesis test carried out to decide if the difference between and is due to sampling variability or effect of A on B. ! X 1 ! X (^) 2 Level 2 of independent variable A administered subjects measured on dependent variable B ! D The subjects are first given Level 1 of the independent variable A The same subjects are then given Level 2 of the independent variable A Subjects are measured on dependent variable B. ( and s 1 are computed from these data) ! X 1 Subjects are measured on dependent variable B. ( and s 2 are computed from these data) ! X (^) 2 ! SD
Both types of t -test have one independent variable , with two levels (the
two different conditions of our experiment).
There is one dependent variable (the thing we actually measure).
Example 1: Effects of personality type on a memory test.
-Independent Variable is “personality type ;
Two levels - introversion and extraversion.
-Dependent Variable is memory test score.
Use an independent-measures t -test: measure each subject's memory
score once, then compare introverts and extraverts.
Example 2: Effects of alcohol on reaction-time (RT) performance.
-Independent Variable is “alcohol consumption ;
Two levels - drunk and sober.
-Dependent Variable is RT.
Use a repeated-measures t -test: measure each subject's RT twice,
once while drunk and once while sober.
Accuracy of Olympic marksmen/markswomen Shots fired between heartbeats versus during a heartbeat (repeated measures design) Rationale behind repeated measures the t-test: EXAMPLE: Experiment on the effect of alcohol on task performance (time in seconds). Measure time taken to perform the task for subjects when drunk, and when (same subjects are) sober. Null hypothesis: alcohol has no effect on time taken: variation between the drunk sample mean and the sober sample mean is due to sampling variation. i.e. The drunk and sober performance times are samples from the same population. From the lecture on sampling distribution:
" X =^ "
N
μ = 63 in. σ = 2 in.
4 Condition A Level 1 Condition A Level 2 Participant With Alcohol Without Alcohol Diff. (D) 1 12.4 10.0 2. 2 15.5 14.2 1. 3 17.9 18.0 -0. 4 9.7 10.1 -0. 5 19.6 14.2 5. 6 16.5 12.1 4. 7 15.1 15.1 0. 8 16.3 12.4 3. 9 13.3 12.7 0. 10 11.6 13.1 -1.
! "^ D = Step 1. Add up the differences: Step 2. Find the mean difference:
"^ D^ =^16
D =
" D
N
Step 3. Estimate of the population standard deviation (the standard deviation of the differences):
SD =
" ( D^ #^ D^ )
2
N # 1
S
D
SD
N
Condition A Level 1 Condition A Level 2 Participant With Alcohol Without Alcohol Diff. (D) 1 12.4 10.0 2.4 0.8 0. 2 15.5 14.2 1.3 -0.3 0. 3 17.9 18.0 -0.1 -1.7 2. 4 9.7 10.1 -0.4 -2.0 4. 5 19.6 14.2 5.4 3.8 14. 6 16.5 12.1 4.4 2.8 7. 7 15.1 15.1 0.0 -1.6 2. 8 16.3 12.4 3.9 2.3 5. 9 13.3 12.7 0.6 -1.0 1. 10 11.6 13.1 -1.5 -3.1 9. 16.0 48. ! D " D
( D " D )^2
"^ D = !
( D " D )^2 =
!
D = 16
… Step 3. Estimate of the population standard deviation (the standard deviation of the differences):
SD =
" ( D^ #^ D^ )
2
N # 1
Step 4. Estimate of the population standard error (the standard error of the differences between two sample means):
S
D
S
D
N
Step 5. Hypothesised difference between the sample means. Our null hypothesis is usually that there is no difference between the two sample means. (In statistical terms, the sample means have come from two identical populations): μ D (hypothesised) = 0 Step 6. Work out t : Step 7. "Degrees of freedom" ( df ) are the number of subjects minus one: df = N - 1 = 10 - 1 = 9
t ( observed ) =
(a) Two-tailed test : if we are
predicting a difference between
Level 1 and 2; find the critical
value of t for a "two-tailed" test.
With df = 9, critical value = 2.26.
(b) One-tailed test : if we are
predicting that Level 1 is bigger
than 2, (or 1 is smaller than 2),
find the critical value of t for a
"one-tailed" test. For df = 9,
critical value = 1.83.
TWO-Tailed: t -observed (2.183) is smaller than t -critical (2.26) We fail to reject the null hypothesis t(9) = 2.183, p > 0. ONE-Tailed: t -observed (2.183) is larger than t -critical (1.83) We reject the null hypothesis t(9) = 2.183, p < 0.
Step 8. Find t -critical value of t from a table
Running SPSS (repeated measures t -test)
Running SPSS (repeated measures t -test) Running SPSS (repeated measures t -test)
Interpreting SPSS output (repeated measures t -test)
