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Statistical Randomization Tests:Issues and Applications
Randomization Tests versus Permutation Tests
Test Statistic Choice
Complete versus Sampled Randomization Distribution
Randomization Test
1. Random assignment of 3 blocks to each condition
2. Compute observed test statistic
3. Create randomization distribution by computing test statistic
for each of the 20 possible randomizations
4. Compare the obtained test statistic to the randomization
distribution
Permutation Test
1. No random assignment
2. Compute observed test statistic
3. Create permutation distribution by computing test statistic
for data permutations - 20 permutations of blocks of 5 observations?- 155,117,520 permutations of individual observations?
4. Compare the obtained test statistic to the permutation
distribution
Type I error control of randomization tests
Type I error is controlled as long as:
1. Randomization is used in some form 2. Randomization distribution mirrors the
randomization used 3. The test statistic choice is not influenced by
knowledge of treatment assignments
Type I error control of permutation tests
Type I error may or may not be controlled
Type I error can be controlled if one can assume exchangability, but this is generally difficult to assume with time series data.
Consequently, it is statistically preferable to use some form of random assignment and a randomization test than to use a permutation test.
What if you can’t randomize?
Some permutation tests work better than others to control the Type I error rate.
Design: ABABABABABABABABABABABABABABAB
Type of Permutation: individual observations Sample Permutations:
ABABABABABABABABABABABABABABAB
BBAAABBAABABABABABABABABABBAAB
AAABABABABABABABBBABABABABABAB
This test will become conservative with positive autocorrelation
What if you can’t randomize?
Design: AAAAABBBBBAAAAABBBBBAAAAABBBBB
Type of Permutation: blocks of 5 observations
Sample Permutations: AAAAABBBBBAAAAABBBBBAAAAABBBBB
AAAAAAAAAABBBBBBBBBBAAAAABBBBB
BBBBBAAAAAAAAAABBBBBBBBBBAAAAA
This tests will become conservative with positive autocorrelation
What if you can’t randomize?
Design: AAAAAAABBBBBBBAAAAAAABBBBBBB
Type of Permutation: start points (say minimum phase length = 5)
Sample Permutations: AAAAAAABBBBBBBAAAAAAABBBBBBB
AAAAAAAABBBBBAAAAAAAAABBBBBB AAAAABBBBBBBBAAAAAAABBBBBBBB
This test will become conservative with positive autocorrelation
Choice of Test Statistic
AB MMT
Anticipated increase in level:
AB MedianMedianT
0
0.2
0.4
0.6
0.8
1
Choice of Test Statistic
AB MMT *
Delayed increase in level:
*BM
0
0.2
0.4
0.6
0.8
1
is mean of last n observations in B
Choice of Test Statistic
AB bbT
Change in slope:
0
0.2
0.4
0.6
0.8
1
Choice of Test Statistic
B
A
s
sT
Change in variation:
0
0.2
0.4
0.6
0.8
1
Complete versus Sampled Randomization Distribution
Complete – Randomization distribution is constructed by systematically enumerating all possible random assignments and computing the test statistic for each.
Sampling – Randomization distribution is constructed by randomly sampling with replacement some large number (e.g., 1000) of the possible assignments, and then computing the test statistic for each sampled assignment.
Suppose you have 20 possible randomizations and the observed test statistic is the largest of the 20.
Complete: the p-value = .05
Sampling: the p-value will be approximately .05. Therefore, you will only get a statistically significant (p ≤ .05) result about half the times you run the test.
Research Applications
The Impact of a Computer Network on Pediatric Pain and Anxiety: A
Randomized Control Clinical Trial
Authors: Holden, G., Bearison, D. J., Rode, D. C., Kapiloff, M. F., Rosenberg, G., & Rosenzweig, J.
Publication Date: 2002
Journal: Social Work and Health Care, 36, 21-33
Treatment: Starbright World – a private computer network that allows hospitalized children to interact with other hospitalized children
Outcomes: Pain intensity, pain aversiveness, and anxiety (self report)
Design: Alternating treatment design replicated across participants with random assignment of conditions (B=SBW, A=Control) to observation periods with the restriction that there couldn’t be more than 6 consecutive periods of the same condition.
Sample: 39 Hospitalized Children 7-18 years old that had at least 8 observations and went through at least one reversal (e.g., ABA)
Randomization Test: separate test on each child for each outcome using the mean difference as the test statistic and Single Case Randomization Test (SCRT) software
Promoting Expressive Writing Among Students with Emotional and
Behavioral Disturbance Via Dialogue Journals
Authors: Regan, K. S., Mastropieri, M. A., & Scruggs, T. E.
Publication Date: 2005
Journal: Behavioral Disorders, 31, 33-50
Treatment: individualized dialogue journals in which students and teachers communicated daily about observed social and behavioral issues
Outcomes: Time on task, number of words written, and writing quality
Design: Multiple-baseline design across participants with randomly ordered intervention onset. Baseline lengths of 4, 8, 12, 16, or 20 observations in a series of 26 observations
Sample: 5 students in 6th grade identified as having EBD
Randomization Test: “These differences were also statistically significant from baseline to intervention phases according to a randomization test (p < .001).” How many randomizations were possible? Minimum p-value?
Extending Classwide Social Skills Interventions to At-Risk Minority
Students: A Preliminary Application of Randomization Tests Combined with
Single-Subject Methodology
Authors: Bardon, L. A., Dona, D. P., & Symons, F. J.
Publication Date: 2008
Journal: Behavioral Disorders, 33, 141-152
Treatment: PATHS a violence prevention program
Outcome: % of intervals cooperative play during classroom play time
Design: Replicated AB design across participants with randomly selected intervention start times. For each participant, onset of the intervention was selected randomly from the 6th to 15th observation in a planned 20 observation sequence. Baseline lengths were 6, 8, and 8. Is this random assignment ideal?
Sample: 3 elementary students with at least 3 office behavioral referrals
Randomization Test: “…the proportion of 1,000 randomly sampled data divisions producing a mean difference in the predicted direction at least as large as the experimentally obtained difference was 0.036”
How many randomizations were possible?