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PSY 1950t-tests, one-way ANOVA
October 1, 2008
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vs
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0
2
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2 12 22 32 42 52 62 72 82 92
sample N
mean sampling statistic
sample SQRT(SS/N) sample SQRT(SS)/Npopulation SQRT(SS/N) population SQRT(SS)/N
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History of the t-test
• William Gosset– Statistician, brewer at Guinness factory
• Which variety of barley is best?– Small samples, no known population – Student. (1908). The probable error of a mean. Biometrika, 6, 1–25.
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From z to tOne sample z-testNull hypothesized
Known 2
sample mean - population mean
standard error
One sample t-test
Null hypothesized Unknown 2
sample mean - population mean
estimated standard error Use s2 for 2
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The Sampling Distribution of s2
• s2 is unbiased estimator of 2
– mean s2 = 2
• But sampling distribution of s2 is positively skewed, especially for small samples
• Because of this, odds are that an individual s2 underestimates 2, especially for small samples
• Thus, on the average, t > z, especially for small samples
• Can’t use z-distribution to determine p for t
• Must devise new distribution that takes into account sample size
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http://www.uvm.edu/~dhowell/SeeingStatisticsApplets/TvsZ.html
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Psychologists are Naughty Brewers
• Pearson to Student/Gosset in 1912:
“only naughty brewers take n so small that the difference is not on the order of the probable error!”
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Assumptions1. Normality (of population, not
sample)2. Independence of observations
(within sample)
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Tails• Two-tailed test
– p <. 025 in both tails– Conservative, conventional
• One-tailed test– p < .05 in predicted tail– A priori, justifiable directional
hypothesis?
• The one-and-a-half tailed test– p <. 05 in predicted tail– p <. 025 in unpredicted tail– Un-ignorable “wrong-tailed” result?
• The lopsided test– p <. 05 in predicted tail– p <. 005 in unpredicted tail
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From 1-sample t to 2-sample t
One sample t-test
Null hypothesized Unknown 2
sample mean - population mean
estimated standard error Use s2 for 2
Two sample t-testNull hypothesized = 1-2 Unknown 2
= 12 + 2
2 sample mean dif - population
mean difestimated standard error
Use s12 and s2
2 for 12
and 22
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Standard Error of the Difference Between Means
• Variances add: the variance of x minus y = the variance of x plus the variance of y– Only true if x and y are uncorrelated
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Assumptions1. Normality (of populations, not
samples)2. Independence of observations (within
and between samples)• Dependence due to groups
• Sampling• Shared history• Social interaction
• Dependence due to time/sequence• e.g., psychophysical variables
• Dependence due to space• e.g., city blocks
3. Homogeneity of variance (of populations, not samples)– Okay so long as one variance isn’t more
than 4 times the other, and samples sizes are approximately equal
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ANOVA• Analysis of variance
– Comparing variance between sample means with variance within samples means• Variancewithin = noise
• Variancebetween = noise + possible signal
• Omnibus test– Are there any differences in means between populations?
– H0: 1 = 2 = 3…
– H1: at least one population mean is different from another
• F-ratio = Variancebetween/Variancewithin
– Variancebetween/variancewithin > 1 reject H0
– Variancebetween/variancewithin ≤ 1 retain H1
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ANOVA
Example: 0,1,2;1,2,3;2,3,4
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Assumptions1. Normality (of populations, not
samples)2. Independence of observations
(within and between samples)3. Homogeneity of variance (of
populations, not samples)– Okay so long as one variance isn’t
more than 4 times another, and samples sizes are approximately equal
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Crawford, J. R., & Howell, D. C. (1998). Comparing an individual’s test score
against norms derived from small samples. The Clinical Neuropsychologist, 12, 482-
486.
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Why develop new statistics?• Clinicians often compare an individual’s score to a normative sample that is treated like a population
• Sometimes normative sample is small– Instruments with poor normative data– Demographic considerations decrease n– Local norms are expensive to collect– Case studies can have small comparison groups
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What’s wrong with the z score?
• Z-scores assume that normalized sample is a population
• With small n, sampling distribution of variance can be skewed
• Leads to a greater likelihood of underestimating population SD and overestimating z
• http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html
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Why use the modified t statistic?
• T-statistic allows clinicians to use a small normative sample to estimate population SD
• Formula is almost the same as z-score formula but allows for wider tails
• t = [X1 – XM2] / [s2 √[(N2 + 1) / N2]]
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When should modified t statistic be used?
• Difference is “vanishingly small” when sample size is greater than 250, and not necessarily large even with smaller samples
• Modified t-test should be used with a sample size of less than 50
• Shouldn’t be used when normative data are skewed