PSY 1950 t-tests, one-way ANOVA October 1, 2008. vs.
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Transcript of PSY 1950 t-tests, one-way ANOVA October 1, 2008. vs.
PSY 1950t-tests, one-way ANOVA
October 1, 2008
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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
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
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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df = n - 1
http://www.uvm.edu/~dhowell/SeeingStatisticsApplets/TvsZ.html
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!”
Assumptions1. Normality (of population, not
sample)2. Independence of observations
(within sample)
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
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
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
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
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
ANOVA
Example: 0,1,2;1,2,3;2,3,4
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
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.
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
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
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]]
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