Kin 304 Tests of Differences between Means: ttests

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Kin 304Tests of Differences between Means
t-tests

• SEM
• Visual test of differences
• Independent t-test
• Paired t-test
• Power and sample size

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Comparison

• Is there a difference between two or more groups?
• Test of difference between means
• t-test
• only two means
• ANOVA - Analysis of Variance
• Multiple means
• ANCOVA
• Includes covariates

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Standard Error of the Mean
Describes how confident you are that the mean of
the sample is the mean of the population
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Visual Test of Significant Difference between
Means
1 Standard Error of the Mean
1 Standard Error of the Mean
Overlapping standard error bars therefore no
significant difference between means of A and B
A
B
Mean
No overlap of standard error bars therefore a
significant difference between means of A and B
at about 95 confidence
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Independent t-test

• Two independent groups compared using an
  independent T-Test (assuming equal variances and
  normality of populations)
• e.g. Height difference between men and women
• The t statistic is calculated using the
  difference between the means in relation to the
  variance in the two samples
• A critical value of the t statistic is based
  upon sample size and probability acceptance level
  (found in a table at the back of a stats book or
  part of the EXCEL t-test report, or SPSS output)
• the calculated t based upon your data must be
  greater than the critical value of t to accept
  a significant difference between means at the
  chosen level of probability

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t statistic quantifiesthe degree of overlap of
the distributions
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Calculation of the t statistic

• The t statistic is calculated as the ratio of the
  difference between sample means to the standard
  error of the difference.

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Assumptions of the t test

• Dependent variable is distributed normally in the
  population
• (robust ok if non-normal)
• Continuity of dependent variable
• (but ordinal scale is ok)
• Randomized samples from populations and of
  subjects to treatments (if not, cannot
  generalize)
• Homoscedasticity
• particularly problematic if sample sizes are not
  equal

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Characteristics of the t distribution

• The difference between sample means, when divided
  by the standard error of the difference between
  means, adheres to a Normal distribution only for
  very large N.
• For smaller N (lt100), the more accurate
  distribution for the difference between means is
  the t distribution

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t statistic
In general
where
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t statistic
If n1 n2
There is a unique t distribution for each sample
size. The distribution is defined by the degrees
of freedom (df), where df (n1-1)(n2-1)
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Critical values of t

• Hypothesis
• There is a difference between means
• Degrees of Freedom 2n 2
• tcalc gt tcrit significant difference

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Example of Independent t-test

• Hypothesis Students with or without eye glasses
  differ in mean exam scores
• Eye Glasses mean X85 s12.5 n8
• No Eye Glasses mean X79 s13.1 n8
• Critical value of t
• df14, alpha0.05 two-tailed2.145
• Therefore there is insufficient evidence to
  reject Ho

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Paired Comparison

• Paired t Test
• Sometimes called t-test for correlated data
• Typically used for measurements repeated on the
  same subject
• Pretest vs. posttest Experiments
• Bilateral Symmetry

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Paired t-test (correlated)

• Hypothesis
• Is the mean of the differences between paired
  observations significantly different than zero?
• df (n-1), where n number of paired
  observations

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9 Subjects All Lose Weight
Mean of differences 1.13
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Calculation of Power or Sample Size

• Use of interactive software
• Needed for sample size calculation
• estimates of the 2 group means
• estimates of the 2 group standard deviations
• alpha and statistical power desired
• If power is to be determined, sample size
  estimates are given instead
• Refer to http//statpages.org/