Testing Statistical Hypothesis for Dependent Samples

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Testing Statistical Hypothesisfor Dependent
Samples
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Testing Hypotheses about Two Dependent Means

• Dependent Groups t-test
• Paired Samples t-test
• Correlated Groups t-test

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Steps in Test of Hypothesis

1. Determine the appropriate test
2. Establish the level of significancea
3. Determine whether to use a one tail or two tail
   test
4. Calculate the test statistic
5. Determine the degree of freedom
6. Compare computed test statistic against a
   tabled/critical value

Same as Before
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1. Determine the appropriate test

• When means are computed for the same group of
  people at two different points in time (e.g.,
  before and after intervention)
• When subjects in one group are paired to subjects
  in the second group on the basis of some
  attribute. Examples
• Husbands versus wives
• First-born children versus younger siblings
• AIDS patients versus their primary caretakers

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1. Determine the appropriate test
Continued

• Researchers sometimes deliberately pair-match
  subjects in one group with unrelated subjects in
  another group to enhance the comparability of the
  two groups.
• For example, people with lung cancer might be
  pair-matched to people without lung cancer on the
  basis of age, education, and gender, and then the
  smoking behavior of the two groups might be
  compared.

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Example Two Interventions in Same Patients

• Suppose that we wanted to compare direct and
  indirect methods of blood pressure measurement in
  a sample of trauma patients. Blood pressure
  values (mm Hg) are obtained from 10 patients via
  both methods
• X1 Direct method radial arterial catheter
• X2 Indirect method the bell component of the
  stethoscope

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2. Establish Level of Significance

• a is a predetermined value
• The convention
• a .05
• a .01
• a .001

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3. Determine Whether to Use a One or Two Tailed
Test

• H0 µD 0
• Ha µD ? 0

Two Tailed Test if no direction is specified
Mean of differences across patients
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3. Determine Whether to Use a One or Two Tailed
Test
Continued

• H0 µD 0
• Ha µD ? 0

One Tailed Test if direction is specified
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4. Calculating Test Statistics
How to calculate standard deviation of differences
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4. Calculating Test Statistics
Continued
Defining Formula Calculating Formula
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4. Calculating Test Statistics
Continued
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4. Calculating Test Statistics
Continued
Calculate totals
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4. Calculating Test Statistics
Continued
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4. Calculating Test Statistics
Continued

• Calculate t-statistic from average of differences
  and standard error of differences

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5. Determine Degrees of Freedom

• Degrees of freedom, df, is value indicating the
  number of independent pieces of information a
  sample can provide for purposes of statistical
  inference.
• Df Sample size Number of parameters estimated
• Df for paired t-test is n minus 1

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6. Compare the Computed Test Statistic Against a
Tabled Value

• a .05
• Df n-1 9
• ta(df 9) 2.26 Two tailed
• ta(df 9) 1.83 One tailed
• Reject H0 if tc is greater than ta

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Alternative Approach

• Estimating Standard deviation of differences from
  sample standard deviations

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Variance / Covariance matrix
X1 X2
X1 X2
Variance ofthe first measure
S21 S12
S12 S22
Variance ofthe second measure
Co-variance of Measures of 1 and 2
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Variance / Covariance matrix

• X1 X2

S21
S22
X1 X2
Standard error of difference can be calculated
from above table
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Alternative Approach for Calculating standard
Error
Standard error of Differences
Correlation between two measures
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Correlation Matrix

• Direct Indirect
• Direct Pearson Correlation 1 .996()
• Sum of Squares and
• Cross-products 4496.100 4611.00
• Covariance 499.567 512.333
• N 10 10
• Indirect Pearson Correlation .996() 1
• Sum of Squares and
• Cross-products 4611.00 4768.00
• Covariance 512.333 529.778
• N 10 10

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Alternative Approach for Calculating standard
Error
Same value as before
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SPSS output for Paired Sample t-test 

Paired Samples Statistics
Paired Samples Correlations
 Paired Samples Test
     
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Take Home Lesson

• How to compare means of paired dependent samples