Calculating differences between groups of scores

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Calculating differences between groups of scores

• Chapter 9

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Comparing group means

• There are instances when you are interested in
  determining if two groups of scores are
  "different"
• E.g. a group of injured athletes who receive an
  anti-inflammatory drug heal faster than those who
  do not
• E.g. Individuals have lower body fat after an
  exercise intervention, in a randomized and
  crossed-over study
• In these cases, you will perform a t-test.
• This test tells you whether the difference you
  observe between two group means (unless the means
  are the exact same score) is meaningful, and not
  due to random chance

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Fundamentals of statistics

• There are certain fundamental issues that are
  necessary to compare group means
• Observations are drawn from ___________________
  populations
• Observations represent ____________ samples from
  populations
• Groups being compared have equal ___________

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Normally distributed populations

• Statistics that operate under the assumption of a
  normal distribution and the variables being
  compared having equal variances are called
  ____________________
• Generally the most common approach
• If tests indicate that the data are not normally
  distributed, then you can use non-_________ tests
  (we will do this later)
• There are certain tests to determine if the data
  are normally distributed but dont worry about
  it!

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• What it looks like when variables are not
  _______________

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Equal variances?

• Another assumption is that the variances (squared
  standard deviations) with the means you are
  comparing are equal.
• If the variables do not have equal variances, you
  can get inaccurate results (makes results look
  different when really they are not)
• If variables do not have equal variances, you
  can
• ____________ the data somehow to make the data
  follow a normal distribution
• http//davidmlane.com/hyperstat/A45619.html

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Real-life Example
This figure demonstrates that 1) People with
higher energy intakes also have more variable
energy intakes, and 2) If you log-transform the
data, the effect goes away (makes the variances
equal)
Rumpler WV, Kramer M, Rhodes DG, Paul DR. The
impact of the covert manipulation of
macronutrient intake on energy intake and the
variability in daily food intake in nonobese
men.Int J Obes (Lond). 2006 May30(5)774-81
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Using t-Tests

• There are generally two different scenarios where
  t-tests are used to detect a difference between
• Two different groups of people or a different
  "treatment" (_______________)
• One group of people tested twice (___________)
• If you are testing the difference between a group
  of men and women for body composition?
• __________________ t-test
• If you are comparing sit-and-reach scores on a
  group of people before and after a flexibility
  program?
• __________________ t-test

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t-Test Fundamentals

• The purpose of a t-test is to reject the Null
  Hypothesis (Ho)
• Accepting the null hypothesis means there is no
  difference between two means
• HA ________ hypothesis reject the null
  hypothesis
• H1 µ1 µ2 gt 0 (rejecting Ho the means are
  different)
• H2 µ1 µ2 lt 0 (rejecting Ho the means are
  different)
• Type ___ error rejecting a true Ho (alpha ?)
• Type ___ error when a false Ho is not rejected
  (Beta ß)

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t-Test Fundamentals

• Next, you must take into account that if you
  reject Ho, there is a chance that you were wrong
  to do so (Type ___ error)
• One of the most common is 0.05 means that 5x/100
  the "real" difference detected between 2 means is
  due to random chance only
• To be REALLY confident, you use a smaller number
  for ? (say, _______)
• Another fundamental concept is "______"
• The differences between 2 means can be positive
  or negative (cover both "directions")
• e.g. a drug that may increase blood pressure vs.
  a drug that can either increase or decrease blood
  pressure

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Calculating Independent t-Test

• Define what the Ho is (no difference between
  means)
• Select an Alpha-level
• Usually __________, but can be lower
• Or you can calculate and exact value
• Calculate degrees of freedom
• ________ for each group
• Calculate the t-value from equations
• Determine whether you want to look at a 1- or
  2-tailed test
• Look up critical values from table, and compare
  them to your calculated t-value

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Critical Values TablegtStandardized table in many
textbooks
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Calculating Independent t-Test

• Ho No drug group that receives drug
• ? 0.05
• Calculate degrees of freedom
• (10 10 2)
• 18
• Calculate t value

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Calculating Independent t-Test
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Calculating Independent t-Test

• 1- or 2-tailed?
• Lets assume 2 (drug may increase or decrease
  blood sugar)
• 6. Critical values from table (2-tailed, df18)
• 2.10 (p0.05) and 2.45 (p0.025)
• t-value of 2.36gt2.10, so p lt 0.05
• Therefore _______________
• t-value of 2.36lt2.45, so p gt 0.025
• Exact p is between 0.05 and 0.025 (actually 0.03)

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Dependent t-test

• Lets assume that the drug/no drug groups are the
  same people in a randomized, cross-over study
• Ho No drug drug (no change within group)
• ? 0.05
• Calculate degrees of freedom
• (n 1) 10-19
• Calculate t-score

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Calculating Dependent t-test
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Calculating Dependent t-test

• t4.0
• Critical value for t at p0.05 is 1.83
• p 0.01 is 3.25
• t of 4.0 gt 3.25, therefore the means are very
  different plt 0.01
• Actual plt0.003

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t-tests using Excel

• Formula
• TTEST(array1,array2,tails,type)
• Array1 is the first data set (highlight A).
• Array2 is the second data set. (highlight B)
• Tails specifies the number of distribution
  tails.
• If tails 1, TTEST uses the one-tailed
  distribution.
• If tails 2, TTEST uses the two-tailed
  distribution.
• Type is the kind of t-Test to perform.
• 1 Paired
• 2 Two-sample equal variance (homoscedastic)
• 3 Two-sample unequal variance (heteroscedastic)
• For our purposes, we will assume the variances
  are equal

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Independent t-test using Excel

• Independent t-test formula
• TTEST(A1A8,B1B8,2,2)
• (GrpA,GrpB,2-tailed test, 2-sample of equal
  variances)
• Returns a value of 0.03 (p-value)
• Less than 0.05 therefore, diabetics getting the
  drug have lower blood sugar
• Dependent t-test formula
• TTEST(A1A8,B1B8,2,1)
• (1 implies a paired t-test)
• Returns a value of 0.003 diabetics have lower
  blood sugar when they are on drug