Section 46-47 Computation of t for Dependent Data; Analysis of Variance

Question 1

t -Test for Dependent Data (Computation)

Answer 1

DEPENDENT DATA are obtained when each score in one set of scores is paired with a score in another set (ex: twins separated into each group), OR when there is REPEATING GROUP DATA (meaning a group is scored once, then the same group is scored again and the two sets of scores are compared) mitigating the differences between the two sets.

• The differences between the computation for INDEPENDENT and DEPENDENT DATA are reflected in the different formulas.
  
  • Additionally, you’ll need to set up a table for the DEPENDENT DATA calculation.
  • See the “t-Test Dependent” Tab in the “Quant Psych Tool” Spreadsheet for full computation, an example, and interpretation.
  • The interpretation of t is the same for both INDEPENDENT and DEPENDENT DATA.
• The LIMITS of the t-test are that it can only compare TWO data sets and no more.
• Make sure you read the Flashcard on Computation of t for INDEPENDENT DATA first as those points apply to this test as well.

Question 2

Introduction to Analysis of Variance (ANOVA)

Answer 2

ANALYSIS of VARIANCE (ANOVA) (sometimes called the F -Test) is used to test the differences among TWO or MORE MEANS.

• The formula for t allows you to enter the values for only two means, limiting the usefulness of the t-Test to studies in which only two means are to be compared.

ANOVA can ALSO be used to test the difference between TWO means (so why use the t-Test at all?).

• In fact, if you analyze two means with the F -Test, the value of F will equal t² (F = t²), so the ⎷F = t and then you can compare the OBSERVED t to the CRITICAL t value as we normally do.
• If you have already mastered the t-Test, you do not need to learn how to use ANOVA to test the difference between two means.

The true power of ANOVA (F) is that it can analyze more than two means in a single test.

• Ex: A new drug for treating migraine headaches was tested on three groups selected at random.
  
  • Group 1 received 250 milligrams
  • Group 2 received 100 milligrams
  • Group 3 received a placebo
• The average reported pain level for the three groups (on a scale from 0 to 20, with 20 representing the most pain) was determined by calculating the means. The means for the groups were as follows:
  
  • Group 1: m = 1.78
  • Group 2: m = 3.98
  • Group 3: m = 12.88
• So there are THREE DIFFERENCES among the means
  
  • m₁₂ = The difference between Groups 1 and 2
  • m₁₃ = The difference between Groups 1 and 3
  • m₂₃ = The difference between Groups 2 and 3

_To REPORT the FINDINGS of the ANOVA (*F* -Test):_

• “The differences among the means are statistically significant at the .01 level (F = 58.769, df = 2, 36)”
• Note that the method of reporting is similar to that for reporting the results of a t -test.
• The NULL HYPOTHESIS for an F - Test says that the SET of THREE DIFFERENCES was created through RANDOM SAMPLING ERROR alone.
  
  • By REJECTING the NULL HYPOTHESIS, we are rejecting the notion that one or more of the differences were created at random, though the test does NOT tell us WHICH of the THREE DIFFERENCES is responsible for the rejection of the null hypothesis.
  • It could be that only one or two of the three differences was responsible for the significance. Procedures for determining which individual differences are significant are described in Sections 49 and 50.
• 4-Means Example: Four methods of teaching computer literacy were used in an experiment, which resulted in four means. This produced SIX DIFFERENCES:

1. m₁₂ = The difference between Methods 1 and 2
2. m₁₃ = The difference between Methods 1 and 3
3. m₁₄ = The difference between Methods 1 and 4
4. m₂₃ = The difference between Methods 2 and 3
5. m₂₄ = The difference between Methods 2 and 4
6. m₃₄ = The difference between Methods 3 and 4

• A single ANOVA can determine whether the null hypothesis for this ENTIRE SET of six differences should be rejected.
  
  • If the result is NOT significant, the researcher is DONE.
  • If the result IS SIGNIFICANT, he or she may test to see which of the six differences are significant by using the techniques described in Sections 49 and 50.
• This all refers to a ONE-WAY ANOVA – (or single-factor ANOVA). This term is derived from the fact that subjects were classified in one way.
  
  • Ex: In the top Example, they were classified only according to the drug group to which they were assigned. In the second example, they were classified only according to the method of instruction to which they were exposed.