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Oneway ANOVA

Oneway ANOVA. Advanced Research Methods in Psychology - lecture - Matthew Rockloff. When to use a Oneway ANOVA 1. Oneway ANOVA is a generalization of the independent samples t-test . Recall that the independent samples t-test is used to compare the mean values of 2 different groups.

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Oneway ANOVA

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  1. Oneway ANOVA Advanced Research Methods in Psychology - lecture - Matthew Rockloff

  2. When to use a Oneway ANOVA 1 • Oneway ANOVA is a generalization of the independent samples t-test. • Recall that the independent samples t-test is used to compare the mean values of 2 different groups. • A Oneway ANOVA does the same thing, but it has the advantage of allowing comparisons between more than 2 groups.

  3. When to use a Oneway ANOVA 2 • In psychology, for example, we often want to contrast several conditions in an experiment; such as a control, a standard treatment, and a newer “experimental” treatment. • Because Oneway ANOVA is simply a generalization of the independent samples t-test, we use this procedure (to follow) to recalculate our previous 2 groups example. • Later, we will do an example with more than 2 groups.

  4. Example 7.1 • Let’s return to our example of the pizza vs. beer diet. • Our research question is: “Is there any weight gain difference between a 1-week exclusive diet of either pizza or beer?”

  5. Example 7.1 (cont.)

  6. Example 7.1 (cont.) • An Oneway-ANOVA is a generalization of the independent samples t-test in which we can specify more than 2 conditions. • If we only specify 2 conditions, however, the results will be exactly the same as the t-test. • The calculations are somewhat different, but the resulting “p-value” will be the same, and therefore the research conclusion will always be the same.

  7. Example 7.1 (cont.) • ANOVA operates on the principle of “partitioning the variance”. • There is a total amount of variance in the set of data previous. • This total variance is found by subtracting each value (e.g., 1,2,2…) from the mean for all 10 people ( ), squaring the result, summing the squares, and dividing by the number of values (i.e., 10):

  8. Example 7.1 - Formula or 

  9. Example 7.1 (cont.) • This total variance (S2t=1.4) can be partitioned, or divided, into 2 parts: • the variance within, and • the variance between.

  10. Example 7.1 – Variance within • The variance within is calculated by averaging the variances within each condition. • For the previous example 

  11. Example 7.1 – Variance within (cont) , where J = number of conditions

  12. Example 7.1 – Variance between • The variance between is calculated by taking the variance of the means of all conditions. • In our example, of course, we only have 2 means: or for a balanced study.

  13. Example 7.1 – Variance between (cont.) In our example:

  14. Example 7.1 (cont.) • Now we can write a formula for the partition of the variance into its components: S2total = S2between+S2within , or 1.4 = 1 + 0.4 • The formula above will allow you to check your hand calculations. • If you’ve done everything right, all variances should “add up” to the total variance.

  15. Example 7.1 – ANOVA table • Next, we need to fill-in the so-called ANOVA table:

  16. Example 7.1 – ANOVA table (cont.) • Here’s what we know so far: • S2between = 1 • S2within = 0.4 • S2total =1.4 • J=2 (because there are 2 conditions) • n=5 (because there are 5 people in each condition) • N=10 (because there are 10 subjects in total)

  17. Example 7.1 – ANOVA table (cont.) • Now we can fill-in the table:

  18. Example 7.1 (cont.) • This is a 2-tailed test because we had no notion of which diet should have greater weight gain. • In the back of a Statistic text we find the critical value of this “F” is 5.32, by looking for a 2-tailed F with 1 and 8 degrees of freedom. • The first, or numerator, degrees of freedom are the degrees of freedom associated with the Mean Squared Between (df=1). • The second, or denominator, degrees of freedom are associated with the Means Squared Within (df=8).

  19. Example 7.1 – Conclusion … • Our calculated F = 20 is higher than the critical value, therefore we reject the null hypothesis and conclude that: there is a significant difference in weight gain between the 2 diets.

  20. Example 7.1 – Conclusion (cont.) • More specifically, we can look at the mean weight gain in each condition (Mpizza = 2 and Mbeer = 4), and conclude that: The beer diet (M = 4.00) has significantly higher weight gain than the pizza diet (M = 2.00), F(1,8) = 20.00, p < .05 (two-tailed).

  21. Example 7.1 - Using SPSS • First, we need to add 2 variables to the SPSS variable view: • IndependentVariable = diet (coded as 1=Pizza and 2=Beer) • DependentVariable = wtgain (or “weight gain”) • As before, personid is added a convenient – although not critical - additional variable.

  22. Example 7.1 - Using SPSS (cont.) • In addition, we must code for the “diet” variable (per above): 

  23. Example 7.1 - Using SPSS (cont.)

  24. Example 7.1 - Using SPSS (cont.) • In the same manner as the independent samples t-test, we enter the data in the SPSS data view:

  25. Example 7.1 - Using SPSS (cont.) • The only “change” in performing the ANOVA procedure is the new syntax: Oneway DependentVariable by IndependentVariable/ranges = scheffe. • In our example, the following syntax is entered:

  26. Example 7.1 – SPSS output viewer • Running this syntax produces the following in the SPSS output viewer:

  27. Example 7.1 – SPSS(cont.) • A warning is given which states that the sub-command “/ranges = scheffe” was not executed. • This procedure is only necessary when there are more than 2 groups, because it helps to test all possible pairs of means between groups. • In our example, we can simply interpret the ANOVA table to determine significant difference between our 2 means for Pizza and Beer. • The warning can be safely ignored.

  28. Example 7.1 – SPSS(cont.) • The ANOVA table is simply a reproduction of the table that was computed by hand. • Unlike the hand calculated results, SPSS provides an exact probability value associated with the F-value.

  29. Example 7.1 – Conclusion • The conclusion can therefore be modified as follows: The beer diet (M = 4.00) has significantly higher weight gain than the pizza diet (M = 2.00), F(1,8) = 20.00, p < .01 (two-tailed).

  30. Example 7.1 – NB: APA style • Notice that the probability given by SPSS was p = .002. • Per APA style, rounded to 2 significant digits the probability becomes p=.00. • Probabilities, however, are never zero, so we must modify this result to the smallest p-value normally expressed in APA style, p < .01.

  31. Thus concludes  Oneway ANOVA Advanced Research Methods in Psychology - Week 6 lecture - Matthew Rockloff

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