Week 21 Flashcards
The larger the noncentrality parameter, , the larger the power.
a) true
b) false
a) true
Explanation … The exact relationship between the power of an ANOVA test and the noncentrality parameter, L-symbol, is complex. At low values of L-symbol the power increases slowly as L-symbol increases. Then, at intermediate values of L-symbol, the power increases much faster, and finally, at high values of L-symbol and as the power approaches its limit of 100%, the power goes up slowly again. There is one thing you can always say, however, and that is that larger values of L-symbol are always associated with larger power.
In a 1-way design with k = 5, n = 5, and Cohen’s f = 2, what is the value of the noncentrality parameter, L-symbol?
a) L-symbol = 0
b) L-symbol = 10
c) L-symbol = 25
d) L-symbol = 50
e) L-symbol = 100
e) L-symbol = 100
Explanation … For this design, the total number of observations is the number of observations per cell (5) times the number of cells (5). Hence N = 25. But the formula for the noncentrality parameter is L-symbol=f^2 N. Therefore L-symbol = 2^2 x 25 = 100.
Consider a 2-way, between-subjects analysis with results shown in the following output table. What is the value of the noncentrality parameter, L-symbol A, for treatment A?
Source SS df MS F A 24 3 8 4 B 50 5 10 5 AxB 30 15 2 1 Error 144 72 2 Total 248 95
a) L-symbolA = (3/72) x 96
b) L-symbolA = (24/144) x 96
c) L-symbolA = (24/224) x 96
d) L-symbolA = (24/224) x 4
b) L-symbolA = (24/144) x 96
Explanation … For this design the total number of observations is N = 96 (because dfTotal = 95), and f2 = SSA/SSError = 24/144. Thus, the value of the noncentrality parameter for factor A is L-symbol = f^2N = (24/144) x 96 = 16.
Source SS df MS F A 24 3 8 4 B 50 5 10 5 AxB 30 15 2 1 Error 144 72 2 Total 248 95
Suppose the same data as for the previous problem were to be re-analyzed in a 1-way analysis with factor A as the only factor. What would the value of the noncentrality parameter for factor A be then?
a) L-symbolA = (3/72) x96
b) L-symbolA = (24/144) x96
c) L-symbolA = (24/224) x 96
d) L-symbolA = (24/248) x4L-symbol = 10
e) L-symbol= 25
c) L-symbolA = (24/224) x 96
Explanation … We can derive a 1-way between-subjects output table from the 2-way table shown earlier by recognizing that the new 1-way analysis ignores factor B. Since factor B is ignored, any SS or df term from the old 2-way analysis immediately becomes incorporated into the unexplained or “Error” terms of the 1-way analysis. Hence in the 1-way analysis SSError = SSError(2-way) + SSB + SSAxB = 144 + 30 + 50 = 224, and dfError = dfError(2-way) + dfB + dfAxB = 72 + 15 + 5 = 92. So the 2-way analysis changes to the 1-way analysis as shown below.
You can see that the 1-way analysis has a larger MSE (circled) than the 2-way analysis and that the F-value in the 1-way analysis is correspondingly smaller than for the 2-way analysis. What has happened is that by ignoring factor B and its effects, the 1-way analysis has foregone a chance to explain some of the variability in dependent variable values in the design. And this increased ignorance about what is going on has appeared in the shape of an increased MSE. Large MSEs are undesired. They are always associated with smaller F values and this means that we are less likely to find significance.
Finally we can calculate the new 1-way noncentrality parameter. The overall number of observations hasn’t changed – it is still N = 96. And SSA hasn’t changed – it is still SSA = 24. But SSError has changed and that means we now have f2 = SSA / SSError = 24/224. So L-symbolA = (24/224) x 96 = 10.3
5) Which is larger, L-symbolA for the 2-way analysis above or L-symbolA for the 1-way analysis
a) L-symbolA for the 1-way analysis
b) L-symbolA for the 2-way analysis
b) L-symbolA for the 2-way analysis
Explanation …To see that the L-symbol-value for the 2-way analysis is larger, you could directly compare the L-symbol-values that were just calculated. Or you could just realize that the power in the 2-way analysis was higher than for the 1-way analysis (look at the F-values!) and that higher power always means higher L-symbol-value.
