The Professor's Response
Hello Professor, the question is in regards to completing an ANOVA table for a two factorial experiment, I understand certain elements to generating values but I am wondering if you can answer this question that was given to me in my course that didn't give much of an explanation on how certain parts were done. Thank You.
a) give the numbers of levels of each factor
b)Complete the ANOVA Table
c) Test to determine whether treatment means differ? using alpha .10
d) Conduct a test of factor interact and mean effects, each at .10 alpha level of significance.
Sources of Variation SS DF MS F
A 3 .75
B .95 1
AB .30
Error
Total 6.5 23
See the professor's answer below.
Hi Anthony,
I can't read your table. Please type it as A: SS = 0.75 DF = ..., B: SS = 0.95 DF = 1, ...
Thanks,
Professor McGuckian
Hi Anthony,
I am going to try to answer your question based on a guess of what your table is supposed to have in it. If I haven't read your message properly, please write me back so that I can correct the work.
a) give the numbers of levels of each factor Factor A has four levels since the degrees of freedom is the number of levels minus one, and B has two levels.
b)Complete the ANOVA Table
Source | SS | DF | MS | F |
A | 2.25 | 3 | 0.75 | 4.00 |
B | 0.95 | 1 | 0.95 | 5.067 |
AB | 0.30 | 3 | 0.10 | 0.5333 |
Error | 3.0 | 16 | 0.1875 | |
Total | 6.5 | 23 |
SS for A: Since MS = SS/DF, I multiplied both sides of this equation by DF to get DF*MS = SS. So, 3*0.75 = 2.25.
SS for Error: I used the fact that SStotal = SSA + SSB + SSAB + SSE, then I solved for SSE since it was the only unknown. Subtract all of the known SS values from SStotal to get the one we don't know, SSE.
DF for AB: This degree of freedom is always the product of the degrees of freedom for A and B.
DF for Error: Use the same subtraction idea that we used for SSE. Subtract all of the degrees of freedom we know from the total DF to get the one we don't know, DF for Error.
All of the MS values: I used the formula SS/DF = MS
All of the F values: I used the MS/MSE = F formula.
c) Test to determine whether treatment means differ? using alpha .10
Here you need to find the SST by combining all of the partitioned treatment SS values (A, B, and AB) into one treatment sum of squares.
So SSA + SSB + SSAB = 2.25 + 0.95 + 0.30 = 3.5 = SST
Next you need the degrees of freedom which is just the sum of the degrees of freedom for the three partitioned pieces: DFA + DFB + DFAB = 3 + 1 + 3 = 7 = DFtreatments
Now get MST = SST/DF = 3.5/7 = 0.5
Then get your test stat F by dividing MST by MSE = F = 0.5/0.1875 = 2.667
Note: the degrees of freedom for your F test stats are always the degree of freedom from the treatment, the degree of freedom from the error. In this example, the critical value will be F 7,16, alpha = 0.10 = 2.12800
Since the test stat is greater than the critical value, we reject the null and we conclude there is a treatment effect.
You can watch this video example from the forum if you'd like for the rest of it: http://www.statsprofessor.com/atpquestion.php?qid=20
d) Conduct a test of factor interact and mean effects, each at .10 alpha level of significance. See the video link above
Hope that helps,
Professor McGuckian