# What is the meaning of × in statistics?

What is the meaning of the symbol × in an ANOVA context?

More specifically what is the meaning of × in the following table?

Study        Condition MeanSource      SS     df MS    F     p
Original     .700      Subjects        0.668  39
Mirror image .621      Item            0.357   2 0.178 12.90 <.001
Unprimed     .567      Item × Subjects 1.078  78 0.014
Total           2.103 119


And how can I calculate SS for the third row in the table?

• I am not sure about the specifics of the software you are using, but it looks to me as if it were representing an interaction term – David Jun 14 at 9:35
• @David I am reading this article. The symbol appears in Table 1. I need to know how to calculate SS for the third row but I have no idea what they mean by Item x Subject – kbr85 Jun 14 at 9:38
• As far as my (limited) knowledge goes: The subject row indicates how variance is "explained" by subject, the item row indicates how variance is "explained" by item. Finally, the row "subject x item" indicates how much variance is "explaind" by considering subject and item simultaneously. For example, if you have subjects 1 and 2, and items A and B, the first row relates to changing from 1 to 2, the second one, to changing from A to B, and the third row, to changing among combinations 1A, 2A, 1B and 2B – David Jun 14 at 9:46
• @David In this case how can I calculate SS? – kbr85 Jun 14 at 9:52

That x is probably representing an interaction term.

You should really give us more information about the data (and the experiment which produced it), but the SS's here seems to be sequential SS's, as $$0.668+0.357+1.078 =2.103$$. Note also that for the df's (degrees of freedom) from the table we have: $$119-78-2-39=0$$, so there is no df's left for error. That means that the interaction term Item × Subjects here is confounded with Error.

The anova table shows that indeed the interaction is used as an estimate of error variance, since $$0.178/0.014 = 12.71429$$ and pf(12.71429, 2, 78, lower.tail=FALSE) returns 1.662293e-05. How can it be allowable to use an interaction as error? Again, it would help to know the specifics of this experiment, but: Assuming this is a randomized experiment, with some Subjects randomized to some treatment group Item. Then the interest is really in comparing the different treatments. The Subjects are just some random persons/animals/whatever used to compare the Items. We are not specifically interested in comparing different Subjects.

The act of randomization means that there should not be systematic differences between the subjects randomized to the different treatments, the subjects are in reality a random factor. The act of randomization is what justifies the use of the interaction as error.

• +1, but it may be helpful to explain why the IxS SS are being used to form the F-statistic for the test. – gung - Reinstate Monica Jun 14 at 14:29

The original paper (cited in the link you provided) tells us that the data were collected using a "repeated-measures ANOVA" experimental design (see page 492, last paragraph). To learn how to calculate the ANOVA table for data from such a study, read a textbook on ANOVA. Probably a first or second course textbook would tell you how to do that. (BTW, the column headings in your ANOVA table do not line up perfectly, making the table a bit confusing. See the table in the original article.) The original paper is here: https://www.researchgate.net/profile/Anna-Lisa_Cohen/publication/24187239_Long-Term_Repetition_Priming_of_Briefly_Identified_Objects/links/00b7d51a505f7c0c8f000000/Long-Term-Repetition-Priming-of-Briefly-Identified-Objects.pdf

• No paper is cited in the question! How do you know the OP refers to this paper? – kjetil b halvorsen Jun 14 at 23:23
• The original paper, mentioned by the OP in a comment, is "A tutorial on a practical Bayesian alternative to null-hypothesis significance testing", not the one you refer to. (Also @kjetilbhalvorsen FYI). – jbowman Jun 14 at 23:34
• @jbowman The paper the op referenced uses data from a named study. The study Iink to is that study - it includes the methodology and the statistical analysis. I have clarified my answer to indicate that. – Joel W. Jun 16 at 2:06