I have some data which in the terms of a 3x2 contingency table look in the following way.
Category 1 Category 2 Total Example type 1 1140 60 1200 Example type 2 840 360 1200 Example type 3 1020 180 1200 Total 3000 600 3600
More specifically, I have data from 60 people all of which were shown 60 ambiguous stimuli (20 per example type). Each of those examples had to be categorized as Category 1 or Category 2.
I have several questions:
1) Is there a specific difference between the data represented in that way (3x2 contingency table) and in a 2x3 manner (where Category 1 and Category 2 are on the rows and Example type 1, 2 and 3 are as columns)?
2) If I am interested in whether there is an association between the two variables, is the chi-square test for association (independence) my best way to go?
3) Given that the survey is designed in this way - all people see 20 examples per row - is the assumption of independence violated?
4) How can I acquire more specific information about eventual interaction - for example, that people behave significantly different only for the Example type 2 cases?
Edit: the percentages in the table are changed to counts and the subtotals and totals are added.
Edit 2: Thank you for your suggestion for using log-linear model. I fitted the model as per the suggestion:
fit <- glm(Freq~category+example+category*example, data=dataDF, family=poisson)
The summary output is the following:
Deviance Residuals:  0 0 0 0 0 0 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 7.03878 0.02962 237.657 < 2e-16 *** categoryCategory 2 -2.94444 0.13245 -22.230 < 2e-16 *** exampleExample type 2 -0.30538 0.04547 -6.716 1.87e-11 *** exampleExample type 3 -0.11123 0.04310 -2.581 0.00986 ** categoryCategory2:exampleExample type 2 2.09714 0.14667 14.298 < 2e-16 *** categoryCategory2:exampleExample type 3 1.20984 0.15518 7.797 6.36e-15 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for poisson family taken to be 1) Null deviance: 2.0336e+03 on 5 degrees of freedom Residual deviance: -7.7272e-14 on 0 degrees of freedom AIC: 58.905 Number of Fisher Scoring iterations: 2
Do I understand correctly, that fitted in this way - if the Null deviance is low and the Residual deviance is high this would indicate that the data do not plausibly emenate from a logistic reegression model with a constant term only.
So, the correct way to extract a p-value from the Null deviance would be:
> 1 - pchisq(2033.6, 5) > 0
and to extract a p-value from the Residual deviance would be:
> 1 - pchisq(0, 0) > 1
We see that this p-value is large, meaning that it is plausible that the data emanate from a model which includes the IVs?
However, isn't it weird that all numbers (the first Deviance Residuals, the degrees of freedom, the residual deviance) as so small and so big?