If two variables are not coherent (they are independent), does not mean to say that a sample automatically produces a perfect crosstab or a relative risk of 1. It will be close, but that is a little vague. An impression of the "neighborhood" can be obtained using simulations by random sampling from the data of a perfect table.
.
| 4.6a |  |
You have three measures by which to assess a cross table: the relative risk, the percentage difference and the chi square.
By chance, there are always small deviations in the data due to variation in the samples. So there are also random deviations from the perfect table in the assessment.
If the deviations are so large that randomness is not an argument, you must conclude that the starting point of the perfect table with the expected frequencies at independence cannot be correct. Then the variables cannot be independent and therefore they will show cohesion.
The question then is how to get that. Conveniently: by using simulations as you have done.
4.6b
|
Survival |
||
Gender |
Survived |
Died |
Total |
Male |
10 |
35 |
45 |
Female |
50 |
55 |
105 |
Total |
60 |
90 |
150 |
From the above cross table, the relative risk for "Died: men compared to women" is equal to 1.48. That's greater than 1, is that too big a difference or not?
With a simulation you can check that and you can conclude whether the gender and survival variables are dependent or not.
4.6c
|
Survival |
||
Gender |
Survived |
Died |
Total |
Male |
14 |
31 |
45 |
Female |
46 |
59 |
105 |
Total |
60 |
90 |
150 |
Another sample provided by the above cross-table. The relative risk is 1.23
Use a simulation to come to a conclusion about the (in) dependence.
4.6d
|
Survival |
||
Gender |
Survived |
Died |
Total |
Male |
42 |
93 |
136 |
Female |
138 |
177 |
315 |
Total |
180 |
270 |
450 |
The above sample is three times the previous one. The relative risk is therefore also 1.23
Investigate whether you come to the same conclusion about (in) dependence..
Conclusion: The size of the sample plays an important role in drawing statistical conclusions.
| 4.6e | Of the variables in the 2x3 table you examined in 4.5, dependence was questionable. |
You'll see that more closely with the chi square. |