Output of chisq.test in r

Question

I have a question regarding the below output from a chi-squares test, which I find to be confusing and contrary to my expected results - my chi-squared value is infinity here :)

I have two questions here

1. I made a data frame showing the relation between smoking and working out. In the column workoutideal, I have tried to convey that smokers don't work out and non smokers work out. In the column workoutmixed, it's any random data.

I expected it to show a strong relation between smoke and workoutideal (I was expecting chi square to be 0), but a weak relation between smoke and workoutmixed (I was expecting any integer value for chi square here). However, what I observe is the exact opposite. Please see my output below:

mydata = data.frame(smoke = c('no','yes','no','no','yes')
                    workoutideal = c('yes','no','yes','yes','no')
                    workoutmixed = c('no','no','yes','yes','yes') )

table(smoke, workoutideal)
     workoutideal
smoke no yes
  no   0   3
  yes  2   0
table(smoke, workoutmixed)
     workoutmixed
smoke no yes
  no   1   2
  yes  1   1
chisq.test(smoke,workoutideal)

    Pearson's Chi-squared test with Yates' continuity correction

data:  smoke and workoutideal
X-squared = 1.7014, df = 1, p-value = 0.1921

Warning message:
In chisq.test(smoke, workoutideal) :
  Chi-squared approximation may be incorrect

chisq.test(smoke, workoutmixed)

    Pearson's Chi-squared test with Yates' continuity correction

data:  smoke and workoutmixed
X-squared = 0, df = 1, p-value = 1

Warning message:
In chisq.test(smoke, workoutmixed) :
  Chi-squared approximation may be incorrect

2. While deciding whether null hypothesis should be accepted or rejected in R, should I look at the X-squared value and accept null hypothesis if it is less than the critical value for it's degrees of freedom and reject otherwise. OR, should I look the p-value and accept null hypothesis if it is higher than 0.05, the significance level and reject otherwise.

Answer 1

You are confused about the nature of hypothesis testing, test statistics, and p-values.

What you might expect from your "ideal" case is that the p-value would be 0, not that the chi-squared test statistic would be 0. Your test statistic would be very large. (The reason why it isn't very large, and your p-value isn't very low, is just that you have few data.)

On the other hand, for your "mixed" case, the opposite should be true: That is, the test statistic should be very low, and the p-value should be close to 1. Which we in fact see, low N notwithstanding.

Regarding question 2, using the critical value for the chi-squared test statistic or using whether the p-value is < alpha will always yield the same decision. This is because the critical value corresponds to the spot where the p-value drops below alpha.

Answer 2

The data seems right. However, doing a chisq test on data with cell counts less than 4 is not recommended. That could be why you got bad results.