Frequency Tables Statistics 2126

Introduction Say we have a question like “Where are you from?” or “What is your favourite hockey team?” How would we find out if we have some sort of effect How would the population be distributed? We have to somehow come up with a ‘chance’ distribution

Say oh rolling a die 60 times 1 2 3 4 5 6 Outcome 15 9 10 12 7 expected

How shall we attack this Observed – expected 5 -1 0 2 -3 -3 We can’t just add these up because we would end up with a sum of 0 Sound familiar? Square ‘em! 25 1 0 4 9 9

We’re always squaring stuff… What if the expected values differed? Well we should weight the squared deviations somehow Divide by the expected 25/10+1/10+4/10+9/10+9/10 = 4.8 Add ‘em all up to get some estimate of the total deviation from the expected

Chi squared

How to read the χ2 table df = number of values we can arbitrarily assign 5 in our case (there must be 60 rolls in total, so the last number is fixed)

Easiest statistic ever… H0 all the frequencies are the same Ha they are not χ2cr = 11.07 χ2obt < χ2cr So we do not reject H0

For a 2 way table… Drug No relapse Yes relapse Desipramine 14 10 Lithium 6 18 Placebo 4 20

So… This is a 3 x 2 table 3 rows and 2 columns In general r x c table We want to test if all of the proportions are equal H0 p1 = p2 = p3 Ha all ps are not equal

So how do we get the expected values? We have to compare the observed value to the expected value But it is hard to get what we would expect by chance We let the data tell us what to do (Row total) x (column total) / (table total)

Row and Column totals No Yes Total 14 10 24 6 18 4 20 48 total

Working out the expecteds (24 x 24) / 72 for the Yes column (24 x 48) / 72 for the No column So we get a table of expected frequencies with 8 in the yes column and 16 in the no column We expect some relapses but if all the drugs are the same the proportion should be the same

Now Do the Test

Degrees of Freedom No Yes D 24 L P 48 At most we can assign two values arbitrarily

So… In general the degrees of freedom for such a table are (r-1)(c-1) In our case (3-1)(2-1) 2cr = 5.99 We reject H0 The relapses are distributed differently than chance