Contingency Table Chi-Square Test Contingency Table Chi-Square Test
Contingency Table Chi-Square Test
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- lets say there are a couple of herbs that believe help prevent the flu.
- so to test this what we do is we wait for flu season and
- randomly assign people to three different groups.
- And over the course of flu season,we have them either in one group taking
- we have one group taking herb one and the second group taking herb two
- And the third group they take a placebo.
- And if you don't know what a placebo is, its something thatto the patient or to the person
- participating in it it feels like its doing something that you told them might help them
- but it does nothing. It could be just a sugar pill so it feels like medicine
- The reason you go through the effort of giving them something is cause often
- times there is something called the placebo effect where people
- get better just because they are told that they are being given something that
- will make them better.
- So this right here could just be a sugar pill.
- And a very small amount of sugar so it really cant affect their
- actual likelyhood of getting the flu.
- over here we have a table and this is actually called a contingency
- table, contingency table,contingency table and it has on
- it on each group the number that got sick and the number
- that did not get sick. And from this we can also calculate the total number
- So in group 1 we had a total of 120 people and group 2 we had
- a total of 30 plus 110 is a 140 people and
- the placebo group, that is the group that just got the sugar pill,
- we had a total of 120 people and we can also tabulate the total number of people that got sick
- so thats 20 plus 30 that is 50 plus 30 is 80
- so that is the total column right over here and the total people that didnt get sick
- over here is 100 and 110 is 210 plus 90 is 300
- ant the total people here are 380 both this column and this row should
- add up to 380. so with this out of the way lets think about how we
- can use this information in the contingency table and our
- knowledge of the chi square distribution
- to come up with some conclusion.
- so lets come up with some null hypothesis. so our null hypothesis
- is that the herbs do nothing.
- the herbs do nothing.do nothing. and we have our alternative
- hypothesis that the herbs do something. herbs do something.
- notice i dont even care whether they actually improve.
- Im just saying they do something. they may even increase your likelyhood
- of getting the flu. all we are not testing if they are actually good.
- we are just saying are they different than doing nothing.
- so like we did with all of our hypothesis tests, lets just assume
- the null . we are going to assume the null and given that assumption,
- figure out the problems. figure out if the likelyhood of getting data like this
- or more extreme is really low.
- and if it is really low, and if it is really low, we reject it.
- and in this test like in every hypothesis test, we need a significance level and the significance level
- we care about for whatever reason is 10% or 0.10 thats the significance level we care about.
- now to do this we need to calculate a chi square statistic for this contingency table. and to do this,
- we do something very similiar with what we did with the restaurant situation. we figure out assuming
- the null hypothesis, the expected results in each of these cells in each of these entries as a cell,
- thats what we do in excel. each of the entries in excel.
- each of the entries in a table.
- we figure out what each of the values would have been if you do assume the null hypothesis, we find the
- square distance from that expected value and you normalize it by the expected value ,take the sum of
- all those differences and if the square differences are really big, the probability of getting it would
- be really small and maybe we can reject the null hypothesis so lets just figure out how you can get the
- expected number .
- so we are assuming that the herbs do nothing. so if the herbs do nothing, we can just figure out that
- this whole population had nothing happen to them and the herbs were useless and so we can use this population
- sample,i shouldnt call this a population sample. we can use this sample right here to figure out the
- expected number of people who would get sick or not sick so over here, we have 80 out of 380 did not
- get sick. i should be careful here. i just said population but we havent sampled the whole universe of
- people taking this herb so this is a sample. i dont want to confuse you . i am using population in more
- of a conversational sense rather than a statistical sense. any way in all of our sample, we are using
- all of the data because we are assuming that there is no difference so we might aswell use all of the
- data to find the frequency of getting sick and not getting sick so 80 divided by 380 didnot get sick
- so thats 21 percent. 21 percent didnot get sick. so 21 percent, thats 21 percent of the total and ten
- if this will be 79 percent if we just subtract so we should divide 300 by 380, we should get 79 percent
- as well so one would expect the 21 percent of your total based on the total sample righ over here that
- 21 percent should be getting sick and 79 percent should not be getting sick. so lets look at this for
- each of the group. if we assume that 21 percent of this 120 people should have gotten sick, what would
- have been the expected value right over here? lets just multiply 21 percent times a 120. so lets just
- multiply that times 120 that gets us to 25.3 people. ill just round it. so the expected ill just write
- this over here. ill do expected in yellow. so the expected right over here . so if you assume that 21
- percent of a group would have gotten sick, you would have expected 25.3 people to get sick in group 1
- in herb 1 group and the remainder will not get sick. so lets just subtract. or i can multiply 79 percent
- times a 120 either one of them will be good. but let me just take 120 - 25.3 and i get 94.7. so you would
- have expected 94.7 to not get sick. so this is expected again. expected. expected. 94.7 to not get sick
- and i also do that for each of those groups. so once again group 2, you would have expected 21 percent
- to get sick. 21 percent of the total people in that group thats 140 that 29.4 and the remainder that
- is 140- 29.4 should not have gotten sick. so that gets us this right here. we have 29.4 should have
- gotten sick if the herbs did nothing. and over here we have 110.6 should not have gotten sick and this
- is pretty close and by just looking at the result it looks like the herb doesnt do too much relative
- to all of the groups combined. and in the placebo group, lets see what happens. we expect 21 percent
- to get sick of our group of 120 . so thats 25.2 so this right here, actually this would be the same number
- over here. i said 21 percent but it is actually 21. something. but the group sizes are the same and we
- should expect the same proportion to get sick ill say 25.3. just to make it consistant. the reason why
- i got 25.2 is because i lost some of the trailing decimals over here . but since i had them over here,
- im gonna use them over here as well and over here, in this group, you would expect 94.7 to get sick.
- lets figure out our chi square statistic. so to figure this out, lets get our statistic, our chi square
- statistic. ill write it like this here for fun. or maybe ill right it as a big X because its really
- , this random distribution is approximately a chi square distribution. so ill write it like that. and
- well talk about the degrees of freedom in a second acually let me write it in a curly axis.
- so you see that some people write with the chi instead of the x. so our chi square statistic over here,
- we are literally going to find the squared distance between the observed and the expected divided by
- the expected which will be 20-25.3 squared over 25.3 plus 30-29.4 squared over 29.4 + 30 -25.3 squared
- over 25.3 and then im gonna have to do these over here so let me just continue it you can ignore this
- h1 over here + 100-94.7 squared over 94.7 +110-110.6 squared over 110.6 and finally 90-94.7 squared over
- 94.7. so we have (20-25.3) squared /25.3+(30-29.4 )squared/29.4 + (30-25.3)squared/25.3+(100-94.7)squared
- /94.7+(110-110.6)squared /110.6+(90-94.7)squared/94.7
- we get 2.53.so our chi square statistic assuming the null hypothesis is correct is equal to 2.53
- next thing we need to do is figure out the degrees of freedom we had while doing this. and ill give you
- the rule of thumb. that is you have the number of rows. so you have the rows and you have the number
- of columns so you have 2 rows and 3 columns you dont count the total. so the degrees of freedom is for
- your contingency table is the number of rows-1 times the number of columns-1.
- in our situation we have two rows and 3 columns so that will be 2-1 times 3-1.
- so that is going to be 2.we have 2 degrees of freedom. the reason why that should have some intuitive
- sense is that if you assume that you know the total. if you know all of this information, over here,
- if you know the total information, actually if you knew the parameters of the population as well, but
- if you knew the total information, and you knew this information or if you knew r-1 of the informations
- in the rows, the last one can be figured out if you subtract it from the otal. for example, in this situation,
- if you knew this, you could easily figure out this. this is not new information.l this is just total
- - 20. same thing, if you knew this one over here, this one is not new information. similarly,if you knew
- these two, this one here id not new information. you can calculate this based on the total and everything
- else. so that's the sense as to why the degrees of freedom are the columns - 1 times the rows - 1 .
- so any way, our chi square statistic has two degrees of freedom. so remember our alpha value is 10 %.
- so we are going to figure out what our critical chi square statistic is that gives us an alpha of 10
- % and if this is more extreme than that, if the probability of getting this is even less than that critical
- statistic, we can reject the null hpothesis, and if it is not more extreme, we will not reject the null
- hypothesis. so what we need to do is to figure out what the chi square distribution is and 2 degrees
- of freedom, what is our critical chi square statistic. so lets just go back, we have two degrees of freedom
- here and we have a care about a significance level of 10 % so our critical chi square value is 4.60.
- another way to visualize this is if you look at a kai square distribution with 2 degrees of freedom,
- that is the blue one over here, at a value of 4.60, the probability of getting something atleast that
- extreme is 10%.this is what we care about. If the kai square distribution that we care about falls into
- this rejection region, then we reject our null hypothesis but our kai square statistic is only 2.53 so
- it is sitting some place right over here. so its actually not that crazy to get it if you assume the null hypothesis. so based
- on the data we have right now, we cant reject the null hypothesis. we dont know for fact that the herbs
- do nothing but we cant say that they do something. so we are not going to reject it.but from this point
- of view , it doesnt seem like the herbs are different from each other and one of the herbs is obviously a placebo.
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