Contingency Tables. Contingency tables are used when we want to looking at two (or more) factors. Each factor might have two more or levels.

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1 Contingency Tables Definition & Examples. Contingency tables are used when we want to looking at two (or more) factors. Each factor might have two more or levels. (Using more than two factors gets complicated, so we won't be looking at this in this class). A factor can be defined as a categorical variable ; examples might be sex, color, age, etc. We already looked at test where we only have one factor - goodness of fit tests. A level cab be defined as the different values for the factor; examples matching the above might be {male, female}, {red, blue, green}, {0-4, 5-9, 10-14, etc.}. For each combination of levels, we have a certain count, which tells us how many individuals (or whatever we re measuring) are in that category. An example (data from 1988, Florida): Safety equipment in use Injury Fatal Non-fatal Total None 1, , ,128 Seat belt , ,878 Total 2, , ,006 We have two factors, injury and safety equipment used, each with two levels. So now that we have an example, what are we interested in? Depends. In the example above, what would be interesting to know? Do seat-belts save lives? Specifically, is p 1 = p 2?, where: p 1 = proportion of fatalities for people not wearing a seat belt (estimated by p 1 ). p 2 = proportion of fatalities for people wearing a seat belt (estimated by p 2 ). Incidentally, we re obviously interested in a one-sided alternative here (why?).

2 But let's do from our old textbook. This one's kind of interesting as it was based on a study done in 1899, and then re-analyzed in 1954 by Goodman and Kruskal. Kruskal is famous for being one of the people to develop the Kruskal-Wallis test, which we discuss in slightly more advanced classes: Hair color dark light Total Eye dark color light Total Now what do we want to know? Does it make sense to figure out if the proportion of people with dark eyes is the same if they have dark hair or light hair? Well, maybe. It might be much more interesting to ask the question like this: Does eye color influence hair color, or vice-versa? Or in statistical language: Is eye color independent of hair color? So, depending on the kind of data, we re either interested in: 1) Comparing proportions 2) Establishing independence/dependence So here's an outline of how to do our test: We phrase our hypotheses accordingly: 1) H 0 : p 1 = p 2, where the p's are the true population proportions, H 1 : p 1 p 2 (or p 1 > p 2, etc.) 2) H 0 : Factor 1 and factor 2 are independent H 1 : Factor 1 and factor 2 are dependent We'll notice that the math will be the same regardless of which set of hypotheses we use. Now let s choose α, just as always Calculate our test statistic: χ 2 * = i=1 c (O i E i ) 2 E i

3 Note that this is identical to that used for the goodness of fit test. But now c is the number of cells in our table (four in both of our examples so far). We ll figure out how to get expected values below. Look up the tabulated χ 2 value Our degrees of freedom are not c-1 anymore. Instead, they are (r-1) x (k-1), where: r = # of rows k = # of columns So for both our examples so far we have: (2-1) x (2-1) = 1 x 1 = 1 Compare our χ 2* with χ 2 table, and if it s larger (or equal to), then reject H 0. State our conclusion in terms of our original hypothesis. So what about our expected values? Let's work with the proportion of people with dark eyes and suppose that hair color has no effect on eye color. This implies that the proportion of people with dark eyes is the same in both columns. Therefore, we figure out what our overall proportion of people with dark eyes using the row totals: Add up all people with dark eyes, and divide this by the total number of people in our sample (in other words, we use the row totals). Now we note that if it doesn't make any difference if you have light or dark hair, then the proportion of people with dark eyes should be the same in both columns. In other words, we now multiply the column totals by the proportion of people with dark eyes. This give us the expected number of people with dark eyes for each column. If we think about this a bit, it gives us the following (easy to remember) formula: Expected value = (Row total) (Column total) (Grand total)

4 So now let's do a few examples: Seat belts and fatalities: State hypotheses: H 0 : The proportion of people killed is the same whether or not they are wearing a seatbelt or: H 0 : p 1 = p 2 (as defined above) H 1 : The proportions are not the same (we ll stick with a two-sided test for the moment) Let's use α =.05 We calculate our expected values from our observed values: Safety equipment in use Injury Fatal Non-fatal Total None 1, , ,128 Seat belt , ,878 Total 2, , ,006 So, our first expected value (fatal, none): (2,111 x 167,128)/580,006 = Our second expected value (non-fatal, none): (577,895 x 167,128)/580,006 = 166, Our third expected value (fatal, seat belt): (2,111 x 412,878)/580,006 = 1, And finally, (non-fatal, seat belt): (577,895 x 412,878)/580,006 = 411,375.28

5 We can put this in a table if we want (to keep it straight): Expected values: Safety equipment in use Injury Fatal Non-fatal Total None , ,128 Seat belt 1, , ,878 Total 2, , ,006 Now let s calculate our χ 2* : χ 2 * = ( ) (165, ,519.72)2 166, ( ) (412, ,375)2 411,375,28 = 2, If we look up our critical value of χ 2 in the table (1 d.f., and α =.05), we get: χ 2.05,1 = 3.84 So we reject H 0 and conclude that seat belts do affect the outcome of a traffic accident. Incidentally, p < 1 x , though I sort of doubt the accuracy of that figure. However, we suspect seatbelts save lives, so what we really wanted was a one sided test: Proceed as above, though now you re alternative hypothesis is H 1 : p 1 > p 2 (p 1 is the proportion of fatalities for folks not wearing seatbelts). Make sure the data deviate from the null hypothesis in the direction of your alternative, otherwise STOP. In other words, make sure p 1 p 2 (notice that p 1 = , p 2 = , so we're okay) Now just use the appropriate column in your χ 2 tables (divide by two): 2 table 2 =.05,1 = 2.71 And again we get to reject

6 Now let s do our second example: Notice that the χ 2 table value is lower, which makes rejection easier and gives us more power (not really needed in this example, but still true!). Hair color dark light Total Eye dark color light Total H 0 : Eye color and Hair color are independent H 1 : They are not independent α =.05 Calculate expected values (I m skipping the details, they re in your text, and I went through them above): Hair color dark light Eye dark color light Calculate χ 2* (the same as usual, I m skipping the details): Our tabulated χ 2 = 3.84 χ 2* = So we reject our H 0, and conclude H 1. Do we want to do anything else?? Yes, we can see which direction our sample deviates from our H 0. In other words, since eye color and hair color are not independent, we can look and see if dark eyes occurs more commonly with dark hair, or if dark eyes occurs more commonly with light hair: If a person has dark hair, the proportion of dark eyes is 729 / 3858 = If a person has light hair, the proportion of dark eyes is 131 / 2945 = From that information, we can conclude that dark hair goes with dark eyes, and therefore light hair with light eyes. (No surprise: blond, blue-eyed, etc. etc. )

7 Note that to calculate the proportions above we used the column totals in the denominator of both calculations. You can also use the row totals - as long as you're consistent. You'd be calculating different proportions, but you should still be able to interpret them: E.g., if you have dark eyes, the proportion of people with dark hair is, and then if you have light eyes the proportion of people with dark hair is, and so on. R x K tables. We refer to any tables bigger than 2 x 2 as R x K This is pretty easy to deal with. With the possible exception of figuring out your hypotheses, you know how to do this. Let's look at three different species of squirrel and compare food preferences: Peanuts Walnuts TOTAL Species A Species B Species C TOTAL An obvious thing to compare is if there's a difference in food preference between the three species. H 0 : The proportions of peanuts and walnuts is the same for all three species of squirrel. H 1 : The proportions are not the same. (Note that we can't do a directional or one sided alternative once we're bigger than 2 x 2 tables). α =.05 χ 2* = (calculated the same way as always!) df = ν = 2 (r-1)(k-1) = 2 x 1 = 2 From our table, the critical value of chi-square with 2 d.f. and an α of.05 = 5.991, so we reject. Our conclusion is that the proportions of blood types is not the same in the two groups (some blood types are more prone to getting ulcers - Biological note: what actually causes ulcers?).

8 IV) Comments: One can calculate things like Odds ratios and relative risk for tables like this. These are actually very important in medical trials. We don't have the time to go into the details, but notice that the relative risk isn't that difficult to calculate: For example, let's look at the risk of dying in a car accident when not wearing a seatbelt as opposed to wearing a seatbelt: ^ RR = ^p 1 ^p 2 = = 7.73 Where ^ RR is the estimated relative risk. This tells us that the risk of dying in a car crash is 7.73 times higher than if you're not wearing a seatbelt!! The odds ratio is a similar measure, but requires rather more explanation. As mentioned, both are used extensively in medical trials (e.g., the risk of getting lung cancer if you smoke is about 40 times that of a non-smoker (figures are approximately correct!)) Finally, a quick word about the assumptions. They are identical to those for a goodness of fit test: Random data Smallest expected value 5