Two Way Table 2 Categorical Variable Analysis Problem

Transcription

1 Two Way Table 2 Categorical Variable Analysis Problem It seems unfortunate that we have to present one the richest concepts, full of a few twists and turns, in this intro. stats. Class this early in the semester. It is a concept which needs much thinking and studying by students. We statistics people are constantly wanting to see if there are any strong relationships between 2 categorical variables, one called the explanatory variable (or x or independent variable), and the response variable (or y or dependent variable). We always start with a 2-way-table, where one categorical variable (traditionally the response) is placed with its categories composing the row titles, and the other categorical variable (traditionally the explanatory) is placed with its categories being the column titles. See an example below. Example: A U.S. Census gathered data relating household income (SALARY) to type of pet (PET) in the home there were only single pet homes used in the study. They wanted to know if there was a relationship between PET and SALARY. The data is categorized as shown below. PET SALARY Dog Cat Bird Horse under $12.5 K $12.5-$24.999K $25K-$39.999K $40K-$59.999K $60K or above Whenever we get a 2-way-table, we should first make a TOTAL row, a TOTAL column and a TOTAL-TOTAL, which will be the sum of either the TOTAL row or TOTAL column (note, if they do not both sum to the same TOTAL-TOTAL, redo your computations for mistakes!). We will need this extra row and column when we find distributions related with this 2-way-table analysis namely the marginal distribution and the categorical distribution. So, below is the same table with the extra row and column added. With these columns added we can now compute these 2 specialized distributions, as well as have the figures necessary to construct segmented bar graphs of these distributions. -1-

2 PET SALARY Dog Cat Bird Horse TOTAL under $12.5 K $12.5-$24.999K $25K-$39.999K $40K-$59.999K $60K or above TOTAL We use the TOTALS row and column to compute the marginal distributions of both variables. In our case above, the marginal for SALARY will use the TOTAL column. So the marginal distribution of SALARY is: under $12.5K 54/400 = $12.5K--$ K 82/400 = $25K--$39.999K 96/400 = $40K--$59.999K 87/400 = above $60K 81/400 = Notice that the denominator of all marginal distributions is the TOTAL-TOTAL, and numerators are each cell number in the TOTAL column. Also notice that (as with any distribution) the sum of all proportions = The marginal of PET comes from the TOTAL row, in the same way as we got the SALARY marginal. The marginal distribution of PET is: dog 100/400 = cat 100/400 = bird 100/400 = horse 100/400 = Below are the segmented bar graphs of each variable. -2-

3 Now, let us compute the conditional distributions. We have to first know which variable is the explanatory (x) and which is the response (y). Normally, the table will indicate which is which for the study, where the x variable categories are the column headings, and y variable categories are the row column headings (on the left side of the table). If we follow that convention, then we will want to construct a visualization of the conditional distribution, called the segmented bar graph, where the bars are each category of x variable, and segments are the categories of the y variable. So, for our case, we will have each bar as a PET, with the proportion of the bar as each level of SALARY. We say that we want the conditional of SALARY to PET, or SALARY vs PET, or SALARY on PET. In general we say we want the conditional of y to x, y vs x, etc. We use each cell value in each PET column as numerator of each proportion, and use the TOTAL at the bottom of the column as the denominator of the proportion. Below is our computation of the conditional distribution of SALARY vs PET. < $12.5K $ K $25-40K $40-60K > $60K Dog 14/100 20/100 24/100 22/100 20/100 = 0.14 = 0.20 = 0.24 = 0.22 = 0.20 Cat 15/100 20/100 23/100 22/100 20/100 = 0.15 = 0.20 = 0.23 = 0.22 = 0.20 Bird 16/100 21/100 24/100 21/100 18/100 = 0.16 = 0.21 = 0.24 = 0.21 = 0.18 Horse 9/100 21/100 25/100 22/100 23/100 = 0.09 = 0.21 = 0.25 = 0.22 = 0.23 Below is a segmented bar graph of this conditional. -3-

4 We notice that these bars are different, enough to say that the variables PET and SALARY are not independent. It seems that the richer salaried people have horses, the poorer salaried people have more birds, etc. Possibly the richer the people are, the more expensive pet maintenance they can afford. If we had wanted the conditional distribution of PET on SALARY, we would have the following, where we end up with the bars of the segmented bar graph being levels of SALARY (the explanatory or x variable) and the PET being the segments (because it is the response or y variable). Notice in this case we use the rows instead of the columns we have numerators which are the cell entries, with the TOTAL on the right being the denominators. The computations for PET vs SALARY is below. Dog Cat Bird Horse < $12K = 14/54 = 15/54 = 16/54 = 9/54 = = =.2963 = $12-25K = 20/82 = 20/82 = 21/82 = 21/82 = = = = $25-40K = 24/96 = 23/96 = 24/96 = 25/96 = = = = $40-60K = 22/87 = 22/87 = 21/87 = 22/87 = = = = > $60K = 20/81 = 20/81 = 18/81 = 23/81 = = = = The segmented bar graph is shown below. -4-

5 Summary of 2 categorical variable analysis: Some things to remember when doing this very computational activity-- when computing marginal distributions, ALWAYS use the TOTAL-TOTAL in the denominators when computing conditional distributions, NEVER use the TOTAL-TOTAL in the denominators conditional distributions are distributions where the denominators are numbers from either the TOTAL row or TOTAL column conditional distributions are where the world is narrowed down to only those subjects who have one and only one condition (or category) of explanatory (x) variable there are as many conditional figures (numbers or bars ) as there are worlds of conditions (or categories) of the explanatory (x) variable two categorical variables are independent if the category of one variable does not influence or change the proportion of participation by the category of the other variable two categorical variables are independent (do not have much, if any, relationship or effect on each other) if the bars look the same --two categorical variables are not independent (do have a relationship) if the bars are different if the bars look the same, then they will be like the bars of the marginal distribution perfect independence is when the conditional bars match the marginal bar (as well as match each other) -5-