STAT10x10.13.05TwoWayTables

# STAT10x10.13.05TwoWayTables - Inference for Two-Way Tables...

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Inference for Two-Way Tables : The finer things in life (Agresti, Categorical Data Analysis is a good additional resource for this topic) Example : Music and Wine . A marketing study examined the effect of playing different kinds of music on the number of bottles of wine purchased. Three conditions were tried and the distribution of the first 100 bottles of wine purchased were recorded. Here were the results : Wine Purchased Music Played French Italian Other TOTAL None 36 13 51 100 French Accordion 51 3 46 100 Italian Strings 36 23 42 100 TOTAL 123 39 139 300 Data is presented in a two-way or contingency table with r =3 rows and c =3 columns . The total number of observations n is 300. Each value is placed in a cell . Two-way tables present relationships between different levels of two different categorical variables. STAT 101-106a Introduction to Statistics 227

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Marginal Distributions : Same table as above presented as percentages : Wine Purchased Music Played French Italian Other TOTAL None 12% 4% 17% 33% French Accordion 17% 1% 15% 33% Italian Strings 12% 8% 14% 33% TOTAL 41% 13% 46% 100% In both tables, the values that occur in the row and column titled “Total” are called the marginal distributions of each variable. If we pay no attention to the kind of music played, 41% of sales are French wine, and 13% of sales are Italian wine. Conditional Distributions The distribution of one variable at a fixed level of the other variable. Within a row (or column), percentages total 100% for each row (or column). STAT 101-106a Introduction to Statistics 228
Relative Risk (a term mostly appropriate to medical trials where tables are commonly used) Measures differences between two levels in one variable at a particular level of another variable Given by the ratio of values in two conditional distributions Example : the relative risk of buying Italian wine given that Italian string music is playing vs. no music is 8 . 1 % 13 % 23 = That is, almost twice as many bottles of Italian wine are sold when Italian string music is on vs. no music as all. The same type of calculation indicates that 8 % 3 % 23 = times as much Italian wine is sold with Italian music vs. French accordion music. STAT 101-106a Introduction to Statistics 229 Wine Purchased Music Played French Italian Other TOTAL None 36% 13% 51% 100% French Accordion 51% 3% 46% 100% Italian Strings 36% 23% 42% 100%

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The real question here : “Does music influence the type of wine being bought”? We need to be able to perform a hypothesis test!! Hypothesis Test of Independence : the Chi-Squared Distribution Recall the four-step general process for hypothesis testing : Step 1 : Formulate two hypotheses The standard hypothesis test for two-way tables : H o : There is no relationship between the variables (variables are independent ) Ha : There is a relationship between the variables (variables are not independent ) If there is no relationship between music and wine purchase, the EXPECTED number of observations in each cell should be the product of the marginal probabilities multiplied by the total number of observations.
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STAT10x10.13.05TwoWayTables - Inference for Two-Way Tables...

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