Nov 17 2004 Lab 8 Econ240A 1 L Phillips Goodness of Fit Chi Square and Contingency Table Analysis I Goodness of Fit For A Variable with the Multinomial Distribution This is an example from the text Chapter 16 problem 16 7 p 550 It uses the data file XR16 07 XR15 07 5th Ed There are 100 trials with outcomes ranging from integer values of 1 through five If each value is equally likely the associated probability will be 1 5 and the expected number in 100 trials will be 20 This is listed in Table 1 For example on the first trial the probability of getting the outcome with value one is given by the multinomial 5 P n1 1 n2 0 n3 0 n4 0 n5 0 n n j j 1 5 p j n j j 1 1 1 0 0 0 0 0 1 5 1 1 5 0 1 5 0 1 5 0 1 5 0 1 5 Table 1 Expected Frequencies For Each Outcome Values 1 5 in 100 Trials Outcome Probability Expected Frequency 1 1 5 20 2 1 5 20 3 1 5 20 4 1 5 20 5 1 5 20 Open the file XR15 07 in Excel Go to cell D1 and type outcome and in cell E1 type count In cells D2 D6 type sequentially 1 5 for each value of the outcome Select cell E2 click the equal sign or type it in the formula bar and select statistical for function class and select countif for function Click on the in the countif box and use the office assistant to read about this function In the dialog box type in 1 for criteria and type in A2 A101 for range The box will indicate a count of 28 Go to cell E3 and repeat for outcome 2 and so on though outcome value 5 Go to cell D7 and type in sum Go to cell E7 and click on and select the sum function For number select E2 E6 You should get 100 the number of trials as a check Table 2 lists the outcome values the expected frequencies and the oberved frequencies recovered from this data file Nov 17 2004 Lab 8 Econ240A 2 L Phillips Goodness of Fit Chi Square and Contingency Table Analysis Table 2 Expected and Observed Frequencies For Each Outcome Values 1 5 Outcome Observed Frequency Expected Frequency 1 28 20 2 17 20 3 19 20 4 17 20 5 19 20 To check how close the observed simulated frequencies come to the expected cell counts for each outcome take the difference between the observed cell count and the expected cell count square this difference and divide by the expected cell count This is the contribution of each outcome to the Chi Square statistic which is the sum over all 5 outcomes Oj Ej 2 Ej This process is displayed in Table 3 j 1 Table 3 Simulated frequencies Compared to Theoretical Outcome Observed Oj Expected Ej 1 28 20 2 17 20 3 19 20 4 17 20 5 19 20 2 3 20 0 45 0 05 0 45 0 05 4 20 Oj Ej 2 Ej 64 20 3 20 9 20 0 45 1 20 0 05 9 20 0 45 1 20 0 05 Oj Ej 8 3 1 3 1 There are four degrees of freedom since there are five outcomes with probability 0 20 where the sum of all five add to one so only four are independent as the fifth probability 5 can be found by subtracting the other four from one This statistic Oj Ej 2 Ej is j 1 distributed as Chi Square with four degrees of freedom In general if you take independently distributed normal variables subtract their mean and divide by their standard deviation i e in z or standardized form and square and sum them they are distributed Chi Square The critical value of Chi square at the 5 level of significance Nov 17 2004 Lab 8 Econ240A 3 L Phillips Goodness of Fit Chi Square and Contingency Table Analysis the problem uses 10 for 4 degrees of freedom is from Table 5 in the text p B 10 9 49 So there is no significant difference between the expected distribution and the observed simulated distribution The Chi Square distribution for 4 degrees of freedom is illustrated in Figure 1 Figure 1 Chi Square Density for 4 Degrees of Freedom 0 20 DENSITY 0 15 0 10 0 05 5 0 00 0 5 9 48 10 15 II The Chi Square Distribution in EViews Chi Square Variable To create such a figure open Eviews go to the file menu and open a new workfile In the box select undated for the data frequency and a range of 1 to 100 observations more if you want a more dense plot In the workfile window select the GENR command and in the window type CHI rchisq 4 to generate a random variable distributed Chi Square with 4 degrees of freedom To get background information go to the EViews help menu select Eviews Help Topics Contents Tab and select Eviews Basics Double click on Using Expressions Nov 17 2004 Lab 8 Econ240A 4 L Phillips Goodness of Fit Chi Square and Contingency Table Analysis and read Scroll down until you get to Mathematical Operators and Functions and click and scroll way down about of the way until you get to Statistical Distribution Functions and read Here you will find the Rosetta Stone for deciphering what we are doing with the Chi Square The guide to doing similar exercises with other distributions is here To calculate the Chi Square density in the workfile window select the GENR command and in the window type density dchisq chi 4 to generate the ChiSquare density with 4 degrees of freedom for our random variable CHI distributed ChiSquare Go to the Quick menu and select graph In the window type in chi density and select scatterplot to obtain Figure 1 I added the critical value for 0 05 in Word III Two Way Contingency Tables in Excel This next example is also from the text Chapter 16 Excel data file XR1625 XR15 24 5th Ed problem 16 25 p 545 This is relevant for commercial advertising on TV Students were surveyed One group was watching a happy program Real People and the other group a sad Sixty Minutes program They were asked what they were thinking during the final commercial with the responses categorized into three a thinking primarily about the commercial b thinking primarily about the program or c thinking about both Open the data file XR15 24 in Stats the special macro supplied by Keller with Data Analysis Plus Note there are 151 observations in columns A and B headed by program and thinking Program is coded 1 and 2 for real People and Sixty Minutes respectively Thinking is coded 1 2 and 3 corresponding to a b and c above Select the two columns of data A2 B152 and go to the Tools menu and select Data Nov 17 2004 Lab …
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