Basic Analysis • Contingency Analysis • Additional Examples of the Contingency Platform
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Additional Examples of the Contingency Platform
This section contains additional examples using the options in the Contingency platform.
Example of Analysis of Means for Proportions
This example uses the Office Visits.jmp sample data table, which records late and on-time appointments for six clinics in a geographic region. 60 random appointments were selected from 1 week of records for each of the six clinics. To be considered on-time, the patient must be taken to an exam room within five minutes of their scheduled appointment time. Examine the proportion of patients that arrived on-time to their appointment.
1.
Select Help > Sample Data Library and open Office Visits.jmp.
2.
Select Analyze > Fit Y by X.
3.
Select On Time and click Y, Response.
4.
Select Clinic and click X, Factor.
5.
Select Frequency and click Freq.
6.
Click OK.
7.
From the red triangle menu next to Contingency Analysis, select Analysis of Means for Proportions.
8.
From the red triangle menu next to Analysis of Means for Proportions, select Show Summary Report and Switch Response Level for Proportion.
Example of Analysis of Means for Proportions
Example of Analysis of Means for Proportions shows the proportion of patients who were on-time from each clinic. From Example of Analysis of Means for Proportions, notice the following:
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The proportion of on-time arrivals is the highest for clinic F, followed by clinic B.
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Clinic D has the lowest proportion of on-time arrivals, followed by clinic A.
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Clinic E and clinic C are close to the average, and do not exceed the decision limits.
Example of Correspondence Analysis
This example uses the Cheese.jmp sample data table, which is taken from the Newell cheese tasting experiment, reported in McCullagh and Nelder (1989). The experiment records counts more than nine different response levels across four different cheese additives.
1.
Select Help > Sample Data Library and open Cheese.jmp.
2.
Select Analyze > Fit Y by X.
3.
Select Response and click Y, Response.
The Response values range from one to nine, where one is the least liked, and nine is the best liked.
4.
Select Cheese and click X, Factor.
A, B, C, and D represent four different cheese additives.
5.
Select Count and click Freq.
6.
Click OK.
Mosaic Plot for the Cheese Data
From the mosaic plot in Mosaic Plot for the Cheese Data, you notice that the distributions do not appear alike. However, it is challenging to make sense of the mosaic plot across nine levels. A correspondence analysis can help define relationships in this type of situation.
7.
To see the correspondence analysis plot, from the red triangle menu next to Contingency Analysis, select Correspondence Analysis.
Example of a Correspondence Analysis Plot
Example of a Correspondence Analysis Plot shows the correspondence analysis graphically, with the plot axes labeled c1 and c2. Notice the following:
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c1 seems to correspond to a general satisfaction level. The cheeses on the c1 axis go from least liked at the top to most liked at the bottom.
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Cheese D is the most liked cheese, with responses of 8 and 9.
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Cheese B is the least liked cheese, with responses of 1,2, and 3.
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Cheeses C and A are in the middle, with responses of 4,5,6, and 7.
8.
From the red triangle menu next to Correspondence Analysis, select 3D Correspondence Analysis.
Example of a 3-D Scatterplot
From Example of a 3-D Scatterplot, notice the following:
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Looking at the c1 axis, responses 1 through 5 appear to the right of 0 (positive). Responses 6 through 9 appear to the left of 0 (negative).
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Looking at the c2 axis, A and C appear to the right of 0 (positive). B and D appear to the left of 0 (negative).
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You can conclude that c1 corresponds to the general satisfaction (from least to most liked).
Example of a Cochran Mantel Haenszel Test
This example uses the Hot Dogs.jmp sample data table. Examine the relationship between hot dog type and taste.
1.
Select Help > Sample Data Library and open Hot Dogs.jmp.
2.
Select Analyze > Fit Y by X.
3.
Select Type and click Y, Response.
4.
Select Taste and click X, Factor.
5.
Click OK.
6.
From the red triangle menu next to Contingency Analysis, select Cochran Mantel Haenszel.
7.
Select Protein/Fat as the grouping variable and click OK.
Example of a Cochran-Mantel-Haenszel Test
From Example of a Cochran-Mantel-Haenszel Test, you notice the following:
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The Tests report shows a marginally significant Chi-square probability of about 0.0799, indicating some significance in the relationship between hot dog taste and type.
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The Cochran Mantel Haenszel report shows that the p-value for the general association of categories is 0.2816, which is much larger than 5%.
Example of the Agreement Statistic Option
This example uses the Attribute Gauge.jmp sample data table. The data gives results from three people (raters) rating fifty parts three times each. Examine the relationship between raters A and B.
1.
Select Help > Sample Data Library and open Attribute Gauge.jmp.
2.
Select Analyze > Fit Y by X.
3.
Select A and click Y, Response.
4.
Select B and click X, Factor.
5.
Click OK.
6.
From the red triangle menu next to Contingency Analysis, select Agreement Statistic.
Example of the Agreement Statistic Report
From Example of the Agreement Statistic Report, you notice that the agreement statistic of 0.86 is high (close to 1) and the p-value of <.0001 is small. This reinforces the high agreement seen by looking at the diagonal of the contingency table. Agreement between the raters occurs when both raters give a rating of 0 or both give a rating of 1.
Example of the Relative Risk Option
This example uses the Car Poll.jmp sample data table. Examine the relative probabilities of being married and single for the participants in the poll.
1.
Select Help > Sample Data Library and open Car Poll.jmp.
2.
Select Analyze > Fit Y by X.
3.
Select marital status and click Y, Response.
4.
Select sex and click X, Factor.
5.
Click OK.
6.
From the red triangle menu next to Contingency Analysis, select Relative Risk.
The Choose Relative Risk Categories window appears.
The Choose Relative Risk Categories Window
Note the following about the Choose Relative Risk Categories window:
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If you are interested in only a single response and factor combination, you can select that here. For example, if you clicked OK in the window in The Choose Relative Risk Categories Window, the calculation would be as follows:
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If you would like to calculate the risk ratios for all (=4) combinations of response and factor levels, select the Calculate All Combinations check box. See Example of the Risk Ratio Report.
7.
Ask for all combinations by selecting the Calculate All Combinations check box. Leave all other default selections as is.
Example of the Risk Ratio Report
To see how the relative risk is calculated, proceed as follows:
1.
Examine the first entry in the Relative Risk report, which is P(Married|Female)/P(Married|Male).
2.
You can find these probabilities in the Contingency Table. Since the probabilities are computed based on two levels of sex, which differs across the rows of the table, use the Row% to read the probabilities, as follows:
P(Married|Female)=0.6884
P(Married|Male) = 0.6121
Therefore, the calculations are as follows:
P(Married|Female)/P(Married|Male) = = 1.1247
Example of a Two Sample Test for Proportions
This example uses the Car Poll.jmp sample data table. Examine the probability of being married for males and females.
1.
Select Help > Sample Data Library and open Car Poll.jmp.
2.
Select Analyze > Fit Y by X.
3.
Select marital status and click Y, Response.
4.
Select sex and click X, Factor.
5.
Click OK.
6.
From the red triangle menu next to Contingency Analysis, select Two Sample Test for Proportions.
Example of the Two Sample Test for Proportions Report
In this example, you are comparing the probability of being married between females and males. See the Row% in the Contingency Table to obtain the following:
P(Married|Female)=0.6884
P(Married|Male) = 0.6121
The difference between these two numbers, 0.0763, is the Proportion Difference shown in the report. The two-sided confidence interval is [-0.03175, 0.181621]. The p-value by the adjusted Wald method corresponding to the confidence interval is 0.1686, which is close to the p-value (0.1665) by Pearson’s Chi-square test. Generally, Pearson’s Chi-square test is more popular than the modified Wald’s test for testing the difference of two proportions.
Example of the Measures of Association Option
This example uses the Car Poll.jmp sample data table. Examine the probability of being married for males and females.
1.
Select Help > Sample Data Library and open Car Poll.jmp.
2.
Select Analyze > Fit Y by X.
3.
Select marital status and click Y, Response.
4.
Select sex and click X, Factor.
5.
Click OK.
6.
From the red triangle menu next to Contingency Analysis, select Measures of Association.
Example of the Measures of Association Report
Since the variables that you want to examine (sex and marital status) are nominal, use the Lambda and Uncertainty measures. All of them are small, so it seems that there is a weak association between sex and marital status.
Example of the Cochran Armitage Trend Test
1.
Select Help > Sample Data Library and open Car Poll.jmp.
For the purposes of this test, change size to an ordinal variable:
2.
In the Columns panel, right-click on the icon next to size and select Ordinal.
3.
Select Analyze > Fit Y by X.
4.
Select sex and click Y, Response.
5.
Select size and click X, Factor.
6.
Click OK.
7.
From the red triangle menu next to Contingency Analysis, select Cochran Armitage Trend Test.
Example of the Cochran Armitage Trend Test Report
The two-sided p-value (0.7094) is large. From this, you cannot conclude that there is a relationship in the proportion of male and females that purchase different sizes of cars.