The 2*2 contingency table is one of the most commonly used ways to summarize categorical data. Let's consider our Remedial Mathematical Program (RMP) example. Categorizing students by their performances on the examination (pass/fail) to two different instructions (CAI-Tutoring) can be summarized in the 2*2 table.
The interest in the 2*2 table is in whether there exists an association between the row variables (usually explanatory variables) and the column variables (usually response variables). (Table 5) summarizes the information from a randomized clinical trial that compares two different instruction methods.
(Table 5) Remedial Mathematics Program (2 * 2)
Chi-Square Statistics
The question of interest is whether the pass probabilities for CAI and Tutoring are the same. In a statistical term, we can address this question by investigating whether there is a statistical association between the instruction methods (explanatory) and the performance of the exam (response variable). The hypotheses of this test are
H0: There is no association between the instruction methods and the performance of the exam.
H1: There is an association between the instruction methods and the performance of the exam.
The chi-square is one way to test the hypotheses. The significant chi-square statistic, i.e., p<.05, only tells the existence of the association between the explanatory variable and the response variable. After identifying the association, you may want to know how strong its association is. The strength of the association is measured by the difference of proportions, the relative risk, and the odds ratio.
Difference of Proportions
The difference of proportions compares the yes (success, pass) probability between two row groups (explanatory variable). It is defined to be
Difference of Proportions (DP) = p1 - p2, where p1 = n11/n1+, p2 = n21/n2+.
In our example, the difference of proportions calculates the difference of the pass probability between CAI group and Tutoring group. The difference of proportion in RMP is 0.846 - 0.914 = -0.068. The yes probability of Tutoring group is larger than that of CAI group.
Relative Risk
The relative risk is the ratio of the yes probability for the two row groups (explanatory variable). It is defined to be
Relative Risk (RR) = p1 / p2.
In our example, the relative risk is 0.846/0.914=0.926. The proportion of passing the exam in CAI group is .926 times that in Tutoring group.
Odds Ratio
The odds ratio is a ratio of two odds. There are two ways to calculate the odds ratio. One is to use the ratio of two odds, and the other is to use the cross-product ratio.
The first way is to use the ratio of two separate odds. The first odds of success in the first row are odds1 and the second odds of success in the second row are odds2. Each odds is defined to be
odds1 = p1 / (1-p1)
odds2 = p2 / (1-p2).
The ratio of odds of success from row 1 and row 2 is called the odds ratio and is defined to be
Odds Ratio (OR) = odds1 / odds2 = {p1 / (1-p1)} / {p2 / (1-p2)}.
In our example, the first odds (odds1) is p1/(1-p1)=0.846/(1-0.846) = 5.494; the second odds (odds2) is p2/(1-p2)=0.914/(1-0.914)=10.628. With two odds, we can calculate the odds ratio between odds1 and odds2. The odds ratio is 5.494/10.628=0.517.
The second way to calculate the odds ratio is to use the cross-product ratio, which is defined to be
Odds Ratio (calculated by the cross-product ratio) = (n11*n22) / (n12*n21).
In our RMP example, the odds ratio calculated by the cross-product ratio is (55*3)/(32*10)=0.516. The cross-product ratio is equivalent to the ratio of two odds (OR=odds1/odds2).
It is concluded that the odds of passing the exam in CAI is almost 0.52 times the odds of passing the exam in Tutoring group. In other words, the odds of passing the exam in Tutoring group is almost twice as high as CAI group. Tutoring seems to be better way to improve the math performance in the Remedial Mathematic Program. Because of convenience, the cross-product odds ratio method is preferred in hand calculation.
In short, the odds ratio compares the odds of success (pass) proportion for CAI Group to the odds of success (pass) proportion for Tutoring Group. In general, the odds ratio compares the odds of success proportion in row 1 (Group 1) to the odds of success proportion in row 2 (Group 2).
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The interest in the 2*2 table is in whether there exists an association between the row variables (usually explanatory variables) and the column variables (usually response variables). (Table 5) summarizes the information from a randomized clinical trial that compares two different instruction methods.
(Table 5) Remedial Mathematics Program (2 * 2)
Chi-Square Statistics
The question of interest is whether the pass probabilities for CAI and Tutoring are the same. In a statistical term, we can address this question by investigating whether there is a statistical association between the instruction methods (explanatory) and the performance of the exam (response variable). The hypotheses of this test are
H0: There is no association between the instruction methods and the performance of the exam.
H1: There is an association between the instruction methods and the performance of the exam.
The chi-square is one way to test the hypotheses. The significant chi-square statistic, i.e., p<.05, only tells the existence of the association between the explanatory variable and the response variable. After identifying the association, you may want to know how strong its association is. The strength of the association is measured by the difference of proportions, the relative risk, and the odds ratio.
Difference of Proportions
The difference of proportions compares the yes (success, pass) probability between two row groups (explanatory variable). It is defined to be
Difference of Proportions (DP) = p1 - p2, where p1 = n11/n1+, p2 = n21/n2+.
In our example, the difference of proportions calculates the difference of the pass probability between CAI group and Tutoring group. The difference of proportion in RMP is 0.846 - 0.914 = -0.068. The yes probability of Tutoring group is larger than that of CAI group.
Relative Risk
The relative risk is the ratio of the yes probability for the two row groups (explanatory variable). It is defined to be
Relative Risk (RR) = p1 / p2.
In our example, the relative risk is 0.846/0.914=0.926. The proportion of passing the exam in CAI group is .926 times that in Tutoring group.
Odds Ratio
The odds ratio is a ratio of two odds. There are two ways to calculate the odds ratio. One is to use the ratio of two odds, and the other is to use the cross-product ratio.
The first way is to use the ratio of two separate odds. The first odds of success in the first row are odds1 and the second odds of success in the second row are odds2. Each odds is defined to be
odds1 = p1 / (1-p1)
odds2 = p2 / (1-p2).
The ratio of odds of success from row 1 and row 2 is called the odds ratio and is defined to be
Odds Ratio (OR) = odds1 / odds2 = {p1 / (1-p1)} / {p2 / (1-p2)}.
In our example, the first odds (odds1) is p1/(1-p1)=0.846/(1-0.846) = 5.494; the second odds (odds2) is p2/(1-p2)=0.914/(1-0.914)=10.628. With two odds, we can calculate the odds ratio between odds1 and odds2. The odds ratio is 5.494/10.628=0.517.
The second way to calculate the odds ratio is to use the cross-product ratio, which is defined to be
Odds Ratio (calculated by the cross-product ratio) = (n11*n22) / (n12*n21).
In our RMP example, the odds ratio calculated by the cross-product ratio is (55*3)/(32*10)=0.516. The cross-product ratio is equivalent to the ratio of two odds (OR=odds1/odds2).
It is concluded that the odds of passing the exam in CAI is almost 0.52 times the odds of passing the exam in Tutoring group. In other words, the odds of passing the exam in Tutoring group is almost twice as high as CAI group. Tutoring seems to be better way to improve the math performance in the Remedial Mathematic Program. Because of convenience, the cross-product odds ratio method is preferred in hand calculation.
In short, the odds ratio compares the odds of success (pass) proportion for CAI Group to the odds of success (pass) proportion for Tutoring Group. In general, the odds ratio compares the odds of success proportion in row 1 (Group 1) to the odds of success proportion in row 2 (Group 2).
www.socialresearchmethods.net
1 comments:
really odd... :-)
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