# Use of the Chi-Square Statistic. Marie Diener-West, PhD Johns Hopkins University

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1 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this site. Copyright 2008, The Johns Hopkins University and Marie Diener-West. All rights reserved. Use of these materials permitted only in accordance with license rights granted. Materials provided AS IS ; no representations or warranties provided. User assumes all responsibility for use, and all liability related thereto, and must independently review all materials for accuracy and efficacy. May contain materials owned by others. User is responsible for obtaining permissions for use from third parties as needed.

2 Use of the Chi-Square Statistic Marie Diener-West, PhD Johns Hopkins University

3 Section A Use of the Chi-Square Statistic in a Test of Association Between a Risk Factor and a Disease

4 The Chi-Square ( X 2 ) Statistic Categorical data may be displayed in contingency tables The chi-square statistic compares the observed count in each table cell to the count which would be expected under the assumption of no association between the row and column classifications The chi-square statistic may be used to test the hypothesis of no association between two or more groups, populations, or criteria Observed counts are compared to expected counts 4

5 Displaying Data in a Contingency Table Criterion 2 Criterion C Total 1 n11 n12 n13 n1c r1 2 n21 n22 n23 n2c r2 3 n r nr1 nrc rr Total c1 c2 cc n. 5

6 Chi-Square Test Statistic The test statistic is: c 2 k = i= 1 ( i E i O - E i ) 2 The degrees of freedom are: (r 1)(c 1) r = # of rows and c = # of columns Where: O i = the observed frequency in the i th cell of the table E i = the expected frequency in the i th cell of the table 6

7 Example: Is Disease Associated With Exposure? The relationship between disease and exposure may be displayed in a contingency table We can see that: 37/54 = 68 % of diseased individuals were exposed 13/66 = 20 % of non-diseased were exposed Do these data suggest an association between disease and exposure? Disease Exposure Yes No Total Yes No Total

8 Observed Counts The observed numbers or counts in the table are: Disease Exposure Yes No Total Yes No Total

9 Test of No Association Question of interest: is disease associated with exposure? Calculate what numbers of exposed and non-exposed individuals would be expected in each disease group if the probability of disease were the same in both groups If there was no association between exposure and disease, then the expected counts should nearly equal the observed counts, and the value of the chi-square statistic would be small In this example, we can calculate: Overall proportion with exposure = 50/120 = 0.42 Overall proportion without exposure = 70/120 = 0.58 =

10 Expected Counts Under the assumption of no association between exposure and disease, the expected numbers or counts in the table are: Disease Exposure Yes No Total Yes 50/120 x 54 = /120 x 66 = No 70/120 x 54 = /120 x 66 = Total

11 Chi-Squared (X 2 ) Statistic χ 2 = i (O i E i ) 2 E i = ( ) ( ) ( ) ( )

12 Test of Association The test statistic is: χ 2 = 29.1 with 1 degree of freedom Assumption: no association between disease and exposure A small value of the χ 2 statistic supports this assumption (observed counts and expected counts would be similar) A large value of the χ 2 statistic would not support this assumption (observed counts and expected counts would differ) What is the probability of obtaining a statistic of this magnitude or larger when there is no association? 12

13 Probability Associated with a X 2 Statistic If the assumption of no association is true, then what is the probability of observing this value of the χ 2 statistic? Table A.8 of the Pagano text provides the probability (area in upper tail of the distribution) associated with values of the chi-square statistic for varying degrees of freedom Degrees of freedom = 1 for a 2x2 table: 13

14 Conclusion: Test of (No) Association For the data in this example, χ 2 = 29.1 with 1 degree of freedom From the chi-squared table, the probability obtaining a statistic of this magnitude or larger when there is no association is < In other words, the probability of obtaining discrepancies between observed and expected counts of this magnitude is < (unlikely to occur by chance alone) Conclude that our finding is unlikely to occur if there is no association between disease and exposure Thus, we conclude that there appears to be an association 14

15 Guidelines for Interpreting the X 2 Statistic The χ 2 statistic is calculated under the assumption of no association Large value of χ 2 statistic small probability of occurring by chance alone (p < 0.05) conclude that association exists between disease and exposure Small value of χ 2 statistic large probability of occurring by chance alone (p > 0.05) conclude that no association exists between disease and exposure 15

16 Short-Cut X 2 Formula for 2x2 Table Suppose: Disease Exposure Yes No Total Yes a b a+b No c d c+d Total a+c b+d n Then we can write: χ 1 2 = n(ad bc) 2 (a+ c)(b+ d)(a+b)(c+ d) 16

17 Short-Cut X 2 Formula for the Example Using this formula for the previous example gives χ [(37)(53) (13)(17)] = 54(66)(50)(70) = 29.1 Same as before! Disease Exposure Yes No Total Yes No Total

18 Helpful Hints Regarding the Chi-Square Statistic The calculations use expected and observed counts or frequencies, not proportions The χ 2 short-cut formula applies only to 2x2 tables Probabilities are available from tables and computing packages 18

19 Review The χ 2 statistic provides a statistical test for ascertaining whether an association exists between disease and exposure A large value of the χ 2 statistic indicates that the observed data are unlikely under an assumption of no association between disease and exposure small probability (pvalue) association A small value of the χ 2 statistic indicates that the observed data are likely under an assumption of no association between disease and exposure large probability (pvalue) no association 19

20 Section B Applications of the Chi-Square Statistic in Epidemiology

21 Applications of the X 2 Statistic in Epidemiology Cohort study (2 samples) Case-control study (2 samples) Matched case-control study (paired cases and controls) 21

22 Cohort Study 2x2 Table Disease Factor Present (D) Absent (D) Total Pres ent (F) a b a+b Abs ent (F) c d c+d Total a+c b+d N 22

23 Cohort Study: Measure of Association Assumptions: The two samples are independent Let a+b = number of people exposed to the risk factor Let c+d = number of people not exposed to the risk factor Assess whether there is association between exposure and disease by calculating the relative risk (RR) 23

24 Cohort Study: Relative Risk We can define the relative risk of disease: p1= P(disease factorpresent) = P(DF) p2 = P(disease factor absent)=p(d F ) RR = p1 p2 = P(DF) P(D F ) is called the relative risk 24

25 Cohort Study: Test of Association For these samples, we can estimate the relative risk as: a RR = a+b c c+d We can test the hypothesis that RR=1 by calculating the chisquare test statistic χ 1 2 = n(ad bc) 2 (a+ c)(b+ d)(a+b)(c+ d) 25

26 Example: Test of Association in a Cohort Study Develop CHD Smoke Yes No Total Yes No Total RR = 1.61 Chi-square statistic = χ 2 1 = 10.1 = 8000(84(4913) 2916(87)) 2 ( )( )( )( ) 26

27 Example: Test of Association in a Cohort Study j Using Table A.8 in the Pagano text, the probability is less than (between and 0.010) This supports an association between exposure and disease 27

28 Case-Control Study 2x2 Table Disease Factor Present (case) Absent (control) Total Pres ent a b a+b Abs ent c d c+d Total a+c b+d N 28

29 Case-Control Study: Measure of Association Assumptions The samples are independent Cases = diseased individuals = a+c Controls = non-diseased individuals = b+d We are interested in whether: P( F D) = P( F D ) We cannot estimate P(D), the prevalence of the disease and, hence, cannot estimate the RR Assess whether there is association between exposure and disease by calculating the odds ratio (OR) 29

30 Case-Control Study: Odds for Diseased Group The odds of exposure for the diseased group is: p 1 1 p 1 = a a+ c c a+ c = a c Disease Factor Present (case) Absent (control) Total Present a b a+b Absent c d c+d Total a+c b+d N 30

31 Case-Control Study: Odds for Non-Diseased Group The odds of exposure for the non-diseased group is: p 2 1 p 2 = b b+d d b+d = b d Disease Factor Present (case) Absent (control) Total Present a b a+b Absent c d c+d Total a+c b+d N 31

32 Case-Control Study: Odds Ratio The odds ratio is: p 1 1 p 1 p 2 1 p 2 And is estimated by OR = a c b d = ad bc 32

33 Case-Control Study: Test of Association We can test whether OR=1 by calculating the chi-square statistic: χ n(ad bc) = (a + c)(b + d)(a + b)(c + d) 33

34 Example: Test of Association in a Case-Control Study Past Smoking Disease CHD Cases Controls Total Yes No Total OR = 1.62 Chi-square statistic = χ 2 1 = 7.69 = 2 600(112(224) 176(88)) ( )( )( )( ) 34

35 Example: Test of Association in a Case-Control Study Using Table A.8 in the Pagano text, the probability is between and 0.01 This supports association between exposure and disease 35

36 Matched Case-Control Study Table Cases Controls Exposed Not Exposed Exposed aa bb Not exposed cc dd 36

37 Matched Study: Measure of Association Assumptions Case-control pairs are matched on characteristics such as age, race, sex Samples are not independent The discordant pairs are case-control pairs with different exposure histories The matched odds ratio is estimated by bb/cc Pairs in which cases exposed but controls not = bb Pairs in which controls exposed but cases not = cc Assess whether there is association between exposure and disease by calculating the matched odds ratio (OR) 37

38 Matched Case-Control Study: Test of Association We can test whether OR = 1 by calculating McNemar s statistic McNemar s test statistic: χ 2 1 = ( bb cc 1) (bb + cc) 2 38

39 Example: Test of Association in a Matched Study Cases Controls Exposed Not Exposed Exposed 2 4 Not exposed 1 3 OR = bb/cc = 4 McNemar s test statistic =χ 2 1 = 0.80 χ 2 1 = ( 4 1 1) (4 + 1) 2 39

40 Example: Test of Association in a Matched Study Using Table A.8 in the Pagano text, the probability is greater than This supports no association between exposure and disease 40

41 Review: Uses of the Chi-Squared Statistic The chi-squared statistic provides a test of the association between two or more groups, populations, or criteria The chi-square test can be used to test the strength of the association between exposure and disease in a cohort study, an unmatched case-control study, or a cross-sectional study McNemar s test can be used to test the strength of the association between exposure and disease in a matched casecontrol study 41

42 Review: Association between Exposure and Disease The χ 2 statistic is calculated under the assumption of no association A large value of the chi-square or McNemar s test statistic small probability supports an association A small value of the chi-square or McNemar s test statistic large probability does not support an association between exposure and disease 42

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