# Section 12 Part 2. Chi-square test

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1 Section 12 Part 2 Chi-square test McNemar s Test

2 Section 12 Part 2 Overview Section 12, Part 1 covered two inference methods for categorical data from 2 groups Confidence Intervals for the difference of two proportions Two sample z-test of equality of two proportions Section 12 Part 2 covers inference methods for categorical data Chi-square test for comparisons between 2 categorical variables (Fisher s exact test) McNemar s Chi-square (Binomial Test) test for paired categorical data PubH 6414 Section 12 Part 2 2

3 Chi-square test The Chi-square test can also be used to test for independence between two variables The null hypothesis for this test is that the variables are independent (i.e. that there is no statistical association). The alternative hypothesis is that there is a statistical relationship or association between the two variables. The Chi-square test can be used to test for equality of proportions between two or more groups. The null hypothesis for this test is that the 2 proportions are equal. The alternative hypothesis is that the proportions are not equal (test for a difference in either direction) PubH 6414 Section 12 Part 2 3

4 Contingency Tables Setting: Let X 1 and X 2 denote categorical variables, X 1 having I levels and X 2 having J levels. There are IJ possible combinations of classifications. Level=1 Level=2 Level=1 Level=2. Level=J Level=I When the cells contain frequencies of outcomes, the table is called a contingency table. PubH 6414 Section 12 Part 2 4

5 Chi-square Test: Testing for Independence Step 1: Hypothesis (always two-sided): H o : Independent H A : Not independent Step 2: Calculate the test statistic: 2 2 ( xij eij) 2 Χ = ~ χ with df = ( I 1)( J e ij 1) Step 3: Calculate the p-value p-value = P(Χ 2 > X 2 ) <- value 2-sided Step 4: Draw a conclusion p-value<α reject independence p-value>α do not reject independence PubH 6414 Section 12 Part 2 5

6 Racial Differences and Cardiac Arrest In a large mid-western city, the association in the incidence of cardiac arrest and subsequent survival was studied in 6117 cases of non-traumatic, out of hospital cardiac arrest. During a 12 month period, fewer than 1% of African- Americans survived an arrest-to-hospital discharge, compared to 2.6% of Caucasians. PubH 6414 Section 12 Part 2 6

7 Racial Differences and Cardiac Arrest Survival to Discharge Race YES NO Total Caucasian African- American Total PubH 6414 Section 12 Part 2 7

8 Racial Differences and Cardiac Arrest Scientific Hypothesis: An association exists between race (African- American/Caucasian) and survival to hospital discharge (Yes/No) in cases of non-traumatic out-of-hospital cardiac arrest. Statistical Hypothesis: H o : Race and survival to hospital discharge are independent in cases of non-traumatic out-of-hospital cardiac arrest. H A : Race and survival to hospital discharge are not independent in cases of non-traumatic out-of-hospital cardiac arrest. PubH 6414 Section 12 Part 2 8

9 Chi-square Test: Testing for Independence 1. Obtain a random sample of n independent observations (the selection of one observation does not influence the selection of any other). 2. Observations are classified subsequently according to cells formed by the intersection rows and columns in a contingency table. Rows (r) consist of mutually exclusive categories of one variable. Columns (c) consist of mutually exclusive categories of the other variable. 3. The frequency of observations in each cell is determined along with marginal totals. PubH 6414 Section 12 Part 2 9

10 Chi-square Test: Testing for Independence 4. Expected frequencies are calculated under the null hypothesis of independence (no association) and compared to observed frequencies. Recall: A and B are independent if: P(A and B) = P(A) * P(B) 5. Use the Chi-square (X 2 ) test statistic to observe the difference between the observed and expected frequencies. PubH 6414 Section 12 Part 2 10

11 χ 2 Distribution The probabilities associated with the chi-square distribution are in appendix D. The table is set up in the same way as the t-distribution. The chi-square distribution with 1 df is the same as the square of the Z distribution. Since the distribution only takes on positive values all the probability is in the right-tail. PubH 6414 Section 12 Part 2 11

12 Chi-square distributions and critical values for 1 df, 4 df and 20 df For Chi-square with 20 df, the critical value (α = 0.05) = Critical value for α = 0.05 and Chi-square with 1 df is 3.84 Critical value for α = 0.05 and Chi-square with 4 df is 9.49 Since the Chi-square distribution is always positive, the rejection region is only in the right tail PubH 6414 Section 12 Part 2 12

13 How to Identify the critical value The rejection region of the Chi-square test is the upper tail so there is only one critical value First calculate the df to identify the correct Chisquare distribution For a 2 X 2 table, there are (2-1)*(2-1) = 1 df R commander: > qchisq(0.95,1) [1] PubH 6414 Section 12 Part 2 13

14 State the conclusion The p-value for P( χ 2 > X 2 ) > 1-pchisq(X 2,1) Reject the null hypothesis by either the rejection region method or the p-value method X 2 > Critical Value or Pvalue < α PubH 6414 Section 12 Part 2 14

15 Racial Differences and Cardiac Arrest Survival to Discharge Race YES NO Total Caucasian African American Total Under the assumption of independence: P(YES and Caucasian) = P(YES)*P(Caucasian) = 108/6117 * 3207/6117 = Expected cell count =e ij = * 6117 = PubH 6414 Section 12 Part 2 15

16 Racial Differences and Cardiac Arrest Survival to Discharge Race YES NO Total Caucasian African American Total Expected Cell Counts = (Marginal Row total * Marginal Column Total)/ n Rule of Thumb: Check to see if expected frequencies are > 2 No more than 20% of cells with expected frequencies < 5 PubH 6414 Section 12 Part 2 16

17 Racial Differences and Cardiac Arrest Step 1: Hypothesis (always two-sided): H o : Independent (Race/Survival) H A : Not independent Step 2: Calculate the test statistic: ( x e ij 2 Χ = eij ij ) 2 = ( ) ( ) ( ) ( ) = = PubH 6414 Section 12 Part 2 17

18 Racial Differences and Cardiac Arrest Step 3: Calculate the p-value p-value = P(Χ 2 >28.42) < > 1-pchisq(28.42,1) [1] e-08 Step 4: Draw a conclusion A significant association exists between race and survival to hospital discharge in cases of non-traumatic out-of-hospital cardiac arrest. PubH 6414 Section 12 Part 2 18

19 Chi-square Test: Testing for Equality or Homogeneity of Proportions Testing for equality or homogeneity of proportions examines differences between two or more independent proportions. In chi-square test for independence, we examine the crossclassification of a single sample of observations on two qualitative variables. The chi-square test can also be used for problems involving two or more independent populations. PubH 6414 Section 12 Part 2 19

20 Chi-square Test: Testing for Equality or Homogeneity of Proportions 30 day outcome Patients with evolving myocardial infarction were assigned independently and randomly to one of four thrombolytic treatments, and then followed to determine 30 day mortality. Streptokinase and SC Heparin Streptokinase and IV Heparin Accelerated t-pa and IV Heparin Accelerated t-pa Streptokinase with IV Heparin Total Survived Died Total Are these four treatment populations equal with respect to 30- day mortality? PubH 6414 Section 12 Part 2 20

21 Chi-square Test: Testing for Equality or Homogeneity of Proportions 30 day outcome Streptokinase and SC Heparin Survived Streptokinase and IV Heparin Example Accelerated t-pa and IV Heparin Accelerated t-pa Streptokinase with IV Heparin Total Died Total Under the assumption of independence: P(Streptokinase and SC Heparin and Survival) = P(Streptokinase and SC Heparin )*P(Survival) = 9796/40845 * 37997/40845 =0.223 Expected cell count =e ij = * = PubH 6414 Section 12 Part 2 21

22 Chi-square Test: Testing for Equality or Homogeneity of Proportions Example 30 day outcome Streptokinase and SC Heparin Streptokinase and IV Heparin Accelerated t-pa and IV Heparin Accelerated t-pa Streptokinase with IV Heparin Total Survived Died Total Under the assumption of independence: Expected Cell Counts = (Marginal Row total * Marginal Column Total)/ n PubH 6414 Section 12 Part 2 22

23 Chi-square Test: Testing for Equality or Homogeneity of Proportions Step 1: Hypothesis (always two-sided): H o : The four treatment options are homogeneous with respect to 30 day survival. H A : The four treatment options are not homogeneous with respect to 30 day survival. Step 2: Calculate the test statistic: 2 2 ( xij eij) 2 Χ = ~ χ with df = ( I 1)( J e ij Step 3: Calculate the p-value = P(Χ 2 > X 2 ) Step 4: Draw a conclusion p-value<α reject independence p-value>α do not reject independence 1) PubH 6414 Section 12 Part 2 23

24 Chi-square Test: Testing for Equality or Homogeneity of Proportions Step 1: Hypothesis (always two-sided): H o : The four treatment options are homogeneous with respect to 30 day survival. H A : The four treatment options are not homogeneous with respect to 30 day survival. Step 2: Calculate the test statistic: Χ 2 = ( x ij e ij e ij ) 2 = with df = (2 1)(4 1) = 3 PubH 6414 Section 12 Part 2 24

25 Chi-square Test: Testing for Equality or Homogeneity of Proportions Step 3: Calculate the p-value p-value = P(Χ 2 >10.85) = > 1-pchisq(10.85,3) [1] Step 4: Draw a conclusion p-value<α=0.05 reject null The four treatment groups are not equal with respect to 30 day mortality. The largest relative departure from expected was noted in patients receiving accelerated t-pa and IV heparin, with fewer patients than expected dying. PubH 6414 Section 12 Part 2 25

26 Chi-square online calculator This website will calculate the Chi-square statistic and p-value for data in a 2 X 2 table. Enter the cell counts in the table. Choose the Chi-square test without Yate s correction to obtain the same results as in the example PubH 6414 Section 12 Part 2 26

27 Chi-Square Testing:Rules of Thumb All expected frequencies should be equal to or greater than 2 (observed frequencies can be less than 2). No more than 20% of the cells should have expected frequencies of less than 5. What if these rules of thumb are violated? PubH 6414 Section 12 Part 2 27

28 Small Expected Frequencies Chi-square test is an approximate method. The chi-square distribution is an idealized mathematical model. In reality, the statistics used in the chi-square test are qualitative (have discrete values and not continuous). For 2 X 2 tables, use Fisher s Exact Test (i.e. P(x=k) ~ B(n,p)) if your expected frequencies are less than 2. (Section 6.6) PubH 6414 Section 12 Part 2 28

29 Tests for Categorical Data To compare proportions between two groups or to test for independence between two categorical variables, use the Chi-square test If more than 20% of the expected cell frequencies < 5, use the Fisher s exact test When categorical data are paired, the McNemar test is the appropriate test. PubH 6414 Section 12 Part 2 29

30 Comparing Proportions with Paired data When data are paired and the outcome of interest is a proportion, the McNemar Test is used to evaluate hypotheses about the data. Developed by Quinn McNemar in 1947 Sometimes called the McNemar Chi-square test because the test statistic has a Chi-square distribution PubH 6414 Section 12 Part 2 30

31 Examples of Paired Data for Proportions Pair-Matched data can come from Case-control studies where each case has a matching control (matched on age, gender, race, etc.) Twins studies the matched pairs are twins. Before - After data the outcome is presence (+) or absence (-) of some characteristic measured on the same individual at two time points. PubH 6414 Section 12 Part 2 31

32 Summarizing the Data Like the Chi-square test, data need to be arranged in a contingency table before calculating the McNemar statistic The table will always be 2 X 2 but the cell frequencies are numbers of pairs not numbers of individuals Examples for setting up the tables are in the following slides for Case Control paired data Twins paired data: one exposed and one unexposed Before After paired data PubH 6414 Section 12 Part 2 32

33 Pair-Matched Data for Case-Control Study: outcome is exposure to some risk factor Control Exposed Unexposed Case Exposed a b Unexposed c d The counts in the table for a case-control study are numbers of pairs not numbers of individuals. PubH 6414 Section 12 Part 2 33

34 Paired Data for Before-After Counts Before treatment After treatment a b - c d The counts in the table for a before-after study are numbers of pairs and number of individuals. PubH 6414 Section 12 Part 2 34

35 Null hypotheses for Paired Data The null hypothesis for case-control pair matched data is that the proportion of subjects exposed to the risk factor is equal for cases and controls. The null hypothesis for twin paired data is that the proportions with the event are equal for exposed and unexposed twins The null hypothesis for before-after data is that the proportion of subjects with the characteristic (or event) is the same before and after treatment. PubH 6414 Section 12 Part 2 35

36 McNemar s test For any of the paired data Null Hypotheses the following are true if the null hypothesis is true: Ho: b = c Ho: b/(b+c) =0.5 Cells b and c are called the discordant cells because they represent pairs with a difference Cells a and d are the concordant cells. These cells do not contribute any information about a difference between pairs or over time so they aren t used to calculate the test statistic. PubH 6414 Section 12 Part 2 36

37 McNemar Statistic The McNemar s Chi-square statistic is calculated using the counts in the b and c cells of the table: 2 χ = ( b c) b + c Rule of thumb: b + c 20 If the null hypothesis is true the McNemar Chi-square statistic = 0. 2 PubH 6414 Section 12 Part 2 37

38 McNemar statistic distribution The sampling distribution of the McNemar statistic is a Chi-square distribution. For a test with alpha = 0.05, the critical value for the McNemar statistic = The null hypothesis is not rejected if the McNemar statistic < The null hypothesis is rejected if the McNemar statistic > PubH 6414 Section 12 Part 2 38

39 P-value for McNemar statistic You can find the p-value for the McNemar statistic using R 1-pchisq(X 2,1) If the test statistic is > 3.84, the p-value will be < 0.05 and the null hypothesis of equal proportions between pairs or over time will be rejected. PubH 6414 Section 12 Part 2 39

40 McNemar test Example Breast cancer patients receiving mastectomy followed by chemotherapy were matched to each other on age and cancer stage. By random assignment, one patient in each matched pair received chemo perioperatively and for an additional 6 months, while the other patient in each matched pair received chemo perioperatively only. PubH 6414 Section 12 Part 2 40

41 Chemo Study Periop. only Periop. + 6 Months Survived 5 years Died within 5 years Survived 5 years Died within 5 years PubH 6414 Section 12 Part 2 41

42 McNemar test hypotheses Scientific Question: Does survival to 5 years differ by treatment group? H O : b = c H A : b not equal c PubH 6414 Section 12 Part 2 42

43 McNemar test Check: b + c 20 Critical value for Chi-square distribution with 1 df = 3.84 Calculate the test statistic 2 χ = ( b c) b + c P-value = P(χ 2 > 6.54) = = (17 5) = 6.54 > 1-pchisq(6.54,1) [1] PubH 6414 Section 12 Part 2 43

44 Decision and Conclusion Decision: Reject Ho By the rejection region method: 6.54 > 3.84 By the p-value method: 0.01 < 0.05 Conclusion: The data provide evidence that an extra 6 months of chemotherapy results in a different survival rate compared to treatment with perioperative chemo alone. (p = 0.01). PubH 6414 Section 12 Part 2 44

45 McNemar test online calculator This website will calculate the McMemar test statistic and p-value PubH 6414 Section 12 Part 2 45

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