# Chi-square test Fisher s Exact test

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1 Lesson 1 Chi-square test Fisher s Exact test McNemar s Test Lesson 1 Overview Lesson 11 covered two inference methods for categorical data from groups Confidence Intervals for the difference of two proportions Two sample z-test of equality of two proportions Lesson 1 covers three inference methods for categorical data Chi-square test for comparisons between categorical variables Fisher s exact test McNemar Chi-square test for paired categorical data Chi-square test The Chi-square test can be used for two applications 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 proportions are equal. The alternative hypothesis is that the proportions are not equal (test for a difference in either direction) Contingency Tables Setting: Let X 1 and X denote categorical variables, X 1 having I levels and X having J levels. There are IJ possible combinations of classifications. Level1 Level LevelI Level1 Level. LevelJ When the cells contain frequencies of outcomes, the table is called a contingency table. 3 4 Step 1: Hypothesis (always two-sided): H o : Independent H A : Not independent Step : Calculate the test statistic: ( x e ) Χ ~ χ with df ( I 1)( J 1) e Step 3: Calculate the p-value p-value P(Χ > X ) <- value -sided Step 4: Draw a conclusion p-value<α reject independence p-value>α PubH do 6414 not Lesson reject 1 independence 5 Racial differences and cardiac arrest. An Example 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 1 month period, fewer than 1% of African- Americans survived an arrest-to-hospital discharge, compared to.6% of Caucasians. 6 1

2 Racial differences and cardiac arrest Survival to Discharge Race Caucasian African- American YES NO 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 Obtain a random sample of n independent observations (the selection of one observation does not influence the selection of any other).. 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. 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 (Χ ) test statistic to observe the difference between the observed and expected frequencies χ Distribution Chi-square distributions and critical values for 1 df, 4 df and 0 df 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. For Chi-square with 0 df, the critical value (α 0.05) 31.4 Since the distribution only takes on positive values all the probability is in the right-tail. Critical value for α 0.05 and Chi-square with 1 df is 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 11 1

3 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 Chi-square distribution For a X table, there are (-1)*(-1) 1 df Use the CHIINV function to find the critical value General formula: CHIINV(alpha, df) State the conclusion The p-value for P( χ > X ) CHIDIST(X,df) Reject the null hypothesis by either the rejection region method or the p-value method X > Critical Value or Pvalue < α Chi-squared Test: Calculating expected frequencies Race Caucasian African- American Survival to Discharge YES NO Under the assumption of independence: P(YES and Caucasian) P(YES)*P(Caucasian) 108/6117 * 307/ Expected cell count e * Calculating expected frequencies Race Caucasian African- American Survival to Discharge YES NO Expected Cell Counts (Marginal Row total * Marginal Column )/ n Rule of Thumb: Check to see if expected frequencies are > No more than 0% of cells with expected frequencies < 5 16 Step 1: Hypothesis (always two-sided): H o : Independent (Race/Survival) H A : Not independent Step : Calculate the test statistic: ( x e ) Χ e ( ) ( ) ( ) ( ) Step 3: Calculate the p-value p-value P(Χ >8.4) Chidist(8.4,1) < Step 4: Draw a conclusion p-value<α reject independence A significant association exists between race and survival to hospital discharge in cases of non-traumatic out-of-hospital cardiac arrest

4 Testing for equality or homogeneity of proportions examines differences between two or more independent proportions. In chi-square test for independence, we examine the cross-classification 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. 30 day outcome Survived Died Example Patients with evolving myocardial infarction were assigned independently and randomly to one of four thrombolytic treatments, and then followed to determine 30 day mortality. and SC and IV Accelerated t-pa and IV 969 Are these four treatment populations equal with respect to 30-day mortality? Accelerated t-pa with IV day outcome Survived Died Example and SC and IV Accelerated t-pa and IV Accelerated t-pa with IV Under the assumption of independence: P( and SC and Survival) P( and SC )*P(Survival) 9796/40845 * 37997/ Expected cell count e 0.3 * day outcome Survived Died Example and SC and IV Accelerated t-pa and IV Accelerated t-pa with IV Under the assumption of independence: Expected Cell Counts (Marginal Row total * Marginal Column )/ n 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 : Calculate the test statistic: ( x e ) Χ ~ χ with df ( I 1)( J 1) e Step 3: Calculate the pvalue P(Χ > X ) Step 4: Draw a conclusion p-value<α reject independence p-value>α do not reject independence 3 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 : Calculate the test statistic: Χ ( x e ) with df ( 1)( 4 1) 3 e 4 4

5 Step 3: Calculate the p-value p-value P(Χ >10.85) chidist(10.85,3) Step 4: Draw a conclusion p-value<α0.05 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. 5 Chi-Square Testing Independence Ho: two classification criteria are independent Ha: two classification criteria are not independent. Requirements: One sample selected randomly from a defined population. Observations cross-classified into two nominal criteria. Conclusions phrased in terms of independence of the two classifications. Equality Ho: populations are homogeneous with regard to one classification criterion. Ha: populations are not homogeneous with regard to one classification criterion. Requirements: Two or more samples are selected from two or more populations. Observations are classified on one nominal criterion. Conclusions phrased with regard to homogeneity or equality of treatment 6 populations. Chi-square test in EXCEL or online calculator There is no Excel function or Data Analysis Tool for the Chi-square test For the Excel examples, you ll need to calculate the expected cell frequencies from the observed marginal totals and then calculate the statistic from the observed and expected frequencies. This website will calculate the Chi-square statistic and p-value for data in a X 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 Chi-Square Testing: Rules of Thumb All expected frequencies should be equal to or greater than (observed frequencies can be less than ). No more than 0% of the cells should have expected frequencies of less than 5. What if these rules of thumb are violated? 7 8 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 X tables, use Fisher s Exact Test (i.e. P(xk) ~ B(n,p)) if your expected frequencies are less than. (Section 6.6) Fisher s Exact Test: Description The Fisher s exact test calculates the exact probability of the table of observed cell frequencies given the following assumptions: The null hypothesis of independence is true The marginal totals of the observed table are fixed Calculation of the probability of the observed cell frequencies uses the factorial mathematical operation. Factorial is notated by! which means multiply the number by all integers smaller than the number Example: 7! 7*6*5*4*3**

6 Fisher s Exact Test: Calculation a c a+c b d b+d a+b c+d n Fisher s Exact Test: Calculation Example If margins of a table are fixed, the exact probability of a table with cells a,b,c,d and marginal totals (a+b), (c+d), (a+c), (b+d) (a + b)! (c + d)! (a + c)! (b + d)! n! a! b! c! d! The exact probability of this table 9! 9! 13! 5! 18! 1! 8! 4! 5! Probability for all possible tables with the same marginal totals Set of 6 possible tables with marginal totals: 9,9,5,13 The following slide shows the 6 possible tables for the observed marginal totals: 9, 9, 5, 13. The probability of each table is also given. The observed table is Table II The p-value for the Fisher s exact test is calculated by summing all probabilities less than or equal to the probability of the observed table. The probability is smallest for the tables (tables I and VI) that are least likely to occur by chance if the null hypothesis of independence is true. 33 I II III IV Pr Pr V Pr 0.13 Pr Pr VI Pr Fisher s Exact Test: p-value Probability I II III IV V VI Table The observed table (Table II) has probability 0.13 P-value for the Fisher s exact test Pr (Table II) + Pr (Table V) + Pr (Table I) + Pr (Table VI) Conclusion of Fisher s Exact test At significance level 0.05, the null hypothesis of independence is not rejected because the p- value of 0.94 > Looking back at the probabilities for each of the 6 tables, only Tables I and VI would result in a significant Fisher s exact test result: p * for either of these tables. This makes sense, intuitively, because these tables are least likely to occur by chance if the null hypothesis is true

7 Fisher s Exact test Calculator for a x table This website will calculate the Fisher s exact test p-value after you enter the cell counts for a x contingency table. Use the p-value for the two-sided test. 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 0% of the expected cell frequencies < 5, use the Fisher s exact test When categorical data are paired, the McNemar test is the appropriate test 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 The McNemar test is only used for paired nominal data. Use the Chi-square test for independence when nominal data are collected from independent groups. 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 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 X 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 41 Pair-Matched Data for Case-Control Study: outcome is exposure to some risk factor Control Exposed Case Exposed a b Unexposed c d a - number of case-control pairs where both are exposed b - number of case-control pairs where the case is exposed and the control is unexposed c - number of case-control pairs where the case is unexposed and the control is exposed d - number of case-control pairs where both are unexposed The counts in the table for a case-control study are numbers of pairs not numbers of individuals. Unexposed 4 7

8 Paired Data for Before-After counts The data set-up is slightly different when we are looking at Before-After counts of some characteristic of interest. For this data, each subject is measured twice for the presence or absence of the characteristic: before and after an intervention. The pairs are not two paired individuals but two measurements on the same individual. The outcome is binary: each subject is classified as + (characteristic present) or (characteristic absent) at each time point. 43 Before treatment Paired Data for Before-After Counts + - After treatment + a - number of subjects with characteristic present both before and after treatment b - number of subjects where characteristic is present before but not after c - number of subjects where characteristic is present after but not before d - number of subjects with the characteristic absent both before and after treatment. a c - b d 44 Null hypotheses for Paired Nominal 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. McNemar s test For any of the paired data Null Hypotheses the following are true if the null hypothesis is true: b c b/(b+c) 0.5 Since cells b and c are the cells that identify a difference, only cells b and c are used to calculate the test statistic. 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 McNemar Statistic McNemar statistic distribution The McNemar s Chi-square statistic is calculated using the counts in the b and c cells of the table: χ ( b c) b + c Square the difference of (b-c) and divide by b+c. If the null hypothesis is true the McNemar Chisquare statistic 0. The sampling distribution of the McNemar statistic is a Chi-square distribution. Since the McNemar test is always done on data in a X table, the degrees of freedom for this statistic 1 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 >

9 P-value for McNemar statistic You can find the p-value for the McNemar statistic using the CHIDIST function in Excel Enter CHIDIST(test statistic, 1) to obtain the p-value. 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. McNemar test Example In 1989 results of a twin study were published in Social Science and Medicine: Twins, smoking and mortality: a 1 year prospective study of smoking-discordant twin pairs. Kaprio J and Koskenvuo M pairs of twins were enrolled in the study. One of the twins smoked, the other didn t. The twins were followed to see which twin died first. For 17 pairs of twins: the smoking twin died first For 5 pairs of twins: the nonsmoking twin died first Data for Twin study in a table Smoking Twin Died 1st Died nd Non-smoking Twin Died 1st 0 5 Died nd 17 0 McNemar test hypotheses H O : The proportion of smoking twins who died first is equal to the proportion of nonsmoking twins who died first H A : The proportion of smoking twins who died first is not equal to the proportion of nonsmoking twins who died first In this study that counts which twin dies first, all data are discordant and are in the b and c cells 51 5 McNemar test Significance level of the test 0.05 Critical value for Chi-square distribution with 1 df 3.84 Calculate the test statistic ( b c) χ b + c P-value 0.01 CHIDIST(6.54, 1) (17 5) Decision and Conclusion for Twin study Decision: The null hypothesis of equal proportions of first death for smoking and nonsmoking twins is rejected By the rejection region method: 6.54 > 3.84 By the p-value method: 0.01 < 0.05 Conclusion: A significantly different proportion of smoking twins died first compared to their nonsmoking twin indicating a different risk of death associated with smoking (p 0.01)

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